GAME THEORY: Analysis of Strategic
Thinking.
Part I: Static Games
Pierpaolo Battigalli
Università Bocconi
Academic Year 2006-2007
Abstract
These notes introduce some notions of the theory of static games:
rationality, dominance, rationalizability and equilibrium (Nash, mixed,
correlated and self-con…rming). For each of these concepts, the interpretative aspect is emphasized. Even though no advanced mathematical knowledge is required, the reader should nonetheless be familiar
with the concepts of set, function, probability, and more generally be
able to follow abstract reasoning and formal arguments.
1
1
Static Games
A static game, or simultaneous moves game, is a mathematical structure
G = hN; (Ai ; ui )i2N i
where
N = f1; 2; :::; ng is the set of individuals or players and we denote by
i 2 N a generic individual
Ai is the set of possible actions for player i and we denote a generic
action using the symbols ai , ai , a0i , etc;
u i : A1
i.1
A2
:::
An ! R is the utility (or payo¤ ) function of player
Where otherwise not explicitly indicated, we assume for simplicity that
all the sets Ai (i 2 N ) are …nite.2
The utility function ui numerically represents the preferences of player i
among the di¤erent actions pro…les a; a0 ; a00 ::: 2 A [a "pro…le" is a speci…cation of a type of object, e.g. actions, for each player; thus an action pro…le
is an n-tuple a = (a1 ; :::; an )]. The strategic interdependence is due to the
fact that the utility that a generic individual, i, can achieve depends not
only on his choices, but also on those of the other individuals. To stress how
the utility of i depends on a variable under i’s control as well as a vector of
1
We denote by R the set of real numbers, by R+ the set of non negative real numbers.
The set of n-dimensional real vectors (n-dimensional Euclidean space) is denoted by Rn ,
the set of non negative n-dimensional real vectors by Rn+ (known as the positive orthant).
Analogously, for any given domain X; the set of functions that map X into R (or R+ ) is
denoted by RX (or, respectively, RX
+)
2
Sometimes static games are also called "normal form games" or "strategic form
games". This terminology is somewhat misleading. The normal, or strategic form of
a game has the same structure of a static game, but the game itself may have a sequential structure. The normal form of shows the payo¤s induced by any combination of
plans of actions of the players. Some game theorists, including the founders Von Neumann
and Morgenstern, argue that from a theoretical point of view all the strategically relevant
aspects of a game are contained in its normal form. Anyway, here by "static game" we
mean a game where players move simultanously.
2
variables controlled by other individuals, we denote by
of individuals di¤erent from i and de…ne
A
i
:= A1
:::
Ai
1
Ai+1
:::
i = N nfig the set
An
(where “:=”is to be read as “equal by de…nition”). We also write the utility
of i as a function of the two arguments ai 2 Ai and a i 2 A i , i.e ui :
Ai A i ! R.
In order to be able to reach some conclusions regarding players’behavior
in a game G, we impose two minimal assumptions (further assumptions will
be introduced later on as necessary):
(1)
Every player i knows Ai , A i and the function ui : Ai A i ! R.
(2)
Every player is rational (see next section for a formal de…nition
of rationality).
It could be argued that assumption (1) (i knows ui ) is tautological. In
fact, it seems almost tautological to assume that every individual knows his
preferences. Often, however, the function ui is derived by another utility
function vi : C ! R and a outcome function g : A ! C, where C is the
set of possible consequences deriving from the interaction, as speci…ed by g.
Even if we were to take for granted that i knows vi , i.e his preferences over
the set of consequences C, i could still not know the outcome function g.
Example 1 (knowledge of the utility and payo¤ functions) Players 1
and 2 work in a team for the production of a public good. Their action is given
by the employed e¤ort ai 2 [0; 1]. The output, y, is of a public good nature
and depends on the e¤orts employed according to a Cobb-Douglas production
function
y = K(a1 ) 1 (a2 ) 2 :
The cost of the e¤ort measured in terms of public good is ci (ai ) = a2i . The
utility function of player i is vi (y; c1 ; c2 ) = y ci . The outcome function
g : [0; 1] [0; 1] ! R3+ is
g(a1 ; a2 ) = (K(a1 ) 1 (a2 ) 2 ; (a1 )2 ; (a2 )2 ):
The payo¤ function is
ui (a1 ; a2 ) = vi (g(a1 ; a2 )) = K(a1 ) 1 (a2 )
2
(ai )2 .
If i does not know all the parameters K, 1 and 2 (as well as the functional
forms above), then he does not know the payo¤ function ui , even if he knows
the utility function vi .
3
2
Rationality and Dominance
Let us start by analyzing a static game from the perspective of decision
theory. We take as given the conjecture that a player holds concerning the
behavior of his opponents. This conjecture is, in general, probabilistic, i.e it
can be expressed by assigning subjective probabilities to the di¤erent action
pro…les of the other players. A player is said to be rational if he maximizes
his expected utility, given his conjecture. The concept of dominance allows to
characterize which actions are compatible with the rationality assumption.
2.1
Expected Utility
The utility function implicitly represents the preferences of i over di¤erent
probability measures, or lotteries, over A. The lotteries that result to be
preferred are those that yield a higher expected utility. We explain this in
details below.
The set of probability measures over a generic domain X is denoted by
(X). If X is …nite3
(
)
X
(X) :=
2 RX
(x) = 1 .
+ :
x2X
Notice that X can be regarded as a subset of (X). As a matter of fact, a
point x 2 X can be characterized by the probability measure such that
(x) = 1. Hence, with a small abuse of notation, we can write X
(X)
(more abuses of notation will follow).
De…nition 1 Consider a probability measure 2 (X), where X is a …nite
set. The subset of the elements x 2 X to which assigns a positive probability
is said to be the support of and it is denoted by4
Supp := fx 2 X : (x) > 0g :
3
Recall that RX
+ is the set of non negative real valued functions de…ned over the domain
n
X. If X is a …nite set, X = fx1 ; :::; xn g, RX
+ is isomorphic to R+ , the positive orthant of
n
the Euclidean space R .
4
If X is a closed and measurable subset of Rm , we de…ne the support of a probability
measure 2 (X) as the intersection of all closed sets C such that (C) = 1. If X is
…nite, this de…nition coincides with the previous one.
4
If X is a set of alternatives over which an individual expresses his preferences, then we can consider (X) as a set of lotteries. In particular, letting
X = A in a static game, we assume that the individual holds a complete system of preferences not only over A, but over (A) as well. Those preferences
can be represented via the utility function ui as follows: is preferred to
if and only if the expected utility of is higher than the expected utility of
, i.e
X
X
(a)ui (a) >
(a)ui (a).
a2A
2.2
a2A
Conjectures
The reason why we need to introduce preferences over lotteries and the notion of expected utility is that an individual i cannot observe the actions of
the other individuals (a i ) before making his own choice (moves are simultaneous). Hence, he needs to form a conjecture over such actions. If i were
certain of the other players’choices, then he could represent his conjecture
simply through an action pro…le a i 2 A i . However, in general i might
be uncertain over the other players’actions and assign a subjective positive
probability to several pro…les a i , a0 i , etc..
De…nition 2 We call conjecture of an individual i a (subjective) probability
measure i 2 (A i ). A deterministic conjecture is a probability measure
i
2 (A i ) that assigns probability one to a particular action pro…le a i 2
A i.
In other words, we call "conjecture" a (probabilistic) belief about the
behavior of other players. We use the term "belief" to refer to a more general
type of uncertainty.
Remark 1 The set of deterministic conjectures of player i can be represented
by the set of other players’action pro…les: A i
(A i ).
Game Theory most interesting aspect consists in determining players’
conjectures, or at least in narrowing down the set of possible conjectures,
combining some general assumptions regarding the way players reason with
assumptions speci…c to the game G. For the moment, we will take the conjectures that a player holds as given.
5
For any given conjecture i , the choice of an action ai corresponds to
the choice of the lottery over A that assigns probability i (a i )to the action
pro…les (ai ; a i ) (a i 2 A i ) and probability zero to the pro…les (a0i ; a i )
with a0i 6= ai . Therefore, if a player i forms the conjecture i and chooses the
action ai , the corresponding (subjective) expected utility is
X
i
ui (ai ; i ) :=
(a i )ui (ai ; a i ):
a
i 2A i
There are many di¤erent ways to represent graphically actions/lotteries,
conjectures, and preferences when the number of available actions is small.
Here we focus our attention to the instructive case for which player i’s opponent (denoted by i) disposes of only two actions, that we indicate with
s (sinistra, or left in Italian) and d (destra, or right in Italian). Consider,
for instance, the function ui represented by the following payo¤ matrix for
player i:
in i
a
b
c
e
f
Matrix
s
4
1
2
4
1
1
d
1
4
2
0
1
Given that i disposes of only two actions, we can represent the conjecture i of i over i with a single number which indicates the probability that
i subjectively assigns to the action right. Any action ai can be characterized
by a function that assigns to any value of i (d), the expected utility of ai .
Since the expected utility function is linear with respect to probabilities, the
function obtained is a simple line. Therefore, any action is represented by a
line, as in …gure 1.
6
b
a
4
e
3
c
c
2
b
a
f
f
1
e
()isµ
()idµ
0
1
1
0
Figura 1
From Figure 1 we can see that only actions a, b and e are justi…able. If
(d) = 0, then a and e are both optimal. If 0 < i (d) < 21 , then a is the
only optimal action (the line aa yields the maximum expected utility). If
i
(d) > 12 , then b is the only optimal action (the line bb yields the maximum
expected utility). If i (d) = 12 , then there are two optimal actions: a and b.
i
2.3
Mixed actions
In principle, a player instead of choosing directly a certain action could delegate his decision to a roulette, or toss a coin. In other words, a player could
simply choose the probabilities with which any given action will be chosen.
De…nition 3 A random choice by player i, also called a mixed action, is a
probability measure i 2 (Ai ). An action ai 2 Ai is also called pure action.
Remark 2 The pure actions can be regarded as a subset of the mixed actions,
i.e Ai
(Ai ):
7
It is should be understood that the random draw of an action of i is
stochastically independent from the other players’actions. Hence, if a player
i forms the conjecture i and chooses the mixed action i , the subjective
probability of the actions pro…le (ai ; a i ) is i (ai ) i (a i ) and i’s expected
utility is
X X
X
i
i
ui ( i ; i ) :=
):
i (ai ) (a i )ui (ai ; a i ) =
i (ai )ui (ai ;
ai 2Ai a
ai 2Ai
i 2A i
If the opponent has only two feasible actions, it is possible to use a graph
to represent the lotteries corresponding to pure and mixed actions. For each
action ai we consider a corresponding point in the Cartesian plane with
coordinates given the utilities that i obtains for each of the two actions of
the opponent. If the actions of the opponents are s and d, we denote such
coordinates ui ( ; s) e ui ( ; d). Any pure action ai corresponds to the vector
(ui (ai ; s); ui (ai ; d)) (i.e the corresponding line in the payo¤ matrix). The same
holds for the mixed actions: i corresponds to the vector (ui ( i ; s); ui ( i ; d)).
The set of points (vectors) corresponding to the mixed actions is simply
the convex hull of the points corresponding to the pure actions. Figure 2
represents such a set for matrix 1. How can we represent conjectures in such a
…gure? It is quite simple. Every conjecture induces some preferences over the
space of payo¤ pairs (lotteries). Such preferences can be represented through
a map of iso-expected utility curves (or indi¤erence curves). Let (x; y) be
the generic vector of utilities corresponding, respectively, to s and d. The
expected utility induced by the conjecture i is i (s)x + i (d)y. Therefore,
i (s)
the indi¤erence curves are lines de…ned by the equation y = k
i (d) x, and
every conjecture i corresponds to a set of parallel lines with negative slope
(or, in the limit, horizontal or vertical slope) given by the orthogonal vector
( i (s); i (d)). The direction of increasing expected utility is given precisely
by such orthogonal vector. The optimal actions (pure or mixed) can be
graphically determined considering, for any conjecture i , the highest line
of iso-expected utility among those who touch the set of feasible payo¤ vectors
8
(the grey area in the …gure).
•(),iu
()13isµ=
d
()23idµ=
b
4
3
()23isµ=
c
2
()13idµ=
a
f
1
e
•(),iu
1
2
3
s
4
Figura 2
If we allow for the possibility that players use mixed actions, then the set
of vectors of possible payo¤s can be represented using a convex polyhedron
(as in Figure 2). To see this, note that if i and i are mixed actions, then
also p i + (1 p) i is a mixed action, where p i + (1 p) i is the
function that assigns to each pure action ai the weight p i (ai ) + (1 p) i (ai ).
Thus all the convex combinations of feasible payo¤ vectors are feasible payo¤
vectors, once we allow for randomization.
However, the idea that individuals use coins or roulettes to make their
choices random may seem weird or unrealistic. Further, as it can be gathered
from the …gure, for any conjecture i and any mixed action i , it always exists
a pure strategy ai that yields the same or a higher expected utility than i
(su¢ ces it to check all possible slopes of the iso-expected utility curves and
verify how the set of optimal points looks like).5 Hence, a player will never
5
This result will be proven rigorously in the next section.
9
be strictly better o¤ by choosing a mixed action. In these notes we adopt
the point of view according to which players never opt for random choices.
Nonetheless, we will show that in order to evaluate the rationality of a given
action it is analytically convenient to introduce mixed actions. We will also
show later on that it is possible to give to mixed actions (i.e to the elements
of (Ai )) a di¤erent interpretation that is not based on random choices.
2.4
Best Replies and Undominated Actions
De…nition 4 A (mixed) action
8
i
2
i
is a best reply to the conjecture
(Ai ), ui (
i;
i
)
ui ( i ;
i
i
if
),
that is
i
2 arg max ui ( i ;
i 2 (Ai )
i
).
The set of pure actions that are best replies to the conjecture
by
ri ( i ) := Ai \ arg max ui ( i ; i ).
i2
i
is denoted
(Ai )
The correspondence ri : (A i ) ) Ai is said best reply correspondence6 . If
for a given action ai there exists a conjecture i such that ai 2 ri ( i ), then
we say that ai is justi…able.
Notice that de…nition 4 does not immediately imply (when the choice of
mixed actions is also available) that the set of pure actions that are best
reply to the conjecture i , i.e ri ( i ), is a non-empty set. In principle, it could
happen that in order to maximize expected utility the use of non degenerate
mixed actions (mixed actions that assign positive probability to more than
one pure action) are necessary. In such event, we would have that ri ( i ) = ;.
However, we will see that ri ( i ) 6= ; for any i (provided that Ai is …nite or,
more generally, that Ai is compact and ui continuous in ai , two properties
that trivially hold if Ai is …nite), and that it is never necessary to use nonpure actions to maximize expected utility.
We say that a player is rational if he chooses a best reply to the conjecture
he holds. The following result shows, as already anticipated, that a rational
6
A correspondence ' : X ) Y is a multi-function that assigns to any element x 2 X a
set of elements '(x) Y .
10
player does not need to use mixed actions. Therefore, it can be assumed
without loss of generality that his choice is restricted to the set of pure
actions.
Lemma 1 Consider a given conjecture i 2 (A i ). A mixed action i is
a best reply to i if and only if any pure action in the support of i is a best
reply to i (i.e, if and only if Supp i
ri ( i )).
Proof (only if) We want to show that if Supp i is not included in ri ( i ),
then i is not a best reply to i . Let ai be a pure action such that i (ai ) > 0
and assume that, for some i , ui ( i ; i ) > ui (ai ; i ). Since ui ( i ; i ) is a
weighed average of the set of values fui (ai ; i )gai 2Ai , there must be a pure
action a0i such that ui (a0i ; i ) > ui (ai ; i ). But then we can construct a mixed
action 0i that satis…es ui ( 0i ; i ) > ui ( i ; i ): for any i 2 Ai , let
8
if ai = ai ,
< 0;
0
0
(a ) + i (ai ); if ai = a0i ,
i (ai ) =
: i i
if ai 6= ai ; a0i .
i (ai );
P
P
Notice that 0i is a mixed action since ai 2Ai 0i (ai ) = ai 2Ai (ai ) = 1.
Moreover it can be easily checked that
ui ( 0i ;
i
)
ui (
i
i;
)=
0
i (ai )[ui (ai ;
i
)
ui (ai ;
(If) Assume now that Supp i
ri ( i ). Then, for any
X
i
ui ( i ; i ) =
)
i (ai ) max ui ( i ;
ai 2ri (
X
ai
i2
i)
i
)] > 0:
i
2
(Ai ),
(Ai )
ui (a0i ;
i (ai ) max
0
ai 2Ai
i
)
X
i (ai )ui (ai ;
i
):
ai 2Ai
The equality holds by assumption. The …rst inequality follows from the
de…nition ri ( i ) = Ai \ arg maxai 2 (Ai ) ui ( i ; i ), given that Ai
(Ai )
implies
max ui ( i ; i ) max ui (ai ; i ):
i2
ai 2Ai
(Ai )
The second inequality holds by de…nition.
In Matrix 1, for instance, any mixed action that assigns a positive probability only to a or b is a best reply to the conjecture ( 12 ; 21 ). Clearly,
ri (( 12 ; 12 )) = fa; bg.
11
Notice that if at least one pure action is a best reply among all pure and
mixed actions, then the maximum that can be attained by constraining the
choice to pure actions is necessarily equal to what could be attained choosing
among (pure and) mixed actions, i.e
ri ( i ) 6= ; ) max ui ( i ;
i2
i
) = max ui (ai ;
ai 2Ai
(Ai )
i
).
This observation along with lemma 1 allows us to conclude the following:
Corollary 2 For any conjecture
i
2
(A i ),
ri ( i ) = arg max ui (ai ;
ai 2Ai
i
)
Hence, it is not necessary to use mixed actions to maximize expected utility.
Moreover, since Ai is …nite, the best reply correspondence yields a non empty
set, i.e ri ( i ) 6= ; for any i 2 (A i ).7
Proof. For any given conjecture i , the expected utility function ui ( ; i ) :
(Ai ) ! R is continuous; further, the domain (Ai ) is a compact subset of RAi . Hence, the function admits a maximum i . By Lemma 1
Supp i
ri ( i ), so that ri ( i ) 6= ;. As we have seen above, this implies
that
max ui ( i ;
i2
(Ai )
i
) = max ui (ai ;
ai 2Ai
and therefore ri ( i ) = arg maxai 2Ai ui (ai ;
i
i
)
).
Recall that, for a given function f : X ! Y and a given subset C X,
we write f (C) := fy 2 Y : 9x 2 C; y = f (x)g to denote the set of images of
the points in C. Analogously, for a given correspondence : X ) Y we
denote by
(C) := fy 2 Y : 9x 2 C; y 2 (x)g
the set of points y that belong to the image (x) of some point x 2 C. In particular, we use this notation for the best reply correspondence. For instance,
ri ( (A i )) is the set of actions that are best reply to some conjecture.
7
The same conclusion holds if A is compact and ui is continuous.
12
A question should spontaneously arise at this point: when is it possible to
state that an action ai does not belong to ri ( (A i ))? In other words, when
can we say that an action would not be chosen by a rational player whatever
conjecture this player might hold? The answer follows from the concept of
dominance that we now introduce.
De…nition 5 A mixed action i dominates another mixed action i if it
yields a higher expected utility whatever the choice of the other players, i.e
8a
We denote by N Di
any mixed action.
i
2 A i , ui ( i ; a i ) > ui ( i ; a i ):
Ai the set of (pure) actions that are not dominated by
Remark 3 If a (pure) action ai is dominated by a mixed action i then, for
any conjecture i 2 (A i ), ui (ai ; i ) < ui ( i ; i ). Hence, from Corollary 1
we can conclude that an action which is dominated by a mixed action cannot
be a best reply to any conjecture.
In Matrix 1, for instance, the action f is dominated by c that at the same
time is dominated by a mixed action that assigns probability 12 to a and b.
Therefore, N Di fa; b; eg. Given that a, b, and e are best replies to at least
one conjecture, we have that fa; b; eg N Di . Hence, N Di = fa; b; eg.
The following lemma states that also the converse of remark 3 holds. It
therefore provides a complete answer to our previous question asking when it
is possible to state that an action would not be chosen by a rational player.
Lemma 3 (see Pearce [14]) Fix an arbitrary action ai 2 Ai . There exists
a conjecture i 2 (A i ) such that ai is a best reply to i if and only if ai
is not dominated by any pure or mixed action. In other words, the set of
undominated (pure) actions and the set of justi…able actions coincide:
N Di = ri ( (A i )) .
