1. Game Theory Georg Groh, WS 08/09 Verteiltes Problemlösen

2. Part 3: Static Games of incomplete Information

3. Bayesian Games and Bayesian Equilibrium Some players do not know payoffs of others: Game of incomplete information Example: Player 1: Build or not build plant, knows building costs (3 or 0) Player 2: Enter market or not Player 2: Don´t know bulding costs of player 1 Player 2: Entering profitable iff player 1 does not build Player 1: Build is dominant if cost is low Don´t build is dominant if cost is high Example: Build Game Enter Build Dont´t build 0, -1 2, 0 2, 13, 0 Don´t Enter Enter Build Dont´t build 3, -1 5, 0 2, 13, 0 Don´t Enter Version 1: Build cost high Version 2: Build cost low

4. Bayesian Games and Bayesian Equilibrium Player 2 assumes: Prior probability for „player 1‘s cost high“ is p1  player 2 enters if p1 > 0.5, stays out if p1 < 0.5 (Iterated deletion of strictly dominated strategies) Example: Build Game Enter Build Dont´t build 0, -1 2, 0 2, 13, 0 Don´t Enter Enter Build Dont´t build 3, -1 5, 0 2, 13, 0 Don´t Enter Version 1: Build cost high Version 2: Build cost low

5. Bayesian Games and Bayesian Equilibrium Now: change low costs for building from 0 to 1.5 (situations 1´ and 2´)  situation more complicated Player 1: „Don´t build“ still dominant if build cost is high, BUT: Player 1: If cost low: optimal strategy depends on probability y that player 2 enters: Building is better than not building if 1.5 y + 3.5(1-y) > 2y + 3(1-y)  if y < 0.5 Each „version“ has mixed NE that we can find with standard techniques from part 1, but „which version applies“ is unknown to player 2  player 2 has incomplete information Example: Build Game Enter Build Dont´t build 0, -1 2, 0 2, 13, 0 Don´t Enter Enter Build Dont´t build 1.5, -13.5, 0 2, 13, 0 Don´t Enter Version 1‘: Build cost high Version 2‘: Build cost low

6. Bayesian Games and Bayesian Equilibrium Harsanyi: Suggests prior move by nature that determines player 1`s „type“ (his costs)  „Model“/Visualize incomplete information game as game of imperfect information Example: Build Game Enter Build Dont´t build (1.5, -1)(3.5, 0)(2, 1)(3, 0) Don´t Enter Cost high [p1]Cost low [1-p1] 1 Build Dont´t build 1 Enter Don´t Enter Enter Don´t Enter Enter Don´t Enter (0, -1)(2, 0)(2, 1)(3, 0) 2222 N stage

7. Bayesian Games and Bayesian Equilibrium Harsanyi: Bayesian EQ (Bayesian NE): NE (not neccessarily SGPE) of imperfect information game; Analyze with techniques from part 2: x: player 1‘s probability of building when cost is low (x=0 if cost is high);; y: player 2‘s probability of entering player 2‘s payoff: y * [p1* 1 + (1-p1)*((-1)*x + (1)*(1-x))] + (1-y) * 0  optimal strategy for player 2 is: y=1 if x 1/[2(1-p1)] y ∈ [0,1] if x = 1/[2(1-p1)] Example: Build Game Enter Build Dont´t build 0, -1 2, 0 2, 13, 0 Don´t Enter Enter Build Dont´t build 1.5, -13.5, 0 2, 13, 0 Don´t Enter Version 1‘: Build cost high Version 2‘: Build cost low

8. Bayesian Games and Bayesian Equilibrium Harsanyi: Bayesian EQ (Bayesian NE): NE (not neccessarily SGPE) of imperfect information game; Analyze with techniques from part 2: x: player 1‘s probability of building when cost is low (x=0 if cost is high);; y: player 2‘s probability of entering  optimal strategy for player 2 is: y=1 if x 1/[2(1-p1)] y ∈ [0,1] if x = 1/[2(1-p1)] Example: Build Game Enter Build Dont´t build (1.5, -1)(3.5, 0)(2, 1)(3, 0) Don´t Enter Cost high [p1]Cost low [1-p1] 1 Build Dont´t build 1 Enter Don´t Enter Enter Don´t Enter Enter Don´t Enter (0, -1)(2, 0)(2, 1)(3, 0) 2222 N Depends on „exogeneous“ probability p1

9. Bayesian Games and Bayesian Equilibrium Player 1: Knows own type  high cost case clear  optimize ONLY version 2‘ : player 1‘s payoff: x(1.5y + 3.5(1-y)) + (1-x)(2y + 2(1-y)) maximize  optimal strategy for player 1 is: x=1 if y 0.5 x ∈ [0,1] if y = 0.5  Find NE (x,y,p1) : Intersect optimal response curves: (0, 1) is NE for any p1 (1, 0) is NE for p1 ≤ 0.5 (1/[2(1-p1)], 0.5) is NE for p1 ≤ 0.5 etc. because of dependence on p1 we have many NE Example: Build Game Enter Build Dont´t build 0, -1 2, 0 2, 13, 0 Don´t Enter Enter Build Dont´t build 1.5, -13.5, 0 2, 13, 0 Don´t Enter Version 1‘: Build cost high Version 2‘: Build cost low

