Sapienza University of Rome. Ph.D. Program in Economics a.y. 2012-2014
Microeconomics 2 – Game Theory Lecture notes
1 . Simultaneous-move games: Solution in (iterated) strict dominance and
rationalizable strategies
1.1 Simultaneous-move games in normal form: definition and notation
1.1.1 Games in pure strategies
1.1.2 Games in mixed strategies
1.2 Alternative approaches to the determination of the equilibrium
1.3 Strict dominance
1.3.1 Strict and weak dominance: definitions
1.3.2 The prisoners’ dilemma: solution in strict dominance
1.4 Iterated strict dominance
1.4.1 Iterated elimination of strictly dominated strategies
1.4.2 Iterated weak dominance
1.5 Rationalizable strategies
1.5.1 Approach and definition
1.5.2 Iterated elimination of non rationalizable strategies
1.5.3 Relation between the set of strictly undominated and rationalizable strategies
The solution, or equilibrium of a game is a prediction that we make as to the choices that the
agents, involved in the strategic situation described by the game, will make. Two keyu
assumptions enter in the formulation of this prediction. We first is that players behave
rationally; we generally attach to this assumption the specific meaning that players choose an
action that maximizes their payoff. But in a context of strategic interaction the best action of
any player would depend on the actions of the other players. We further assume, therefore,
that each player formulates a conjecture of the other players’ choices and evaluates, on this
basis, the expected payoff of his alternative strategies. Players’ rationality and beliefs of other
players’ choices have a decisive role in the solution concepts of simultaneous-move games
that we are going to examine in these Lecture Notes. In increasing order of the deductive and
introspective capacities asked of the players, these are basically the following:
- strict dominance;
- iterated strict dominance;
- rationalizable strategies surviving the iterated elimination of non rationalizable strategies;
- Nash equilibrium.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.1
This Lecture Note is dedicated to the presentation and the critical implications of the solutions
based on the application of the notions of dominance and rationalizability (sections 1.3-1.5).
Nash equilibrium is analyzed in Lecture Note n. 2; extensions and refinements of Nash
equilibrium in static games are presented in separate Lecture Notes. Sections 1.1 and 1.2
contain some preliminary notions: in the former the definition of simultaneous-move games,
both in pure and mixed strategies, is presented and notation established; in the latter
alternative conceptual approaches to the determination of the equilibrium of the game are
briefly illustrated.
1.1 Simultaneous-move games in normal form: definition and notation
A simultaneous-move or static game describes a situation of strategic interaction in which the
agents involved – henceforth, to be named the players – are supposed to take decisions not
knowing the actions simultaneously chosen by the other players. The interpretation to be
given of the term “simultaneous” should not be reserved to situations in which the players
literally take decisions at the same time, but more generally to situations of ignorance of the
choices made by the other players, possibly at different moments in time.
The lack of knowledge of the other players’ moves characterizes the game as one of imperfect
information, as opposed to situations of perfect information, which obtain if each player
knows, when called to play, the decisions taken by all the other players in the preceding
rounds of the game.
1.1.1
Games in pure strategies
The assumption that players move simultaneously leads to the description of the game in
terms of the normal or strategic form, which consists in the indication of: (1) the number of
players; (2) the strategies available to each player; (3) the payoff received by each player for
each possible combination of strategies chosen by the players. In standard notation:
-
I is the number of players; players are specified by the subscript i  1,..., I ; a standard
compact notation distinguishes player i from all other players, formally indicated as i ;
-
S i is the space of strategies available to player i. The strategy space may consists of a


finite number Ki of elements, si  si1 ,..., si ki ,..., si Ki  Si , representing Ki distinct pure
strategies, or of an infinite number of elements representing a continuum of possible pure
strategies, i.e. of possible values of player i ’s decision variable. In this latter case the strategy
space is represented by the interval Si   si , si  . 1 In static games strategies coincide with
actions; the two terms will therefore be used interchangeably. In dynamic games a strategy is
1
The strategic interaction problem may call for the presence of several variables capable of assuming a continuum of
values.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.2
a rule of action indicating the specific action of the player at each round of the game in which
he must take a decision; a strategy is therefore an ordered sequence of actions;
-
S   Si  Si  Si , with Si   S j , is the strategy space of the game and
j i
iI
s   s1 ,...si ,..., sI    si , si   S is a profile of strategies, where the compact notation i for
all players different from player i has been used;
-
ui  si , si  : S 
is player i’s payoff resulting from all possible profiles of actions of all
players, formally a mapping from the strategy space of the game into the real line. Strategic
interaction among players shows up in this definition of the payoff function. The payoff
function is assumed to be, in general, an ordinal utility function; unless otherwise indicated,
this is the interpretation to be given to the players’ payoffs in the matrix games used to
illustrate different game situations. When the players are firms, involved for instance in an
oligopoly game, payoff functions are better considered to represent money profits.
A game is finite if the strategy space S is finite; this requires that both the number of players
and the strategy space of each player be finite. In several game situations of specific interest
to economic theory, the strategy space of the players is naturally modeled with continuous
variables, such as prices and quantities, for instance, in the classic oligopoly models of
Cournot and Bertrand. The presentation in this Note basically refers to finite games; separate
consideration will be given, when appropriate, to continuous games.
Definition 1.1. The normal form representation of a game G consists in the
specification of the number of players, the pure strategy space and the payoff function
of each player:
(1.1)
G   I , Si , ui  si , si  i 1
I
We say that the game is of complete information if all players know the structure of the game,
namely all its constituent elements: the number of players, the strategy space of every player
and everybody’s payoff functions. This last element is particularly relevant: incomplete
information typically occurs when the payoff function of one or several players is not known
by all the other players. A Bayesian approach to the description of the game is in that case
necessary; the notion of equilibrium must be correspondingly extended.
