Games of Incomplete Information
EC202 Lectures V & VI
Francesco Nava
London School of Economics
January 2011
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
1 / 22
Summary
Games of Incomplete Information:
De…nitions:
Incomplete Information Game
Information Structure and Beliefs
Strategies
Best Reply Map
Solution Concepts in Pure Strategies:
Dominant Strategy Equilibrium
Bayes Nash Equilibrium
Examples
EXTRA: Mixed Strategies & Bayes Nash Equilibria
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Incomplete Information (Strategic Form)
An incomplete information game consists of:
N the set of players in the game
Ai player i’s action set
Xi player i’s set of possible signals
A pro…le of signals x = (x1 , ..., xn ) is an element X =
j 2N Xj
f a distribution over the possible signals
ui : A
X ! R player i’s utility function, ui (ajx )
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Bayesian Game Example
Consider the following Bayesian game:
Player 1 observes only one possible signal: X1 = fC g
Player 2’s signal takes one of two values: X2 = fL, R g
Probabilities are such that: f (C , L) = 0.6
Payo¤s and action sets are as described in the matrix:
1n2.L
y1
z1
Nava (LSE)
y2
1,2
0,4
z2
0,1
1,3
1n2.R
y1
z1
EC202 – Lectures V & VI
y2
1,3
0,1
z2
0,4
1,2
Jan 2011
4 / 22
Information Structure
Information structure:
Xi denotes the signal as a random variable
belongs to the set of possible signals Xi
xi denotes the realization of the random variable Xi
X i = (X1 , ..., Xi 1 , Xi +1 , ..., Xn ) denotes a pro…le
of signals for all players other than i
Player i observes only Xi
Player i ignores X i , but knows f
With such information player i forms beliefs regarding the realization of
the signals of the other players x i
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Beliefs about other Players’Signals [Take 1]
In this course we consider models in which signals are independent:
f (x ) = ∏j 2N fj (xj )
This implies that the signal xi of player i is independent of X
i
Beliefs are a probability distribution over the signals of the other players
Any player forms beliefs about the signals received by the other players by
using Bayes Rule
Independence implies that conditional observing Xi = xi the beliefs of
player i are:
fi (x i jxi ) = ∏j 2N ni fj (xj ) = f i (x i )
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Extra: Beliefs about other Players’Signals [Take 2]
Also in the general case with interdependence players form beliefs about
the signals received by the others by using Bayes Rule
Conditional observing Xi = xi the beliefs of player i are:
fi (x i jxi ) = Pr(X
i
Pr(X
i
=
= x i jXi = xi ) =
= x i \ Xi = xi )
=
Pr(Xi = xi )
=
Pr(X i = x i \ Xi = xi )
=
∑y i 2X i Pr(X i = y i \ Xi = xi )
=
f (x i , xi )
∑y i 2X i f (y i , xi )
Beliefs are a probability distribution over the signals of the other players
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Strategies
Strategy Pro…les:
A strategy consists of a map from available information to actions:
αi : Xi ! Ai
A strategy pro…le consists of a strategy for every player:
α(X ) = (α1 (X1 ), ..., αN (XN ))
We adopt the usual convention:
α i (X i ) = (α1 (X1 ), ..., αi
Nava (LSE)
1 (Xi 1 ), αi +1 (Xi +1 ), ..., αN (XN ))
EC202 – Lectures V & VI
Jan 2011
8 / 22
Bayesian Game Example Continued
Consider the following game:
Player 1 observes only one possible signal: X1 = fC g
Player 2’s signal takes one of two values: X2 = fL, R g
Probabilities are such that: f (C , L) = 0.6
Payo¤s and action sets are as described in the matrix:
1n2.L
y1
z1
y2
1,2
0,4
z2
0,1
1,3
1n2.R
y1
z1
y2
1,3
0,1
z2
0,4
1,2
A strategy for player 1 is an element of the set α1 2 fy1 , z1 g
A strategy for player 2 is a map α2 : fL, R g ! fy2 , z2 g