10. Bayesian Games and Bayesian Equilibrium „Free Rider Problem“ Simultaneous moves by players 1 and 2: Contribute to public good or not Player i‘s cost of contributing is ci (secret knowledge of player i). (Player i‘s „type“) Each player‘s benefit from public good is 1 (common knowledge) Both players believe it is common knowledge that the ci are drawn from a strictly increasing, continuous cumulative distribution function on [c,c] where c < 1 < c (so P(c)=0 and P(c)=1) Providing a Public Good under Incomplete Information Contrib 1-c1, 1-c2 Don´t Contrib Contrib Don´t Contrib 1-c1, 1 0, 01, 1-c2 P ccc

11. Bayesian Games and Bayesian Equilibrium Pure strategy: Function si(ci) from [c,c] to Action set {0,1} (Contribute or not) Payoff: ui(si, sj, ci) = max(0, max (s1,s2) – ci si) (does not directly depend on cj (j≠i)) Bayesian EQ is pair of strategies (s1*(.), s2*(.)) so that for all players i and every possible ci, strategy si*(ci) maximizes the expectation value E cj ( ui(si, sj*(cj), ci) ) Providing a Public Good under Incomplete Information Contrib 1-c1, 1-c2 Don´t Contrib Contrib Don´t Contrib 1-c1, 1 0, 01, 1-c2

12. Bayesian Games and Bayesian Equilibrium Let zj = Probability([sj*(cj)=1]) be the EQ-probability that player j contributes Player i: Maximize own payoff: Contribute if ci < 1(1-zj) (his benefit times prob that other doesn‘t contribute)  si*(ci) = 1 if ci 1-zj ((( if ci = 1-zj : player is indifferent but P(.) is continuous  probability of a single point-set is zero )))  types of player i (or j resp.) who contributes lie in [c,ci*] (or [c,cj*] resp.) (player contributes only if cost is sufficiently low) Because zj = Probability(c ≤ cj ≤ cj*) = P(cj*) we have ci*=1- P(cj*) ( and cj*=1- P(ci*) analogously )  c1* and c2* must satisfy c* = 1 – P(1-P(c*))  if unique c* exists  ci*= cj*= c*=1-P(c*) Example: P uniform on [0,2] (P(c)=c/2) then unique c* is 2/3 Providing a Public Good under Incomplete Information Contrib 1-c1, 1-c2 Don´t Contrib Contrib Don´t Contrib 1-c1, 1 0, 01, 1-c2

13. Bayesian Games and Bayesian Equilibrium From prev. slide: Example: P uniform on [0,2] (P(c)=c/2) then unique c* is 2/3 If player does not contribute: Expected payoff is P(c*)=1/3 if player contributes (cost c*): Expected payoff is 1-c*=1/3  indifference player does not contribute if cost is in [2/3, 1] even though his cost is still less than his benefit (of 1) and although the probability that the other doesn‘t contribute is 1-P(c*)=2/3 If we change example from c=0 to c≥1-P(1)  game has two NE: One player never contributes, other contributes for all c ≤1 if player 1 is the one that never contributes: c1*=1-P(1) < c and c2* = 1 Never contributing player does so because his min cost c exceeds the gain 1(1-P(1)) from increased supply of good Always contributing player: if he would not contribute, probability for getting the public good is 0 Providing a Public Good under Incomplete Information Contrib 1-c1, 1-c2 Don´t Contrib Contrib Don´t Contrib 1-c1, 1 0, 01, 1-c2  NE, ok because the other always contributes if his c2<1

14. Bayesian Games and Bayesian Equilibrium In the examples: „type“ = cost Also possible: more complex types (including player‘s belief about other player‘s payoff functions etc.) Assume player‘s types {Θ i } (i ∈ [1,I]) are drawn from distribution p(Θ 1, Θ 2,..., Θ I } For simplicity: Assume the Θ i are drawn from a discrete set Θ i Θ i is observed by player i only Formal definition of Bayesian EQ

15. Bayesian Games and Bayesian Equilibrium p(Θ -i | Θ i ) = p(Θ 1, Θ 2,..., Θ i-1, Θ i+1,...,Θ I | Θ i ) conditional probability of opponent‘s types Pure strategies s i (Θ i ), mixed strategies σ i (Θ i ) Payoff function u i (s 1,..., s I, Θ 1,..., Θ I ) Formal definition of Bayesian EQ

16. Bayesian Games and Bayesian Equilibrium Bayesian EQ of game of incomplete information with finite number of types for each player i, prior distribution p, pure Strategy Spaces S i is Nash EQ of the „expanded game“ where each player‘s pure strategy space is the set S i Θ i of maps from Θ i to S i. Given strategy profile s(.) and alternative s i ´(.) ∈ S i Θ i ; Let (s i ´(.), s -i (.)) denote the profile where i plays s i ´(.) and others play s(.) ; Let (s i ´(Θ i ), s -i (Θ -i )) = (s 1 (Θ 1 ), s 2 (Θ 2 ),...,s i-1 (Θ i-1 ), s i ´(Θ i ), s i+1 (Θ i+1 ),... s I (Θ I )) denote the value of this profile at (Θ i, Θ -i ) Then: profile s(.) is in a pure strategy Bayesian EQ if for all players i which is equivalent to players i maximizing their expected utility conditional on Θ i for each Θ i : Formal definition of Bayesian EQ

