Game Theory
8 – Games of Incomplete information
GAMES OF INCOMPLETE INFORMATION
Review of lecture seven
• Games and supergames
• Repeated Prisoner Dilemma
• Finite number of stages: stage game with one
NE
• Indefinite number of stages and probability of
continuation
• The “shadow of the future” and the
inducement to cooperate
2
GAMES OF INCOMPLETE INFORMATION
Games and information (1)
• Static games of complete information
– Are generally represented in normal form
– In the case of two players the game is a
table (matrix) where rows and columns
represent the strategies of players
– Cells represent payoffs, known to all players
– Solution: NE
3
GAMES OF INCOMPLETE INFORMATION
Games and information (2)
• Dynamic games of complete and perfect
information
– Are generally represented through a game tree
– Each node represents an information set, i.e. a
situation where a player’s choice has to be done
– Branches represent moves
– A strategy is a set of instructions at each node
– All payoffs are known to all players
– Subgames and backwards induction
– Solution: SPNE
4
GAMES OF INCOMPLETE INFORMATION
Games and information (3)
• Dynamic games of complete and imperfect
information
– Some players do not know the node where they are
when they are asked to play
– Information sets are not all singleton
– A new definition of subgames is needed
– All payoffs are known to all players
– Generally strategies are envisaged through the normal
form
– Solution: SPNE (selecting those NE that satisfy
subgame perfection)
5
GAMES OF INCOMPLETE INFORMATION
Games and information (4)
Games studied so far assume that all players
know:
• Their own and others possible moves and
strategies
• Their own and others payoffs (and utility
functions when utility are different from
outcomes)
• When this is true we are dealing with games
of complete information
6
GAMES OF INCOMPLETE INFORMATION
Private information
• In real interactions people know their
preferences more than others’
• For instance an individual knows better her
own intentions and desires than intentions
and desires of her opponents in interaction
• In the reality no interaction exists without the
presence of private information
7
GAMES OF INCOMPLETE INFORMATION
Asymmetric information
• In strategic models private information is the
same as asymmetric information
• In turn that means that players do not know
perfectly strategies and preferences of
competitors
• It stems from this that the assumptions of
complete information are violated
• To deal with real situations it is necessary to
abandon the realm of complete information
8
GAMES OF INCOMPLETE INFORMATION
Incomplete information
• Strategic models of incomplete information
assume that players may not know payoffs
and utilities of some other players
• That implies that players may not know which
strategy other players will choose in all
circumstances
• Game theory has produced strategic models
that formalize this lack of knowledge
9
GAMES OF INCOMPLETE INFORMATION
“Types” of players and the move of “Nature”
• John Harsanyi (1967-1968) has proposed that
players may be a priori of some types, within a
defined set of types
• At the beginning of the game a casual move by a
fictitious player called Nature assigns the
effective type to each player
• This information is given privately to each single
player that knows exactly his/her own type
• Moreover players have common knowledge of
the distribution of probability of the types
10
GAMES OF INCOMPLETE INFORMATION
Players and types
–Each type of a player is characterized by
its preferences, i.e. a utility function
–Each player knows its own type but has
only a random knowledge of others’ types
–Harsanyi’s insight transforms a game of
incomplete information among players in
a game of imperfect information among
types
11
GAMES OF INCOMPLETE INFORMATION
Bayesian games
• Games of incomplete information introduce
beliefs as a necessary concept to perform the
analysis
• Beliefs are probabilities that some players assign
to the event of tackling this or that type of some
other players, while the distribution of probability
that each player has about the types of her
competitors is called her system of beliefs about
the other players
12
GAMES OF INCOMPLETE INFORMATION
Bayesian games
• In dynamic games it may happen that the
probabilities vary during the game as a
consequence of some previous moves of
some players
• When this happens, beliefs have to be
updated through the Bayes rule (see later…)
• This is why such games of incomplete
information are called Bayesian games
13
GAMES OF INCOMPLETE INFORMATION
Bayesian games in normal form
• Players have private information, and players’ actions are taken
simultaneously. In this case the system of beliefs of players are not
updated during the game (and they CANNOT be updated
actually…)
• Knowing exactly her type, each player chooses the strategy that
gives her the best expected gain, taking into account the possible
types of other players and their probability of realization
• When strategies are chosen the game ends
• A NE is reached if no player, observing the outcome, regrets her
choice (no other strategy would have given more) given the
existing system of beliefs.