1.1.2 Games in mixed strategies
Not all games in normal form have solution in pure strategies. Take the following game in
bimatrix form, known as Matching Pennies (see Fig. 1.1). Each player’s strategy space
contains two actions Si  Heads, Tails , for short Si  H , T  , i  1, 2 . The payoffs reflect a
game in which each player has a penny and must choose how to display it. If the two pennies
match with heads or tails facing up, player 1, the row player, wins player 2’s penny; if the
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.3
pennies do not match, player 2 wins player 1’s penny.2 Since each player wins what the other
player looses, this is a competitive or zero-sum game. It is immediately clear that the game has
no solution in pure strategy; each player would try to outguess the other, but with no element
on the basis of which to make one’s choice of strategy.
2
Heads
Tails
Heads
+1, -1
-1, +1
Tails
-1, +1
+1, -1
1
Figure 1.1 – Payoff matrix of the Matching Pennies game
The uncertainty each player faces in the choice of strategy has led to the idea that players may
randomize between their pure strategies S i and thus adopt a mixed strategy, which represents
a probability distribution over the set of pure strategies. Although the mixed extension of the
game has a crucial role in the proof of existence of a equilibrium solution, the traditional view
of mixtures as conscious randomization carried out by each player has, nonetheless, come
under heavy criticism. To suppose, in fact, that rational economic agents, firms in particular,
should delegate the determination of their behavior to the outcome of a pure chance
mechanism is disturbing and utterly unrealistic.
Harsanyi (1973) and Aumann (1987) have, however, convincingly argued for a different view
of mixing, namely for an approach which reflects the uncertainty that each player has on the
strategy choice of the other player - more generally, of the other players.3 “According to this
view, players do not randomize; each player chooses a definite strategy. But other players
need not know which one, and the mixture represents their uncertainty, their conjecture about
this choice” (Aumann and Brandenburger, 1995, p. 1162, emphasis added). This uncertainty
can be analytically expressed in terms of a probabilistic conjecture, or belief, concerning the
opponent’s behavior. Let us then indicate with  2 the probability that player 1 assigns to
player 2 playing strategy s12   Heads  and with 1   2  the probability the he assigns to
player 2 playing strategy s22  Tails  . Let us similarly indicate with  1 and 1   1  the
conjecture of player 2 as to the strategy choices s11   Heads  and s12  Tails  of player 1.
The probability distributions 1 ,1  1  and  2 ,1   2  define a new set of possible
strategies that each player attributes, by way of conjecture, to the opponent. We will refer to
these strategies as mixed strategies to distinguish them from the pure strategies Heads and
Tails, of which they are a probability combination. For short, as in most textbooks, we will
2
In the Matching Pennies game payoffs are naturally taken to be a quantity of money, rather than the utility of winning
or losing a penny.
3
Actually Harsanyi’s so called purification theorem has the still further implication that players play pure strategies
and mixed strategies represent the probabilities with which the pure strategies are effectively played.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.4
write that  i is the mixed strategy of player i, omitting to say that this is a conjecture made by
player j.


Definition 1.2. The mixed strategy  i   i1 ,...,  i ki ,...,  i Ki of player i is a probability
distribution over the set of pure strategies S i . The set of mixed strategies of player i
(1.2)
Ki




  Si    i    Si   i k  0,  i k  1
k 1




is therefore a K i  1 dimensional simplex of probability distributions over the set of
 
pure strategies S i ;  i si k   ik is the probability assigned by the mixed strategy  i to
the pure strategy si k ;   Si  is a non empty, closed, bounded and convex subset of a
K i dimensional vector space.
The mixed strategy  i is then a vector with as many component as there are pure strategies.
When the strategy space contains only two pure strategies, as in the case of the Matching
Pennies game, it is sufficient to indicate the first component of the vector  i1 ; we will directly
set  i   i1 .
The set of mixed strategies   Si  of the game Matching Pennies is represented in Fig. 1.2 by
the segment AB. The points A and B correspond to the mixed strategies  i  1, 0  and
1 B
A
1
Fig. 1.2 – The set of mixed strategies of the Matching Pennies
 i   0,1 , that is respectively to the case of playing with certainty strategy Heads and to the
case of playing Tails for certain. Pure strategies thus correspond to degenerate mixed
strategies, i.e. to mixed strategies that assign probability one to just one pure strategy and
probability zero to all the others.
With the same notation as used above with regard to pure strategy games, let
  S  i      S j  e   S     Si     Si  be the sets of mixed strategies of the players
j i
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.5
different from player i and the space of mixed strategies of the game.4 A profile of mixed
strategies is therefore a vector  i ,  i     S  .
To complete the description of a mixed strategy game we must define players’ payoffs.
Returning to Matching Pennies, let us determine the expected payoffs of player 1, first
separately for his two pure strategies Heads and Tails and then for the mixed strategy over
Heads and Tails that represents player 2’s conjecture.
Each pure strategy is now a lottery over the consequences  1, 1 with probabilities
 2 ,1   2  . The expected payoffs are
u1  Heads,  2   u1  Heads, Heads   2  u1  Heads, Tails 1   2  
(1.3)
  1  2   11   2 
u1 Tails,  2   u1 Tails, Heads   2  u1 Tails, Tails 1   2  
=  1  2   11   2 
These payoffs correspond to von Neumann and Morgenstern’s expected utilities. The
expected payoff to player 1’s mixed strategy 1 ,1  1  is therefore the weighted average of
the expected payoffs of the two pure strategies Heads and Tails:
(1.4)
u1 1 ,  2   u1  Heads,  2  1  u1 Tails,  2 1  1  
  1 1 2   1 1 1   2    11  1   2   11  1 1   2 
The expected payoff of player 2 can, obviously, be determined in a wholly similar way. The
payoff function of player i is then a mapping from the mixed strategy space of the game to the
real line.