Even without rigorously proving lemma 3, we can understand why it is
true by observing Figure 2. The set of justi…able actions corresponds to the
13
extreme points (the "kinks") of the “North-East frontier”of the convex polyhedron. An action ai is not justi…able if and only if the corresponding point
lies (strictly) below and to the left of such frontier. But for any point below
and to the left of the North-East frontier, there exists at least a corresponding
point on the frontier that dominates such point.8
We can thus conclude that a rational player always chooses undominated
actions and that any undominated action is justi…able as a best reply to some
conjecture.
There are interesting situations of strategic interaction where an action
dominates all others. In those cases, a rational player should choose that
action. This consideration motivates the following:
De…nition 6 An action ai is said to be dominant if it dominates every other
action, i.e if
8ai 2 Ai n fai g ; 8a
i
2 A i , ui (ai ; a i ) > ui (ai ; a i ).
You should try to prove the following as an exercise:
Remark 4 Suppose that (pure) action ai is dominant. Then (1) ai dominates every (pure or) mixed action i 6= ai , and (2) ai is the only action
with the property of being a best reply to every conjecture.
A classic example is provided by the well known Prisoners’ Dilemma,
where each player has dominant action ("confess") and the choice of such
actions leads to a Pareto-ine¢ cient outcome. The following example is of
a rather similar nature and leads to the same conclusion: in situations of
strategic interaction the individual rationality (assuming that players are only
motivated by their material self-interest) may lead to socially undesirable
outcomes.
Example 2 (Public Goods). In a community composed of n individuals is
possible to produce a quantity g of a good using an input X according to the
production function g = minfkX; gg where the input X is measured in value
8
As Figure 2 suggests, a similar result relates undominated mixed actions and mixed
best replies: the two sets are the same and correspond to the "North-East frontier" of the
convex polihedron. But we take the perspective that players do not actually randomize,
and therefore we are only interested in characterizing the justi…able pure actions.
14
(say in Euros) and g is the maximum quantity that can be produced. The
utility that in monetary terms each individual can achieve from g is exactly
g ; the production parameter k satis…es 1 > k > n1 . A generic individual
i can freely choose how many Euros to contribute to the community for the
production of the public good. The community members cannot sign a binding
agreement over such contributions as no authority with coercive power to
enforce it exists. Let Wi be player i’s wealth. The game can be represented
asP
follows: Ai = [0; Wi ], ai 2 Ai is the contribution of i; ui (a1 ; :::; an ) =
k( nj=1 ai ) ai is the payo¤ function. It can be easily checked that ai = 0 is
g
.
dominant for any player i. However, the Pareto-optimum is aPi = nk
In some games there exist actions that are best reply to some conjectures
only because such conjectures assign zero probability to some vectors of the
other players’actions. For instance, action e in matrix 1 is justi…able as a
best reply only if i is certain that i chooses s.
We say that a player i is cautious if his conjectures assign positive probability to every a i 2 A i . Let
0
(A i ) :=
i
2
(A i ) : 8a
i
2 A i;
i
(a i ) > 0
be the set of such conjectures.9 A player is rational and cautious if he chooses
actions in ri ( 0 (A i )). These considerations motivate the following de…nition and result.
De…nition 7 A mixed action i weakly dominates another mixed action i
if it yields at least the same expected utility no matter what the other players’
choice and strictly more for at least one a i , i.e if
8a
9a
i
i
2 A i , ui ( i ; a i ) ui ( i ; a i );
2 A i , ui ( i ; a i ) > ui ( i ; a i ):
We denote by N W Di
Ai the set of (pure) actions that are not weakly
dominated by any mixed action.
Lemma 4 (See Pearce [14]) Fix an arbitrary action ai 2 Ai . There exists a
conjecture i 2 0 (A i ) such that ai is a best reply to i if and only if ai is
not strictly dominated by any pure or mixed action. Equivalently,
9
0
(A i ) is the relative interior of
A
(A i ) in R+ i .
15
ri
0
(A i ) = N W Di .
The intuition is similar to the one for Lemma 3. By looking at Figure
2 we can see that the set of (pure or mixed) actions that are not weakly
dominated corresponds to the part with strictly negative slope of the NorthEast frontier of the compact and convex polyhedron. Such set coincides with
the points that maximize the expected utility for at a least a conjecture
that assigns positive probability to all actions of i, that is a conjecture
that is represented by a set of parallel lines with strictly negative slope (not
horizontal or vertical).
De…nition 8 A (pure) action ai is said to be weakly dominant if it weakly
dominates all other (pure) actions, i.e if for any other action b
ai 2 Ai nfai g
the following conditions hold:
8a
9b
a
i
i
2 A i ; ui (ai ; a i ) ui (b
ai ; a i );
2 A i ; ui (ai ; b
a i ) > ui (b
ai ; b
a i ):
Remark 5 If ai is weakly dominant, then ai weakly dominates any mixed
action i 6= ai and it is the only action with the property of being a best reply
to any conjecture i 2 0 (A i ).
If a player who is rational and cautious disposes of a weakly dominant
action, then he will choose such action. There exist signi…cant economic
examples in which the individuals dispose of weakly dominant actions.
Example 3 (Second Price Auction). An art merchant wants to auction
a work of art (e.g. a painting) with the aim of earning the highest amount of
money. However, he does not know how much such work of art is worth to the
potential buyers. The buyers are collectors who buy the art work with the only
objective to keep it, i.e they are not interested in what its future price might be.
The authenticity of the work is not an issue. The potential buyer i is willing
to spend at most vi > 0 to acquire the work. Such valuation is completely
subjective, that is to say that if i were to know the other players’valuations, he
16
would not change his own 10 . Following the advice of W.Vickrey, Nobel Prize
for Economics,11 the merchant decides to adopt the following auction rule:
the art work will go to the player who submits the highest o¤er, but the price
paid will be equal to the second highest o¤er (in case of a draw among the
maximum o¤ers, the work will be assigned by a random draw). This auction
rule induces a game among the buyers i = 1; :::; n where Ai = [0; +1), and
8
vi max a i ,
se ai > max a i ,
<
0,
se ai < max a i ,
ui (ai ; a i ) =
:
1
(v
max a i ) se ai = max a i ,
1+jarg max a i j i
where jXj indicates the cardinality of set X (the number of elements in X).
It turns out that o¤ering ai = vi is a weakly dominant action (can you
prove it?). Hence, if the potential buyer, i, is rational and cautious, he will
o¤er exactly vi . Doing so, he will expect to make some pro…ts. In fact, being
cautious, he will assign a positive probability to the event max a i < vi . Since
by o¤ering vi he will obtain the object only in the event that the price paid
is lower than his valuation vi , the expected utility from this o¤er is strictly
positive.
10
If a buyer were to take into account a potential future resale, things might be rather
di¤erent. In fact, other potential buyers could hold some relevant information that a¤ects
the estimate of how much the art work could be worth in the future. Similarly, if there
were doubts regarding the authenticity of the work, it would be relevant to know the other
buyers’valuations.
11
See [16].
17
3
Rationalizability and Iterated Dominance
Our analysis, so far, has been based on a set of minimal epistemic assumptions:12 every player i knows the set of possible actions (A1 , A2 ,..., An ) and
his payo¤ function ui . From this analysis we have deduced a basic principle
of rationality that substantially belongs to the decision theory approach: a
rational player should not choose those actions that are dominated by mixed
actions. This principle of rationality can be interpreted in a descriptive way
(assuming that a rational player will not choose dominated actions) as well
as in a prescriptive way (a player should not choose dominated actions).
The concept of dominance is su¢ cient to obtain interesting conclusions
regarding those interactive situations for which guessing the other players’
actions in order to take the correct decision is not necessary. But as we
consider strategic reasoning, this rationality principle is just a starting point:
reasoning in a strategic fashion means taking into account that the other
players will also try to anticipate each others moves in order to pursue their
goals.
Strategic reasoning is based on the knowledge of the game, the knowledge
of the other players’knowledge of the game and so on and so forth. In that
respect, we will assume, at least for the moment, complete information, i.e
common knowledge of the game and of all the utility (or payo¤s) functions. In
Example 1 the complete information assumption implies that the parameters
K, 1 , 2 are common knowledge: both players know them, both players know
that both players know them, and so on and so forth.
The complete information assumption is su¢ ciently realistic for some
economic situations for which the consequences of players’actions are purely
monetary outcomes and players are risk neutral.13 The complete information
assumption is also a useful simpli…cation in that it allows to focus on other
aspects of strategic reasoning than players’knowledge of the game and of the
knowledge of other players. Later on we will also analyze strategic reasoning
in games with incomplete information and we will see that the techniques developed for games complete information can be extended to the more general
case.
12
We call "epistemic" those assumptions that regard the knowledge, the conjectures and
more generally the individuals’beliefs of players.
13
In some cases (for instance, oligopolistic models) risk attitudes are irrelevant in the
strategic analysis and risk neutrality can therefore be assumed with no loss of generality.
18
3.1
Assumptions on players’beliefs
The analysis will proceed by steps. At each further step we characterize
more restrictive assumptions about players’ beliefs. Such assumptions can
be formally represented as events, i.e as subsets of a space
of “states
of the world” where every state ! 2
is a conceivable con…guration of
actions, beliefs, beliefs concerning other players’ beliefs and so on. This
formal representation has the advantage to make the analysis more rigorous
as well as the language more expressive and precise. However, it calls for
the preliminary introduction of some rather complex material, which is not
strictly needed for what follows.14 We therefore opt for a compromise. On the
one hand, in order to make the presentation more transparent and to avoid
long verbal sentences, we use symbols to denote the assumptions concerning
players’behavior and beliefs; we refer to these assumptions as "events" and
denote a conjunction of assumptions by the symbol of intersection between
events. On the other hand, rather than providing an explicit mathematical
de…nition for such events, we only de…ne mathematically the set of actions
(and conjectures) that are compatible with them.
We denote, for instance, by the letter Ri the event "player i is rational",
whereas R = R1 \ ::: \ Rn denotes the event “all players are rational”. In the
previous section we showed that N D1 ::: N Dn is the set of action pro…les
compatible with R.
The rationality assumption simply establishes a relation between conjecture (beliefs about the behavior of other players) and actions. This relation
is represented by the best reply correspondence. Now we proceed introducing
some assumptions concerning players’beliefs about each other
Denote by E a generic event regarding players’ actions and beliefs (for
example, we may have E = R). We then represent with the symbol B(E)
the event “everybody believes E”, meaning "everybody assigns probability
one to E". Further, we write B(B(E)) = B 2 (E) (“everybody believes that
everybody believes E”), B(B(B(E))) = B(B 2 (E)) = B 3 (E) and so on.
Consider the conjunction of the following assumptions: R, B(R), B 2 (R),
B 3 (R), ... . Is it possible to provide a characterization in terms of actions?
That is to say, which action pro…les are compatible with R \ B(R), R \
TK
k
B(R) \ B 2 (R), ..., R \
k=1 B (R) ,...? In order to answer this question,
14
The interested reader can refer to the survey by [3].
19
it is useful to introduce a mapping, which is derived from the best reply
correspondence and assigns to any "Cartesian" subset C of A a subset of
action pro…les that are "justi…ed" or "rationalized" by C.
3.2
The Rationalization Operator
Fix some set X and a collection C of subsets of X. We call "operator"
(on X; C) a mapping from C to itself, that is a mapping that associetes to
each subset of X in the collection another subset of X in the collection. In
particular we are going to consider the set X = A of action pro…les, and
the collection of all "Cartesian" subsets of A: Consider a non-empty subset
C = C1 ::: Cn , where Ci Ai for every i. We de…ne the following sets:15
i (C i )
:= ai 2 Ai : 9
(C) :=
i
2
(C i ); ai 2 ri ( i ) = ri ( (C i ));
1 (C 1 )
:::
n (C n ).
(C) is the set of possible action pro…les that could be chosen if every i
were rational and certain that all other players would choose actions in C i .
Therefore, we say that (C) is the set of possible action combinations that are
"justi…ed", or "rationalized", by C and we call the mapping “rationalization
operator”.16
Remark 6 Lemma 3 implies that i (A i ) is the set of undominated actions
for player i. Hence, (A) = N D1 ::: N Dn .
Remark 7 The rationalization operator is monotone: for any pair of subsets
of A, E = E1 ::: En and F = F1 ::: Fn , if E F then (E)
(F ).
To see this note that, for every i, E i F i implies (E i )
(F i ) which
in turn implies i (E i ) = ri ( (E i )) ri ( (F i )) = i (F i ).
3.3
Rationalizability
Given that the set of all possible action pro…les compatible with R (rationality) is (A), if every player believes R, then only those action pro…les that
15
For a given set X (for instance, X = A i ) and a subset Y
X, we denote by (Y )
the subset of probability measures over X that assign probability one to Y .
16
The rationalization operator represents an example of justi…cation operator, a concept
which was …rst introduced by Milgrom and Roberts (1991).
20
are rationalized by (A) can be chosen. It follows that the action pro…les
compatible with R \ B(R) is ( (A)) = 2 (A). Iterating the procedure, we
can deduce that the set of action combinations that are compatible with
R \ B(R) \ B 2 (R) is ( 2 (A)) = 3 (A). In general we have
Assumptions over behavior and beliefs Assumptions over actions
R
(A)
2
R \ B(R)
(A)
2
3
R \ B(R) \ B (R)
(A)
:::
:::
TK
K+1
k
(A)
R\
k=1 B (R)
:::
:::
Notice that, as we should expect, the sequence of subsets f k (A)gk 1 is
k
weakly decreasing, i.e k+1 (A)
(A) (k = 1; 2; :::). This fact can be easily
derived from the monotonicity of the rationalization operator: by de…nition
k 1
we have (A) A; if k (A)
(A), the monotonicity of implies k+1 (A)
= ( k (A))
( k 1 (A)) = k (A). The statement follows by induction.
De…nition 9 (Bernheim [5], Pearce
T [14]) An action pro…le a 2 A is said to
be rationalizable if a 2 1 (A) := k 1 k (A).
Theorem 5 (a) If, for every player i 2 N , Ai is …nite, then there exists a
positive integer K such that K+1 (A) = K (A) = 1 (A) 6= ;. (b) If, for every
player i 2 N , Ai is a compact subset of a Euclidean space17 and the function
ui is continuous, then the set of action pro…les that are rationalizable, 1 (A)
,is non-empty, compact and satis…es 1 (A) = ( 1 (A)).
Proof. (a) If Ai is …nite, for any conjecture i , the set of best replies
ri ( i ) is non-empty. Then for any product set C which is non-empty, (C)
is non-empty (for every i, there exists at least a conjecture i 2 (C i ) and
k
;=
6 ri ( i )
(A) 6= ; for every k.
i (C i )). It follows by induction that
k
The sequence of the subsets f (A)gk 1 is weakly decreasing. Also, if
k
(A) = k+1 (A), then k (A) = ` (A) for every ` k. Given that A is …nite,
k
the inclusion k+1 (A)
(A) can be strict only for a …nite number of steps
17
We call "Euclidean space" a Cartesian space Rm ; endowed with the Euclidean distance
and the associeted topology (collection of open sets). A subset of a Euclidean space is said
to be compact if it is closed and bounded.
21
P
K (in particular, K
n, where jAi j indicates the number of
i2N jAi j
possible actions for player i. This is so as in the …rst steps we eliminate at
least an action for at least a player i and not all possible actions of i can be
eliminated). All the above implies that K+1 (A) = K (A) = 1 (A) 6= ;.
(b) (This part contains more advanced mathematical arguments and can
be omitted.) Since ui : Ai A i ! R is continuous, then also the expected
utility function ui (ai ; i ) is continuous in both arguments. Given that Ai is
compact, the maximum principle implies that the best reply correspondence
ri : (A i ) ) Ai is upper hemicontinuous with non empty values ( (A i )
is assumed to be endowed with the topology of weak convergence of meaA i , the set (C i ) is
sures)18 . For every non empty and compact C i
compact. It follows that its image through the correspondence ri is a nonempty and compact set. Therefore for any non-empty and compact product
set C A, we have that (C) = 1 ( (C 1 )) ::: n ( (C n )) is a non-empty,
compact set. It follows by inducton that f k (A)gk 1 is a weakly decreasing
sequence of non-empty
and compact sets. Hence, for any K = 1; 2; :::, the
TK k
intersection k=1 (A) = K (A)Tis non-empty and compact. By the …nite
intersection property,19 1 (A) = k 1 k (A) is also non-empty and compact.
k
1
Since 1 (A)
(A), the monotonicity of implies that ( 1 (A))
(A).
1
1
We now show that
(A)
( (A)).
Let ai be a rationalizable action. Then for every k = 1; 2; ::: there exists a
conjecture i;k 2 ( k i (A)) such that ai 2 ri ( i;k ), where k i (A) denotes the
projection on A i of k (A). Since (A i ) is compact, we can assume without
loss of generality that the sequence of conjectures f i;k gk 1 is converges to
a limit i 2 (A i ) (any sequence in a compact set admits a convergent
subsequence, hence, even if the original sequence does not converge we can
consider any convergent subsequence). Such limit, i , necessarily assigns
probability one to 1i (A). To see this, note that
8k,8K
k,
i;K
18
(
k
i (A))
= 1,
Let C
Rm be a compact and consider the measures de…ned over the borelians of
C. A sequence of measures k convergesR in the weak
R sense to a measure if for every
continuous function f : C ! R , limk!1 f d k = f d holds.
19
The …nite intersections property for compact sets asserts that if an in…nite sequence
k
of compacts
fC k ; k 2 IgTis such that for any …nite sub-collection
T
T fCk ; k 2 F g we have
k
k
that k2F C 6= ;, then k2I C 6= ;. Moreover, the set C := k2I C is an intersection
of close sets and hence is closed. Given that C is a closed set contained in a compact set
(C C k for every k 2 I), C must also be compact.
22
hence
i
(
k
i (A))
= lim
K!1
i;K
(
k
i (A))
= 1 and
i
(
1
i (A))
= lim
k!1
i
(
k
i (A))
= 1:
Given that the best reply correspondence ri is upper hemicontinuous, ai is
also a best reply to the limit conjecture i . Hence, ai 2 i 1i (A) .
It is possible to give an alternative and useful characterization of rationalizable actions. Consider the following:
De…nition 10 A set C = C1
C
(C).
:::
Remark 8 By Theorem 5, the set
the best reply property.
Cn
1
A has the best reply property if
(A) of rationalizable action pro…les has
Theorem 6 Under the assumptions of Theorem 5, an action pro…le a 2 A
is rationalizable if and only if a 2 C for some Cartesian subset C A with
best reply property.
Proof. (Only if) If a is rationalizable then a 2 1 (A) = ( 1 (A)).
Hence, a belongs to a set with the best reply property.
(If) Let a 2 C
(C). Notice that since C
(C), C
A and
is monotone, it follows that C
(C)
(A). Applying once more
2
2
2
monotonicity, we obtain (C)
(C)
(A), so that C
(A). A
k
k
simple proof by induction can be used to show that C
(C)
(A) for
1
any k.20 Hence, a 2 C
(C).
3.4
Iterated Dominance
The equivalence between best replies and undominated actions allows us
to characterize the actions that survive the iterated dominance procedure.
In order to give the precise de…nition of this procedure, we …rst de…ne the
concept of dominance within a subset of action pro…les.
20
The reader should try to write formally and extensively the proof by induction.
23
De…nition 11 Consider a nonempty Cartesian subset of A, ; 6= C = C1
::: Cn A. An action ai 2 Ci is dominated in C if there is a mixed action
(Ci ) such that
i 2
8a
i
2 C i , ui (ai ; a i ) < ui ( i ; a i ).
We denote by N Di (C) Ai the set of actions that are not dominated in C
and with N D(C) = N D1 (C) ::: N Dn (C)
A the set of undominated
action pro…les in C.
Using the standard notation regarding the iteration of operators, we can
represent the iterated dominance procedure through the following sequence of
sets N D = N D(A), N D(N D(A)) = N D2 (A), ..., N Dk (A), ... . Essentially,
the idea is to eliminate all dominated actions, thus obtaining N D(A). Then
we move on to eliminate all dominated actions in the restricted game set of
action pro…les N D(A), obtaining N D2 (A), and so on.
De…nition 12
T a 2 A kis a pro…le of iteratively undominated actions if a 2
1
N D (A) := k 1 N D (A).