• Sometimes in this case we talk of Bayesian Nash Equilibrium (BNE)
14
GAMES OF INCOMPLETE INFORMATION
Example: an international contest
• An international community C
• A state K is dissatisfied with the
international regime that C has
established to maintain
• K has two strategies: break (b) or not
break (n) the agreement
• In the same time C may decide to react
somehow (r) against K’s threat to the
established regime, or to show
indifference (i) to K’s attitude
• Four possible outcomes:
• br  international crisis (CR)
• bi  new regime (NR)
• nr  preemptive reaction (PR)
• ni  status quo (SQ)
C
K
r
i
b
CR
NR
n
PR
SQ
Let us suppose K is uncertain on C’s
determination to protect the ongoing
international regime
In our wording K does not know C’s
type with certainty
K knows that C may be of two types:
Prudent C : SQ > NR > CR > PR;
Determined C : PR > CR > SQ > NR
Nature informs K that the distribution
of probability on C’s type is
½ prudent, ½ determined
15
C
GAMES OF INCOMPLETE INFORMATION
K is a “one type player”
K
C may acquire two possible types: let us call them Cp
and Cd
Both have common knowledge of this
Moreover suppose K orders NR>SQ>PR>CR
What K will do against a prudent C? and what against a
determined C?
A strategy profile of the game picks out a strategy for all
types of players involved in the game, i.e. K, Cp, Cd
This profile can be written (s, v, w) where s may be (b)
or (n), while v and w may be (r) or (i).
Eight possible profiles exist of this kind: (b,r,r) (b,r,i),
(b,i,r), (b,i,i), (n,r,r), (n,r,i), (n,i,r), (n,i,i)
System of beliefs of player K about C: p(Cp)=p (p=1/2 in
this case)
r
i
b
CR
NR
n
PR
SQ
16
GAMES OF INCOMPLETE INFORMATION
BNE
• In a one-shot PD, how can you control if (DEF;
DEF) is a NE? Given DEF played by column player,
is DEF the Best Reply for row player? And once
established this, is DEF the best reply by column
player to this? If yes to both situations, you have
a NE (best reply to each other)
• In a BNE you look for the same: a pair of
strategies (one for each type, not player!) that
are best reply to each other GIVEN a particular
system of belief
17
C
GAMES OF INCOMPLETE INFORMATION
Determining BNE (1)
r
i
b
CR
NR
n
PR
SQ
K
The Bayesian normal form splits into two
matrices, as if the game was played by three
players: K , Cp, Cd
Cp
K
b
n
Cd
r
i
uK(br) , uCp(br)
uK(bi) , uCp(bi)
K
uK(nr) , uCp(nr)
uK(ni) , uCp(ni)
r
i
b
uK(br) , uCd(br)
uK(bi) , uCd(bi)
n
uK(nr) , uCd(nr)
uK(ni) , uCd(ni)
To find BNE we need to give values to payoffs
respecting the established orderings
18
C
GAMES OF INCOMPLETE INFORMATION
Determining BNE (2)
•
•
•
•
•
Reminding that
K orders NR>SQ>PR>CR
Cp orders SQ > NR > CR > PR
Cd orders PR > CR > SQ > NR
let us put
K
n
i
b
CR
NR
n
PR
SQ
K
uk(bi)=4, uk(ni)=2, uk(nr)=1, uk(br)=0
uCp(ni)=4, uCp(bi)=2, uCp(br)=1, uCp(nr)=0
uCd(nr)=3, uCd(br)=2, uCd(ni)=1, uCd(bi)=0
Cp
b
r
Cd
r
i
0,1
4,2
1,0
2,4
K
r
i
b
0,2
4,0
n
1,3
2,1
19
GAMES OF INCOMPLETE INFORMATION
Determining BNE (3)
Cd
Cp
r
b 0,1
K
n 1,0
i
4,2
2,4
r
b 0,2
K
n 1,3
i
4,0
2,1