Player 1 cannot act upon 2’s private information
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
9 / 22
Dominant Strategy Equilibrium
Strategy αi weakly dominates αi0 if for any a
ui ( α i ( x i ) , a i j x )
i
and x 2 X:
ui (αi0 (xi ), a i jx ) [strict somewhere]
Strategy αi is dominant if it dominates any other strategy αi0
Strategy αi is undominated if no strategy dominates it
De…nitions (Dominant Strategy Equilibrium DSE)
A Dominant Strategy equilibrium of an incomplete information game is a
strategy pro…le α that for any i 2 N, x 2 X and a i 2 A i satis…es:
ui (αi (xi ), a i jx )
ui (αi0 (xi ), a i jx ) for any αi0 : Xi ! Ai
I.e. αi is optimal independently of what others know and do
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
10 / 22
Interim Expected Utility and Best Reply Maps
The interim stage occurs just after a player knows his signal Xi = xi
It is when strategies are chosen in a Bayesian game
The interim expected utility of a (pure) strategy pro…le α is de…ned by:
Ui (αjxi ) = ∑X i ui (α(x )jx )f (x i jxi ) : Xi ! R
With such notation in mind notice that:
Ui (ai , α i jxi ) = ∑X i ui (ai , α i (x i )jx )f (x i jxi )
The best reply correspondence of player i is de…ned by:
bi (α i jxi ) = arg maxai 2A i Ui (ai , α i jxi )
BR maps identify which actions are optimal given the signal and the
strategies followed by others
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
11 / 22
Pure Strategy Bayes Nash Equilibrium
De…nitions (Bayes Nash Equilibrium BNE)
A pure strategy Bayes Nash equilibrium of an incomplete information
game is a strategy pro…le α such that for any i 2 N and xi 2 Xi satis…es:
Ui (αjxi )
Ui (ai , α i jxi ) for any ai 2 Ai
BNE requires interim optimality (i.e. do your best given what you know)
BNE requires αi (xi ) 2 bi (α i jxi ) for any i 2 N and xi 2 Xi
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
12 / 22
Bayesian Game Example Continued
Consider the following Bayesian game with f (C , L) = 0.6:
1n2.L
y1
z1
y2
1,2
0,4
z2
0,1
1,3
1n2.R
y1
z1
y2
1,3
0,1
z2
0,4
1,2
The best reply maps for both player are characterized by:
b2 (α1 jx2 ) =
y2 if x2 = L
z2 if x2 = R
b1 ( α 2 ) =
y1 if α2 (L) = y2
z1 if α2 (L) = z2
The game has a unique (pure strategy) BNE in which:
α1 = y1 , α2 (L) = y2 , α2 (R ) = z2
DO NOT ANALYZE MATRICES SEPARATELY!!!
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
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Relationships between Equilibrium Concepts
If α is a DSE then it is a BNE. In fact for any action ai and signal xi :
ui ( α i ( x i ) , a i j x )
ui (αi (xi ), α i (x i )jx )
∑X i ui (α(x )jx )fi (x i jxi )
Ui (αjxi )
Nava (LSE)
u i ( ai , a i j x ) 8 a i , x
i
)
ui (ai , α i (x i )jx ) 8α i , x
i
)
∑X i ui (ai , α i (x i )jx )fi (x i jxi ) 8α
Ui (ai , α i jxi ) 8α
EC202 – Lectures V & VI
i
)
i
Jan 2011
14 / 22
BNE Example I: Exchange
A buyer and a seller want to trade an object:
Buyer’s value for the object is 3$
Seller’s value is either 0$ or 2$ based on the signal, XS = fL, H g
Buyer can o¤er either 1$ or 3$ to purchase the object
Seller choose whether or not to sell
B nS.L
3$
1$
sale
0,3
2,1
no sale
0,0
0,0
B nS.H
3$
1$
sale
0,3
2,1
no sale
0,2
0,2
This game for any prior f has a BNE in which:
αS (L) = sale, αS (H ) = no sale, αB = 1$
Selling is strictly dominant for S.L
O¤ering 1$ is weakly dominant for the buyer
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
15 / 22
BNE Example II: Entry Game
Consider the following market game:
Firm I (the incumbent) is a monopolist in a market
Firm E (the entrant) is considering whether to enter in the market
If E stays out of the market, E runs a pro…t of 1$ and I gets 8$
