EC941 - Game Theory
Lecture 3
Francesco Squintani
Email: f.squintani@warwick.ac.uk
1
Structure of the Lecture

Bayesian Games and Nash Equilibrium

Bayesian Games and Information

Cournot Duopoly with Private Information

Public Good Provision with Private Information
2
A Bayesian Game: Battle of the
Sexes with Private Information





Two players decide whether to go to a Bach or
Stravinsky concert.
Player 1 prefers Bach and player 2 Stravinsky.
Player 1 prefers to be with player 2.
Player 2 knows player 1’s preferences.
Player 1 thinks that with probability ½ player 2 wants
to got out with her, and with probability ½ player 2
wants to avoid her.
3
Information Structure of the
Battle of the Sexes
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1
We think of there being two states, one in which the players’
payoffs are given in the left table and one in which these
payoffs are given in the right table.
4
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1


Player 2 knows the state—she knows whether she
wishes to meet or avoid player 1—whereas player 1
does not; player 1 assigns probability 1/2 to each state.
From player 1’s point of view, player 2 has two possible
types, one whose preferences are given in the left matrix,
and one with preferences given in the right matrix.
5
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1

Player 1 does not know player 2’s type. To choose her
strategy, she forms a belief on the types’ strategies.

Given these beliefs and her belief about the types’
likelihood, she calculates the payoff of her strategies.
6
Bayesian Nash Equilibrium of
the Battle of the Sexes
A Bayesian Nash equilibrium is a triple of strategies, one
for player 1 and one for each type of player 2:
1.
player 1’s strategy is optimal, given the strategies of
the two types of player 2 and player 1’s beliefs on
player 2’s type.
2.
the strategy of each type of player 2 is optimal, given
player 1’s strategy.
7
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1

(B, (B, S)) is a Bayesian Nash equilibrium.
The first component is the strategy of player 1 and the
second is the pair of strategy of the two types of player 2.
8
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1
1.
2.
Given the strategies (B, S) of player 2, and the belief that
each one of the two types is equally likely, player 1’s
strategy B is optimal.
Given that player 1 chooses B, B is optimal for the type
who wishes to meet player 1 and S is optimal for the type
who wishes to avoid player 1.
9
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1

1.
2.
There are no other (pure-strategy) Nash equilibria.
If player 1 plays S, then the two types of player 2 respond
(S, B). The type who wants to meet 1 chooses S and the
type who wants to avoid 1 chooses B.
But given that the two types of player 2 respond (S, B),
player 1’s best response is B.
10
Bayesian Games
Definition A Bayesian game consists of
 a set of players I
 a set of states W
and for each player i:
 a set of strategies Si
 a set of types Ti, a partition of W
 for each type ti, a belief p(.|ti), a probability distribution
over the set of states included in ti
 a payoff function ui over pairs (s, w), where s is a strategy
profile and q is a state.
11


Each state is a complete description of the players’
relevant characteristics, including their preferences and
their information.
At the start of the game, a state w realizes.
The players do not observe it.

Each player i has a type ti , a set of the possible states.

The types of player i are disjoint sets, and cover the set
of states W. The set of types Ti is a partition of W.
12

The set Ti reflects the quality of player i’s information.

If Ti is composed of sets of single states, then player i
can always distinguish which state has occurred.


If Ti contains a single set, W, then player i has not any
information about the state.
If Ti contains more than one set, but not as many as W,
then player i has partial information.
13
Battle of the Sexes





Players: The pair of people.
States: The set of states is W ={meet, avoid}.
Strategies: The strategies of each player are Si={B, S}.
Types: Player 1 has a single type t1= W.
Player 2 has two types {meet} and {avoid}.
Beliefs: Player 1 assigns probability 1/2 to each state.
Player 2 knows the state.
14
Payoffs: The payoffs ui (s, meet) of each player i for all possible
strategy pairs s are given in the left panel, and the payoffs
ui (s, avoid) are given in the right panel.
B
B
S
S
2, 1
0, 0
B
B
0, 0
1, 2
2 2
S
S
2, 0
0, 2
0, 1
1, 0
1
15
Bayesian Nash Equilibrium
Definition A Bayesian Nash Equilibrium of a Bayesian
game G=(I, W, S, T, u) is a Nash equilibrium of the
strategic game defined as follows.
 Players The set of all pairs (i, ti) where i is a player in the
Bayesian game and ti is one of the signals that i may
receive.
 Strategies The set of strategies of each player (i, ti) is the
set of strategy of player i in the Bayesian game.
 Preferences The payoff function of each player (i, ti) is
given by:
∑w∈W p(w | ti) ui((si, s-i (w)), w).
16
Bayesian Nash Equilibria of the
Battle of the Sexes

The mixed strategy equilibria of the battle of the sexes
are calculated as follows.