Among eight profiles (brr), (bri), (bir), (bii), (nrr), (nri), (nir), (nii)
• for Cp strategy r is dominated
• for Cd strategy i is dominated
 profiles (svw) with r as the second letter and with i as the third
are to be eliminated
Only (bir) and (nir) survive for C
20
GAMES OF INCOMPLETE INFORMATION
Determining BNE (4)
Only (bir) and (nir) survive for C
As to K, it has to calculate its payoffs playing b and playing n, and to weight
both with the probability ½
Cd
Cp
r
b 0,1
K
n 1,0
i
4,2
2,4
r
b 0,2
K
n 1,3
i
4,0
2,1
uK(bir) = 4 ½ + 0 ½ = 2
uK(nir) = 2 ½ + 1 ½ = 1.5
 The only BNE of the game is {(bir)} when p(Cp)=½
the state K will break (b) the international regime and the
international community will react following the character of its
type (will be indifferent if prudent, will react if determined)
21
GAMES OF INCOMPLETE INFORMATION
BNE
•  The only BNE of the game is {(bir)} when
p(Cp)=½
Given this system of beliefs, no player regrets her
choice! The best reply to (ir) given p=1/2 is b for
player K; on the other side, given b, the best reply
to that of the two types of C is precisely (ir). Nash
Equilibrium! But Bayesian one! This pair of
strategies is a best reply to each other ONLY for
that specific system of beliefs (p=1/2)
22
C
GAMES OF INCOMPLETE INFORMATION
BNE conditions
r
i
b
CR
NR
n
PR
SQ
K
To be an equilibrium (s*, v*, w*) must satisfy the following conditions:
uK (s*,v*)p + uK (s*,w*)p  uK (s,v*)p + uK (s ,w*)p
• K has only one strategy to play, as it may assume only one type, and
its utility is expected because of K’s uncertainty whether to tackle Cp
or Cd
uC (s*,v*)  uC (s*,v)
uC (s*,w*)  uC (s*,w)
• C plays two strategies, one for each type it can take, and its utility is
certain, as C is certain about K’s nature
23
GAMES OF INCOMPLETE INFORMATION
Determining BNE (5)
The only BNE of the game is {(bir)} when p(Cp)=½
• The result depends on the values given to utilities and to players’ beliefs (in
this case only player K’s beliefs about the nature of player C)
• In the example beliefs are given as data of the problem
• In real cases the interests involved in K’s action and the probabilities of C’s
reaction have to be uncovered by empirical analysis
Let us call π the probability that K assigns to a prudent C, so that its belief
about the event that C is determined is 1−π
Then, substituting the now unknown π to ½
uK(bir) = 4π + 0(1−π)
uK(nir) = 2π + (1−π)
 uK(bir) > uK(nir) if and only if π > ⅓
 Breaking the international agreement is the better strategy only when the
probability of encountering a prudent C is sufficiently high (more than ⅓ with
the chosen payoffs ). If π < ⅓ then the only BNE of the game becomes {(nir)}
24
GAMES OF INCOMPLETE INFORMATION
Example: a dating riddle (when no
dominated strategies are available)
• The scenario: A boy - Player 1; and a girl – Player 2
• Player 2, as always happens with girls, know player’s 1
preferences, while player 1 is unsure (as all boys are…)
• Specifically player 1 think that with probability ½ player 2 wants
to go out with her, and with probability ½ player 2 want to avoid
him….