If E enters, E incurs a cost of 1$ and pro…ts of both I and E are 3$
I can deter entry by investing at cost f0, 2g depending on type fL, H g
If I invests: I ’s pro…t increases by 1 if he is alone, decreases by 1 if he
competes and E ’s pro…t decreases to 0 if he elects to enter
E nI .L
In
Out
Nava (LSE)
Invest
0,2
1,9
Not Invest
3,3
1,8
E nI .H
In
Out
EC202 – Lectures V & VI
Invest
0,0
1,7
Not Invest
3,3
1,8
Jan 2011
16 / 22
BNE Example II: Entry Game
Let π denote the probability that …rm I is of type L and notice:
αI (H ) = Not Invest is a strictly dominant strategy for I .H
For any value of π, αI (L) = Not Invest and αE = In is BNE:
uI (Not, In jL) = 3 > 2 = uI (Invest, In jL)
UE (In, αI (XI )) = 3 > 1 = UE (Out, αI (XI ))
For π high enough, αI (L) = Invest and αE = Out is also BNE:
uI (Invest, Out jL) = 9 > 8 = uI (Not, Out jL)
UE (Out, αI (XI )) = 1 > 3(1
E nI .L
In
Out
Nava (LSE)
Invest
0,2
1,9
Not Invest
3,3
1,8
E nI .H
In
Out
EC202 – Lectures V & VI
π ) = UE (In, αI (XI ))
Invest
0,0
1,7
Not Invest
3,3
1,8
Jan 2011
17 / 22
Extra: Mixed Strategies in Bayesian Games
Strategy Pro…les:
A mixed strategy is a map from information to a probability
distribution over actions
In particular σi (ai jxi ) denotes the probability that i chooses ai if his
signal is xi
A mixed strategy pro…le consists of a strategy for every player:
σ(X ) = (σ1 (X1 ), ..., σN (XN ))
As usual σ i (X i ) denotes the pro…le of strategies of all players, but i
Mixed strategies are independent (i.e. σi cannot depend on σj )
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
18 / 22
Extra: Interim Payo¤ & Bayes Nash Equilibrium
The interim expected payo¤ of mixed strategy pro…les σ and (ai , σ i ) are:
∑ ∑ ui (ajx ) ∏ σj (aj jxj )f (x i jxi )
Ui (σjxi ) =
X
Ui (ai , σ i jxi ) =
X
i
a 2A
i
a i 2A
∑
j 2N
∑
i
ui (ajx ) ∏ σj (aj jxj )f (x i jxi )
j 6 =i
De…nitions (Bayes Nash Equilibrium BNE)
A Bayes Nash equilibrium of a game Γ is a strategy pro…le σ such that for
any i 2 N and xi 2 Xi satis…es:
Ui (σjxi )
Ui (ai , σ i jxi ) for any ai 2 Ai
BNE requires interim optimality (i.e. do your best given what you know)
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
19 / 22
Extra: Computing Bayes Nash Equilibria
Testing for BNE behavior:
σ is BNE if only if:
Ui (σjxi ) = Ui (ai , σ i jxi ) for any ai s.t. σi (ai jxi ) > 0
Ui (σjxi )
Ui (ai , σ i jxi ) for any ai s.t. σi (ai jxi ) = 0
Strictly dominated strategies are never chosen in a BNE
Weakly dominated strategies are chosen only if they are dominated
with probability zero in equilibrium
This conditions can be used to compute BNE (see examples)
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
20 / 22
Extra: Example I
Consider the following example for f (1, L) = 1/2:
1n2.L
T
D
X
1,0
0,1
Y
0,1
1,0
1n2.R
T
D
W
0,0
1,1
Z
1,1
0,0
All BNEs for this game satisfy:
σ1 (T ) = 1/2 and σ2 (X jL) = σ2 (W jR )
Such games satisfy all BNE conditions since:
U1 (T , σ2 ) = (1/2)σ2 (X jL) + (1/2)(1
σ2 (W jR )) =
= (1/2)(1 σ2 (X jL)) + (1/2)σ2 (W jR ) = U1 (D, σ2 )
u2 ( X , σ 1 j L ) = σ 1 ( T ) = 1 σ 1 ( T ) = u2 ( Y , σ 1 j L )
u2 (W , σ1 jR ) = (1 σ1 (T )) = σ1 (T ) = u2 (Z , σ1 jR )
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
21 / 22
Extra: Example II
Consider the following example for f (1, L) = q
1n2.L
T
D
X
0,0
2,0
Y
0,2
1,1
1n2.R
T
D
2/3:
W
2,2
0,0
Z
0,1
3,2
All BNEs for this game satisfy σ1 (T ) = 2/3 and:
σ2 (X jL) = 0 (dominance) and σ2 (W jR ) =
3
5
2q
5q
Such games satisfy all BNE conditions since:
U1 ( T , σ 2 ) = 2 ( 1
q ) σ 2 (W jR ) =
= q + 3(1 q )(1 σ2 (W jR )) = U1 (D, σ2 )
u2 (X , σ1 jL) = 0 < 2σ1 (T ) + (1 σ1 (T )) = u2 (Y , σ1 jL)
u2 (W , σ1 jR ) = 2σ1 (T ) = σ1 (T ) + 2(1 σ1 (T )) = u2 (Z , σ1 jR )
Nava (LSE)
EC202 – Lectures V & VI
Jan 2011
22 / 22