Theorem 7 (Pearce [14]) For any k = 1; 2; ::: we have that k (A) = N Dk (A).
Therefore, an action pro…le is rationalizable if and only if it is iteratively undominated.
We …rst prove a preliminary result regarding the sequence of subsets
f (A)gk 1 . For any non-empty product set C = C1 ::: Cn A let
[
arg max ui (ai ; i );
8i 2 N , i (C) =
k
i2
(C) =
(C
1 (C)
i)
:::
ai 2Ci
n (C).
Notice that since we require that ai 2 Ci , i (C) is obtained by constrained
maximization. It follows that (C)
C for any C. The rationalization
operator instead does not satisfy this property (if C, for instance, is the set
of dominated action pro…les, then (C) \ C = ;). However, iterating both
operators starting from the set A of all action pro…les, we obtain the very
same result:
24
Lemma 8 For any k = 1; 2; :::,
k
(A) =
k
(A).21
The lemma can be reformulated as follows. We say that an action ai
is iteratively justi…able if (1) it is justi…able (2) is justi…able in the reduced
game G1 obtained via the iterated elimination of non justi…able actions,
(2) it is justi…able in the reduced game G2 obtained through the iterated
elimination of all non justi…able actions in G1 , and so on. The actions that are
iteratively justi…able are exactly those that we obtain iterating the operator
. Hence, Lemma 8 states that, for every player, the set of rationalizable
actions coincides with the set iteratively justi…able actions.
Proof of Lemma 8. By de…nition we have (A) = (A). Suppose
that k 1 (A) = k 1 (A) (inductive hypothesis). We want to show that such
assumption implies k (A) = k (A).
For any player i and any integer m = 0; 1; 2; :::, we denote by m
i (A) and
m
m
22
(A) on Ai and A i respectively. De…ne m
i (A) the projection
i (A) and
m
(A)
analogously.
Using
these
de…nitions
we
obtain
i
[
k 1
k
k
arg max ui (ai ; i ).
i (A) = i
i (A) , i (A) =
ai 2 ik 1 (A)
i2
( k i 1 (A))
First of all, notice that, by the inductive hypothesis k 1 (A) = k 1 (A),
player i’s conjecture assigns probability one to k i 1 (A) if and only if it
also assigns probability one to k i 1 (A). Consider then any such conjecture
k 1
k 1
i
2
i (A) . The set of unconstrained best replies to
i (A) =
i
conjecture
satis…es:
ri ( i )
i(
k 1
i (A))
=
k
i (A)
k 1
(A)
i
where the …rst inclusion holds by de…nition and the second one because
k
k 1
(A)
(A). By the inductive hypothesis ik 1 (A) = ki 1 (A), therefore
k 1
ri ( i )
(A). Since all best replies to i satisfy the constraint ai 2
i
k 1
(A) and given that the set ri ( i ) of unconstrained best replies is noni
empty,23 it means that this set must coincide with set of constrained best
21
As should be clear from the proof, the Lemma holds also for all in…nite games with
Ai compact and ui continuous for any i.
22
Given C X Y the projection of C on X is the set fx 2 X : 9 y 2 Y; (x; y) 2 Cg.
23
This is the only point where we make use of the assumption of the game being …nite.
If Ai were in…nite, we could have that ri ( i ) is empty and the set of constrained best
25
k 1
replies: If ai 2 ki 1 (A)nri ( i ), then there is some ai 2 ri ( i )
(A) such
i
i
i
that ui (ai ; ) < ui (ai ; ). Hence,
[
k 1
k
arg max ui (ai ; i ) = ki (A):
(A)
=
(A)
=
i
i
i
ai 2 ki 1 (A)
i2
( k i 1 (A))
At this point, the theorem follows pretty straightforwardly. Lemma 3
implies that the iteratively undominated actions coincide with the iteratively
justi…able ones. By Lemma 8 the interatively justi…able actions coincide with
the rationalizable actions. The details are as follows:
Proof of Teorem 7. Lemma 3 implies that (A) = N D(A). Assume
that k 1 (A) = N Dk 1 (A) (inductive hypothesis) and consider the game
Gk 1 where the set of actions of each player i is ki 1 (A), and the utility
functions are obtained from the original game by restricting the domain to
k 1
(A). The set of undominated action pro…les in Gk 1 is N D( k 1 (A)) =
N D(N Dk 1 (A)) = N Dk (A). By Lemma 3, N D(N Dk 1 (A)) = (N Dk 1 (A)).
From the inductive hypothesis and Lemma 8 we obtain (N Dk 1 (A)) =
( k 1 (A)) = ( k 1 (A)) = k (A). Hence N Dk (A) = k (A).
Up to now we have considered a procedure of iterated elimination which is
maximal, in the sense that at any step we eliminate all the dominated actions
(in the restricted game that results from previous iterated eliminations) for
all of the players. However, it is possible to show that for the purpose of
computing the set of action pro…les that are iteratively undominated (and
therefore rationalizable), it is su¢ cient to iteratively eliminate some actions
which are dominated for some player until in the restricted game so obtained
there are no dominated actions left . Formally, we have the following result:
k
Theorem 9 Let fAk gK
::: Akn gK
k=0 = fA1
k=0 be a sequence of non-empty
0
subsets of A such that (i) A = A; (ii) for any k = 1; :::; K, N D(Ak 1 )
Ak
Ak 1 and (iii) N D(AK ) = AK . Then, AK is the set of rationalizable
action pro…les.24
replies is non-empty. Therefore the statement in the lemma holds for all games in which
for any player i and any belief i , ri ( i ) 6= ;. In particular, it holds for any game with Ai
compact and ui continuous for any i.
24
The result does not hold for any in…nite game. It does, nonotheless hold, if A is
compact and the functions ui are continuous (see Dufwenberg e Stegeman, 2002).
26
Proof (sketch). Using Lemma 3 and a proof similar to the one of Lemma
8, we obtain that for any k = 0; 1; :::; K, N D(Ak ) = (Ak ) = (Ak ). In
particular, N D(AK ) = (AK ). Since AK = N D(AK ) = (AK ), the set
AK has the best reply property and by Theorem 2 all of his elements are
rationalizable.
We now show that, for any k = 1; 2; :::; K, k (A)
Ak . We know that
(A) = N D(A) A1 . Assume that k 1 (A) Ak 1 . The monotonicity of
implies k (A)T= ( k 1 (A))
(Ak 1 ) = N D(Ak 1 ) Ak . It follows by
induction that k 1 k (A) AK .
27
4
Nash Equilibrium
A Nash25 equilibrium is a situation in which every player is rational and holds
correct conjectures about the actions of the other players.
De…nition 13 An action pro…le a = (ai )i2N is a Nash equilibrium if, for
every i 2 N , ai 2 ri (a i ).
Remark 9 An action pro…le a is a Nash equilibrium if and only if the singleton fag has the best reply property. Hence, by Theorem 6, every Nash
equilibrium is rationalizable and if there is a unique action pro…le which is
rationalizable, then it is necessarily the unique Nash equilibrium.
As it becomes obvious from the analysis of simple games (such as “Matching Pennies”), not all games possess Nash equilibria. The following classic
theorem provides su¢ cient conditions for the existence of at least one Nash
equilibrium.
Theorem 10 Consider a game G = hN; (Ai ; ui )i2N i. If, for every player
i 2 N , Ai is a (non-empty ) compact and convex subset of a Euclidean space,
the utility function ui is continuous and, for every a i 2 A i , the function
ui ( ; a i ) is quasi-concave, then G has a Nash equilibrium.
Proof (sketch). From the maximum principle, the best reply correspondences ri are upper hemicontinuous with non-empty values. Further, for any
a i , ri (a i ) is convex, as Ai is convex and ui ( ; a i ) is quasi-concave. It follows
that the correspondence r : A ) A as de…ned by r(a) = r(a 1 ) ::: r(a n )
maps from compact and convex domain into itself and it is upper hemicontinuous with non-empty and convex values. Then, by the Kakutani Theorem, r admits a …xed point, i.e. there exists an a 2 A such that a 2 r(a).
By de…nition, every …xed point of r( ) is a Nash equilibrium.
25
John Nash, who has been awarded the Nobel Prize for economic science (joint with
John Harsanyi e Reinhard Selten) in 1994, has been the …rst to de…ne such a concept.
All others equilibrium concepts used by non-cooperative game theory can be considered
as generalizations or “re…nements” of the equilibrium proposed by Nash. Perhaps, this is
the reason why all other equilibrium concepts that have been proposed do not take their
name after the one of the researchers who introduced them in the literature.
We should also add that Nash has analyzed the equilibrium concept for the mixed
extension of a game, and has proved the existence of mixed equilibria for all games with
a …nite number of pure actions (see De…nition 12 and Theorem 7).
28
4.1
Equilibria in Symmetric Games
As one can see from the sketch of proof of Theorem 10, rather advanced
advanced mathematical tools are necessary to prove a relatively general existence theorem. Here, we analyze the more speci…c case in which all players
are in a symmetric position and we show using more elementary tools that
under some regularity conditions, there exists an equilibrium in which all
players choose the same action.
De…nition 14 A static game G is symmetric if the players have the same
sets of actions, denoted byB (hence, 8i 2 N , Ai = B), and if there exists
a function v : B n ! R which is symmetric with respect to the arguments
2; :::; n such that 8i 2 N , 8a 2 A = B n
ui (a) = v(ai ; a i ):
A Nash equilibrium a of a symmetric game G is symmetric if 8i; j 2 N ,
ai = aj .
For example, the three-player game in which
aj + ak
1
(ai )2 +
2
2
ui (a1 ; a2 ; a3 ) = ai
is a symmetric game. The function v(ai ; a i ) is given by the right hand side
of the above equation. Another example of a symmetric game is Cournot
oligopoly in which all …rms have the same cost function.
Theorem 11 Let G = hN; (Ai ; ui )i2N i be a symmetric static game in which
8i 2 N , Ai = [0; 1], ui is continuous, and 8a i 2 A i , ui ( ; a i ) is strictly
quasi-concave. Then, G possesses a symmetric Nash equilibrium.
Proof. Since ui is continuous and strictly quasi-concave in ai , for any
possible conjecture regarding a i there exists one and only one best reply.
Let v be as in the previous de…ntion of a symmetric game. In this proof we
denote by r(b) the unique best reply to the symmetric conjecture (b; :::; b) 2
[0; 1]n 1 .26 In other words
r(b) = arg max v(ai ; b; :::; b):
ai 2[0;1]
26
Before we denoted with r : A ) A the joint best response correspondence, a formally
di¤erent mathematical object.
29
We now prove that the function r( ) is continuous, that is, for any sequence of actions bk k 1 such that limk!1 bk = b the corresponding sequence of best replies r(bk ) k 1 converges to r(b ) [limk!1 r(bk ) = r(b )].
Notice …rst that since the sequence r(bk ) k 1 is contained in the compact
set [0; 1], it must have at least one accumulation point ^b 2 [0; 1]. Therefore,
there exists a subsequence bk` ` 1 such that lim`!1 r(bk` ) = ^b. The de…nition of best reply implies
8ai 2 B, 8`
1, v(r(bk` ); bk` ; :::; bk` )
v(ai ; bk` ; :::; bk` ):
Taking the limit for ` ! 1 for both sides and for any given ai , and given
the continuity of v (which is implied by the continuity ui ), we get
8ai 2 B, 8`
1, v(^b; b ; :::; b )
v(ai ; b ; :::; b ):
Hence, ^b is a best reply to b . Since the best reply is unique (by strict quasiconcavity), we must have ^b = r(b ). This is true for every accumulation
point. Therefore, the sequence r(bk ) k 1 has only one accumulation point,
r(b ), which is equivalent to say that limk!1 r(bk ) = r(b ), as desired.
Let us de…ne now an auxiliary function f : [0; 1] ! [0; 1] as follows:
f (b) = b r(b). Function f is continuous because it is a sum of continuous
functions. Besides, f satis…es: f (0) 0 and f (1) 0. It follows that there
exists a point b 2 [0; 1] such that f (b ) = 0, that is b = r(b ). Such action
b is a best reply to the symmetric conjecture (b ; :::; b ) 2 [0; 1]n 1 . This
means that (b ; :::; b ) 2 [0; 1]n is a symmetric Nash equilibrium.
4.2
Interpretations of the Nash equilibrium concept
Nash equilibrium is the most well known and applied equilibrium concepts
in economic theory, besides the competitive equilibrium. Indeed, we can
argue that in principle any economic situation (and more generally any social
interaction) can be represented as a non-cooperative game. The property
according to which every action is a best reply to the other players’actions
seems to be essential in order to have an equilibrium in a non-cooperative
game.
Nonetheless, we should refrain from accepting this conclusion acritically.
Why does the Nash equilibrium represent an interesting theoretical concept?
30
When should we expect that the actions simultaneously chosen by the players constitute an equilibrium? Why should players hold correct conjectures
regarding each others behavior?
We propose a few di¤erent interpretations of the Nash equilibrium concept, each addressing the questions above. Such interpretations can be classi…ed in two subgroups: (1) a Nash equilibrium represents “an obvious way
to play”, (2) a Nash equilibrium represents a stationary (stable) state of an
adaptive process. In some cases, we will also introduce a corresponding generalization of the equilibrium concept which is appropriate under the given
interpretation and will be analyzed in the next section.
(1) Equilibrium as an “obvious way to play”: Let us assume complete information, i.e common knowledge of the game G (this assumption is
su¢ cient to justify the considerations that follow, but it is not strictly necessary). It could be the case that from the (common) knowledge of G and
shared assumptions about behaviors and beliefs, or given some prior events
that occurred before the game, the players can positively conclude that the
action pro…le a represents an “obvious way to play” G. If a represents an
obvious way to play, every player i expects that everybody else makes his
part in choosing a i . Moreover, i needs to have no incentive to choose an
action di¤erent from ai (if i were to have an incentive to choose ai 6= ai , not
only ai would not be chosen, but also the other players would have no reason
to believe that a is the obvious way to play). Hence, a must be a Nash
equilibrium.
What can make the action pro…le a an obvious way to play?
(1.a) Deductive interpretation: The players consider G as an interactive decision problem to which they want to assign a rational solution. If
such solution exists and is unique, then it corresponds to an obvious way to
play.
As an example of solution of a game we can consider the case in which G
has only a single pro…le a of rationalizable actions (this is the case for many
relevant examples analyzed by economic theory, such as many oligopoly models). Then, if all players are rational (that is, they maximize their expected
utility) and their conjectures are derived from the assumption that there is
common certainty of rationality, they will choose exactly the actions in a .
Moreover, a is necessarily the unique Nash equilibrium of the game (Remark
9).
(1.b) Non-binding agreement: Suppose that the players are able to
31
communicate before playing the game and that they reach an agreement to
choose the actions speci…ed in the action pro…le a . Suppose also that such
agreement is not legally binding for the parties, it is simply a "gentlemen
agreement" based on players honoring their words. Further, players attach
little value to honoring their words: if i believes that a di¤erent action yields
him a higher utility, he will not choose ai . All players are perfectly aware
of this. Therefore, the agreement is credible, or "self-enforcing" only if no
player has an incentive to deviate from the agreement, i.e only if a is a Nash
equilibrium.
Is it really necessary that players agree on a speci…c action pro…le? Could
not the player agree on making the action pro…le actually played (for instance,
the concert in the "Bach or Stravinsky" game) dependent on some exogenous
variable, such as the weather? In some cases, this would allow to reach, in
expectation, fairer outcomes. We will go back to this point in the next
section.
(2) Equilibrium as a stationary state of an adaptive process.27 .When
introductory economic textbooks explain why in a competitive market the
price should reach the equilibrium level that equates demand and supply,
they almost inevitably rely on informal dynamic justi…cations. They argue,
for instance, that if there is an excess demand the sellers will realize that
they are able to sell their goods for a higher price; conversely, if there is an
excess supply the sellers will lower their prices to be able to sell the residual
unsold goods. Essentially, such arguments rely more or less explicitly on the
existence of a feedback e¤ect that pushes the prices towards the equilibrium
level. These arguments cannot be formalized in the standard competitive
equilibrium model, where market prices are taken as given parameters by
the economic agents. They nonetheless provide an intuitive support to the
theory.
Similar arguments can be used to explain why the actions played in a
game should eventually reach the equilibrium position. In some sense, the
conceptual framework provided by game theory is better suited to formalize
this kind of arguments. Indeed, any "endogenous" variable in a game is
directly or indirectly determined by the players’actions, according to precise
rules that are part of the game (contrary to the prices in a competitive market
27
To go deeper on this topic the reader can refer to the survey by et al. (1992). In
the already mentioned paper by Migrom e Roberts (1991), the connections between the
concept of rationalizability and adaptive processes are explored.
32
model, where the price-formation mechanism is not speci…ed). Assuming that
the given game represents a interactive situation that players face repeatedly,
we can formulate assumptions regarding how players modify their actions
taking into account the outcomes of previous interactions, thus representing
formally the feedback process.
We can distinguish two di¤erent types of adaptive dynamics: learning
and evolutionary dynamics. Here, we will present only a general and brief
description of both.
(2.a) Learning dynamics. Assume that a given game G is repeated
over time and that players are interested in maximizing their current expected
payo¤ (a reason for this may be that they do not value the future or that they
believe that their current actions do not a¤ect in any way future payo¤s).
Players have access to information regarding previous outcomes. Based on
such information, they modify their conjectures regarding their opponents
behavior in the current period. Let a be a Nash equilibrium. If in period t
every player i expects his opponents to choose a i , then ai is one of his best
replies. It is therefore possible that i chooses ai . If this happens, what players
observe at the end of period t will con…rm their conjectures, which then will
remain unchanged for the following period. So even a small inertia (that
is a preference, ceteris paribus, to repeat the previous action) will induce
players to repeat in period t + 1 the previous actions a . Analogously, a
will be played also in period t + 2 and so on. Hence, the equilibrium a is a
stationary state of the process.
We just argued that any Nash equilibrium is a stationary state of plausible learning processes (we presented the argument in an informal way, but
a formalization is possible). Can we assert that (i) such processes do always
converge to a Nash equilibrium? and (ii) for any plausible learning process
every stationary state is a Nash equilibrium? It is not possible, in general,
to give a¢ rmative answers. First, we need to be more speci…c regarding
the dynamics of the repeated interaction. For instance, we need to specify
exactly what players are able to observe regarding the outcomes of previous
interactions: is it all the actions chosen by the other players? Is it some
variables that depend on such actions? Is it only their own payo¤s? We
also need to specify whether game G is played always by the same players,
or in every period players are randomly drawn from a large population of
individuals (think about the “tra¢ c game" for instance). Specifying precisely these assumptions, we can reach the conclusion that the process does
33
not always converge to a stationary state. Moreover, as we will see in the
next section, there can exist stationary states that do not satisfy the Nash
equilibrium condition of De…nition 13. There are two reasons for this: (1)
Players’conjectures can be con…rmed by observed outcomes even if they are
not correct. (2) If players are randomly drawn from large populations, then
the state variable of the dynamic process is given by the fraction of individuals in the population that choose the various actions. But then it is
possible that in a stationary state two di¤erent individuals, playing in the
same role, choose di¤erent actions. Such situations look like "mixed equilibria", in which players choose randomly among some of their best replies to
their probabilistic conjectures, and such conjectures are correct. We analyze
notions of equilibrium corresponding to situations (1) and (2) in the next
section.
(2.b) Evolutionary dynamics. In the analysis of adaptive processes
in games, the analogy with evolutionary biology has often been exploited.
Consider, for instance, a symmetric two-person game. In every period two
individuals are drawn from a large population. Then they meet, they interact,
they obtain some payo¤ and …nally they split. Individuals are compared to
animals, or plants, whose behavior is determined by a genetic pattern transmitted to the o¤springs. If a is more successful than b, then the individuals
“programmed to play a”reproduce themselves faster than those programmed
to play b, and therefore the ratio between the fraction of individuals playing
a and the fraction of individuals playing b increases.
In evolutionary biology this theoretical approach based on game theory
has been highly successful and has allowed to explain phenomena that appeared as paradoxical within a more naive evolutionary framework. In the
social sciences, evolutionary dynamics are used as a metaphor to represent,
at an aggregate level, learning phenomena such as imitation, in which the
individuals modify the way they play not on the basis of their direct experiences alone, but also observing the behavior of others in the same circumstances. Behaviors that turn out to be more successful are imitated spread
more rapidly.