• That is, player 1 thinks that with probability ½ he is playing the
game of the left, and with probability ½ the game on the right
1
Player 2 ( ½ )
Player 2 ( ½ )
S
O
S
O
S
2,1
0,0
S
2,0
0,2
O
0,0
1,2
O
0,1
1,0
1
25
GAMES OF INCOMPLETE INFORMATION
Example: a dating riddle
• For this situation we define a BNE to be a triple
of actions, one for player 1 and one for each
type of player 2, with the property that:
• The action of player 1 is optimal, given the
actions of the two types of player 2 (and
player 1’s belief about the state)
• The action of each type of player 2 is
optimal, given the action of player 1
26
GAMES OF INCOMPLETE INFORMATION
Example: a dating riddle
• Solving the game…
• Player 1 does not know player’s 2 type, so to choose an action rationally he
needs to form a belief about the action of each type. Given these beliefs
and his belief about the likelihood of each type, he can calculate her
expected payoff to each of her actions. Let’s see the calculus of player 1
(S,S)
(S,0)
(O,S)
(O,O)
S
2
1
1
0
0
0
1/2
1/2
1
• (S,(S,O)) with p=1/2, where the first component is the action of player
1 and the other component is the pair of actions of the two types of
player 2, is a BNE
Player 2 ( ½ )
Player 2 ( ½ )
S
O
S
O
1
S
2,1
0,0
O
0,0
1,2
1
S
2,0
0,2
O
0,1
1,0
27
GAMES OF INCOMPLETE INFORMATION
Example: a dating riddle
• That is, in a BNE, player’s 1 action is a best
response to the pair of actions of the two types
of player 2, while the action of the type of
player 2 who wishes to meet player 1 is a best
response in the previous left table to the action
of player 1, and the action of the type of player
2 who wishes to avoid player 1 is a best
response in the previous right table to the
action of player 1
28
GAMES OF INCOMPLETE INFORMATION
Example: a dating riddle
• Why should player 2, who knows whether she
wants to meet or avoid player 1, have to plan
what to do in both cases? She does not have to
do so! But we, as analysts, need to consider
what she would do in both cases
• Thus the equilibrium action of player 2 for each
of her possible types may be interpreted as
player 1’s correct belief about the action that
each type of player 2 would take, not as a plan
of action for player 2
29
GAMES OF INCOMPLETE INFORMATION
Incomplete information and estensive form
• Even in some dynamic setting, we can have different
types but NO possibility to update beliefs by players
according to actions undertaken during the game
whenever the “type players” do not move FIRST!
• Let us consider an electoral college where plurality rule is
in effect
• In a multi-partisan system, party T is the usual winner
presenting a popular local candidate
• Considering the opportunity to interrupt this tradition,
party S may choose to present one of its national leader
(s1) or to give up (s2)
• In case (s1) is played, T may decide to answer presenting
itself a national leader (t1) or to continue with the local
candidate (t2)
30
GAMES OF INCOMPLETE INFORMATION
“choosing the candidate” game (1)
For T the best outcome is the status quo (it
wins with no costs)
Behaving as usual against the challenge is
the second (the contest is uncertain but a
national leader is saved for other
situations)
The last result is accepting the challenge
(the result is uncertain but a national
leader is lost for other situations)
For S the best result is challenging with no
reaction by T (high probability of winning)
The second best is not to challenge at all (it
looses but the national leader is saved
The last result is when its challenge is
accepted (no better chance to win and one
national leader lost)
T
t1
-2, -2
s1
S
t2
s2
2, -1
-1, 1
The solution it immediate by
backward induction:
• SPNE (s1 ; t2)
• The game develops with a
challenge ignored
• The outcome is (2,-1)
31
GAMES OF INCOMPLETE INFORMATION
“choosing the candidate” game (2)
Let us now suppose that the
challenging party S is uncertain about
party T preferences
1. T is considered preferring to loose
the college than to spend a
national leader
2. Or T is considered preferring to
win the college than to go without
a leader in other competitions
In other words T may be represented
by the type TR (ready to loose the
college) or by the type TD (determined
to get the seat)
Uncertainty can be represented by an
initial move of the Nature choosing
the type TR or TD of party T
N’s move is private information of
player T
t1
T
-2, -2
s1
S
TR
t2
s2
2, -1
-1, 1
N
T
t1
-2, -1
s1
TD
t2
S
s2
2, -2
-1, 1
32
GAMES OF INCOMPLETE INFORMATION
“choosing the candidate” game (3)
Passing to the normal form, the game is among three types of
players: S, TR and TD. As you can see we have two proper subgames
(plus the subgame of the entire game) As usual for three players
games the Bayesian normal form of the entire game implies two
matrices
t -2, -2
T 1
TD
TR
s1
S
t2 2, -1
TR
t1
t2
t
t
1
2
s2 -1, 1
T t1 -2, -1
s1 -2 , -1 2 , -2
s1 -2 , -2 2 , -1
N
s1
S
S
TD
t2 2, -2
s2 -1 , 1 -1 , 1
s2 -1 , 1 -1 , 1
S s
2 -1, 1
There are eight strategy profiles that are:
(s1,t1,t1); (s1,t1,t2); (s1,t2,t1); (s1,t2,t2); (s2,t1,t1); (s2,t1,t2); (s2,t2,t1); (s2,t2,t2)
Note that playing t1 for TR or t2 for TD wouldn’t be consistent with SPNE.