2
Heads
Tails
Heads
1  2
1  1   2 
Tails
1  1    2
1  1   1   2 
1
Fig 1.3 – Probability distribution induced by the mixed strategies  1 and  2
The definition (1.4) of the expected payoff of player 1 – and similarly of player 2 – shows that
the players’ mixed strategies induce a probability distribution over the cells of the payoff
matrix of the game, as indicated in Fig. 1.3. If, as apparently obvious, each player attributes
The definition of the mixed strategy space of players j  i is based on the assumption that each player’s
randomization be independent of that of all other players – or better, that each player believes that the mixed strategy
choices of the other players are independent, i. e. uncorrelated. This possibility and the consequences thereof are, for an
important aspect, analyzed at the end of this Lecture Note.
4
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.6
probability
1
2
playing Heads, the probability on each of the cells of the payoff matrix would
be equal to
1
4
. We are now in a position to give a formal definition of a mixed strategy game,
which parallels Definition 1.1.
Definition 1.3 The normal form representation of a mixed strategy game  consists in
the specification of the number of players, of each player’s mixed strategy space and
expected payoff function
   I ,   Si  , ui  i ,  i  i 1
I
(1.5)
To sum up the notation used in the mixed strategy extension of the game, we have:
- I  1, 2,..., I is the finite number of payers;

- Si  si1 ,..., si k ,..., si Ki
 is the finite set of pure strategies or actions of player i ;
Ki




-   Si    i    Si   i  0,  i k  1 is a K i  1 dimensional simplex of probability
k 1




 
distribution over the set of pure strategies S i ;  i k si k is therefore the probability assigned
by the mixed strategy  i to the pure strategy si k ;   Si  is a non empty, closed, bounded
and convex subset of a K i dimensional vector space;
 
-   Si    j i  S j and   S     Si     Si  ; as the Cartesian product of non empty,
closed, bounded and convex sets,   S     Si     Si  is a non empty, closed, bounded
and convex subset of a
 Ki
dimensional vector space;
i
-  i ,  i     S  is a profile of mixed strategies of all players;
- ui  i ,  i  is the payoff to player i from the strategy profile  i ,  i  .
1.2 Alternative approaches to the determination of the equilibrium
The equilibrium of a game is a profile of strategies such that no player has the incentive to
unilaterally deviate from his choice. How is this choice determined in a situation of strategic
interaction, of which all players are fully aware?
It is commonly assumed, in a context of individual decision-making, that agents behave
rationally, namely that the action chosen by each agent is at least as good, according to the
agent’s preferences,5 as any other available action.6 We will assume this principle to hold also
5
This notion is sometimes referred to as “instrumental rationality”.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.7
in game theoretic situations.7 But in a strategic context the best action of any player generally
depends on the actions that the other players will choose. Each player will then be compelled
to anticipate these actions and base his decision on a conjecture, or belief, of the other
players’ choice of action.
In his PhD dissertation John Nash (1950) suggests two approaches to the determination of an
equilibrium profile of strategies and thus to the formation of conjectures. The first is by
deduction and introspection, that is through a process of deductive reasoning based on the
assumption of rationality and common knowledge of rationality and of the game structure.
This is the approach followed in this Lecture Note and throughout the course. It will be amply
illustrated in the study of Nash Equilibrium.
The second approach is based on the “mass-action” interpretation of the equilibrium points of
the game. Nash presents this approach envisaging a large population of potential players for
each position of the game; with reference to a two-person game, we thus assume a large
population of participants in the position of each of the two players. A participant is selected
at random from each population and plays the game with no possibility of communication
with the other player. The repeated play by randomly selected players, so that the probability
that the same players meet again can be ignored, determines an “average playing” of the
game. Hence, “the probability that a particular [set] of strategies will be employed in a
playing of the game should be the product of the probabilities indicating the chance of each of
n pure strategies to be employed in a random playing” (Nash, PhD dissertation, 1950, p. 22).
One natural specification of each player’s conjecture about the play of his opponent is that it
converges to the probability distribution corresponding to the average playing of the game. In
such idealized setting of play, the equilibrium of the game is a steady state of the process of
repeated playing in a random matching of players (Osborne, 2004, p. 22).
The idea underlying Nash’s mass-action interpretation of equilibrium points of a game is that
players learn each other’s strategies from their experience playing the game. An alternative
approach to Nash’s mass-action model is to assume that players start from some initial,
possible unexplained prior conjecture, which they subsequently revise in response to
information received. This model envisages a personal learning process which, if convergent,
leads to a steady state solution of the game. Cournot (1838, Ch. VII) relies on such a process
for the attainment of the duopoly equilibrium in his quantity strategy game; he assumes, in
particular, that each player adjusts his strategy assuming that the other firm will maintain the
level of output chosen in the previous period.
6
Bounded rationality departs from this principle. As remarked by Simon (1955), economic agents do not have the
knowledge and the means to make optimizing choices and settle, therefore, for satisfycing decisions, on the basis of
routines that have proved in the past to deliver satisfactory results.
7
We will shortly see how the principle of rational choice is adapted in the theory of evolutionary games aiming to
portray the process of natural selection among animals and plants.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.8
Convergence fundamentally depends on the model of learning. Opposite results can easily
emerge with reasonable changes of assumptions. Cournot’s duopoly model offers examples of
convergent a well as of divergent results.