As argued for learning dynamics, although Nash equilibria represent stationary states of evolutonary dynamics, it is not guaranteed that the process
always converges to a stationary state. Further, there may be stationary
states that do not satisfy the best reply property of De…nition 13. Indeed,
in this context a state is represented by the fractions in the population that
use di¤erent actions. It may be the case that in a stationary state distinct
34
individuals of the same population do di¤erent things.
These consideratons motivate the de…nition of more general equilibrium
concepts whereby players may hold probabilistic, rather than deterministic
conjectures about the opponents’behavior. For this reason we refer to these
more general concepts as "probabilistic equilibria".
[NOTE: I HAVE NOT YET CHECKED THE TRANSLATION
FROM ITALIAN TO ENGLISH OF THE FOLLOWING SECTIONS!!!]
35
5
Probabilistic Equilibria
Up to now we have considered mixed actions within dominance relations: if
a (pure) action is dominated by a mixed action, then a rational player should
not choose it. We already noticed (Lemma1), though, that a player has no
incentive to choose a mixed action. If there were any small cost involved
in tossing a coin or in making the choice dependent on a roulette, then no
rational player would choose a mixed action.
We adopt the point of view that (rational) players do not choose mixed
actions. We aim, nonetheless, to show that is possible to reconcile this point
of view with an interpretation of the mixed actions that renders them relevant
independently of the equivalence results stated in Lemma 2 and Theorem 2.
5.1
Mixed Equilibria
Let us introduce the following game between owners and thieves. The owners
(population 1) own a ‡at (each owns one) where they store goods worth a
total value of V . They have the option whether to install or not an alarm
system, costing them c < V (we do not consider the insurance option as
available to the owners). The alarm system is not detectable by the thieves.
The thieves (population 2) can decide whether to attempt a robbery or not.
In the event that no alarm system is installed, the thieves successfully seize
the goods and resell them to some dealer, making a pro…t of V =2. Conversely,
if the alarm system is installed, the attempted robbery is detected and the
police is automatically alerted. The thieves in such event need to leave all
goods in place and try to escape. The probably they get caught is 12 , in which
case the thieves are sanctioned with a monetary …ne P and then released. 28
If the one described were a simple game between one owner and one thief,
it could be represented using the matrix below:
PnL
Robbery
No
Alarm: V c,
P=2 V c, 0
No
0, V =2
V, 0
Matrix 2
It is easy to check that Matrix 2 possess no Nash equilibria.
28
Prisons do not exist. If they cannot pay the amount P , they are in‡icted an equivalent
corporal punishment
36
However, the game between owners and thieves is somehow more complex.
A large population of owners and a large population of thieves exist. For
simplicity we take the two populations as having the same size. Thieves
randomly distribute themselves across ‡ats. The probability that a robbery
is attempted in any given ‡at is equal to the fraction of thieves that decide
to attempt a robbery. From the thieves’ perspective, the probability that
a ‡at is equipped with the alarm system is given by the overall fraction of
owners buying the alarm system.
Assume that the game is repeated over time and that c represents not
just the initial installation cost but also the alarm system maintenance cost.
The fractions of individuals that choose the di¤erent actions evolve according
to some adaptive process that we only roughly describe. At the end of each
period is possible to access (reading them on the newspapers) the statistics
regarding the numbers of attempted robberies that resulted in a success or a
failure. Players are fairly inert in that they tend to replicate their previous
period actions. However, they occasionally decide to revise their choices on
the basis of the previous period statistics.
An owner not equipped with an alarm system decides to install one if
and only if the attempted robberies are in a proportion bigger than c=V .
Conversely, an owner equipped with an alarm system will decide to get rid
of it if and only if the attempted robberies are in a proportion lower than
c=V . The percentage of attempted robberies only slowly changes and the
owners identify the probability of being robbed in today’s period with the
one of the previous period. c=V is the probability that makes the owner
indi¤erent between his two actions. In such a case, the owner deciding to
revise his choice will opt for the conservative choice and will stick to the
previous period action.
Analogously, a thief that was active in the previous period decides not
to attempt a robbery in the current period if and only if the percentage of
‡ats equipped with an alarm system (this is also the fraction of unsuccessful
robberies) is bigger than V =(V + P ). A thief that was not active in the
previous period attempts a robbery in the current one if and only if the
percentage of unsuccessful robbery is lower than V =(V + P ).
This process leads to an equilibrium in which the percentage of installed
alarm system is V =(V + P ) and the percentage of attempted robberies is
c=V . Denote by 1 the percentage of installed alarm system and by 2 , the
percentage of attempted robberies. It can be veri…ed that if 2 > c=V then
1 will grow, while if 2 < c=V
1 will decrease. Similarly, if 1 > V =(V +P )
37
then 2 will decrease, while if 1 < V =(V + P ) then 2 will grow. If 1 =
V =(V + P ) and 2 = c=V then the state of the system is not altered.
In this example we have interpreted the mixed action, j , both as statistical distribution of the actions in populationj and as a belief of the individuals
of population i over the action that would be chosen by an individual of population j with whom they are matched. In a stable environment every belief,
j , is correct in the sense that it corresponds to the statistical distribution of
the actions in population j. Also, j , is such that any individual in population i is made indi¤erent among the actions that are chosen by a positive percentage of individuals. The mixed action pair ( 1 ; 2 ) = (V =(V + P ); c=V )
is said to be a mixed equilibrium of the matrix game above. We now present
a general de…nition of mixed equilibrium.
De…nition 15 The mixed extension of a game G = hN; (Ai ; ui )i2N i is a
game G = hN; ( (Ai ); ui )i2N i where
X
Y
ui ( ) =
ui (a1 ; :::; an )
j (aj );
j2N
(a1 ;:::;an )2A
for any i and
=(
1 ; :::;
n)
2
(A1 )
:::
(An ).
Notice that the utility functions of the mixed extension are obtained
calculating the expected utility corresponding to a vector of mixed actions,
assuming that the actions of di¤erent players are statistically independent.
De…nition 16 (see Nash, 1950) A mixed action pro…le is a mixed equilibrium of the game G if is a Nash equilibrium of the mixed extension of
G.
Theorem 12 Any …nite game possesses at least a mixed equilibrium29 .
Proof. It is easy to verify that the mixed extension G of a …nite game, G,
satis…es all the assumptions of Theorem 5: for any i, (Ai ) is a convex and
compact set and ui is a multi-linear function and therefore both continuous
and (weakly concave) in i .
29
The same holds for any compact and continuous game.
38
The following result introduces an alternative characterization of mixed
equilibria, which is not based on the mixed extension of the game. With
a small abuse of notation we denote by ri ( i ) the set of best replies to
i
i
the
obtained as a product measure from
(a i ) =
i (i.e
Q conjecture
(a
)).
j6=i j j
Theorem 13 A mixed action pro…le ( 1 ; :::; n ) is a mixed equilibrium if
and only if Supp i ri ( i ), for any i 2 N (that is any action played with
positive probability is a best reply to the mixed action pro…les played by the
opponents)
Proof It follows directly from the de…nition of mixed equilibrium and
from Lemma 1 1.
Corollary 14 All actions that are played with positive probability in a mixed
equilibrium are rationalizable
Proof. By Theorem 8 13, Supp 1 ::: Supp n possesses the best reply
property. Hence, Theorem 2 implies that any action played with positive
probability is rationalizable.
Theorem 7 along with Corollary 2 provides us with an algorithm to compute all mixed equilibria in a …nite two persons game. Let [uk`
i ] be the payo¤
(or utility) matrix for playeri. Player 1 chooses the rows (indexed by k) and
player 2 the columns (indexed by l)
Step 1: Eliminate all iteratively dominated actions (by corollary 1 such
actions are played with zero probability in equilibrium). The order of elimination is irrelevant (Theorem 4)
Step 2: For any pair of non-empty subsets A1
A1 and A2
A2 ;
compute the set of mixed equilibria, ( 1 ; 2 ), such that Supp 1 = A1 and
Supp 2 = A2 . Such set, which could be empty, it is computed as follows (let
us consider the non trivial case in which the sets contain at least two actions).
Assume to simplify notation that A1 = f1; :::; m1 g and A2 = f1; :::; m2 g and
denote by m
i the generic probability that action m by player i is played.
39
Solve the following systems of linear equations and inequalities with unknown
1 and 2 :
m1
X
uk`
2
k
1
=
k=1
m1
X
uk`
2
m1
X
k
1
`=1
k
1,
` = 2; :::; m2 ;
uk1
2
k
1,
` = m2 + 1; :::; jA2 j;
u1`
1
`
2,
k = 2; :::; m1 ;
u1`
1
`
2,
k = m1 + 1; :::; jA1 j:
k=1
uk`
1
`
2
`=1
m2
X
uk1
2
(1)
k=1
k=1
m2
X
m1
X
=
m2
X
(2)
`=1
uk`
1
`
2
m2
X
`=1
The subset of equations in (1) determines the set of mixed actions for
player 1 that make player 2 indi¤erent between the actions in the subset A2 .
The subset of inequalities determines the set of mixed actions of player 1
that make the action a2 = 1 (and so all the actions in A2 ) weakly preferred
to the actions that do not belong to the subset A2 . For any 1 that satis…es
the system (1), player 2 has no incentive to ”deviate” from a mixed action
with support A2 . Similar considerations hold for system (2). Such system
determines the set of mixed actions of player 2 that make player 1 indi¤erent
among all actions in A1 and at the same time such actions weakly preferred
to all the others. The following fundamental remark for the computation of
two players mixed equilibria holds true:
The indi¤erence conditions for player 1 determine the equilibrium randomization(s) of player 2, The indi¤erence conditions for player 2 determine
the equilibrium randomization(s) of player 1
In the previous example (Owners and Thieves) the equilibrium is so determined:
First, it happens that the best reply to a sure belief is unique (for instance,
if we were to believe that the probability of a robbery was zero, then the
unique best reply would be not to install the alarm). However, no pairs of
pure actions constitute an equilibrium. Hence, the equilibrium (which we
know to exist by Theorem 6) is necessarily mixed. Let us compute then the
40
indi¤erence conditions that characterize an equilibrium with supports A1 =
fAlarm; N og, A2 = fRobbery; N og. Let 1 = Pr(Alarm), 2 = Pr(Robbery)
Indi¤erent conditions for i = 1 (Owner):
V
c = V (1
2 ):
Indi¤erent conditions for i = 2 (Thief):
V
(1
2
Solving the system we get
5.2
1
=
1)
V
,
V +P
P
2
2
=
1
= 0:
c
V
.
Correlated Equilibria
The mixed equilibria identify probability distributions over the set of action
pro…les A that are in some sense stables. Such distributions are obtained
as a product of the marginal distributions corresponding to the equilibrium
mixed actions: if ( 1 ; :::;Qn ) is a mixed equilibrium, then the probability of
any pro…le (a1 ; :::; an ) is ni=1 i (ai ).
If we follow the interpretation of mixed equilibrium as a stable vector of
statistical distributions of actions in n populations whose individuals are randomly matched (or more generally grouped) with the individuals of the other
populations to play the game G, then it makes sense to consider the product
distributions over A, that is the distributions that satisfy the property of
statistical independence.
Conversely, now we introduce a di¤erent concept of probabilistic equilibrium which, in a strictly mathematical sense, generalizes the concept of
mixed equilibrium and induces a correlated distribution over A, that is a
probability distribution in which the actions of di¤erent players are not necessarily independent. As we will see, the interpretation that we give to this
equilibrium concept is rather di¤erent from the one proposed for the mixed
equilibrium.
Let us consider once again the case in which players are able to communicate and sign a non binding agreement before the start of the game. If said
agreement consists only in playing a certain pro…le a then, as noted earlier, such pro…le must be a Nash equilibrium. However, players could reach
more sophisticated agreements where the chosen actions are made dependent
41
on random variables even if such variables do not have any direct e¤ect on
utilities.
B
S
B 3,1 0,0
S 0,0 1,3
Matrix 3
Assume for the sake of an illustration that Row and Column want to
agree on how to play the Battle of the Sexes the next day and that the
next day no further communication is possible. There exists two simple
agreements that yield a consistent behavior: (B; B) and (S; S). However,
the …rst favors Row and the second Column and none of them wants to give
up. How to sort this dead end? Column can make the following proposal that
would ensure in expected terms a fair distributions of the gains achievable
through cooperation: “If tomorrow’s weather is good then both of us choose
B, if instead it is bad then we both choose S.” Notice that the weather
forecasts are uncertain: there is a 50% probability of good weather and 50%
of bad weather. The agreement generates an expected gain of 2 for both
players. Row immediately understand that the idea is very smart. In fact,
the agreement is credible as both players have an incentive to respect it if
they expect the other to do the same. For instance, if Row expects that
Column respects the agreement and waking up he observes that the weather
is good, then he should play B based on the presumption that Column will
do the same. Analogously, if he observes that the weather is bad he should
expect Column to play S and thus he should do the same.
In other words, a sophisticated agreement can use exogenous and not
directly relevant factors to coordinate players beliefs and behavior. In said
agreement the conjecture of the players regarding others behavior (and so
their best replies) depend on the observation of such factors.
Clearly, it is not always possible to condition the choice on some commonly observable factor. Further, even if that were possible, players could
still perceive as more reasonable to base their respective choices on random
variables only partially correlated.
De…nition 17 (see [1] and [2]) A probabilistic non binding agreement or
correlated equilibrium, is a structure h ; p; ( i ; i )i2N i, where ( ; p) is a probability space (p 2 ( )), and the functions i : ! Ti and i : Ti ! Ai are
42
such that,
8i 2 N; 8ti 2 Ti ; 8ai 2 Ai , if p (f! 0 : i (! 0 ) = ti g) > 0 then
X
X
p(!jti )ui ( i (ti ); i ( i (!)))
p(!jti )ui (ai ; i ( i (!))) :
!
!
In the above de…nition
served by player i, whereas
agreement on player i,
p(!jti ) =
represents the random variable (signal) obrepresents the decision criteria imposed by the
i
i
p(!)=p (f! 0 :
0;
i (!
0
) = ti g) ; se
se
i (!)
= ti
(!)
=
6
ti
i
is the probability of state ! conditional on observing ti (which is well de…ned
if the probability ti is positive) and
i(
i (!))
:= ( 1 ( 1 (!)); :::;
i 1 ( i 1 (!)); i+1 ( i+1 (!)); :::; n ( n (!)))
is the action pro…le that would be chosen by the other players (according
to the agreement) if state ! is observed. The de…nition establishes that no
players has an incentive to deviate from what it is prescribed to him in the
agreement, whatever the observed realization of the random variable might
be. This is so as the expected utility achievable through a deviation is lower
or equal than the conditional expected utility that could be obtained following
the agreement.
Clearly, a correlated equilibrium induces a probability distribution 2
(A) over players action pro…les:
8a 2 A, (a) = p (f! : 8i 2 N; i ( i (!)) = ai g) :
(3)
The distribution
will be a product measure if and only if the random
variables i (i 2 N ) are mutually independent. In this case the marginals of
over the sets Ai (i 2 N ) form a mixed strategy equilibrium.
Remark 10 If h ; p; ( i ; i )i2N i is a correlated equilibrium, then the structure hA; ; ( 0i ; 0i )i2N i where satis…es3), 0i (a1 ; :::; an ) = ai and 0i (a0i ) = a0i
for all i 2 N , (a1 ; :::; an ) 2 A and a0i 2 Ai , is a correlated equilibrium. The
probability measure
2 (A) is said correlated equilibrium in canonical
form.
43
Remark 11 Any correlated equilibrium in canonical form satis…es the following system of linear inequalities in the probabilities (a):
X
8i 2 N; 8ai ; a0i 2 Ai ,
(ai ; a i ) [ui (ai ; a i ) ui (a0i ; a i )] 0:
a
i 2A i
Hence, the system of correlated equilibria in canonical form is a convex and
compact polyhedron.
Theorem 15 If is a correlated equilibrium in canonical form, then any
action to whom assigns a positive marginal probability is rationalizable.
Proof. Let Ci = SuppAi be the set of i’sactions to which assigns a
positive probability. We show that C = C1 ::: Cn is a set with the best
reply property. This implies that any action ai 2 Ci is rationalizable (by
Theorem 2). We denote the marginal and conditional probabilities using a
very natural notation: (ai ), (a i jai ), (aj jai ).
For any i and ai , if ai 2 Ci = SuppAi then (by de…nition) (ai ) > 0 and
the conditional probability ( jai ) 2 (A i ) is well de…ned. Also, (aj jai ) >
0 only if (aj ) > 0. Hence, Supp ( jai )
C i . Finally, given that is a
correlated equilibrium it must be the case that
X
X
(a i jai )ui (ai ; a i )
(a i jai )ui (a0i ; a i );
8a0i 2 Ai ,
a
i 2A i
that is to say , ai 2 ri ( ( jai )). It follows that C possesses the best reply
property.
The concept of correlated equilibrium can be generalized assuming that
di¤erent players can assign di¤erent probabilities to the states ! 2 . In that
case the equilibria are referred to as subjective correlated equilibria (Brandenburger e Dekel, 1987). It can be shown that any action ai is rationalizable
if and only if there exists a subjectiive correlated equilibrium in which ai is
chosen in at least a state ! (see section 7.3.5).
44
5.3
Conjectural Equilibria
We conclude this section on probabilistic equibria introducing another generalization of the Nash equilibium concept. In the previous section we mentioned the possibility of interpreting the equilibrium concept as the stationary
state of a learning process as well as the fact that stationary states that are
not Nash may exist. Consider the following simple example.
1/2 c
d
a
2,0 2,1
b
0,0 3,1
Matrix 4
The game in Matrix 4 has a unique rationalizable outcome, and so a
unique Nash equilibrium (and a unique degenerate mixed equilibrium that
coincides with the pure strategy equilibrium). If First (player 1) knew the
payo¤ function of Second (player 2) and were to believe that Second is rational, then he could expect him to play action d and would choose b.
Instead, let us assume the following (i) information is incomplete: every
player knows his payo¤ function but not necessarily his opponents’one; (ii)
the game is repeated over time and at the end of each period players observe
what their payo¤ is but they do not observe directly the opponent’s action;
(iii) in any period players form probabilistic conjectures over the opponents
actions that are revised according to observed actions; when players observe
something that was expected with probability one they do not revise their
conjectures. Also let us assume for simplicity that players maximize the
current expected payo¤ without worrying about future payo¤s.
As far as Second is concerned, the situation is rather simple: he will
always choose his dominant action d. Conversely, the situation for First is
not so simple. If in any given period t First expects with probability bigger
than 31 that Second chooses c, then First will choose a. In such a case his
payo¤ is equal to 2 independently of the actual choice on the part of Second
and so First is not able to infer anything regarding said choice from observing
the realization u1 = 2. In other words, First observes something that he was
expecting with certainty (to gain a payo¤ of 2) and thus does not revise his
probabilistic conjecture over Second’s action. Thus also in period t+1 he will
repeat the same choice a and according to the same reasoning his conjecture
remains una¤ected. We can conclude that the outcome (a; d) is a stationary
45
state of the learning process, even if it is not an equilibrium in the sense of
De…nitions 12 (Nash equilibrium) or 14 (mixed equilibrium).30
The example just described provides a situation in which each player
chooses a best reply to his conjectures and the information obtained regarding the players after that the choices have been made does not induce them
to change their conjectures, even if such conjectures may be incorrect. Situations of this kind are known as “conjectural ”or “self-con…rming”equilibria.31
5.3.1
Conjectural Equilibria with Pure Actions
As it is apparent from the previous example, in order to verify if a certain
situation is a conjectural equilibrium, we need to formulate some assumptions regarding what players are able to observe ex post regarding the others
behavior. We represent the possible observations that any player i might
incur into with a signal function, i : A ! Mi , where Mi is a set of "messages" that i could receive at the end of each game period. Assuming that
i remembers his choice, what i can infer when at the end of the period he
receives message mi after having played ai is that the players must have
played an action pro…le from the set fa i : i (ai ; a i ) = mi g (In the previous example we had that 1 (:) = u1 (:), M1 = f0; 2; 3g, mi means "You
got mi euros", fa2 : 1 (a; a2 ) = 2g = fc; dg, fa2 : 1 (b; a2 ) = 0g = fcg,
fa2 : 1 (b; a2 ) = 3g = fdg). If i had initially assigned probability one to
the set fa i : i (ai ; a i ) = mi g and then observed mi after having chosen ai ,
then the conjecture of i would be con…rmed and i would stick to his initial
conjectures. Having said that, we are now ready to introduce a more formal
de…nition of conjectural equilibrium:
De…nition 18 Let = ( i : A ! Mi )i2N be a pro…le
of signal functions.