Player 1 is indifferent between S and B if and only if
2sm(B) + 2sv(B) = 1- sm(B) + 1- sv(B).
Player 2 of type m is indifferent between S and B if and
only if s1(B) = 2(1- s1(B)), i.e. s1(B) = 2/3.
Player 2 of type v is indifferent between S and B if and
only if 2s1(B) = 1- s1(B), i.e. s1(B) = 1/3.


17

If type m is indifferent between S and B, then
s1(B) = 2/3, and thus type v chooses S.

If type v is indifferent between S and B, then
s1(B) = 1/3, and thus type m chooses S.

Hence there are two mixed strategy Bayesian Nash
Equilibria:
s1(B) = 1/3, sm(B) = 0, sv(B) = 2/3.
s1(B) = 2/3, sm(B) = 2/3, sv(B) = 0.
1.
2.
18
More Information May Hurt

A decision-maker in a single-person decision problem
cannot be worse off if she has more information.
If she wishes, she can ignore the information.

In a game, the same is not true: if a player has more
information, and the other players know that she has
more information, then she may be worse off.
19



Consider the following Bayesian game, where each of
the two state is equally likely.
If the players know the state of the world, they play
(B,B) in state 1 and (A,A) in state 2, with payoff 1.
If they do not know, their expected payoff is 3/2
regardless of their choice.
B
B
A
A
1, 1
0, 3
B
B
3, 0
2, 2
1,2
1,2
A
A
2, 2
0, 3
3, 0
1, 1
20
Common Knowledge

There is a subtle difference between a situation in
which all players know the payoffs, and a situation in
which the payoffs are common knowledge.

When payoffs are common knowledge, all players know
them, and know that all players know them and so on.

Lack of common knowledge may lead to players being
unable to coordinate on the Pareto optimal outcome,
even if they all know the payoffs.
21



Consider the Bayesian game below.
There are 3 states: 1, 2, 3. Each player has 2 types:
player 1’s types are {1} and {2, 3}, and player 2’s types
are {1, 2} and {3}.
The types’ beliefs on the states are in parenthesis.
2
(3/4)
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
2
(1/4)
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
22


The information is such that player 1 cannot distinguish
between states 2 and 3, but is informed when the state
is 1. Player 2 cannot distinguish between states 1 and 2,
but is informed when the state is 3.
In practice, player 1 knows the payoffs, player 2 may
know (state 3) or may not know it (states 1, 2), and
player 1 may know (state 1) or may not know (states 2,
3) whether player 2 knows player 1’s payoffs.
2
(3/4)
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
2
(1/4)
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
23


In state 3, both players know the payoffs of the game,
player 1 knows that player 2 knows the payoffs, but
player 2 does not know whether player 1 knows that
player 2 knows the payoffs.
If players were perfectly informed, they would play
(A,A) in state 1. But in states 2 and 3, they would try to
coordinate on (B,B).
2
(3/4)
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
2
(1/4)
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
24



We now show that the unique profile of strategies that
survives iterated deletion of strictly dominated
strategies is (AA, AA): all types of both players play A.
Intuitively, the players should play A in state 1, and
possibly in state 2.
It is counterintuitive that they play (A,A) in state 3,
where they both know payoffs, and they even know
that they know them.
2
(3/4)
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
2
(1/4)
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
25



In any rationalizable strategy profile s, type {1} of player
1 plays A, as this strategy is strictly dominant.
Consider type {1,2} of player 2.
With probability 3/4 she faces type {1} who plays A:
u{1,2}(B|s) < 0(3/4) + 2(1/4)
< 1(3/4) + 0(1/4) < u{1,2}(A|s).
Hence type {1,2} plays A.
(3/4)
2
(1/4)
2
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
26



Consider type {2,3} of player 1.
With probability 3/4 she faces type {1,2} who plays A:
u{2,3}(B|s) < 0(3/4) + 2(1/4)
< 1(3/4) + 0(1/4) < u{2,3}(A|s).
Hence type {2,3} plays A.
Clearly, then, type {3} plays A.
2
(3/4)
B
B
A
A
2, 2 0, 0
3, 0 1, 1
1
2
(1/4)
B
B
A
A
B
B
2, 2 0, 0
0, 0 1, 1
(3/4)
A
1
A
2, 2 0, 0
0, 0 1, 1
(1/4)
27
Cournot’s Duopoly Game with
Imperfect Information






Two firms compete a-la Cournot.
Their costs are linear: Ci (qi) = ciqi, i=1,2.
The demand is linear:
P (Q) = a – Q if a > Q, P(Q) = 0 if a < Q.
Both firms know that firm 1’s unit cost is c.
Firm 2 knows its own unit cost.
Firm 1 believes that firm 2’s cost is cL with probability θ
and cH with probability 1 − θ; with 0 < θ < 1, cL < cH.
28