Therefore…only the two profiles (s1,t2,t1) and (s2,t2,t1) survive
33
GAMES OF INCOMPLETE INFORMATION
“choosing the candidate” game (4)
In this case the probabilities δ that S assigns to TR and 1−δ that S assigns to TD are
unknown
We need to see if values exist of δ such that one or the other survived profiles
(s1,t2,t1) and (s2,t2,t1) are not dominated for S
T prob δ
The expected utilities of party S for the two profiles are
t
t
R
S
us(s1,t2,t1) = 2δ−2(1−δ) = 4δ−2
us(s2,t2,t1) = −δ−(1−δ) = −1
us(s1,t2,t1) > us(s2,t2,t1) if and only if δ > ¼
1
2
s1
-2 , -2
2 , -1
s2
-1 , 1
-1 , 1
TD prob 1−δ
S
t1
t2
s1
-2 , -1
2 , -2
s2
-1 , 1
-1 , 1
 The game has two BNE equilibria:
1. {(s1,t2,t1) , δ > ¼} (when S believes that the incumbent party has at least 25%
probability of being TR , i.e. disposed to loose the college, S runs a national
leader and T answers accordingly to its character)
2. {(s2,t2,t1) , δ < ¼} (S continues to present the local candidate and T reacts as
before, if S believes that the incumbent party has at least 75% probability of
being TD , i.e. resolute to keep the college seat)
34
GAMES OF INCOMPLETE INFORMATION
Bayesian games with updating of beliefs
• Consider the following game: the gift game.
Friend tend to keep desirable objects in his
pocket to offer you as a gift, the Enemy no
(such as rocks or frogs…). In this variant of
the game, player 2 prefers to accept a gift
only from a Friend.
• Player 1 however can be of two different
types: Friend or Enemy
35
GAMES OF INCOMPLETE INFORMATION
Bayesian games with updating of beliefs
A
1
NF
GF
0, 0
1, 1
2
(q)
Friend
(p)
Enemy
(1-p)
R
(1-q)
-1, 0
A
1, -1
0, 0
NE
1
GE
R
-1, 0
36
GAMES OF INCOMPLETE INFORMATION
Bayesian games with updating of beliefs
• This game is a dynamic one where player 2
can update his/her beliefs about player 1
type according to what player 1 does…
37
GAMES OF INCOMPLETE INFORMATION
Bayesian games where the updating
of beliefs is possible
• There is indeed a difference, in the previous game,
between p (i.e., the initial belief about player 1’s type) and
q (player 2’s updated belief about player 1’s type, after that
player 2 observes the strategy of player 1)
• For example, suppose that player 1 behaves according to
strategy NF, GE; thus player 2 now expects a gift from an
enemy with (an updated) probability equal to 1, i.e.,
p=q=0.
• In general, player 2 has an updated belief about player 1’s
type, conditional on arriving at player 2’s information set
(that is, conditional on receiving a gift in our example)
• How to include such possibility into an equilibrium?
38
GAMES OF INCOMPLETE INFORMATION
A Perfect Bayesian Equilibrium
• PBE is a solution concept that incorporates sequential
rationality and consistency of beliefs.
• Sequential rationality: players maximize their payoffs from
each of their information sets (on or off the equilibrium
path!)
• How to reach that? Consistency of beliefs! In a PBE player’s
2 updated beliefs should be consistent with nature’s
probability distribution and player 1’s strategy
• In general consistency between nature’s probability
distribution (p in the previous example), player 1’s strategy,
and player 2’s updated belief (q in the previous example)
can be evaluated by using Bayes rule
39
GAMES OF INCOMPLETE INFORMATION
Bayes rule
• Bayes rule gives the conditional probability of an event
when another event has been observed, i.e., it gives us
a criterion to determine how new information should
change our beliefs about a given event
• Let p(A) and p(B) two a priori probability of the events
A and B
• Let us write p(A|B) the probability of the event A when
B has been observed
• Bayes rule is a formula for determining p(A|B)
• For example, suppose that the probability to meet a
Friend in the previous example determined by Nature
is ½. And let’s further suppose that the two types of
Player 1 adopt the following strategy (NE GF). Before
that strategy, p(FRIEND)=1/2. Now the update
probability of p(FRIEND|GF)=1
40
GAMES OF INCOMPLETE INFORMATION
A Perfect Bayesian Equilibrium: definition
• Consider a strategy profile for the players, as
well as beliefs over the nodes at all information
sets. These are called a PBE if: 1) each player’s
strategy specifies optimal actions, given his
beliefs and the strategies of the other players
and 2) the beliefs are consistent with Bayes rule
wherever possible
• In essence a PBE is a coherent story that
describes beliefs and behavior in a game
41
GAMES OF INCOMPLETE INFORMATION
A Perfect Bayesian Equilibrium:
how to find it!