Q
i
A pro…le of actions and conjectures (ai ; )i2N 2 i2N Ai
(A i ) is a conjectural equilibrium (or self-con…rming) if for any i 2 N the following
30
To the previous reasoning one could object that if First is patient then he will want
to experiment action b so as to be able to observe (indirectly) Second’s behavior, even
if that implies an expected loss in the current period. The objection is only partially
valid. It can be shown that for any "degree of patience" (discount factor) there exists a
set of initial conjectures that induce First to choose always the "safe" action a rather than
experimenting using b. It is true, however, that the more First values future payo¤s, the
less is it plausible that the process will be "trapped" in (a; d).
31
For an analysis of how this concept has originated and his relevance for the analysis
of adaptive processes in a repeated iteration context see the survey by Battigalli et al.
(1992).
46
conditions hold:
(1) ( rationality) ai 2 ri ( i ),
(2) ( conjectures’con…rmation property)
1:
5.3.2
i
a
i
:
i (ai ; a i )
=
i (ai ; a i )
Conjectural Equilibria with Mixed Actions: the anonymous
interaction case
We motivated the concept of mixed equilibrium as a possible stationary state
of adaptive processes in situations of anonymous repeated interaction in
which in any period the individuals are randomly drawn from large populations, grouped to play a given game and then splitted. The concept of
conjectural equilibrium can also be generalized to capture stable states of
repeated anonymous interactions. Since it does not have to be the case that
all individuals from i-th population have the same beliefs, we should allow
for some heterogeneity. In general, it is allowed that di¤erent individuals
belonging to the same population choose in a stable state di¤erent actions
that are justi…ed by di¤erent beliefs.
Let us consider here the case in which all individuals of the i-th population
choose the same action and also hold the same belief. It can be shown that
this restriction does not cause a real loss of generality.
In the case of anonymous interaction, the message that i receives at the
end of the game, given that he plays action ai , depends on how the players
with whom he is matched play. However, given that the matching is random,
the received message is also random. Then, it must be the case, for the beliefs’
con…rmation condition to hold, that the subjective probability distribution
over the messages coincides with objective one, which is determined by the
fractions of the individuals of the other populations that choose the various
actions.
Let us try to clarify this point by means of an example. Consider the
following variant of the above example:
1/2 c
d
a
2,3 2,1
b
0,0 3,1
Matrix 4bis
We will refer to the individuals of population 1 as "Misters Row" and to
the individuals of population 2 as "Madams Column". Assume that 25% of
47
=
the madams choose c so that 75% choose d. Then the probability that a
Mr. Row receives the message "You got zero euros" when he chooses b is
1
. Mr. Row does not know this percentages and could, for instance, believe
4
that c happens with probability 12 , which would induce him to choose a;
alternatively a probability of 51 , would induce him to play b. It may happen,
for instance, that half of Misters Row believe that Pr(c) = 21 and the other
half believe Pr(c) = 15 The former ones will choose a, the latter ones b. If
the percentages of Madams Column that play c and d remain unaltered to
25% and 75% respectively, then those who play b will observe"Zero euros"
25% of the times and "Three euros" 75% of the times. They will then realize
that their beliefs were not correct and it is natural to believe that they will
keep on revising them until eventually they will coincide with the objective
proportions, 25%-75%. These Misters Row will continue to play b. The other
half of Misters Row that believe that Pr(c) = 12 and choose a, do not observe
anything new and therefore keep on believing and doing the same things.
But is it possible that the proportions of Madams Column stay constant?
It is. Suppose that 25% of Madams Column believe that Pr(a) = 25 and the
rest believe that Pr(a) = 15 . The former ones will choose c (expecting an
expected value of 56 > 1), the latter ones will choose d (as 1 > 35 ). The ones
choosing d do not receive any new information and keep on doing the same
thing. The ones that choose c half of the times will observe "Three euros",
and the other half "Zero Euros". Their belief is not con…rmed but given
that they get three euros with a higher percentage than expected, they will
continue choosing c. Their beliefs will revise until they will coincide with the
actual proportions, 50%-50%.
We then have a stable situation characterized by the following fractions
, or mixed actions: 1 (a) = 21 , 2 (c) = 14 . Even though stable, such situation does not correspond to a mixed equilibrium. In fact, the necessary
indi¤erence conditions for a mixed equilibrium require 1 (a) = 13 , 2 (c) = 31 .
We move to a more general de…nition. Denote by Pr(mi ; ai ; i ) the probability of message mi given the action ai and the belief i , i.e.
X
i
Pr(mi jai ; i ) =
(a i ).
a
Analogously, Pr(mi ; ai ;
i)
i : i (ai ;a i )=mi
is the probability of mi given ai and the mixed
48
action pro…le
i
Pr(mi jai ;
i)
X
=
a
i : i (ai ;a i )=mi
Y
j (aj ).
j6=i
De…nition 19 Let
= ( i : A ! Mi )i2N be a signal functions
pro…le.
Q
i
A pro…le of mixed actions and beliefs ( i ; ( ai )ai 2Ai )i2N 2 i2N (Ai )
[ (A i )]Ai is a -anonymous conjectural (or self-con…rming)equilibrium if
for any i 2 N and any ai 2 Ai the following conditions hold:
(1) ( rationality) if i (ai ) > 0 then ai 2 ri ( iai ),
(2) ( beliefs’con…rmation property) Pr( jai ; i ) = Pr( jai ; i ).
[Notice that the pro…le ( i ; ( iai )ai 2Ai )i2N speci…es for any action a corresponding belief and that condition (2) requires such belief to be con…rmed.
Indeed, it could be su¢ cient to specify a belief (requiring it to be con…rmed)
only for those actions that are played with positive probability, i.e. those
played by a positive fraction of individuals. We adopted this slightly redundant convention, only as to be able to simplify a bit the notation]
It can be checked as an exercise that the game in matrix 4bis, assuming
that the players observe ex post only their own payo¤, admits three types of
anonymous conjectural equilibrium:
1. a is chosen by all the individuals in population 1: 2 can take any value
since the belief of 1 is no matter what "con…rmed" (1, choosing a, does
1
, 0
1,
not receive new information): 1 (a) = 1, 1a (c)
2 (c)
3
1
2
2
(a)
=
1,
(a)
;
c
d
3
2. d is chosen by all individuals in population 2: 1 can take any value
since the belief of 2 is no matter what "con…rmed" (2, choosing d, does
1
not receive new information)): 0
1, 1a (c)
, 1b (c) = 0,
1 (a)
3
1
2
;
2 (c) = 0, d (a)
3
3. all actions are chosen by a positive fractions of individuals: the beliefs
of those who choose "informative" actions (b for i = 1 and c for i = 2)
1
1
are correct; 31
1, 0 < 2 (c)
, 1a (c)
, 1b (c) = 2 (c),
1 (a)
3
3
1
2
2
.
c (a) = 1 (a), d (a)
3
Notice that the equilibria of type 1 include the Nash equilibrium (a; c),
those of type2 include the Nash equilibrium (b; d), and those of type 3 include
the mixed equilibria 1 (a) = 31 , 2 (c) = 13 .
49
5.3.3
A comment regarding the observability of actions, Nash and
Conjectural equilibria
In simultaneous moves games, if at the end of the game it is possible to
perfectly observe the opponents’ actions, then a conjectural equilibrium is
necessarily a Nash equilibrium. Indeed, in this case, the beliefs result to be
con…rmed if and only if they are correct.32
However, static games are also used to represent the equilibria of corresponding dynamic games. In this case the set of choices does not represent
simply the set of actions, but also a more complex set of strategies, i.e. of
plans that specify which actions to choose for any plausible instance of play.
Even if at the end of the game it is possible to observe all the actions that
have actually been taken, it is however not possible to observe how the opponents would have played in circumstances that did not occur. Hence, it is
impossible, at least for some player, to observe the strategies of other players.
For this reason, in dynamic games it is easier to …nd conjectural equilibria
that do not correspond to Nash equilibria.
32
In the case of anonymous interaction, we can consider at least two scenarios: (a) the
individuals observe the statistical distribution of the actions in previous periods, (b) the
individuals observe the long run frequencies of the opponents actions. In both cases we
can say that a belief is correct if it corresponds to the observed sequences.
50
6
Learning Dynamics,Equilibria and rationalizability
We have up to now avoided entering into the details regarding the learning
dynamics. The analysis of such dynamics requires the use of mathematical
tools whose knowledge we cannot take for granted here (di¤erential equations,
di¤erence equations, stochastic processes). Nonetheless, we can state some
elementary results regarding such dynamics asking ourselves the following
question: when can we say that a trajectory, that is a …nite sequence of
t
action pro…les (at )1
t=1 (with a 2 A), is compatible with an adaptive learning?
We can start by providing a qualitative answer.
Let us consider a …nite game that is played repeatedly. Assume that all
opponents’actions are observable and consider the point of view of a player
i that observes that a certain pro…le a i has not been played for a long time,
say for at least T periods. Then it is reasonable to assume that i assigns to
a i a very small probability. If T is su¢ ciently large, and a i was not played
in the periods t^, t^ + 1, ..., t^ + T , ..., t^ + T + k, then the probability of a i
in t > t^ + T will be so small that ai ; the best reply to i’s belief in t will
also be the best reply to beliefs that assign to a i probability zero. In other
words, i will choose in t > t^ + T only those actions that are best replies to
< t , i.e. only actions on the set
beliefs i such that Supp i
a i : t^
33
ri
< t . Notice that this argument assumes only that i
a i : t^
is able to compute the best replies and it is therefore compatible with a high
degree of incomplete information regarding the rules of the game and the
opponents preferences.
6.1
Compatibility with adaptive learning in the case
of perfectly observable actions
Building on this intuition, we can use the rationalization operator to de…ne
the compatibility of a trajectory (at )1
t=1 with the adaptive learning under the
assumption that the opponents’actions are perfectly observable. Recall that
33
Remember that Ai is …nite and ui (ai ; i ) is continuous (in fact linear) with respect to
probabilities ( i (a i ))a i 2A i . It follows that if ai is a best reply to a belief that assigns
a "very small" probability to a i , then ai is a best reply also to a slightly di¤erent belief
that assigns probability zero to a i .
51
for any given product set C1 Q::: Cn
A we de…ned the set of pro…les
"justi…ed" by C, i.e. (C) = i2N ri ( (C i )). Now we are interested in
considering the action pro…les that are chosen in a given interval of time.
Such set is not, in general, a Cartesian product. For this reason, it is useful
to generalize the de…nition of as follows: Let C
A (not necessarily a
product set).
Y
(C) =
ri
projA i C ;
i2N
where projA i C = fa i : 9ai 2 Ai ; (ai ; a i ) 2 Cg is the set of the other players’action pro…les that are compatible with C. (In case C is a product set we
obtain the de…nition we provided earlier.) Notice that, even with this more
general de…nition, , the operator is monotone: E F implies (E)
(F ).
According to the reasoning developed above, if players base their beliefs
on past observations, they choose rationally. Then, if for a "very long" period
of time only action pro…les in C are observed, it must be that the players’
current action pro…le belongs to (C). This explains the following de…nition.
De…nition 20 A trajectory (at )1
t=1 is compatible with adaptive learning
(given that the others’ actions are perfectly observable) if for any t^ there
exists a T such that, for any, t > t^ + T , at 2
a : t^
<t .
We de…ne in which sense a trajectory (at )1
t=1 generates a"limit distribution" 2 (A).
De…nition 21
2
(i)
(A) is the limit distribution of (at )1
t=1 if for any a 2 A
lim
t!1
jf : 1
< t; a = ag j
= (a)
t
(ii) if (a) = 0, then 9t^ : 8t > t^, at 6= a:
The de…nition requires that the "long run frequency" of any a is (a),
and furthermore that if (a) = 0 there exists a time starting from which the
pro…le a is no longer chosen. Observe that if (a ) = 1 for a given a we
obtain as a particular case the de…nition of convergence of a trajectory to
t
the pro…le a : (at )1
t=1 converges to a , and we write a ! a , if it exists a T
t
such that, for any t > t^, a = a (this is the standard notion of convergence
that we use for spaces given by isolated points). Also, notice that not all
trajectories admit a limit distribution.
52
We are now able to introduce a few elementary results that relate adaptive
learning with the equilibrium concepts introduced earlier. The …rst results
concerns the distributions of a correlated equilibrium and identi…es a su¢ cient condition for the compatibility with adaptive learning.34
t 1
Proposition 16 Let (at )1
t=0 be a trajectory. If the limit distribution of (a )t=0
t 1
(if it exists) is a correlated equilibrium in canonical form, then (a )t=0 is compatible with adaptive learning.
Proof. Let
be the limit distrbution of (at )1
t=0 and …x any given
^
^
time t and pro…le a. If a is never chosen from t onwards, that is if a 2
=
a : t^
< t^ for any t > t^, then [Def. 21 (i)]
(a) = lim
t!1
jf : 1
< t; a = ag j
t^
= lim = 0,
t!1 t
t
i.e a 2
= Supp . Then for any a 2 Supp there exists a Ta0 such that, 8t >
< t . Since A is …nite we can de…ne T 0 = maxa2Supp Ta0
t^+Ta0 , a 2 a : t^
and conclude that, 8t > t^ + T , Supp
a : t^
< t . Moreover, if
00
00
t
^
(a) = 0 there exists a Ta such that 8t > t + Ta , a 6= a [Def. 21 (ii)].
Let T 00 = maxa2AnSupp Ta00 , then 8t > t^ + T 00 , at 2 Supp . Taking T =
max(T 0 ; T 00 ), we get
8t > t^ + T , at 2 Supp
It follows from the monotonicity of
(Supp )
a : t^
<t :
that
( a : t^
< t ).
Let us assume now that the distribution is a correlated equilibrium in
canonical form. Then Supp
(Supp ) (the proof is analogous to the one
of Theorem 15). We can conclude that
8t > t^ + T , at 2 Supp
(Supp )
( a : t^
< t ).
Therefore there exists a su¢ ciently large T for which 8t > t^ + T , at 2
( a : t^
< t ).
The next two results show that in the long run adaptive learning induces
behaviors that are compatible with the solution concepts we studied, even if
compatibility with adaptive learning does not require complete information.
34
See de…nition 17 and remark 10.
53
Proposition 17 Let (at )1
t=0 be a trajectory which is compatible with adaptive
learning. Then (1)
8k
tk , at 2
0; 9tk ; 8t
k
(4)
(A),
so that in the long run only those actions that are rationalizable can be chosen;
(2) if at ! a , then a is a Nash equilibrium.
Proof.
(1) First recall that since A is …nite there exists a K such that 8k K,
k
(A) = 1 (A). Then, (4) implies that from some time tK onwards only
rationalizable actions are chosen. We now prove (4) by induction. The
assertion trivially holds for k = 0 as given that 0 (A) = A we just need to
choose t0 = 1. Let us now assume that the assertion is true for a given k.
By the assumption of compatibility with adaptive learning, there exists a Tk
such that
8t > tk + Tk , at 2 (fa : tk
< tg):
By the inductive assumption
tk implies a 2
fa : tk
From the monotonicity of
we have
(fa : tk
< tg)
k
< tg
k
(A). Hence,
(A):
( k (A)) =
k+1
(A):
Taking tk+1 = tk + Tk + 1, from the previous inclusions we get
8t
tk+1 , at 2 (fa : tk
< tg)
k+1
(A)
as desired.
(2) If at ! a , there exists a t^ such that 8t > t^, at = a . Taking into
account the compatibility of (at )1
t=1 with adaptive learning, we can deduce
that there exists a T such that
8t > t^ + T , a = at 2 ( a : t^
Since a 2 (fa g), a is a Nash equilibrium.
54
< t ) = (fa g).
6.2
Compatibility with adaptive learning in the case
of imperfectly observable’actions
The de…nition of compatibility with adaptive learning needs to be modi…ed
when players cannot perfectly observe the actions previously chosen by their
opponents. Assume that if an action pro…le a = (aj )jseN is chosen player
i only observes a signal (or message) mi = i (a). As in the section on
conjectural equilibria 5.3, the funtions pro…le
= ( i : A ! Mi )i2N is a
primitive element of the analysis.
Take a trajectory (at )1
t=1 . The set of signals observed by player i from time
t^ (included) to time t (excluded) is mi : 9 ; t^
< t; mi = i (a ) . From
0
i s perspective, the set of other players’action pro…les that could have been
chosen in the same time interval is a i : 9 ; t^
< t; i (a ) = i (ai ; a i ) .
Taking that into account, we can state the following de…nition:
De…nition 22 A trajectory (at )1
-compatible with adaptive learning
t=1 is
if for any t^ there exists a T such that, for any t > t^ + T , and any i 2 N ,
< t; i (a ) = i (ai ; a i ) .
ati 2 i a i : 9 ; t^
Remark 12 The imperfect observability implies that, in general, two players
i and j obtain di¤erent information regarding what a third player, k, could
have done. That is why the justi…ability operator could not be employed
to de…ne the -compatibility with adaptive learning. If any signal function
is injective, then we have perfect observability over other players’ previous
actions and we obtain the de…nition of the previous section (de…nition 20).
We can now proceed to generalize the results that relate compatibilitry
with adaptive learning and equilibrium. Consistently with the motivation
provided in section ?? the relevant equilibrium concept for this case is the
conjectural equilibrium.
t 1
Proposition 18 Take a trajectory (at )1
t=0 . If the limit distribution of (a )t=0
(if it exists) corresponds to a -anonymous conjectural equilibrium , then
(at )1
-compatible with adaptive learning.
t=0 is
Proof. [it is indeed a conjecture to be proved]
Proposition 19 If (at )1
-compatible with adaptive learning and at !
t=1 is
a then a is (part of) a -conjectural equilibrium .
55
Proof. If at ! a , there exists a t^ such that 8t > t^, at = a . Taking
into account the -compatibility of (at )1
t=1 with adaptive learning, we deduce
that 9T; 8t > t^ + T; 8i 2 N ,
ai = ati 2 i a i : 9 ; t^
< t; i (a ) =
= i a i : i (ai ; a i ) = i (ai ; a i ) :
i (ai ; a i )
Hence, for any player i there exists a belief i such that: (1) ai 2 ri ( i ),
= 1. The pro…le (ai ; i )i2N is a
(2) i a i : i (ai ; a i ) = i (ai ; a i )
conjectural equilibrium.
56
7
Games with incomplete information and asymmetric information
Let G = hN; (Ai ; ui )i2N i be a static game where the payo¤s functions ui :
A ! R are derived from a outcome function g : A ! C and a utility functions vi : C ! R. In order to give some meaningful sense to the strategic
analysis of the game, we typically assume that G is common knowledge, or
otherwise stated that there is complete information. The reason is apparent
for "deductive" solutions concepts, such as rationalizability, that characterize the set of choice that are compatible with common knowledge of the
game, rationality and common knowledge of rationality. Nonetheless, also
interpretation of the Nash equilibrium solution concept (or more generally of
correlated equilibrium) as a “self-enforcing” agreement makes more sense if
we assume complete information. Assume for instance that before the game
is played the players agreed by means of a non binding agreement to play
the action pro…le a which yields a Nash equilibrium. Then, nobody has
an incentive to deviate if he expects that the others respect the agreement .
But why should we expect that the agreement is respected? If the others’
payo¤ functions are unknown, we could suspect that some other player has
an incentive to deviate and hence does not respect the agreement. In some
games the incentive to deviate if we suspect that the opponent might not
respect the agreement is very strong. With this respect, consider the Pareto
e¢ cient agreement (a; c) of the game illustrated in the …gure below:
a
b
c
d
100,100 0,99
99,0
99,99
The reasoning we just presented shows that even if the opponents’ payo¤
functions are known, when we suspect that they may not be known by someone else, we could suspect that an opponent will not respect the agreement
as he himself suspects that someone else may not respect it. Vice versa, if
information is complete, the fact that there are no unilateral incentives to
deviate from a Nash equilibrium a is common knowledge and this makes
more credible that a can e¤ectively be a “self-enforcing”agreement.35
35
Vice versa, recall that the interpretation of a Nash equilibrium (pure or mixed) as a
stationary state of an adaptive process does not require nor complete information or the
less stringent assumption of common knowledge of the game.