Players: Firm 1 and firm 2.
States: {L, H}.
Strategies: Each firm’s strategies are all non-negative
numbers.
Types: Firm 1 has one type: {L,H}.
Firm 2 has two types: {H} and {L}.
Beliefs: Firm 1 assigns probability θ to state L and
probability 1 − θ to state H. Firm 2 knows the state.
Payoff functions: The firms’ payoffs are their profits:
u1(q1, q2;w) = qi [a – (q1 + q2,{w})] - cqi
u2,{q}(q1, q2;w) = q2,{w}[a –(q1+ q2,{w})] − cq q2,{w}.
29
Bayesian Nash Equilibrium of
the Cournot’s Duopoly Game

To find the best response functions of player 2’s types
w ∈ {L, H}, differentiate p2,{w} with respect to q2,{w}, to
find the first order condition:
dp2,{w}(q1, q2,{w}) /dq2,{w}
= a – q2,{w} – (q1+ q2,{w}) – cw = 0.

Best Response functions:
b2,{w} (q1) = [a – q1 – cw]/2.
30

For player 1, the expected profit is
p1(q1, q2) = q1 [a – (q1 + θq2L +(1-θ)q2H )] – cq1

The best response function is found as follows:
dp1(q1, q2)/dq1 = a – q1–(q1+θq2L +(1-θ)q2H ) – c1 = 0.
b1(q2) = [a – (θq2L+(1-θ)q2H ) – c]/2.

The Bayesian Nash Equilibrium is found by solving the
system defined by the best response functions:
q*1= [a – 2c + θcL +(1-θ)cH]/3,
q*2L= [2a + 2c - 3cL - θcL - (1-θ)cH]/6,
q*2H= [2a + 2c - 3cH - θcL - (1-θ)cH]/6.
31
Public Good Provision

A public good is provided to a group of n people if at
least one person pays the cost of the good, c.

Each person’s valuation of the good is vi. If the good is
not provided, then each individual’s payoff is 0.

Each individual i knows her own valuation vi, but does
know anyone else’s valuation.

It is common knowledge that valuations are
independently and identically distributed, with
cumulative distribution function F.
32

All n individuals simultaneously submit contributions of
either c or 0 (no intermediate contributions allowed).

If all individuals submit 0, then the good is not
provided and each individual’s payoff is 0.

If at least one individual submits c then the good is
provided, each individual i who submits c obtains the
payoff vi - c, and each i who submits 0 obtains vi.
33
Bayesian Game Representation

Players: The set of n individuals.

States: The set of all profiles (v1, . . . , vn) of valuations,
where 0 < vi < 1 for all i.

Strategies: Each player’s set of strategies is {0, c}.

Types: Each player i’s has a type which consists of her
valuation vi and of the set of all profiles v-i of
opponents’ valuations.
34

Beliefs: Each type of player i assigns probability
F(v1)F(v2) · · · F(vi −1)F(vi+1) · · · F(vn) to the event
that the valuation of every other player j is at most vj.

Payoff functions: Player i’s payoff in state (v1, . . . , vn) is
ui(v1, . . . , vn) = 0 if no one contributes,
ui(v1, . . . , vn) = vi if i does not contribute but some
other player does,
ui(v1, . . . , vn) = vi − c if i contributes.
35
Bayesian Nash Equilibria
The game has a symmetric Nash equilibrium in which every
player i contributes if and only if vi ≥ v∗.

Consider player i. Suppose that every other player j
contributes if and only if vj ≥ v∗.

The probability that at least one of the other players
contributes is 1− (F(v∗))n−1.
36

Player i’s expected payoff is [1 − (F(v∗))n−1] vi if she does
not contribute and vi − c if she contributes.

The conditions for player i to not contribute when vi < v∗
and contribute when vi ≥ v∗ are:
(1−(F(v∗))n−1)vi > vi−c if vi< v∗,
(1−(F(v∗))n−1)vi ≤ vi−c if vi ≥ v∗:
I.e. vi(F(v∗))n−1 < c if vi < v∗, vi(F(v∗))n−1 ≥ c if vi ≥ v∗.

Hence, in equilibrium, v∗(F(v∗))n−1 = c.
37

As the number of individuals increases, is the good more
or less likely to be provided in this equilibrium?

The probability that the good is provided is the probability
that at least for one i, vi ≥ v∗, which is equal to 1 − (F(v∗))n.

In equilibrium, this probability is equal to 1 − cF(v∗)/v∗.

The value of v∗ increases as n increases.

As n increases the change in the probability that the good
is provided increases if F(v∗)/v∗ decreases in v∗,
whereas it decreases if F(v∗)/v∗ increases in v∗.
38
Summary of the Lecture

Bayesian Games and Nash Equilibrium

Bayesian Games and Information

Cournot Duopoly with Private Information

Public Good Provision with Private Information
39
Preview of the Next Lecture

Juries and Information Aggregation

Auctions with Private Information
40