A final mention is in order to evolutionary game theory. Evolutionary game theory originated
as an application of the mathematical theory of games to biological contexts. In line with
Nash’s mass-action interpretation, it studies the behavior of a large population of agents who
repeatedly engage in strategic interaction. In evolutionary game theory strategies are
generically inherited traits that control an individual’s action – something akin to a computer
program. Payoffs are specified in units of fitness, which represent the capacity of a specie to
reproduce itself and thus survive in the Darwinian process of natural selection. The
mathematical biologist John Maynard Smith (1972) introduced the notion of evolutionary
stable strategy (ESS), as a tool of explaining the existence of an equilibrium outcome of
animal conflict, in the sense that, if every member of the population follows an ESS, no
mutant can successfully invade and radically change the equilibrium of the specie. An explicit
dynamic foundation, in terms of differential equations, of this equilibrium concept was later
provided by the model of the replicator dynamics. Replicator dynamics reflects the rule that
the fitter players will generate more replicas of themselves than the less fit, which will be thus
culled out of the population.
The approach and the tools elaborated by evolutionary game theory have been of interest to
the development of game theory in general, in particular in the applications to the problem of
equilibrium selection in the presence of multiple equilibria.
1.3 Strict dominance
We start the search of equilibrium solutions of a game on the basis of a negative criterion: no
rational player would ever choose an action that is payoff dominated by another action
available to him. This criterion requires each player to consider only his own payoffs; no
knowledge is required of the structure of the game; no conjecture need be formulated as to
choices of the other players.
1.3.1 Strict and weak dominance: definitions
Definition 1.4 The pure strategy si  Si is a strictly dominant strategy for player i in the
game G   I , Si , ui  si , si   if for all si  si we have
i 1
I
(1.6)
ui  si , si   ui  si, si  for si  Si
A double condition identifies a strictly dominant strategy for player i: i) it offers the highest
payoff among all strategies available to the player and ii) this highest payoff is for all possible
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.9
strategies of the other players. A weaker definition of dominance results if the strict inequality
sign in (1.6) is changed into a greater than or equal sign, as in the following definition
Definition 1.5 The pure strategy si  Si is a weakly dominant strategy for player i in the
game G   I , Si , ui  si , si   if for all si  si we have
i 1
I
ui  si , si   ui  si, si  for si  Si
(1.7)
Weak dominance requires therefore that strategy si have a payoff no less than that of all other
available strategies si and at least one payoff strictly greater.
The converse of the notion of dominant strategy is the notion of a dominated strategy.
Definition 1.6 The pure strategy si  Si is strictly dominated for player i in the game
G   I , Si , ui  si , si  i 1 if there exists a strategy si  Si such that
I
(1.8)
ui  si, si   ui  si , si  for si  Si
Definition (1.6) obviously implies that strategy si strictly dominates strategy si . We can at
this point restate the definition of a strictly dominant strategy as a strategy that strictly
dominates all other strategies si  Si . The extension of Definition 1.6 to the case of weakly
dominated strategy parallels that of a weakly dominant strategy.
We will examine in the next section an example of application of the notion of dominance to
determine the solution of a most renown game, the Prisoners’ Dilemma.
These definitions of strictly (weakly) dominant strategy and strictly (weakly) dominated
strategy apply also to the mixed extension of a game.
Definition 1.7 The mixed strategy  i    Si  is strictly dominated for player i in the
game    I ,   Si  , ui  i ,  i  
I
i 1
(1.9)
if there exists a strategy  i    Si  such that
ui  i,  i   ui  i ,  i  for  i    Si 
We then have:
Definition 1.8 The mixed strategy  i    Si  is strictly dominant for player i in the
game    I ,   Si  , ui  i ,  i  
I
i 1
if it strictly dominates all other strategies in   Si  .
An implication of Definition (1.7) is that a pure strategy may not be strictly dominated by any
other pure strategy, but by a mixed strategy of other pure strategies.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.10
Consider, for example, the following game (Mas-Colell 1995, p.241) in which player 1 has
three pure strategies Top, Middle, Bottom  , for short T , M , B  and player 2 has two pure
strategies  Left , Right  , for short  L, R  . The payoffs of the two players are indicated in the
normal form representation of the game in Fig. 1.4.
Left
Right
Top
10, 1
0, 4
Middle
4, 2
4, 3
Bottom
0, 5
10, 2
5, 3
5, 3
1
Top  12
2
Bottom
1 1
Fig. 1.4 – Game with pure strategy M dominated by the mixed strategy  1   , 0, 
2 2
Note that strategy M is not dominated by either pure strategy T or B: it has a greater payoff
than strategy T (B) if player 2 plays strategy R (L). M is, however, strictly dominated by the
1 1
mixed strategy  1   , 0,  with a payoff of 5 whatever the strategy played by player 2.
2 2
This is not the only mixed strategy of player 1 which dominates the pure strategy M . It can be
easily checked that all mixed strategies with 1 Top    4,6  strictly dominate M.
1.3.2 The prisoners’ dilemma: solution in strict dominance
The Prisoners’ Dilemma game is depicted in Fig. 1.5. The story behind the game is the
following. Two suspects of having committed a crime are taken into jail by the police and
held in separate cells, so that communication among the prisoners is made impossible. The
District Attorney has only indirect evidence, which will lead the jury to a sentence of only 3
month jail unless the prisoners admit of having committed the crime they are accused of. To
obtain a confession, the Attorney meets each prisoner separately and informs him of the
consequences of his possible actions. If he is the only one to confess, he will be rewarded
with a light sentence of, say, 1 month jail; but if he negates participating to the crime while
the other prisoner confesses, he will be heavily punished with a 12 month sentence to jail. If
they both confess, the jury will have mercy on them and sentence both to 6 month jail.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.11
The prisoners’’ Dilemma lends itself to a presentation in the terms of a game in normal form.