• Two additional terms are useful: we call an
equilibrium separating if the types of a player
behave differently
• We call an equilibrium pooling if the types
behave the same
42
GAMES OF INCOMPLETE INFORMATION
An application
A
1
NF
0, 0
There are four
potential equilibria:
two separating
equilibria (featuring
strategy GF NE, or
strategy NF GE) and
two pooling
equilibria (featuring
strategy NF NE, or
strategy GF GE)
GF
1, 1
2
(q)
Friend
(p)
Enemy
(1-p)
R
(1-q)
-1, 0
A
1, -1
0, 0
NE
1
GE
R
-1, 0
43
GAMES OF INCOMPLETE INFORMATION
A Perfect Bayesian Equilibrium:
how to find it!
•
•
•
•
– Steps for calculating PBE:
Starts with a strategy for player 1
If possible, calculate updated beliefs (q in the example) for
player 2 by using Bayes rule. In the event that Bayes rule
cannot be used, you must arbitrarily select an updated
belief; here you will generally have to check different
potential values for the updated belief with the next steps of
the procedure;
Given the updated beliefs, calculate player 2’s optimal action
Check whether player 1’s strategy is a best response to player
2’s strategy. If so, CONGRATULATIONS: you have just found a
PBE!
44
GAMES OF INCOMPLETE INFORMATION
An application
• Let’s apply our procedure:
• Separating with NF GE:
• given this strategy for player 1, it be must be that
q=0. Thus, player 2’s optimal strategy is R. But then
the enemy type of player 1 would strictly prefer not
to play GE. Therefore, the is no PBE in which NF GE
is played
• Separating with GF NE:
• given this strategy for player 1, it be must be that
q=1. Thus, player 2’s optimal strategy is A. But then
the enemy type of player 1 would strictly prefer to
play GE rather than NE. Therefore, there is no PBE in
which GF NE is played
45
GAMES OF INCOMPLETE INFORMATION
An application
• Pooling with GF GE:
• here Bayes rule requires that q=p, so player 2
optimally selects A if and if only p>1/2. When
p>1/2 there is therefore a PBE in which q=p and
(GF GE, A) is played - PBE: (GF GE, A) , p=q; p>1/2
• Why an equilibrium? Given the strategy (GF GE)
played by player 1, the best reply for player 2 to
that GIVEN the belief specified (p=q; p>1/2) is A.
And given the strategy adopted by player 2 (A),
the strategy (GF GE) is the best reply to that for
both players!
46
GAMES OF INCOMPLETE INFORMATION
An application
• Pooling with GF GE:
• On the other hand, in the event that p<1/2,
player 2 must select R, in which case neither type
of player 1 wishes to play G in the first place.
Thus there is no PBE of this type when p<1/2.
47
GAMES OF INCOMPLETE INFORMATION
An application
• Pooling with NF NE:
• in this case Bayes rule does not determine q. Why? Cause in
this case both types of player 1 play N, and player 2 observes
just that! Player 2 cannot update q according to Bayer rule,
given that G is not played!
• Still, regardless of player 1’s strategy, player 2 will have
some updated belief q at his information set
• This number has meaning even if player 2 believes that
player 1 adopts the strategy NF, NE. In this case, q
represents player 2’s belief about the type of player 1
when the “surprise” of a gift occurs (i.e., off the
equilibrium path)
48
GAMES OF INCOMPLETE INFORMATION
An application
• Pooling with NF NE:
• But notice that the types of player 1 prefer not giving gifts only
if player 2 selects R. In order for R to be chosen, player 2 must
have a sufficiently pessimistic belief regarding the type of player
1 after the “surprise” in which a gift is given. Strategy R is
optimal as long as q<1/2. Thus, for every q<1/2 there is a PBE in
which player 2’s belief is q and the strategy profile (NF NE, R) is
played. In this equilibrium player 2 believes that a gift signals
the presence of the enemy (a misanthrope?)
