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Games of Incomplete InformationEC202 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:
Definitions:
Incomplete Information GameInformation Structure and BeliefsStrategiesBest Reply Map
Solution Concepts in Pure Strategies:
Dominant Strategy EquilibriumBayes Nash Equilibrium
Examples
EXTRA: Mixed Strategies & Bayes Nash Equilibria
Nava (LSE) EC202 — Lectures V & VI Jan 2011 2 / 22
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 profile of signals x = (x1, ..., xn) is an element X = ×j∈NXj
f a distribution over the possible signals
ui : A×X→ R player i’s utility function, ui (a|x)
Nava (LSE) EC202 — Lectures V & VI Jan 2011 3 / 22
Bayesian Game Example
Consider the following Bayesian game:
Player 1 observes only one possible signal: X1 = {C}
Player 2’s signal takes one of two values: X2 = {L,R}
Probabilities are such that: f (C , L) = 0.6
Payoffs and action sets are as described in the matrix:
1\2.L y2 z2 1\2.R y2 z2y1 1,2 0,1 y1 1,3 0,4z1 0,4 1,3 z1 0,1 1,2
Nava (LSE) EC202 — Lectures V & VI 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 profileof 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 ofthe signals of the other players x−i
Nava (LSE) EC202 — Lectures V & VI Jan 2011 5 / 22
Beliefs about other Players’Signals [Take 1]
In this course we consider models in which signals are independent:
f (x) = ∏j∈N 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 byusing Bayes Rule
Independence implies that conditional observing Xi = xi the beliefs ofplayer i are:
fi (x−i |xi ) = ∏j∈N\i fj (xj ) = f−i (x−i )
Nava (LSE) EC202 — Lectures V & VI Jan 2011 6 / 22
Extra: Beliefs about other Players’Signals [Take 2]
Also in the general case with interdependence players form beliefs aboutthe signals received by the others by using Bayes Rule
Conditional observing Xi = xi the beliefs of player i are:
fi (x−i |xi ) = Pr(X−i = x−i |Xi = xi ) =
=Pr(X−i = x−i ∩ Xi = xi )
Pr(Xi = xi )=
=Pr(X−i = x−i ∩ Xi = xi )
∑y−i∈X−i Pr(X−i = y−i ∩ Xi = xi )=
=f (x−i , xi )
∑y−i∈X−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 7 / 22
Strategies
Strategy Profiles:
A strategy consists of a map from available information to actions:
αi : Xi → Ai
A strategy profile 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−1(Xi−1), αi+1(Xi+1), ..., αN (XN ))
Nava (LSE) EC202 — Lectures V & VI Jan 2011 8 / 22
Bayesian Game Example Continued
Consider the following game:
Player 1 observes only one possible signal: X1 = {C}
Player 2’s signal takes one of two values: X2 = {L,R}
Probabilities are such that: f (C , L) = 0.6
Payoffs and action sets are as described in the matrix:
1\2.L y2 z2 1\2.R y2 z2y1 1,2 0,1 y1 1,3 0,4z1 0,4 1,3 z1 0,1 1,2
A strategy for player 1 is an element of the set α1 ∈ {y1, z1}
A strategy for player 2 is a map α2 : {L,R} → {y2, z2}
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 α′i if for any a−i and x ∈ X:
ui (αi (xi ), a−i |x) ≥ ui (α′i (xi ), a−i |x) [strict somewhere]
Strategy αi is dominant if it dominates any other strategy α′i
Strategy αi is undominated if no strategy dominates it
Definitions (Dominant Strategy Equilibrium DSE)A Dominant Strategy equilibrium of an incomplete information game is astrategy profile α that for any i ∈ N, x ∈ X and a−i ∈ A−i satisfies:
ui (αi (xi ), a−i |x) ≥ ui (α′i (xi ), a−i |x) for any α′i : 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 profile α is defined by:
Ui (α|xi ) = ∑X−i ui (α(x)|x)f (x−i |xi ) : Xi → R
With such notation in mind notice that:
Ui (ai , α−i |xi ) = ∑X−i ui (ai , α−i (x−i )|x)f (x−i |xi )
The best reply correspondence of player i is defined by:
bi (α−i |xi ) = argmaxai∈Ai Ui (ai , α−i |xi )
BR maps identify which actions are optimal given the signal and thestrategies followed by others
Nava (LSE) EC202 — Lectures V & VI Jan 2011 11 / 22
Pure Strategy Bayes Nash Equilibrium
Definitions (Bayes Nash Equilibrium BNE)A pure strategy Bayes Nash equilibrium of an incomplete informationgame is a strategy profile α such that for any i ∈ N and xi ∈ Xi satisfies:
Ui (α|xi ) ≥ Ui (ai , α−i |xi ) for any ai ∈ Ai
BNE requires interim optimality (i.e. do your best given what you know)