The number I of players is equal to 2; the strategy space of each players contains two actions
 Confess, Don ' t Confess  , for short  C, NC  ;8 the payoffs reflect to story behind the game.
Confess
Don’t
Confess
Confess
-6, -6
-1, -12
Don’t
Confess
-12, -1
-3, -3
2
1
Fig. 1.5 – Payoff matrix of the Prisoners’ Dilemma
The first step in search of a solution is to analyze the payoff matrix of the game to see if one
of the two strategies is strictly dominant say for player 1. From definition 1.1, we must,
therefore, consider only the payoffs of this player. At the risk of being overfastidious, these
payoffs are reproduced in Fig. 1.6. It is immediately evident that the strategy Confess strictly
dominates the alternative strategy Don’t Confess. This means that the rational choice of player
1, based only on the knowledge of his own payoffs (and not of the entire game) and with no
need to make any conjecture as to the decision of the other prisoner, is to play Confess, which
offers him a higher payoff than Don’t Confess whichever the choice of the other player. Since
player’s 2 rational choice is, for the same reasons, Confess, the equilibrium strategy profile of
the game is  Confess, Confess  .
Confess
Don’t
Confess
Confess
-6
-1
Don’t
Confess
-12
-3
2
1
Fig. 1.6 – Payoff of player 1 in the Prisoners’ Dilemma
Needless to say, the solution in strict dominance has a paradoxical flavor: the strategy profile
 Don ' t Confess, Don ' t Confess  is preferable for both prisoners. The conflict between the game
theoretic solution, rigorously based on the application of the exclusive principle of rational
behavior, and the efficient solution, which Pareto dominates the game theoretic solution, is
indicative of a situation that may arise in very different and very real contexts, typically in
problems of industrial organization. This circumstance paves the way to heavy criticism of game
theory solutions concepts and to proposals aimed at finding ways to reconcile equilibrium with
efficient solutions. We will dedicate attention to some of these problems further on in the course.
Fink for Confess and Quiet or Mum for Don’t Confess are also used to indicate the two strategies of the Prisoners’
Dilemma.
8
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1.12
1.4 Iterated strict dominance
The simple criterion that rational players would never use a strictly dominated strategy has led
us to the determination of a unique solution of the Prisoners’ Dilemma. Our purpose is now to
understand if substantially the same idea – that a rational player will never play a strictly
dominated strategy – opens the way to the determination of a solution in games in which the
direct elimination of strictly dominated strategies is, by itself, not sufficient to do it. The
technique consists in considering the possibility that the iterated elimination of strictly
dominate strategies may reduce the payoff matrix to a single surviving cell as in the
Prisoners’ Dilemma. As we will see, this is indeed possible in some instances, but the process
of iterated elimination requires a substantial change in the deductive capacities of the players:
it is not enough for each player to be individually rational; we have to ask players to know
that all players are rational – a property known as common knowledge of rationality (CKR).
We will separately examine the problems that may arise when the process of iterated
elimination is extended to weakly dominated strategies.
To avoid repeating a long sentence, we will distinguish between the solution approach based
on the existence of a strictly dominant strategy and the approach based on the iterated
elimination of strictly dominated strategies in terms of direct dominance and iterated
dominance.
1.4.1 Iterated elimination of strictly dominated strategies
Consider the game depicted in Fig. 1.6, in which both players have three strategies,
S1  Top, Middle, Bottom  , for short S1  T , M , B  , and S2   Left , Center , Right  , for short
S2   L, C , R  .
2
Left
Center
Right
Top
0, 1
3, 0
1, 2
Middle
2, 2
1, 0
2, 1
Bottom
1, 0
0, 1
1, 0
1
Fig. 1.6
While no pure strategy of player 2 is dominated by any other of his pure strategies, strategy
Bottom of player 1 is strictly dominated by strategy Middle. The elimination of strategy
Bottom of player 1 renders, in turn, strategy Center of player 2 strictly dominated by both his
other two strategies. The further elimination of this strategy induces player 1 to discard
strategy Top, which has now become strictly dominated by strategy Middle. The final step of
the process is now for player 2 to eliminate strategy Right that gives him a payoff of 1, as
opposed to the payoff of 2 of strategy Left: By successive elimination of strictly dominated
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.13
strategies of the two players only the strategy profile  Middle, Left  remains. This profile of
strategies is the solution of the game.
We have carried the process of iterated elimination of strictly dominated strategies in an
utterly mechanical way. However, what is the reasoning that justifies the players to
successively discard some strategies without directly knowing the previous decisions of the
other player? Here come the important logical underpinnings of iterated dominance, which
make the distinction with regard to the direct dominance approach of the Prisoners’ Dilemma.
First, the direct dominance approach demands the players to know only their own payoffs:
indirect dominance assumes, on the contrary, that players know the entire structure of the
game, in particular the complete payoff matrix. This assumption implies that the structure of
the game is common knowledge: everybody knows that everybody knows that everybody
knows … and so on. Second, it is necessary to explain why player 2 decides to eliminate his
pure strategy Center, which is not strictly dominated by either pure strategy Left or Right. To
this end, we must assume not only, as we have already done, that the structure of the game is
common knowledge, but also that player 2 knows that player 1 is rational and that, as a
consequence, he will not play his strictly dominated strategy Middle. The further step in the
process of iterated elimination requires that player 1 knows that player 2 knows that he is
rational. The final step requires, in turn, that player 2 knows that player 1 knows that he
(player 2) knows that player 1 is rational. The term for this assumption is common knowledge
of rationality: everybody knows that everybody knows that everybody knows (… and so on)
that players are rational. If the strategy space of every player is finite the process of iterated
elimination of strictly dominated strategies necessarily ends in a finite, though perhaps very
long, number of steps. If the strategy space is infinite, iterated dominance leads to a regress to
infinity, as will become apparent in the subsequent application to Cournot’s duopoly game.