57
7.1
Games with Incomplete information
How to represent a game with incomplete information? The problem can be
viewed as follows: the shape of the outcome function g : A ! C and the one
of the utility functions vi : C ! R depend on a vector of parameters which
is not commonly known. Hence, the very same payo¤ functions ui = vi g
depend on a parameter which is not commonly known.36 To represent this
relation we write
ui :
A!R
where is the set of values of that are jointly considered as feasible.
In order to represent what a player knows regarding , we assume that
is decomposable into subvectors i , i 2 N , and that i knows the true value
of i . To simplify the analysis we assume that joining the knowledge of all
the players it is possible to fully determine (it is possible to prove that this
assumption is, in some sense, innocuous). It makes therefore sense to write
= ( 1 ; :::; n ).
7.1.1
Private and interdependent values
Notice that, in general, a player i’s payo¤ function, ui , may depend on all
the vector , and not just on i (see example1). However, the situation is
simpler for the case in which the shape of the function g is commonly known
and all the uncertainty lies on the utility functions vi : C ! R. In that case
any players as a matter of fact knows his utility function vi (given that it
represents his preferences over the choices in C). To represent the opponents’
ignorance over such function we can write vi ( i ; c) where i is known only to
i. We then obtain parameterized payo¤ functions ui ( i ; a) = vi ( i ; g(a)), in
which ui depends only on i . In this case we say that we have a private values
game, while in the more general case we say that we have an interdependent
value game.
36
The set of the opponents’actions could as well be unknown. This latter uncertainty
can nonetheless be reconducted to the one analyzed in the text. We omit this aspect for
the sake of simplicity.
58
7.1.2
An elementary representation of incomplete information and
rationalizability
We conclude the introduction of the preliminary tools by describing a strategic interaction where there is common knowledge of the game with the following mathematical structure:
hN; (
i ; Ai ; ui
A ! R)i2N i
:
where i represents the set of values of the parameters i that are considered
as feasible (from i’s opponents), = 1 :::
::: An .
n and A = A1
It seems quite natural to assume that all the sets and the functions speci…ed
are commonly known. Otherwise, it would mean that some aspects of the
uncertainty had been omitted in the formal representation.
The speci…cation of the structure above is su¢ cient to reach some results
regarding the outcomes of the strategic interaction that are compatible with
rationality and the standard assumptions over players’beliefs (for instance,
di¤erent degrees in the common knowledge of rationality). Consider the
following examples
Example 4 Consider a situation of play with incomplete information and
interdependent values. The payo¤ functions can be of two types ui ( 0 ; ) or
ui ( 00 ; ). Player 1 (Row) knows the true payo¤ functions while player 2 (Column) does not know them. We can represent the situations using two matrixes, corresponding to the two possible values of the parameter , assuming
that Row knows the true payo¤ matrix.37
0
1
G :
a
b
c
d
4,0 2,1
3,1 1,0
00
a
b
c
d
2,0 0,1
0,1 1,2
Assume that the following assumptions hold: R1 , R2 , B1 (R2 ), B2 (R1 ), B1 (B2 (R1 )).38
Obviously the e¤ect of such hypothesis on Row’s choices depends on the realizations of the parameter (which is known only to him). As theorist,
we will not make any assumption on the realization of , but rather we will
37
In this case, we can omit the use of the indexes as only player 1 has private information
and therefore there isn a o
one to one correspondence between
and 1 . We have that:
0
00
0 00
0
0 ^
00
^
=
;
with = ( 1 ; 2 ) and
= ( 001 ; ^2 ).
1 =
2 =
2 ,
1; 1 ,
38
The meaning of these symbols is explained in section 3.
59
explain what may happen (consistently with the mentioned assumptions) for
any possible value of 2 .
R1 implies that Row chooses a if = 0 ( a is dominant in matrix 0 ).
R2 \B2 (R1 ) implies that Column chooses d. Indeed, B2 (R1 ) implies that Column is certain that Row would choose a if he were = 0 . Hence, Column
assigns probability zero to the pair ( 0 ;b). Also, notice that d is “dominant”
when the feasible set is restricted to f( 0 ; a); ( 00 ; a); ( 00 ; b)g.
R1 \ B1 (R2 ) \ B1 (B2 (R1 )) implies that Row expects d and as a consequence
chooses b if = 00 .
To sum it up, the aforementioned assumptions imply that the pro…le that is
chosen is (a,d) if = 0 and (b,d) if = 00 . [Obviously the set of actions
that result to be compatible with rationality and with the various epistemic
assumptions can depend only on what a player knows. Then, in this case the
set of possible choices by Column (the singletonfdg) is independent of .]
Example 5 Player 1 and 2 receive an envelope with a money prize. The
possible prizes consist in k thousands of euros, with k = 1; :::K. It is possible that both players receive the same amount. Every player knows his own
received amount and can either propose to exchange his envelope with his opponent’s one (action E), or retain his own envelope (action R). The choices
are taken simultaneously and the exchange takes place only if it is proposed by
both. Proposing the exchange implies that a small administrative cost " needs
to be incurred. The players’utility is given by the amount of money they end
up with at the conclusion of the game. For this example, i = f1; :::; Kg and
ui ( ; a) is given by the following table:
ai naj
O
T
O
T
"
j
i
i
"
i
Therefore a necessary condition for i to o¤er the exchange is that he assigns
a positive probability to the joint event [ j > i ] \ [aj = O]. It can be shown
that the assumptions of rationality and common knowledge in rationality imply that i keeps his own envelope, whatever the content. Let us consider, for
simplicity, only the case K = 3 (the general result can be obtained by induction):
Ri implies that i keeps the envelope if i = 3, as in that case, by proposing
60
the exchange he could obtain at best 3 ".
Ri \ Bi (Rj ) implies that i keeps the envelope if i
2, as i in that case is
certain that j would not exchange the envelopes if j = 3. Hence i is certain
that by o¤ering the exchange he would obtain at best 2 ".
Ri \ Bi (Rj ) \ Bi (Bj (Ri )) implies that i keeps the envelope if i 1, whatever
the value of i , as i is certain that j could (perhaps) exchange the envelope
only if j = 1. Hence i is certain that by exchanging the envelopes he would
obtain at best 1 ".
Intuitive solutions to the previous examples can be given generalizing
the concept of rationalizability. There are many ways we can de…ne rationalizability in games of incomplete information. In general, we label as
“rationalizable ” any choice that is compatible with the assumptions of rationality and common knowledge of rationality, which we denote using the
symbol R \ CB(R),39 for any given background common knowledge. The
rationalizability in games of complete information, for instance, identi…es
the choices that are compatible with the assumption R \ CB(R) given the
common knowledge of the game.
In games of incomplete information we may wonder what are the choices
compatible with the assumption R \ CB(R) given the common knowledge of
hN; ( i ; Ai ; ui )i2N i. The way the examples have been resolved suggests how
to obtain those choices : we eliminate step by step the pairs ( i ; ai ) such that
ai is not a best reply to any probabilistic belief i 2 ( i A i ) which
gives probability zero to the pairs that have been eliminated in the previous
steps. The lemma by Pearce (Lemma ??), implies that the pairs in question
are those ( i ; ai ) such that ai is dominated by a mixed action given i and
the previous steps of elimination.
We can extend this solving procedure extending the notion of rationalization operator. Let i
Ai be the Q
set of pairs for player i that
i
have not yet been eliminated, and let
=
i
j6=i j . We extend the de…nition of best reply correspondence in this very natural way: for any belief
i
2 ( i A i ) and any private information i 2 i
X
i
ri ( i ; i ) = arg max
( i ; a i )ui ( i ; i ; ai ; a i ).
(5)
ai 2Ai
39
Recall that CB(R) =
T
k 1
i ;a i
B k (R) indicates the common belief in rationality.
61
We can then de…ne
i(
as follows: let
( i ; ai ) 2
Y
( ) =
i ):
i(
i)
=
i
=
Ai : 9
Q
i
i
i2N
2
(
Q
i2N (
i ); ai
i
Ai )
2 ri ( i ; i ) :
i2N
We obtain the following representation table in which it is implicitly assumed
that there is common knowledge of hN;
Q( i ; Ai ; ui )i2N i [with a slight abuse
of notation we write
A instead of i2N ( i Ai )]:
Assumptions over behavior and beliefs Implications for the pairs ( i ; ai )
R
(
A)
2
R \ B(R)
(
A)
2
3
R \ B(R) \ B (R)
(
A)
:::
:::
TK
k
K+1
(
A)
R\
k=1 B (R)
:::
:::
In example 4 we get 3 (
A) = f( 0 ; a); ( 00 ; b)g fdg. In example 5 we
get K (
A) = ( 1 fT g) ( 2 fT g) [after K steps, where K is the
number of possible prizes, only the pairs ( i ; T ) do survive].
All we have stated so far regarding rationalizability is easily generalizable. In particular, the equivalence with a procedure of iterated elimination of dominated strategies still holds.
We say that the (mixed) acQ
tion i dominates ai given i in = i2N i (we write i domj( i ; )ai ) if
Supp i fa0i : ( i ; a0i ) 2 i g and
X
0
0
8( i ; a i ) 2
i,
i (ai )ui ( i ;
i ; ai ; a i ) > ui ( i ;
i ; ai ; a i ):
a0i
We then proceed to de…ne the set of undominated pro…les N D( ) as follows
N Di ( ) = f( i ; ai ) 2 i : @ i ;
Y
N D( ) =
N Di ( ):
i domj( i ;
)ai g ;
i2N
Using a straightforward extension of Lemma ?? we can prove the following
result:
62
Theorem 20 For any k = 1; 2; ::: we have that k (
A) = N Dk (
A).
Therefore a pro…le of actions and private information is rationalizable if and
only if it survives the iterated elimination of those pairs ( i ; ai ) such that ai
is dominated given i .
7.2
Equilibrium and Beliefs: the simplest case
In examples 4 and 5 the repeated elimination of dominated strategies lead
us to something that in this context can be interpreted as an equilibrium
: for any player i and any realization i of the parameter privately known
by i, we speci…ed a choice, which in general we can denote by i ( i ). (In
example 4 1 ( 0 ) = a, 1 ( 00 ) = b, 2 = d, in example 5 i ( i ) = T .) The
pro…le of choice functions that is obtained is such that there exists no player
i (whatever the private information i he may hold) that has an incentive to
deviate from the choice i ( i ) if he expects that any other player j chooses
according to j ( j ).
The interpretation to give to the choice function of player j is that it
represents the belief of player i over how j would behave as a function of his
private information j . Then, we can say that in this context an equilibrium
is a pro…le of choice functions such that any player, no matter what his
private information is, chooses a best reply to his beliefs and these beliefs are
correct.
However, in general it is not possible to determine whether a pro…le of
choice functions ( 1 ; :::; n ) possesses the property that no player i has no
incentive to unilaterally deviate from i (what the others expect him to
choose).
Example 6 Consider the following variant of game G1 :
0
G
2
a
b
00
c
d
4,0 2,1
3,1 1,0
a
b
c
d
1,1 0,0
0,1 2,0
How can we determine whether d is a best reply to the choice function
0
00
1 ( ) = a,
1 ( ) = b? It all depends on the probability that 2 (Column)
assigns to 0 and 00 . If Pr2 [ 0 ] 21 then d is a best reply, otherwise it is not.
Let us focus on the case for which Pr2 [ 0 ] < 12 . In an “equilibrium” ( 1 ; 2 ),
0
1 ( ) = a needs to hold, as a is dominant for Row given the information
63
0
. Since in matrix 00 Column’s payo¤ is independent of Row’s action, if
Pr2 [ 0 ] < 12 , whatever the value of 1 ( 00 ), Column’s best reply to the choice
function 1 is c (c expected payo¤ is 1 Pr2 [ 0 ] > 12 , while d expected payo¤
is Pr2 [ 0 ] < 21 ). Then, for the case in which Pr2 [ 0 ] < 12 , the equilibrium
functions’pro…le seems to be 1 ( 0 ) = 1 ( 00 ) = a, 2 = c.
From the above it follows that in order to be able to meaningfully talk
about equilibrium behavior, we need to enrich the structure hN; ( i ; Ai ; ui )i2N i
with the probabilistic beliefs of every player regarding his oppponts’private
information. We denote by pi 2 ( i ) such beliefs. At this stage it seems
legitimate to ask: What does determine pi ? What does player j (j 6= i)
know regarding pi ? If pi is simply a subjective probability, then it does not
seem very plausible to assume that j knows pi , and it is even less plausible
to assume that the beliefs’ pro…le (pi )i2N is common knowledge (or, to be
more precise, that there are common and correct beliefs regarding (pi )i2N ).
Then, even though by …xing a belief pro…le (pi )i2N we can determine whether
each action i ( i ) ( i 2 i ) is a best reply to the pro…le of choice functions
i = ( j )j6=i , it is not clear whether the best reply property is su¢ cient to
be able to assert that there exists no incentive to deviate. How can i “be
con…dent”that j will follow his prescribed choice j if i does not know pj and
therefore cannot know whether j in turn satis…es the best reply property?
In some circumstances, it is possible to attach to pi an objective meaning .
Assume that for any role i speci…ed in the game there exists a large population
of agents that can play in that role. Any of them is characterized by a
given realization i of the private information (for instance, his preferences
or his productivity). Denote by qi 2 ( i ) the statistical distribution of i
within population i. If players are randomly chosen from the corresponding
populations, then the probability
opponents characterized by the
Q of meeting
40
private information ( j )j6=i is j6=i qj ( j ). If the statistical distributions are
Q
commonly known , then it is reasonable to assume that pi ( i ) = j6=i qj ( j )
and that the pro…le of “objective” beliefs (pi )i2N is itself commonly known.
We can then analyze the structure hN; ( i ; Ai ; ui ; pi )i2N i assuming that is
common knowledge and we can meaningfully de…ne as an equilibrium a pro…le
40
Clearly, we are assuming that any j is …nite. The generalization to the in…nite case,
though conceptually trivial, requires the use of some notions from measure theory.
64
of choice functions ( 1 ; :::;
8i 2 N; 8
where
i(
i
2
such that
X
pi (
i , i ( i ) 2 arg max
n)
ai 2Ai
i2
i )ui ( i ;
i ; ai ;
i(
i )),
i
(6)
41
i ) = ( j ( j ))j2N .
Example 7 (First Price Auctions with private and interdependent values)
A given object is put on sale by means of an auction. The set of participant
is N , the monetary value for the object for i 2 N is i . We assume for
simplicity that the set of possible values is given by i = [0; 1] and that i
is distributed according to a uniform distribution which is independent from
the one of i . Then, the independent value paradigm holds. The object is
assigned to the player that makes the highest o¤er. In case of a draw, it is
randomly assigned among the set of players holding the highest bid. Finally,
whoever is awarded the object pays his bid (First Price Auction). The payo¤
function that results is:
ui ( ; a) =
(
i
1
, se ai = maxj aj
ai ) j arg max
j aj j
0,
se ai < maxj aj
The private value assumption is therefore satis…ed. We look for...[to be written]
7.3
The General Case: Bayesian Games and Bayesian
Equilibrium
In order to provide a more general de…nition of equilibrium in games of
incomplete information it is necessary to consider the case in which the beliefs
of a generic player i over the other players’private information are not known
to them. This means that i is not characterized by his private information
i
0
i
i alone but also by the belief p . From j s point of view the pair ( i ; p ) 2
( i ) is unknown. According to the subjective formulation, known as
i
"Bayesian", a decision maker forms probabilistic beliefs over all the relevant
41
Further, even if the structure N; ( i ; Ai ; ui ; pi )i2N obtained through the statistical
distributions were not of common knowledge (perhaps because the statistical distributions
are not known), we could nonetheless motivate an equilibrium so de…ned as a possible
stationary state of an adaptive process.
65
variable that are unknown to him. In our example j has probabilistic beliefs
over ( i ; pi ).
Since more complications arise, to keep the exposition simple we restrict
our attention to the two players’case; then i is the only opponent j has and
vice versa. The beliefs of j are therefore joint beliefs q j 2 ( i
( j )) from
j
which the beliefs p 2 ( i ) can be recovered by computing the marginal
distributions over i . The beliefs q j are called second order beliefs, whereas
the beliefs pj are called …rst order beliefs. If we could further add that the
pro…le of second order beliefs (q 1 ; q 2 ) is of common knowledge, then we would
obtain a “closed model” and we could analyze its equilibria. This can be a
useful simpli…cations for some applied analysis (in the same way as it can be
sometimes justi…ed the assumption of common knowledge over the …rst order
beliefs). However, since we are referring to subjective beliefs, in general, we
have to allow for the possibility that a player’s second order beliefs are not
known to his opponent. Thus, we should consider the third order beliefs
rj 2 ( i
( j)
( j
( i ))),42 from which the …rst and second
order beliefs can be recovered by marginalization. At this point the level of
formalization is already quite complex. Furthermore, for analogous reasons
to the one discussed above, in general we can not stop to the third order
beliefs. How should we proceed then? Is it possible to use a formal and
compact representation for the games of incomplete information which is not
too complex but at the same time allows de…ning in a meaningful way what
an equilibrium is?
7.3.1
Bayesian Games
The solution relies on the adoption of a more abstract and self-referential
approach.43 Starting from the structure hN; ( i ; Ai ; ui )i2N i, we consider a
richer structure in which a set “states of the world” is de…ned (for the
sake of simplicity, we assume this set to be …nite). Any state of the world
! characterizes each player knowledge and "interactive" beliefs. This can
be formalized mathematically introducing, for any player i, the functions
! Ti and #i : Ti ! i , and a probability measure pi 2 ( ). It is
i :
42
Clearly, the symbol rj used here is not to be confused with the symbol indicating the
best reply correspondence.
43
This solution has been proposed in a fundamental contribution by John Harsanyi [9],
that lead to the award to this author of the 1994 Nobel Prize for Economics, jointly with
John Nash and Reinhard Selten.
66
also implicitly assumed that all these elements are common knowledge.
To start grasping the meaning of the above functions let us introduce
the following metaphor. Imagine that an ex ante state in which all players
are equally ignorant were to exist. Each player i is endowed with a priori
(subjective) probability measure pi 2 ( ). Before choosing an action, each
player i receives a “signal” ti regarding the state of the world, from which
both his private information regarding the payo¤ funtions i = #i (ti ) and the
state of the world to which the set of counterimages i 1 (ti ) = f! : i (!) = ti g
belongs can be deduced. To make our notation more compact we denote by
[ti ], the event “i0 s signal is ti ”. For simplicity we also assume that pi assigns
positive probability to all signals,44 i.e. pi [ti ] > 0 for any ti 2 Ti .45 Then for
any signal ti a corresponding conditional probability measure, pi [ jti ], results
to be well de…ned. Player i can make use of this measure to estimate the expected payo¤ for any of his possible actions, given his "signal".46 The beliefs
of player i in state of the world ! are given by the distribution pi [ j i (!)].
Since i and pi are common knowledge, the function ! 7 ! pi [ j i (!)] 2 ( )
is also common knowledge. This allows us to conclude that for any state of
the world and any player, besides his own private information also his beliefs
over the others’private information and beliefs result to be fully determined.
To verify this claim let us focus, as we did earlier, to the case of two bidders
(this is done only to simplify notation) and let us start deriving the beliefs
over the others’"signals “, given “signal”ti = i (!):
8tj 2 Tj , pi [tj jti ] = pi [
1
j
(tj )jti ].
We can then derive the …rst order beliefs:
8
j
2
j,
p1i [ j jti ] = pi [#j 1 ( j )jti ],
44
It can be shown that this comes w.l.o.g provided that all subjective probabilities
p1 ; :::; pn are di¤erent one from the other.
45
We denote by pi (!) the probability density of a single state of the world and with pi [ ]
and pi [ j ] the events absolute and conditional probabilities. For instance, we have that
pi [t i jti ] gives the probability that event [t i ] = f! : i (!) = t i g occurs conditional on
the event [ti ] = f! : i (!) = ti g.
46
Recall that the conditional probability is de…ned as follows:
8E
, pi [Ejti ] =
67
pi [E \ i 1 (ti )]
:
pi [ i 1 (ti )]
where #j 1 ( j ) = ftj : #j (tj ) = j g. The functions tj 7 ! p1j [ jtj ] 2 ( i )
(j = 1; 2) are also common knowledge. Hence, we can derive the second
order beliefs as follows:
X
8( j ; p1j ) 2 j
( i ), p2i [ j ; p1j jti ] =
pi [tj jti ]:
tj :#j (tj )=
1
j ;pj [
jtj ]=p1j
[It can be veri…ed that
8
j
2
j,
p1i [ j jti ] =
X
p1j
p2i [ j ; p1j jti ],
that is p11 [ jti ] is the marginal distribution over the set j of the joint distribution p2i [ jti ] 2 ( j
( i )).]