• PBE: (NF NE, R) , q<1/2
49
GAMES OF INCOMPLETE INFORMATION
Job-Marketing Signaling
• Which role of formal education in the
marketplace? To the extent that highly
productive people are more likely than lessproductive people to get degrees, than
rather than helping people become smart,
universities exist merely to help people who
are already smart to prove that they are
smart!
50
GAMES OF INCOMPLETE INFORMATION
Job-Marketing Signaling
• The signaling role of education: a worker (W) and a firm
(F). The worker can be of two types: high or low type.
Firm must decides whether to employ the worker in an
important managerial job (M) or in a much less
important clerical job (C). M produces a benefit of 10 to
both types of worker, however they have different
education costs: the high type to get an education must
pay 4 units of utility, the low type 7. C produces a benefit
of 4.
• Importantly, education is of no direct value to the firm;
the firm’s payoff does not depend on whether the
worker gets an education
51
GAMES OF INCOMPLETE INFORMATION
Job-Marketing Signaling
10, 10
M’
M
N
F
4, 4
10, 0
C’
WH
E
(p)
C
High (1/3)
Low (2/3)
M’
(1-q)
N’
C’
F
(q)
(1-p)
4, 4
6, 10
WL
E’
0, 4
M
3, 0
C
-3, 4
52
GAMES OF INCOMPLETE INFORMATION
Job-Marketing Signaling: comments
• Two PBNE: (NN’, CC’, p=1/3, q<2/5) and (EN’, MC’,
p=0, q=1)
• Insights:
• First: the only way for the high-type worker to get
the job that she deserves is to signal her type by
getting an education. Otherwise the firm judges
the worker to be a low type
• Second: the value of education as a signaling
device depends on the types’ differential
education costs, not on any skill enhancement
that education delivers.
53
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
• Why PBE should be considered as a
refinement of a SPNE?
• Consider once again the gift game.
However, in this variant of the game player
2 always prefer opening gifts, no matter
what
54
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
A
1
NF
GF
0, 0
1, 1
2
(q)
Friend
(p)
Enemy
(1-p)
R
(1-q)
-1, 0
A
1, 0
0, 0
NE
1
GE
R
-1, -1
55
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
• Because the game has sequential decisions, it seems
appropriate to look for SPNE. But note that the game
has no proper subgame, so every NE is subgame
perfect.
• In particular, (NF, NE, R) is a SPNE (one of the two: the
other is???)
• We can easily see it by analyzing the Bayesian normal
form of the game. In this equilibrium, both types of
player 1 choose not to give a gift and player 2 plans
to refuse gifts. But this SPNE has a big problem.
Which one?
56
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
• (NF, NE, R) prescribes behavior for player 2 that is
clearly irrational conditional on the game
reaching his information set. Regardless of player
1’s type, player 2 prefers to accept any gift
offered!
• However this preference is not incorporated into
the SPNE because: 1) player 2’s information set is
not reached on the path induced by (NF, NE, R),
and 2) player 2’s information set does not
represent the start of a subgame
57
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
• Therefore, the concept of subgame perfection does
not sufficiently capture sequential rationality (that
players maximize their payoffs from each of their
information sets)
• On the contrary Perfect Bayesian equilibrium (PBE)
does just that! The key to this equilibrium concept is
that it combines a strategy profile with a description
of beliefs that the players have at each of their
information sets. The beliefs represent the players’
assessments about each other’s type, conditional on
reaching different points in the game
58
GAMES OF INCOMPLETE INFORMATION
SPNE vs. PBE
• Let’s go back to the example: (NF, NE, R) cannot be a PBE!
• We already discussed that also in this scenario player 2 will
have some updated belief q at his information set
• Given the belief q, we can determine player 2’s optimal action
at his information set. It is easy to show that action A is best
for player 2 whatever is q (i.e., there is no value of q that
could be consistent with (NF, NE, R) , that is, that could induce
player 2 to play R)
• Thus, sequential rationality requires that player 2 select A,
and therefore (NF, NE, R) is a SPNE but NOT a PBE.
59