The process of iterated elimination of strictly dominated strategies can be formalized
following the description given above, with player 1 beginning the process and players taking
 
turns in the successive steps.9 Let S10  S1 and  S10    S1  be player 1’s initial pure and
mixed strategy spaces. Define the first found of elimination of strictly dominated strategies in
terms of the surviving undominated strategies
(1.10)

S11  s1  S10 there does not 1    S1  s.t. u1 1, s2   u1  s1, s2  , s2  S2

Moving to the following round, define
(1.11)


S22  s2  S20 there does not  2    S2  s.t. u2  s1,  2 ,   u2  s1, s2  , s1  S11
Proceeding in this way we can define the subset of pure strategies surviving iterated
elimination respectively of player 1 and player 2 as S1n 1 and S 2n . At each step of the process
9
A more elegant formal description, independent of the player who starts the process, has the players to move
simultaneous at each step.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.14
the surviving subset of uneliminated pure strategies is strictly contained in the preceding one;
we then have S1n1  S1n3    S10 and S2n  S2n2    S20 . Indicating with S1 and S2 ,
obviously non empty, the final stages of the process, we can conclude that the game admits a
solution in iterated dominance if and only if both S1 and S2 contain only one element.
Nothing, of course, guarantees that the sets S1 and S2 contain only one element, as the
matrix game depicted in Fig. 1.7 of the following Section 1.5.1 shows, since none of the pure
strategies of the players is strictly dominated. In this case, we have Si  Si , i  1, 2 .
1.4.2
Iterated weak dominance
The process of iterated dominance applies to the elimination of strictly dominated and not to
weakly dominated strategies. The basic motivation for this exclusion is that, when there are
more than one pure strategy with is weakly dominated by another pure strategy, the outcome
of the process of iterated elimination may be path dependent, that is it may depend on the
order of elimination to the weakly dominated strategies.
Consider the game of Fig. 1.7 (Mas-Colell, p. 238).
2
Left
Right
Top
5, 1
4, 0
Middle
6, 0
3, 1
Bottom
6, 4
4, 4
1
Figura 1.7
While neither strategy of player 2 is dominated, the strategies Top and Middle of player 1 are
weakly dominated by strategy Bottom. We can start the process of iterated elimination by
either one. Suppose we start by eliminating strategy Top – suppose, more correctly, that
player 2, knowing that player 1 is rational, assumes that he will not play Top. The sequence of
iterated elimination proceeds at this point by player 1 assuming that the rational player 2 will
in turn eliminate the now weakly dominated strategy Right. The remaining profile of
strategies is then  Bottom, Left  , which would accordingly be the solution of the game. It can
easily be checked that if the process of iterated elimination of weakly dominated strategies
starts with the elimination of strategy Middle, the resulting equilibrium profile of strategies
would be  Bottom, Right  .
1.5 Rationalizable strategies
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.15
The notion of rationalizable strategies was independently proposed by Bernheim (1984) and
Pearce (1984) in a context of diffused criticism of Nash equilibrium and its various
refinements. The nature of the criticism implied by the rationalizable strategies approach will
be considered further on in the course. Our concern here is with the definition and the
relationship with dominance.
1.5.1 Approach and definition
As above underlined, the rationale for the elimination – direct and iterated - of strictly
dominated strategies is the criterion that no rational player would choose a strategy that would
deliver a lower payoff than another, pure or mixed, strategy available to him, regardless of the
strategies chosen by the other players. To this negative approach of eliminating “poor”
strategies, rationalizable strategies oppose a positive approach to the solution of a game,
namely of searching for “good” strategies. These are the strategies that a player would be
justified playing, by an appropriate chain of reasoning, when the structure of the game and the
players’ rationality are common knowledge. By implication, since a player would never
choose a strategy which is not “good”, non rationalizable strategies can be eliminated, in a
way similar to iterated dominance, when looking for a solution of the game.
Definition 1.9 In the game    I ,   Si  , ui  i ,  i  
I
i 1
the mixed strategy  i    Si 
is rationalizable if it is a best response to some conjectures (beliefs) that player i as to
the strategies  i    Si  of the other players. Formally:  i is rationalizable if there
exists  i    Si  such that
(1.12)
ui  i ,  i   ui  i,  i  for  i   i    Si 
By implication, the mixed strategy  i    Si  is never a best response if there is no
conjecture of player i as to the strategies  i    Si  of the other players for which
 i is a best response.
There are aspects concerning the role of conjectures that deserve being stressed:
i)
player i is supposed to formulate a conjecture as to the possible choices of the other
players and determine his best response on that basis; these conjectures correspond to
the definition we have adopted for mixed strategies   i - an expression of player i’s
uncertainty as to the strategy choices of his rivals;
ii)
 i is rationalizable if there is some conjecture that makes it a best response. As regards
the practical approach to follow to test for razionalizable strategies in typical two-player
bi-matrix games, it is convenient to proceed as follows. Start from the pure strategies of
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.16
iii)
player 1 and suppose (conjecture) that also the other player uses a pure strategy (in other
terms, that he plays a degenerate mixed strategy). If a pure strategy of player i is not a
best response to any pure strategy of another player, it will not be best response to any
non degenerate mixed strategy of the other player;
The assumption of common knowledge of the structure of the game and of the
rationality of the players has the important implication that each player’s conjecture
about the other players’ strategies should not be arbitrary. He should expect the other
players to use only strategies that are best response to some of their conjectures. The
argument of common knowledge of rationality leads now to a potential regress to
infinity about the reciprocal conjectures of the players and supplies the basis for the
process of iterated elimination of strategies that are never a best response (Fudenberg
and Tirole, 1998, p. 49).