It should be apparent by now that is possible to iterate the argument and
compute the functions that assign third, fourth order beliefs and so forth.
To sum up we can say that all the information and beliefs of player i are
determined via ti according to the function ti 7! (#i (ti ); p1i [ jti ]; p2i [ jti ]; :::).
For this reason we say that ti is the type of player i.47 The information and the
beliefs of any player i in a given state of the world ! are those corresponding
to the type ti = i (!). Sometimes we refer to private information over the
payo¤ functions i = #i (ti ) as payo¤-type of player i.
We are ready to introduce the following general de…nition of a game with
incomplete information:
De…nition 23 A Bayesian Game (with incomplete information) is a structure BG = hN; ; ( i ; Ti ; Ai ; i ; #i ; pi ; ui )i2N i with i : ! Ti , #i : Ti ! i ,
8ti 2 Ti , pi [ti ] := pi [ i 1 (ti )] > 0, ui :
A ! R for any i 2 N .
In what follows in order to make our language more compact, we will use
the expression “type ti chooses ai ” to mean that if player i were of type ti
he would choose action ai .
7.3.2
Bayesian Equilibria
In general, players’choices depend not only on their private information regarding the payo¤ functions, but also on their types. The following de…nition
of equilibrium follows:
47
The expression “type à la Harsanyi” is also used.
68
De…nition 24 A Bayesian Equilibrium is a pro…le of choice functions ( i :
Ti ! Ai )i2N such that
X
8i 2 N; 8ti 2 Ti , i (ti ) 2 arg max
p[t i jti ]ui (#i (ti ); # i (t i ); ai ; i (t i )):
ai 2Ai
t
i 2T i
Example 8 We now illustrate the concepts of Bayesian game and equilibrium by elaborating on the example provided by game G2 . Assume that Row
does not know Column …rst order beliefs. From his point of view such beliefs can assign either probability 13 or 43 to 0 . The two possibilities are
regarded as equally likely by Row and all this information is common knowledge. The model so described can be represented with the following Bayesian
game:
= f ; ; ; g, 1 = f 01 ; 001 g, T1 = ft01 ; t001 g, 1 ( ) = 1 ( ) n= to01 ,
00
0
1
00
0
00
^2 ,
2 =
1 ( ) = 1 ( ) = t1 , #1 (t1 ) = 1 , #1 (t1 ) = 1 , 8! 2 , p1 (!) = 4 ,
T2 = ft02 ; t002 g, 2 ( ) = 2 ( ) = t02 , 2 ( ) = 2 ( ) = t002 , p2 ( ) = 83 , p2 ( ) = 16 ,
p2 ( ) = 81 , p2 ( ) = 31 , and where the functions ui are as in G2 . The probabilistic structure is described in the table below:
0
; t01
; t001
00
t02
; 41 ; 38
; 14 ; 18
t002
; 14 ; 61
; 14 ; 13
To verify that this Bayesian game represents the game outlined above we
compute the following:
3=8
3
p2 ( )
=
=
p2 ( ) + p2 ( )
3=8 + 1=8
4
p2 ( )
1=6
1
p12 [ 0 jt002 ] = p2 [t01 jt002 ] =
=
=
p2 ( ) + p2 ( )
1=6 + 1=3
3
1
p1 [t02 jt01 ] = p1 [t02 jt001 ] = :
2
p12 [ 0 jt02 ] = p2 [t01 jt02 ] =
This means that in any state of the world Row considers as equally likely Column’s belief that assigns probability 34 to 0 and the belief 13 to 0 . We proceed
now with the speci…cation of the Bayesian equilibria. Applying dominance we
get 1 (t01 ) = a in any possible equilibria. Hence, for any equilibrium 1 the
expected payo¤ accruing to type t02 if he chooses c is
p2 [t01 jt02 ]u2 ( 0 ; a,c) + p2 [t001 jt02 ]u2 ( 00 ;
69
00
1 (t1 ); c)
=
3
4
0+
1
4
1
1= ;
4
the expected payo¤ for t02 if he chooses d is
p2 [t01 jt02 ]u2 ( 0 ; a,d) + p2 [t001 jt02 ]u2 ( 00 ;
It follows that in equilibrium
actions of type t002 are
0
2 (t2 )
00
1 (t1 ); d)
=
3
4
1
3
1
p2 [t01 jt002 ]u2 ( 0 ; a,d) + p2 [t001 jt002 ]u2 ( 00 ; 1 (t001 ); d) =
3
00
2 (t2 )
00
1 (t1 ); c)
=
1
2
1
p1 [t02 jt001 ]u1 ( 00 ; b,d) + p1 [t002 jt001 ]u1 ( 00 ; b,c) =
2
7.3.3
00
1 (t1 )
3
0= :
4
2
3
2
1+
3
2
1= ;
3
1
0= :
3
0+
= c. We can now determine the equilib-
p1 [t02 jt001 ]u1 ( 00 ; a,d) + p1 [t002 jt001 ]u1 ( 00 ; a,c) =
Therefore,
1
4
= d. The expected payo¤s for the two
p2 [t01 jt002 ]u2 ( 0 ; a,c) + p2 [t001 jt002 ]u2 ( 00 ;
It follows that in equilibrium
rium choice for type t001 :
1+
1
2
1
2+
2
0+
1
1= ;
2
0 = 1:
= b.
Bayesian Equilibrium and Nash Equilibrium
The concept of Bayesian equilibrium for a game of incomplete information
BG can be restated in an equivalent fashion as a Nash equilibrium of two
games with complete information that are associated to the original game:
the ex ante strategic form and the interim strategic form.
The ex ante strategic form refers to the metaphor that was previously
introduced to explain the elements of the Bayesian game: if an ex ante state
exists in which all players are “ignorant”, then at such state a player can
formulate a contingent plan of action, or strategy, i : Ti ! Ai , that speci…es
the action to take for any possible signal ti that player i might receive. The
expected payo¤ corresponding to strategy i if it is believed that the other
players do follow the strategies pro…le i is
X
pi (!)ui (#i ( i (!)); # i ( i (!)); i ( i (!)); i ( i (!)))(7)
U ( i; i) =
!2
=
X
ti 2Ti
pi [ti ]
t
X
i 2T
i
pi [t i jti ]ui (#i (ti )); # i (t i );
70
i (ti );
i (t i )):
Let i = ATi i (this is the set of functions with domain Ti and codomain Ai ).
The ex ante strategic form of BG is the staic game hN; ( i ; Ui )i2N i, where
Ui is de…ned by (7).
Remark 13 A pro…le ( 1 ; :::; n ) is a Bayesian equilibrium of BG if and
only if it is a Nash equilibrium of the game hN; ( i ; Ui )i2N i.
The interim strategic form is based on a di¤erent metaphor. Assume that
for any role i in the game there exists a set of potential players Ti . A potential
player ti is characterized by the payo¤ function ui (#i (ti ); ) :
A!R
i
and the beliefs pi [ jti ] 2 (T i ). In the event that he is the one selected to
play the game, he will assign probability p[t i jti ] to the event that is going
to face exactly a pro…le of “opponents”t i . Denote by Ati the set of actions
available to ti . Clearly, Ati = Ai (i 2 N , ti 2 Ti ). If any player tj chooses
the action atj 2 Atj , ti ’s expected payo¤ is computed as follows:
X
(8)
pi [t i jti ]ui ((#i (ti ); # i (t i ); ati ; (atj )j6=i ):
uti ((atj )j2N;tj 2Tj ) =
t
i 2T i
The interim strategic form of BG is the game
*
+
[
Ti ; (Ati ; uti )i2N;ti 2Ti
i2N
where uti is de…ned by (8).
Remark 14 A pro…le ( 1 ; :::; n ) is a Bayesian equilibrium of BG if and
only if the corresponding pro…le (ati )i2N;ti 2TS
such that ati = i (ti ) (i 2 N ,
i
ti 2 Ti ) is a Nash equilibrium of the game
i2N Ti ; (Ati ; uti )i2N;ti 2Ti :
7.3.4
Bayesian equilibria, “common prior” and correlated equilibria
Even though it may seem unnatural, the de…nition of Bayesian game admits
as a particular case that more than a type per player exists and at the same
time that the sets i are all singletones. In other words, we can de…ne a
Bayesian game with many types even starting from a a game with complete
information! An even more special case can be analyzed: the one in which,
besides observing complete information, all players are characterized by the
71
same a priori distribution (8i 2 N , pi = p), which we refer to as “common prior”. What can we say about such Bayesian equilibria of a game of
complete information?
To make our exposition more precise, let us introduce some notation:
^ = hN; (Ai ; u^i )i2N i we call “Bayesian elaboration” of G
^ any
given a game G
game BG = hN; ; ( i ; Ti ; Ai ; i ; #i ; pi ; ui )i2N i such that, for any i 2 N the
set i contains a single element ^i and for any pro…le a 2 A, ui (^; a) = u^(a).
The following remark follows directly from the de…nitions given::
Remark 15 Let BG be a Bayesian elaboration of a game of complete in^ characterized by a common a priori distribution. Then, any
formation G
^ 48
Bayesian equibrium BG corresponds to a correlated equilibrium of G.
7.3.5
Subjective correlated equilibria and rationalizability
Removing the “common prior”assumption we obtain the notion of subjective
correlated equilibria: that is an equilibrium of any given Bayesian elaboration
^
BG of a static game G.
Theorem 21 An action pro…le (ai )i2N of the incomplete information game
^ is rationalizable if and only if it is selected in a subjective correlated equiG
librium (more explicitly, if and only if a Bayesian elaboration BG, an equilibrium ( i )i2N and a state of the world ! in BG such that ai = i ( i (!))
exist for any i 2 N ).
^ We
Proof. (If) Let be an equilibrium of a Bayesian elaboration of G.
show that any image of is a pro…le of rationalizable actions. De…ne the sets
Ci = i (Ti ) (i 2 N ). It is easy to verify that the product set C = C1 ::: Cn
has the best reply property. Therefore, any element in C is rationalizable
(Theorem 2). By construction, C is the set of action pro…les that are selected
by the equilibrium in some state of the world !.
(Only if part) We show that it exists a a subjective correlated equilibrium
in which every rationalizable pro…le is played in some state of the world. Let
C = 1 (A) be the set of rationalizable pro…le. Then C = (C) (Theorem 1)
and for any i 2 N and ai 2 Ci there exists a belief ai 2 (C i ) that justi…es
ai (i.e. such that ai 2 ri ( ai )). Let us specify the Bayesian elaboration as
48
In a way, we could say that Harsanyi [9] had implicitly invented the correlated equilibria before Aumann [1], but without realizing that!
72
follows :
= A, Ti = Ai , 8i 2 N , 8ai 2 Ai = Ti , pi [ jai ] = ai , pi (a) =
1
p [a i jai ]. Then the pro…le of identity functions (8i 2 N , 8ai 2 Ti = Ai ,
jAi j i
i (ai ) = ai ) is an equilibrium of the Bayesian elaboration so constructed.
[Notice that the "type” ti = ai is not to be interpreted as the action that i
necessarily has to play but rather as his belief over the opponents and the
best reply to such beliefs, which i freely chooses to adopt in equilibrium.]
7.4
Games with Incomplete Information and with Asymmetric Information
Consider the following setting, which generates a game with asymmetric information: the payo¤s of the players do not depend just on their choices,
but also on random variable (or move of nature), which we denote by ! 2 ,
that takes place before players make their choices. We also denote by pi the
subjective probability measure of i over the possible moves by nature. Each
player receives a signal regarding the move by nature and then chooses an action ai 2 Ai (simultaneously to the other players). We denote by i : ! Ti
player i ’s signal function and we assume that each signal is associated with a
positive probability: 8i 2 Ti , pi [ i 1 (ti )] > 0. In this context, it is natural to
consider the case that an "objective" probability measure p 2 ( ) over the
moves by nature exists and it is commonly known by the players. More generally, we assume that the pro…le of subjective measures (pi )i2N is commonly
known. The payo¤ of player i is determined by the function vi :
A ! R.
An equilibrium of a game of asymmetric information is a strategy pro…le
( i : Ti ! Ai )i2N , such that the strategy of each player is a best reply to
the strategy pro…le of the other players. More explicitly, we de…ne the payo¤
function in strategic form
X
Ui ( 1 ; :::; n ) =
pi (!)vi (!; 1 ( 1 (!)); ; :::; n ( n (!))):
!
An equilibrium of a given game of asymmetric information is a Nash equilibrium of the corresponding game in strategic form hN; ( i ; Ui )i.
From a mathematical-formal perspective this model is equivalent to the
one of a Bayesian game. At the stage in which all players have received their
private signals regarding the move by nature the situation is indeed very
similar to the one of a game of incomplete information: the way payo¤s depend on actions is not perfectly known; each player has private information
73
about that, along with probabilistic beliefs regarding the others’private information and beliefs. There are many ways to establish an explicit relation
between the two models. One way is to obtain a corresponding Bayesian
game de…ning i = Ti so that #i becomes an identity function over Ti , and
X
ui ( ; a) =
p[!j ]vi (!; a);
!
1
( )]
, 1 ( 1 ; ; ; : n ) = f! 0 : ( 1 (! 0 ) = 1 ; :::; n (! 0 ) = n g.49
where p[!j ] = p[f!g\
p[ 1 ( )]
Keeping in mind remark 13, it is easy to verify that a pro…le of functions
( i )i2N is a Bayesian equilibrium of the game BG so obtained if and only if
it is an equilibrium of the asymmetric information game.
Recall that in order to introduce the constituent elements of the model
of the Bayesian game we used the metaphor of the ex ante stage. The use of
such metaphor and the mathematical-formal analogy between Bayesian and
asymmetric information games (and the corresponding solution concepts)
has induced many scholars not to distinguish between these two interactive
decision problems. We should stress however that they are two di¤erent
problems. In the asymmetric information problem the ex ante stage does
not exist, it is rather a useful stratagem: only represents a set of possible
states of the world, where by “state of the world” we mean a con…guration
of information and subjective beliefs. The so called a priori distributions
pi 2 ( ) are simply a useful mathematical arti…ce to determine (along with
the functions #i and i ) the players’ interactive beliefs in a given state of
the world.50 Instead, in those interactive decision problems that we called
49
The value of p[!j ] se p[ ] = 0 is irrilevant for the computation of the equilibria.
Essentially, nothing would have changed if we had only speci…ed the functions i :
Ti ! (T i ) that for any type ti determine a probability measure over the others’types.
At the end of the day, we are simply interested in the types’ beliefs. Moreover, it is
always possible to specify a distribution pi that yields the beliefs pi [ jti ] = i (ti ) for any
type ti 2 Ti . Actually, there exist an in…nite number of them, as it is apparent from the
following construction. Let 2 (Ti ) be any strictly positive distribution and determine
pi as follows
50
8t
=
8!
2
(ti ; t i ) 2 T , pi [t] = (ti )
pi [t]
1
(t), pi (!) =
j 1 (t)j
i (ti )[t i ]
(where (!) = ( 1 (!); :::; n (!)) and j j denotes the cardinality, or numerousness, of a
set). Then, pi [t i jti ] = i (ti )[t i ], that is pi [ jti ] = i (ti ).
74
“games of asymmetric information”the ex ante stage is real and the players’
a priori probability measures represent the expectation they hold at that
stage.
The di¤erences in the interpretation of the formal model are not innocuous. The interpretation (asymmetric information vs. incomplete information) determines to what extent given assumptions are meaningful and plausible. For instance, for the case of asymmetric information over an initial
random move, it is meaningful and plausible to assume that there exists an a
priori common distribution , that is an objective distribution over the moves
by nature. Instead, for the case of incomplete information it is not even so
clear what is the sense of the common prior assumption and certainly there
is no reason to consider it plausible. Furthermore, we will see that the notion of rationalizability that is appropriate for a situation illustrated by an
incomplete information game it is di¤erent from the one that is appropriate
for a situation represented by a game of asymmetric information over an
initial move by nature. This even though the two situations are formally
represented by the very same mathematical model.
7.5
Rationalizability in Bayesian Games
The mathematical structure hN; ( i ; Ai ; ui )i2N i does not specify the interactive beliefs of the players, that is the beliefs over the others private information and beliefs. We noticed though how such speci…cation is needed in order
to meaningfully de…ne the equilibrium concept. Therefore a richer structure
BG = hN; ; ( i ; Ti ; Ai ; i ; #i ; pi ; ui )i2N i, known as Bayesian game has been
analyzed.
We also pointed out that from an equilibrium analysis perspective it
makes no di¤erence whether a mathematical structure BG represents a game
situation where there is no common knowledge of the payo¤ functions (incomplete information), or another one where there is asymmetric information
over an initial move by nature. The interpretation we give may make the
assumptions (for instance the one of an apriori common distribution) over
the interactive beliefs represented by such structure more or less plausible.
Nonetheless, the computation of the equilibria is not a¤ected.
The interpretation we give instead matters when it comes to de…ne the
appropriate concept of rationalizability, given the common knowledge of BG.
The crucial point is the following: the rationality of a player needs to be
evaluated in according to his type. In other words we consider the so called
75
interim stage and we ask ourselves: is the action ai justi…able for type ti ?51
Conversely, in a game with asymmetric information over an initial move
by nature, rationality needs to be assessed at the ex ante stage. We ask
ourselves: is strategy i justi…able? By answering to those di¤erent questions
we get two di¤erent concept of rationalizability for Bayesian games52 .
De…nition 25 Let BG be a Bayesian game. We say that an action ai is
interim rationalizable for type ti if ai is rationalizable for ti in the interim
strategic form of BG. We say that a function i : Ti ! Ai is ex ante
rationalizable if i is a rationalizable strategy in the ex ante strategic form
of BG.
Remark 16 If a strategy
for any type ti the action
is justi…able in the ex ante strategic form, then
i (ti ) is justi…able in the interim strategic form
i
Proof. The …rst thing to observe is that, for type ti according to the
formula (8) the actions of the other types t0i of player i are completely irrelevant for the determination of the interim payo¤. Hence, we can re-de…ne the
belief for type ti in the interim strategic form as a probability distribution
over the action pro…les of the types of players j 6= i. But such action pro…les
do coincide with the strategy pro…les of the opponents of player i in the ex
ante strategic form:
Y
Y
ATj =
(atj )j6=i;tj 2Tj 2
j =
i:
j6=i
j6=i
Hence, the set of beliefs for any player ti in the interim strategic form coincides with the sets of beliefs of player i in the ex ante strategic form. To be
consistent with the notation employed for the games of complete information,
we denote by Ui ( i ; i ) e uti (ai ; i ) the expected payo¤s of player i and of
type ti in the corresponding strategic forms (given the belief i 2 ( i )).
51
Recall that we say that a choice is “justi…able” if it is a best reply to some belief. A
choice is "rationalizable" if it survives the iterated elimination of non justi…able choices.
52
It can be shown by means of examples that the concept of interim rationalizability
de…ned in the text is problematic if we consider a more general setting than the one that
has been analyzed. That is if once the private information have been determined, it still
persist some uncertainty regarding the rules of the game (formally, if = 0
1 :::
n,
where the "residual" uncertainty regards 0 2 0 ). Dekel et al (2005) have shown how
to modify the concept of rationalizability such that it represents the outcomes that are
compatible with the assumptions of rationality given the common knowledge of BG.
76
We now prove that i is a best reply to the belief i if and only if for
any type ti the action i (ti ) is a best reply to i . This implies the assertion.
Consider a belief i 2 ( i ). For any strategy i we have that:
X
i
Ui ( i ; i ) =
( i )Ui ( i ; i )
i
X
=
i
(
i)
i
X
=
pi [ti ]
ti
X
=
X
X
pi [ti ]
ti
i
X
t
(
i)
t
i
pi [ti ]uti ( i (ti );
i
X
i
i
pi [t i jti ]ui (#i (ti ); # i (t i );
i (ti );
i (t i ))
pi [t i jti ]ui (#i (ti ); # i (t i );
i (ti );
i (t i ))
)
ti
It follows that
i
2 arg max Ui ( i ;
i
) = arg max
i
i
X
pi [ti ]uti ( i (ti );
i
)
ti
if and only if
8ti 2 Ti ,
i (ti )
2 arg max uti (ai ;
ai
i
):
The average of the expected payo¤s for the di¤erent types of i is maximized
if and only if the expected payo¤ of any type of i is maximized.