BNE requires αi (xi ) ∈ bi (α−i |xi ) for any i ∈ N and xi ∈ 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:
1\2.L y2 z2 1\2.R y2 z2y1 1,2 0,1 y1 1,3 0,4z1 0,4 1,3 z1 0,1 1,2
The best reply maps for both player are characterized by:
b2(α1|x2) ={y2 if x2 = Lz2 if x2 = R
b1(α2) ={y1 if α2(L) = y2z1 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 13 / 22
Relationships between Equilibrium Concepts
If α is a DSE then it is a BNE. In fact for any action ai and signal xi :
ui (αi (xi ), a−i |x) ≥ ui (ai , a−i |x) ∀a−i , x−i ⇒
ui (αi (xi ), α−i (x−i )|x) ≥ ui (ai , α−i (x−i )|x) ∀α−i , x−i ⇒
∑X−i ui (α(x)|x)fi (x−i |xi ) ≥ ∑X−i ui (ai , α−i (x−i )|x)fi (x−i |xi ) ∀α−i ⇒
Ui (α|xi ) ≥ Ui (ai , α−i |xi ) ∀α−i
Nava (LSE) EC202 — Lectures V & VI 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 = {L,H}Buyer can offer either 1$ or 3$ to purchase the object
Seller choose whether or not to sell
B\S .L sale no sale B\S .H sale no sale3$ 0,3 0,0 3$ 0,3 0,21$ 2,1 0,0 1$ 2,1 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
Offering 1$ is weakly dominant for the buyerNava (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 profit of 1$ and I gets 8$
If E enters, E incurs a cost of 1$ and profits of both I and E are 3$
I can deter entry by investing at cost {0, 2} depending on type {L,H}If I invests: I’s profit increases by 1 if he is alone, decreases by 1 if hecompetes and E’s profit decreases to 0 if he elects to enter
E\I .L Invest Not Invest E\I .H Invest Not InvestIn 0,2 3,3 In 0,0 3,3Out 1,9 1,8 Out 1,7 1,8
Nava (LSE) EC202 — Lectures V & VI Jan 2011 16 / 22
BNE Example II: Entry Game
Let π denote the probability that firm 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|L) = 3 > 2 = uI (Invest, In|L)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|L) = 9 > 8 = uI (Not,Out|L)UE (Out, αI (XI )) = 1 > 3(1− π) = UE (In, αI (XI ))
E\I .L Invest Not Invest E\I .H Invest Not InvestIn 0,2 3,3 In 0,0 3,3Out 1,9 1,8 Out 1,7 1,8
Nava (LSE) EC202 — Lectures V & VI Jan 2011 17 / 22
Extra: Mixed Strategies in Bayesian Games
Strategy Profiles:
A mixed strategy is a map from information to a probabilitydistribution over actions
In particular σi (ai |xi ) denotes the probability that i chooses ai if hissignal is xi
A mixed strategy profile consists of a strategy for every player:
σ(X ) = (σ1(X1), ..., σN (XN ))
As usual σ−i (X−i ) denotes the profile 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 Payoff & Bayes Nash Equilibrium
The interim expected payoff of mixed strategy profiles σ and (ai , σ−i ) are:
Ui (σ|xi ) = ∑X−i
∑a∈A
ui (a|x) ∏j∈N
σj (aj |xj )f (x−i |xi )
Ui (ai , σ−i |xi ) = ∑X−i
∑a−i∈A−i
ui (a|x) ∏j 6=i
σj (aj |xj )f (x−i |xi )
Definitions (Bayes Nash Equilibrium BNE)A Bayes Nash equilibrium of a game Γ is a strategy profile σ such that forany i ∈ N and xi ∈ Xi satisfies:
Ui (σ|xi ) ≥ Ui (ai , σ−i |xi ) for any ai ∈ 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 (σ|xi ) = Ui (ai , σ−i |xi ) for any ai s.t. σi (ai |xi ) > 0
Ui (σ|xi ) ≥ Ui (ai , σ−i |xi ) for any ai s.t. σi (ai |xi ) = 0
Strictly dominated strategies are never chosen in a BNE
Weakly dominated strategies are chosen only if they are dominatedwith 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:
1\2.L X Y 1\2.R W ZT 1,0 0,1 T 0,0 1,1D 0,1 1,0 D 1,1 0,0
All BNEs for this game satisfy:
σ1(T ) = 1/2 and σ2(X |L) = σ2(W |R)
Such games satisfy all BNE conditions since:
U1(T , σ2) = (1/2)σ2(X |L) + (1/2)(1− σ2(W |R)) == (1/2)(1− σ2(X |L)) + (1/2)σ2(W |R) = U1(D, σ2)u2(X , σ1|L) = σ1(T ) = 1− σ1(T ) = u2(Y , σ1|L)u2(W , σ1|R) = (1− σ1(T )) = σ1(T ) = u2(Z , σ1|R)
Nava (LSE) EC202 — Lectures V & VI Jan 2011 21 / 22
Extra: Example II
Consider the following example for f (1, L) = q ≤ 2/3:
1\2.L X Y 1\2.R W ZT 0,0 0,2 T 2,2 0,1D 2,0 1,1 D 0,0 3,2
All BNEs for this game satisfy σ1(T ) = 2/3 and:
σ2(X |L) = 0 (dominance) and σ2(W |R) =3− 2q5− 5q
Such games satisfy all BNE conditions since:
U1(T , σ2) = 2(1− q)σ2(W |R) == q + 3(1− q)(1− σ2(W |R)) = U1(D, σ2)
u2(X , σ1|L) = 0 < 2σ1(T ) + (1− σ1(T )) = u2(Y , σ1|L)u2(W , σ1|R) = 2σ1(T ) = σ1(T ) + 2(1− σ1(T )) = u2(Z , σ1|R)
Nava (LSE) EC202 — Lectures V & VI Jan 2011 22 / 22