Definition 1.10 In the game    I ,   Si  , ui  i ,  i  
I
i 1
the strategies   Si  that
survive the iterated elimination of never best response strategies represent the set
  Si  R of rationalizable strategies.
A convenient analytical definition of a best response strategy is by means of the
correspondence BRi  i  :   Si     Si  , that is the mapping from some conjecture
 i    Si  to  i    Si  .10
Definition 1.11 A best response strategy  i is an element of the best response
correspondence BRi  i 
(1.13)
 i  BRi  i    i    Si  u  i ,  i   u  i,  i   i   i and some  i    Si 
We will analyze the properties of best response correspondences in Lecture Nore n. 2 when
we will examine the problem of existence of a Nash equilibrium.
It is time to work out an example of the determination of best response strategies and of the
set of rationalizable strategies. Consider Bernheim’s game depicted in Fig 1.7
Both players have 4 pure strategies: S1  a1 , a2 , a3 , a4  and S2  b1 , b2 , b3 , b4  . In order to
determine which strategies of player 1 are best response to some strategies of player 2 we
proceed as indicate in point ii) above. Suppose that player 1 conjectures that player 2 is going
to play for certain strategy b1 , i.e. that he adopts the degenerate mixed strategy
 2  1, 0, 0, 0 ; comparing the payoffs of the different strategies of player 1 in the column
corresponding to player 2 playing b1 (respectively 0,5, 7, 0 ) we immediately see that
10
The reason for the definition of the mapping as a correspondence, i.e. a multivalued function, will become apparent
in the construction of the best response function in mixed strategies.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.17
strategy a3 is player 1’s best response. We signal this in the payoff matrix by a bar over the
highest payoff. As we continue in this way, modifying the conjecture of player 1 as to the
pure strategy played for certain by player 2, we realize that all his pure strategies are best
response to some pure strategy of his rival. We do the same for player 2, indicate his best
response pure strategies by a bar under the highest payoff and notice that strategy b4 is never
a best response. In fact, as indicated in the last column of the payoff matrix of Fig. 1.7,
strategy b4 is also strictly dominated by an equal probability mix of strategies b1 and b3 . We
proceed, therefore, to the elimination of this strategy. When this is done, we realize that
strategy a4 is no longer a best response, which was previously justified by the presence of b4 .
This is a short and imprecise wording for the longer, but correct statement: player 1 correctly
conjectures that strategy b4 is never best response; player 2, whose rationality is by
assumption common knowledge, will therefore never use it; at this point player 2 correctly
conjectures, on the same basis, that player 1 will never choose strategy a4 , which ceases to be
a best response. The set of rationalizable strategies of the two players are, therefore,
S1  a1 , a2 , a3  and S2  b1 , b2 , b3  ; the set of rationalizable strategies of the game is the
Cartesian product S R  S1  S 2 .
1
b
2 1
 12 b3
b1
b2
b3
b4
a1
0, 7
2, 5
7, 0
0,3
3,5;3,5
a2
5, 2
3, 3
5, 2
0,1
2,5; 2
a3
7, 0
2, 5
0, 7
0,1
3,5;3,5
a4
0, 0
0, -2
0, 0
10, 1
0; 0
1
Fig. 1.7 – Bernheim’s game
As stated at the beginning of this Section, rationalizable strategies are strategies that a player
would be justified playing, by an appropriate chain of reasoning, when the structure of the
game and the players’ rationality are common knowledge. The chain of reasoning that leads
to the conclusion that a strategy is rationalizable deserves to be spelled out in detail: important
differences emerge among rationalizable strategies from this point of view.
Consider strategy a1 of player 1. This strategy is rationalizable by the conjecture that player 2
will play strategy b3 , which is in turn rationalizable for player 2 by the conjecture that player
1 will play strategy a3 . Proceeding further in our chain of reasoning, a3 is razionalizable by
player 1’s conjecture that player 2 will play b1 , which is rationalizable by player 2’s
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.18
conjecture that player 1 plays a1 . At this point, we are back where we started and the loop
leading to rationalization continues to infinity. In extreme synthesis:
a1 is rationalizable by b3 ;
b3 is rationalizable by a3 ;
a3 is rationalizable by b1 ;
b1 is rationalizable by a1 ;
a1 is rationalizable by b3 ; and so on in an infinite regress with players’ conjectures
always disproved.
Consider, on the contrary, strategy a2 , which is justifiable by player 1’s conjecture that player
2 will play strategy b2 , which is turn razionalizable for player 2 by the conjecture that player
1 will play strategy a2 . In this case, the loop is immediately closed; both players’ conjectures
are verified. This property of conjectures to be verified constitutes the equilibrium assumption
that leads to the definition of Nash equilibrium, which is a subset of rationalizable strategies.
In Definition 1.9 we have associated the set of rationalizable strategies to the set of strategies
surviving the elimination of never best response strategies. From Definition 1.10 we have
further drawn the conclusion that the set of rationalizable strategies coincides with the set of
best response strategies. In games with a finite set of pure strategies, as in Bergheim’s game,
there is no analytical way to identify razionalizable strategies: we have to check which
strategies are actually best response to some best responses of the other players. In the mixed
extension of a game, best response correspondences can be formally determined as will be
shown in Lecture Note n. 2, where examples of derivation will be given and, in connection
with the proof of existence of Nash equilibrium, properties of mixed best response strategies
are defined.