Corollary 22 If a strategy i is rationalizable in the ex ante strategic form,
then for any type ti the action i (ti ) is rationalizable for ti in the interim
strategic form.
From the previous considerations it should be clear that interim rationalizability is the appropriate solution concept if we interpret the mathematical
structure BG as a game situation of incomplete information, where the ex
ante stage is only a useful stratagem and therefore the states ! represent
only possible con…gurations of private information and beliefs. Ex ante rationalizability is instead the appropriate solution concept if we interpret BG
as a game situation of asymmetric information over some initial move by
nature (which a¤ects the payo¤s only through the parameter ).
Corollary 22 states that ex ante rationalizability is at least a strong solution concept as interim rationalizability. Actually, it is a stronger (restrictive)
solution concept in a strict sense. The reason might be understood already
77
from the proof of remark 16. Assume that, for any ti , the action i (ti ) is
justi…able. Then it exists a belief pro…le ( ti )ti 2Ti such that i (ti ) is a best
reply to ti for any ti . Note that it is not guaranteed that there exists a
unique belief i such that i (ti ) is a best reply to ti for any ti . The di¤erence between the two solutions concept is clearly illustrated by the following
numerical example.
Example 9 Let = f! 0 ; ! 00 g; player 1 (Row) knows the true state, whereas
player 2 (Column) does not and considers the two states equally likely: p2 (! 0 ) =
p2 (! 00 ) = 21 . The payo¤ functions are as in the matrixes below:
!0
a
m
b
c
3,3
2,0
0,0
! 00
a
m
b
d
0,2
2,2
3,2
c
3,0
2,0
0,3
d
0,2
2,2
3,2
First, note that any action by Column is justi…able. In particular, c is justi…able by the belief that Row chooses a if ! 0 and b if ! 00 . It is easy to verify
that for any of the two types of Row all the actions are justi…able by some
belief of his opponent. Hence, if choices are evaluated at the interim stage it
is not possible to exclude any action. Instead if choices are evaluated at the
ex ante stage, then the strategy 1 (! 0 ) = a; 1 (! 00 ) = b, which we denote by
ab can excluded . Indeed, if Row believes 1 (c) 21 , then in both states b is
not a best reply; else if Row believes 1 (c) 12 , then in both states a is not
a best reply. The same argument shows that ba is not justi…able either; the
strategy by Row that are ex ante justi…able are : aa,am,ma,mm,bb,bm,mb.53
Given any belief 2 such that 2 (ab) = 0, the action c yields an expected
payo¤ less or equal than 23 ; action d instead yields a payo¤ 2 > 23 . Then,
the only rationalizable action for Column in the ex ante srategic form is d.
It follows that the only rationalizable choice in the ex ante strategic form for
Row is bb.
7.6
The Electronic Mail Game
Consider the following incomplete information game where player 1 (Row)
knows the true payo¤ matrix, whereas 2 (Column) does not know it. The
53
Given that Row’s payo¤ does not depend on !, those strategies that select di¤erent
actions for the two states are justi…able only by beliefs that make Row indi¤erent between
these two actions; the strategies am and ma are among the best replies to the belief
1
(c) = 23 , the strategies bm and mb are among the best replies to the belief 1 (c) = 31 .
78
probabilities and the payo¤s are as below:
(1
)
:
a
b
a
M,M
-L,1
L > M > 1,
b
1,-L
0,0
1
<
2
: a
b
a
0,0
-L,1
b
1,-L
M,M
If we had complete information, (a; a) would be the dominant strategy
equilibrium for matrix , and (b; b) would be the Pareto e¢ cient for the
matrix . The second matrix has also another equilibrium , (a; a).
It can be easily veri…ed that the incomplete information game has only a
pro…le that is interim rationalizable: (aa; a). As a matter of fact, for type
of Row a is dominant. If column believes that Row is rationale so that type
chooses a, then the expected utility of a2 = a is bigger than the one for
a2 = b: More precisely let 2 2 (aa; ab; ba; bb) be Column belief over Row
strategy and let u2 ( 2 ; a2 ) be the expected utility of the action a2 2 fa; bg
based on such belief. Given that 2 (ba) = 2 (bb) = 0 (as
certainly chooses
a), we obtain
u2 ( 2 ; a)) = (1
)u2 ( ; a; a) +
(1
)M
>
(1
)L + M
(1
)u2 ( ; a; b) +
2
= u2 ( ; b)):
2
2
(aa)u2 ( ; a; a) +
2
(aa)u2 ( ; a; b) +
2
(ab)u2 ( ; b; a)
(ab)u2 ( ; b; b)
Hence, the only action that results rationalizable for Column is a. This
implies that also for type
the only interim rationalizable choice is a.
Consider now the following more complex variant of the game.54 . Obviously players would …nd convenient to communicate and coordinate on (a; a)
if =
and on (b; b) if = . Let us then consider the following form
of communication via electronic mail. Row and Column sit in front of their
respective computers. If the payo¤-type of Row is , his computer, C1 , automatically sends a message to the computer of Column, C2 . Additionally, if
54
See [15] (or the textbook byOsborne and Rubinstein [13], pp 81-84). The analysis in
terms of interim rationalizability is not contained in the original work. The same holds
for many papers that analyze similar games. All these papers only consider the Bayesian
game and, occasionally, ex ante rationalizability. However, it is not di¢ cult to adapt the
argument used to the case of interim rationalizability.
79
computer Ci receives a message it automatically sends a con…rmation message to computer C i . However, any time a message is sent there exists a
probability " that the message gets lost and does never reach the receiver.
A corresponding Bayesan game unfolds. A generic state of the world is
given by a pair of numbers ! = (q; r) where q = t1 is the number of sent
messages by C1 and r = t2 is the number of messages received (and therefore
also sent) by C2 . Hence, the set of states of the world is given by
= f(0; 0); (1; 0); (1; 1); (2; 1); (2; 2); (3; 2); (3; 3); :::g
= f(q; r) 2 N N : q = r or q = r + 1g :
If the state is (q; q 1) it means that the last message (among those sent
by C1 and C2 ) has been sent by C1 . If the state is (r; r) (with r > 0) it
means that the last message sent by C1 has reached C2 , but the con…rmation
by C2 has not reached destination. The signal functions are 1 (q; r) = q,
is #1 (0) = , #1 (q) =
if
2 (q; r) = r. The function that determines
q > 0.55 The a priori (common) distribution is given by p(0; 0) = (1
),
p(r + 1; r) = (1 ")2r ", p(r + 1; r + 1) = (1 ")2r+1 " for any r 0 (that
is, p(q; r) = (1 ")q+r 1 " for any (q; r) 2 n f(0; 0)g). This information is
summed up in the following table:
t1 =q n
0,
1,
2,
3,
4,
5,
...
t2 =r
0
1
"
–
–
–
–
...
1
–
(1
(1
–
–
–
...
")"
")2 "
2
–
–
(1
(1
–
–
...
")3 "
")4 "
3
–
–
–
(1
(1
–
...
")5 "
")6 "
4
–
–
–
–
(1
(1
...
")7 "
")8 "
5
–
–
–
–
–
(1
...
")9 "
The resulting beliefs, or conditional probabilities, are then determined as
55
2
is a singleton, hence #2 is a constant.
80
...
...
...
...
...
...
...
...
follows:
p1 [0j0] = 1,
p1 [rjr + 1] =
(1
1
p2 [0j0] =
1
p2 [r + 1jr + 1] =
(1
1
(1 ")2r "
=
;
2r
2r+1
") " + (1 ")
"
2 "
,
+ "
(1 ")2r+1 "
1
=
(r > 0).
2r+1
2r+2
")
" + (1 ")
"
2 "
In other words, any player –having received a certain number of messages –
computes that the probability that his con…rmation message does not reach
the other computer is 2 1 " > 21 :
If " is very small, the Bayesian game is in some sense "similar" to the
game with complete information over . First notice that in all states (q; r)
with r > 0 both players know that the true state is . Moreover, for any
n and any > 0, there always exists an " su¢ ciently small such that, given
= , the probability that there exists mutual knowledge of degree n of
the true value of is bigger than 1
.56 Given that (b; b) is an equilibrium
of the complete information game in which =
, we could be induced to
think that if " is very small, then action b is interim rationalizable for some
type of Row and Column. However, the following result holds:
Proposition 23 In the Electronic Mail game there exists a unique pro…le
which is interim rationalizable: any type of any player chooses action a.
The crucial point in the proof is to realize that when a player i knows
that =
but is sure that i plays a whenever he has not received his
last message, then i prefers to play a. On the other hand it is easy to show
that when i does not know that =
then a is the only action that is
56
For instance, consider n = 2. For any q
2, in state (q; r) (r = q, or r = q 1)
any player knows that =
and any player knows that the other knows, that is there is
mutual knowledge of degree two over the true payo¤ matrix. The probability of being in
one of these state conditional on the event =
is
1
p(f(1; 0); (1; 1)g
=1
To guarantee that such probability is bigger than 1
81
2" + "2 :
we need " < 1
p
1
.
rationalizable. It follows by induction that a is the only action which is
rationalizable in all states. Here is the formal argument:
Proof. Clearly, the only rationalizable action for t1 = 0 is a, as this type
of Row knows to be in the matrix that has a as dominant. A similar argument
to the one used for the simple case above shows that the only rationalizable
action for t2 = 0 is a.
Consider now the rationalizable choices for the types ti = r > 0, that
is for those types for which player i knows that = . We …rst show two
intuitive steps:
(i) If the only rationalizable action for type t2 = r 1 is a, then the
only rationalizable action for type t1 = r is a. As a matter of fact, any
rationalizable action for t1 = r needs to be justi…ed by a rationalizable belief
over Column (see Theorem 5). By assumption, such belief assigns probability
one to the set of pro…les 2 2 fa; bgT2 such that 2 (r 1) = a. Hence, the
expected utility for type t1 = r when he chooses a is at least 0 (this value
is realized if type t2 = r also chooses a), whereas the expected utility from
choosing b is at most p1 [r 1jr]L + p1 [rjr]M (this value is realized if type
t2 = r chooses b). Since p1 [r 1jr] = 2 1 " > 12 and L > M , we have that
0 > p1 [r 1jr]L + p1 [rjr]M:
Exactly the same argument (just reverse the roles and modify indexes
accordingly) shows that:
(ii) If the only rationalizable action for type t1 = r is a, the only rationalizable action for type t2 = r is also a.
From steps (i) and (ii) it follows that, for any r > 0; if the only rationalizable pro…le in state (r 1; r 1) is (a; a) then the only rationalizable pro…le
in state (r; r 1) is (a; a); if the only rationalizable pro…le in state (r; r 1) is
(a; a) then the only rationalizable pro…le in state (r; r) is (a; a). Since we have
shown that the only rationalizable pro…le in state (0; 0) is (a; a), it follows
by induction that (a; a) is the only rationalizable pro…le in any state.
7.7
Conjectural Equilibria in Incomplete Information
Games
As previously argued, with the concept of conjectural (or self-con…rming)
equilibrium we aim at representing outcomes that are stable with respect to
learning processes in situations of repeated interaction, keeping into account
players’ information restrictions. The appropriate de…nition of conjectural
82
equilibrium is going to depend on the scenario we assume and in particular on the interpretation of the mathematical structure we use to represent
incomplete information.
The …rst question we need to ask us is whether the set of players that
do interact with each others and their characteristics are (a) …xed once and
for all, or (b) randomly determined in each period (via draws from large
populations). In case (b) we need to ask ourselves which information players
are able to obtain regarding the statistical distributions of the actions and
characteristics in the populations drawn in previous periods. In both cases,
as we have done earlier, we assume for the sake of simplicity that the players
objective is to maximize the expected payo¤ in the current period, without
worrying about future payo¤s.
Case (a): long run interaction. This is the easiest case to analyze. It
is su¢ cient to consider the structure hN; ( i ; Ai ; ui :
A ! R)i2N i. The
pro…le of parameters (or characteristics) is …xed once and for all; obviously,
we continue to assume that any player i knows only i , that the sets of
possible value is j (j 6= i) and that ex post he gets further information
according to a signal function i . It can be assumed that the signal that
is received may depend not only on the chosen action but also from the
parameter , that is i :
A ! Mi ; for instance, the signal could be the
57
payo¤ obtained by the player. Any player i holds a complete belief i over
the others’private information and beliefs; we need to askQ
ourselves whether
i
for any given , a pro…le of actions and beliefs (ai ; )i2N 2 i2N Ai
( i
A i ) satis…es the conditions of rationality and beliefs con…rmation. These
conditions are obtained from a simple generalization of de…nition 18.
De…nition 26 Let = ( i :
A ! Mi )i2N beQa pro…le of signal functions. A pro…le of actions and beliefs (ai ; i )i2N 2 i2N Ai
( i A i)
is a -conjectural equilibrium (or self-con…rming) given
if for any i 2 N
the following conditions hold:58
(1) ( rationality) ai 2 ri ( i ; i ),
(2) ( beliefs’con…rmation) i (f( i ; a i ) : i ( i ; i ; ai ; a i ) = i ( ; a )g) =
1:
57
However, we need to be aware that the payo¤ ui does not necassarily represent a
material gain which can be cashed. Therefore in the theoretic analysis we are restricted
to assume that the realization of ui is observable by i.
58
Recall that ri ( i ; i ) is the set of actions of i that maximize his expected payo¤ given
the belief i and the private information i .
83
The following remark highlights the fact that in the private value case (ui
not depending on i ) we get the notion of conjectural equilibrium already
introduced in section 5.3, coherently with the assertion made there according
to which the concept of conjectural equilibrium presumes only that a player
does know his own payo¤ function:
Remark 17 Let hN; ( i ; Ai ; ui :
A ! R)i2N i be a private value model ;
then a is part of a -conjectural equilibrium given if and only if a is part
of a conjectural equilibrium of the game G( ) = hN; (Ai ; ui ( ; ) : A ! R)i2N i
considering the pro…le of signal functions ( i ( ; ) : A ! Mi )i2N .
Case (b): anonymous short run interaction. Consider the scenario that
we used to motivate the simplest de…nition of Bayesian equilibrium. Assume that the individuals belonging to population i are heterogeneous. The
percentage of individuals with characteristic i (of which they are aware) is
qi ( i ) > 0 (we consider …nite games). From each population i an individual is randomly drawn. Then the individuals that have been drawn interact
(via simultaneous moves) and obtain payo¤ which are given by the functions
ui :
A ! R (i 2 N ). This is precisely the scenario that motivated
the de…nition of "simple" Bayesian game (with independent types), that is a
structure hN; ( i ; qi ; Ai ; ui )i2N i, where qi 2 ( i ), ui :
A ! R (i 2 N ).
In this case we do not assume that hN; ( i ; qi ; Ai ; ui )i2N i is common
knowledge, but rather that any individual knows his payo¤ function. We
imagine that the interaction is repeated, each time with di¤erent co-players
randomly drawn from the respective populations. We assume that the way
of playing does stabilize, at least in a statistical sense, that is for any j, j ,
aj , the percentage of individuals in the sub-population j with characteristic
j that choose action aj remains constant; we denote such percentage by
j ( j j j ) and we write
i = ( j ( j j ))j6=i; j 2 j to indicate the entire pro…le
of percentages (frequencies) regarding the populations di¤erent from i. Given
the random matching used, we get that the long Q
run frequency of the others’
actions and characteristics ( i ; a i ) is given by j6=i j (aj j j )qj ( j ). Hence,
in the long run, an individual belonging to population i and with characteristic i that (always) chooses ai , will observe that the frequency with which
he observes a given message mi is
X
Y
Pr(mi jai ; i ; i ; q i ) =
j (aj j j )qj ( j ):
(
i a i ): i ( i ;
84
i ;ai ;a i )=mi
j6=i
The belief
probability
i
2
(
A i ) assigns to message mi (given ai and
i
Pr(mi jai ; i ;
i
X
)=
(
i a i ): i ( i ;
i
(
i)
the
i ; a i ).
i ;ai ;a i )=mi
The belief results to be con…rmed in the long run if the subjective probability
of each message is equal to the observed frequency. If the action is a best
reply to the belief, given the characteristic i , and the belief is con…rmed,
then the individual has no reason to change his action. These considerations
lead us to the de…nition of anonymous conjectural equilibrium:
De…nition 27 Let = ( i :
A ! Mi )i2N be a pro…le of signal functions.
Q
2 i2N [ (Ai )
A pro…le of mixed actions and beliefs i ( j i ); ( i(ai ; i ) )ai 2Ai
i2N: i 2
i
[ ( i A i )]Ai ] i (a mixed action for any i and i , and a belief for any i, i
e ai ) is a -conjectural equilibrium (or self-con…rming) of the basic Bayesian
game hN; ( i ; qi ; Ai ; ui )i2N i if for any i 2 N , i 2 i , ai 2 Ai the following
conditions hold:
(1) ( rationality) if i (ai j i ) > 0, then ai 2 ri ( i ; i ),
(2) ( beliefs’con…rmation) Pr( jai ; i ; i ) = Pr( jai ; i ; i ; q i ):
Remark 18 Even in this case the condition of beliefs’ con…rmation can be
regarded as a bit redundant as we require that a con…rmed belief exists for any
pair ( i ; ai ), and not just for those such that i (ai j i ) > 0: Such redundancy
is completely innocuous and it is used exclusively to make the notation more
transparent.
Remark 19 The de…nition needs to be modi…ed and made more stringent if
we assume that the distributions qj are known. For instance, for the case of
two players we need to require that marg j i = qj .
Remark 20 If the signal functions are injective, we obtain as a particular
case (we did not report it in the previous sections) the Bayesian mixed equilibrium (randomized). Moreover, if any i ( j i ) assigns probability one to a
certain action i ( i ) 2 Ai , then we obtain as a particular case the so called
"basic"de…nition of Bayesian equilibrium [eq. (6)].
85
References
[1] AUMANN, R.J. (1974): “Subjectivity and Correlation in Randomized
Strategies,”Journal of Mathematical Economics, 1, 67-96.
[2] AUMANN, R.J. (1987): “Correlated Equilibrium as an Expression of
Bayesian Rationality,”Econometrica, 55, 1-18.
[3] BATTIGALLI, P. e G. BONANNO (1999): “Recent Results on Belief,
Knowledge and the Epistemic Foundations of Game Theory,” Research
in Economics (Ricerche Economiche), 53, 149-226.
[4] BATTIGALLI, P., M. GILLI and M.C. MOLINARI (1992): “Learning and Convergence to Equilibrium in Repeated Strategic Interaction,”
Ricerche Economiche, 46, 335-378.
[5] BERNHEIM, D. (1984): “Rationalizable Strategic Behavior,” Econometrica, 52, 1002-1028.
[6] BRANDENBURGER, A. e E. DEKEL (1987): “Rationalizability and
Correlated Equilibria,”Econometrica, 55, 1391-1402.
[7] DEKEL, E., D. FUDENBERG e S. MORRIS (2005): “Interim Rationalizability,”mimeo.
[8] DUFWENBERG, M. e M. STEGEMAN (2002) “Existence and Uniqueness of Maximal Reductions Under Iterated Strict Dominance,”Econometrica, 70, 2007-2023.
[9] HARSANYI, J. (1967-68): “Games of Incomplete Information Played by
Bayesian Players. Parts I, II, III,” Management Science, 14, 159-182,
320-334, 486-502.
[10] HARSANYI, J. (1995): “Games with Incomplete Information,”American Economic Review, 85, 291-303.
[11] MILGROM, P e J. ROBERTS (1991), “Adaptive and Sophisticated
Learning in Normal-Form Games,” Games and Economic Behavior,
3,82-100.
[12] NASH, J. (1950): “Equilibrium Points in N-Person Games,”Proceedings
of the National Academy of Sciences, 36, 48-49.
86
[13] OSBORNE, M. and A.RUBINSTEIN (1994): A Course in Game Theory. Cambridge MA: MIT Press.
[14] PEARCE, D. (1984): “Rationalizable Strategic Behavior and the Problem of Perfection,”Econometrica, 52, 1029-1050.
[15] RUBINSTEIN, A. (1989): “The Electronic Mail Game: Strategic Behavior Under Almost Common Knowledge,” American Economic Review, 79, 385-391.
[16] VICKREY, W. (1961). “Counterspeculation, Auctions, and Competitive
Sealed Tenders,”Journal of Finance 16, 8-37.
87