1.6 Relation between the set of strictly undominated and rationalizable strategies
It can be readily verified that in the game depicted in Fig. 1.7 the set of razionalizable
strategies coincides with the set of iterated undominated strategies. While this is a result of
general validity in 2-person games, it need not hold in games with more than 2 players. Let us
start by proving the following result
Proposition 1.1 A strictly dominated strategy can never be a rationalizable strategy.11
11
The proof of the converse proposition, namely that a rationalizable strategy cannot be strictly dominated, can be
found in Fudenberg and Tirole (1991, pp. 50-52).
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.19
Proof. By direct comparison of definitions we have: from definition 1.9, the pure strategy si
is not razionalizable if there exists  i    Si  such that ui  si,  i   ui  i ,  i  for all
 i  si    Si  . But this condition is identical to the condition, given in Definition 1.7, of a
strictly dominated strategy.
We can draw a first conclusion, namely that the set of razionalizable strategies S R is weakly
contained in the set of undominated strategies SUD : S R  SUD . In games with 3 or more
players, on the contrary, the relation between rationalizable and strictly undominated
strategies may depend on the existence of a possible correlation among the choices of the
opponents of a given player i . It is a standard assumption in game theory that the mixed
strategies   i , with i  2 , are statistically independent, in the sense that each player
chooses his mixed strategy independently of the simultaneous choices of the other players.
The following example, taken from Brandenburger (1992), illustrates the pointConsider the 3-player game in which the strategy spaces of the players are: S1  U , D  ,
S2   L, R  and S3   A, B, C  . We can represent the game in normal form by means of 3
payoff matrices, in which player 1 is the raw player, player 2 the column player and player 3
chooses the matrix. In each cell of the matrices (see Fig. 1.8), the payoffs of the three players
are indicated in the proper order: first that of player 1, then that of player 2 and finally that of
player 3.
2
Left
Right
Up
1,1,1
1,0,1
Down
0,1,0
0,0,0
1
2
Left
Right
Up
2,2,.7
0,0,0
Down
0,0,0
2,2,.7
1
Player 3: plays A
2
Left
Right
Up
1,1,0
1,0,0
Down
0,1,1
0,0,1
1
Player 3: plays B
Player 3: plays C
Fig. 1.8 – Payoff matrices of the 3-players game
We first check for strictly dominated strategies. To this end we consider the payoffs of player 1
from all the possible combinations of strategies of player 2 and 3. The result is presented in Fig.
1.9.
2+3
1
 L , A   L , B   L , C   R , A   R , B   R, C 
Up
1
2
1
1
0
1
Down
0
0
0
0
2
0
Fig. 1.9 – Payoffs of player 1 to the possible combinations of strategies of players 2 and 3
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.20
The payoff matrix depicted in Fig. 1.9 shows that neither strategy of player is strictly dominated.
The same is true for player 2, since the payoffs to his strategies Left and Right to all the possible
combinations of strategies of players 1 and 3 are identical to the payoff matrix of Fig. 1.9.
3
1+2
1
2
U , L  U , R   D, L   D, R 
A
1
1
0
0
B
.7
0
0
.7
C
0
0
1
1
.5
.5
.5
.5
A  12 C
Fig. 1.10 – Payoffs of player 3 to the possible combinations of strategies of players 1 and 2
The payoff matrix of Fig. 1.10 is constructed in the same way as that of Fig. 1.9; it shows the
payoffs of player 3 to all possible combinations of strategies of players 1 and 2. Inspection
reveals that no pure strategy is no strictly dominated either by another pure strategy or by non
degenerate mixed strategy. Note, in particular, that strategy B is not dominated, for instance, by
an equal chance mixture of strategies A and C.
However, strategy B is not a best response to any possible conjecture of independent choices of
players 1 and 2, as the bar over the payoffs show. Actually, only the pure strategies A and C
can be best response to the possible combinations of strategies of players 1 and 2. Take, for
instance, the independent mixed strategies of players 1 and 2 1   2  12 which induces a
probability distribution of
1
4
over the columns of the matrix 1.10. the resulting payoffs to
strategies A , B and C of player 3 are respectively .5,.3.5,.5 .
We conclude that the set of razionalizable strategies is strictly contained in the set of
undominated strategies.
Assume, on the contrary, that the mixed strategies of players 1 and 2 are correlated, either on
account of preplay communications between the players or by the observation of a common
signal and are, e.g.,  U , L     D, R   12 and  U , R     D, L   0 . The expected payoffs
to strategies A , B and C of player 3 are now respectively .5,.7,.5 . So that we can conclude
that, if correlated randomization is admitted, the set of razionalizable strategies does coincide
with the set of undominated strategies.
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
1.21
References
Aumann, R. (1987), “Correlated Equilibrium as an Expression of Bayesian Rationality”,
Econometrica, vol 55, pp. 1-18
Aumann, R. and A. Brandenburger (1995), “Epistemic Conditions for Nash Equilibrium”,
Econometrica, vol. 63, n. 5, pp. 1161-1180
Brandenburger, A. (1992), “Knowledge and Equilibrium in Games”. Journal of Economic
Perspectives, vol. 6, no. 4, pp. 83-101
Fudenberg, D. and J. Tirole (1991), Game Theory, the MIT Press, Cambridge, Mass, USA, pp.
Harsanyi, J. (1973), “Games with Randomly Disturbed Payoffs: A New Rationale for Mixed
Strategy Equilibrium Points”, International Journal of Game Theory, vol. 2, pp.1-23,
Maynard Smith, J. (1972), On Evolution, Edinburgh University Press, Edinburgh
D. Tosato – Game Theory – Lecture Notes – a.y. 2013.2014
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