17. Bayesian Games and Bayesian Equilibrium Let u i = q i (Θ i - q i - q j ) Usual Cournot: strategies = choose output ( s i = q i ) If linear demand aq+b then type Θ i = |b-c i | : type is related to costs Public knowledge : Θ 1 = 1 Public knowledge: Firm 1 believes: p(Θ 2 = 3/4)=1/2 ; p(Θ 2 = 5/4)=1/2 firm 1‘s output q 1 firm 2‘s output: if Θ 2 = 3/4 : q 2 L ; if Θ 2 = 5/4 : q 2 H  firm 2‘s EQ choice q 2 (Θ 2 ): q 2 (Θ 2 ) =argmax_q 2 {q 2 (Θ 2 - q 1 - q 2 )}  q 2 (Θ 2 )= (Θ 2 - q 1 )/2  q 2 L = (3/4 - q 1 )/2 ; q 2 H = (5/4 - q 1 )/2  firm 1‘s EQ choice q 1 (Θ 1 )= q 1 (1)= q 1 : q 1 =argmax_q 1 { ½ (q 1 (1 - q 1 - q 2 H )) + ½ (q 1 (1 - q 1 - q 2 L )) }  q 1 = ¼ (2- q 2 H- q 2 L )  3 equations for 3 variables  solve : q 1 =1/3; q 2 L =11/24; q 2 H =5/24 (unique EQ) Example: Cournot game with incomplete information

18. (small) Part 4: Dynamic Games of incomplete Information

19. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games Now: consider dynamic games of incomplete information; What we saw in (static+complete  dynamic+complete): need for SGPE Here: Information incompleteness (no knowledege of other player‘s types)  no proper definition of subgame  no check for NE possible as in SGPE definition (the only proper subgame of a dyn. game of incomplete info is the whole game  each NE is SGPE)  Several EQ refinement concepts: perfect Bayesian EQ, Sequential EQ, Trembling hand EQ Preliminaries

20. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games Easiest variant: signaling games: leader (with private type Θ) + follower (type is common knowledge) Strategy for player 1: σ 1 (.| Θ) over actions a 1 Strategy for player 2: σ 2 (.| a 1 ) over actions a 2 Player 1‘s payoff: u 1 ( σ 1, σ 2, Θ) = ∑_ a 1 ∑_ a 2 σ 1 ( a 1 | Θ) σ 2 ( a 2 | a 1 ) u 1 ( a 1, a 2, Θ) Player 2‘s „ex ante“ payoff of strategy σ 2 (. | a 1 ) when player 1 plays σ 1 (. | Θ) is ∑_ Θ p(Θ) ∑_ a 1 ∑_ a 2 σ 1 ( a 1 | Θ) σ 2 ( a 2 | a 1 ) u 2 ( a 1, a 2, Θ) Player2 observes player 1‘s move  update probability distribution to posterior distribution μ(. | a 1 ) over the possible types {Θ} Notions / Analysis

21. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games From prev slide: Player2 observes player 1‘s move  update probability distribution to posterior distribution μ(. | a 1 ) over the possible types {Θ} update: How?  In Baysian EQ: player 1‘s strategy is : σ 1 *(.| Θ) ; Knowing σ 1 * and the action a 1 of player 1 and p(.)  use Bayes rule to compute μ(. | a 1 ) in Perfect Baysian EQ: Player 2: maximize his payoff conditional on a 1 for each a 1 where the cond. payoff to strategy σ 2 (. | a 1 ) is ∑_ Θ μ( Θ | a 1 ) u 2 ( a 1, σ 2 (. | a 1 ), Θ) = ∑_ Θ ∑_ a 2 μ( Θ | a 1 ) σ 2 ( a 2 | a 1 ) u 2 ( a 1, a 2, Θ) Notions / Analysis

22. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games A perfect Baysian EQ of a signaling game is a strategy profile σ* and posterior beliefs μ(. | a 1 ) such that (P1) ∀ Θ σ 1 *(.| Θ) ∈ argmax_ α1 u 1 ( α 1, σ 2 *, Θ) (P2) ∀ α 1 σ 2 *(.| a 1 ) ∈ argmax_ α2 ∑_ Θ μ(Θ|a 1 ) u 2 ( a 1, α 2, Θ) (B) μ(Θ|a 1 ) = p(Θ) σ 1 *( a 1 | Θ) / ∑_ Θ‘ p(Θ‘) σ 1 *( a 1 | Θ‘) Interpretation of conditions: (P1) player 1 takes into account effect of a 1 on player 2‘s actions (P2) player 2 reacts optimally given posterior beliefs about Θ (B) Application of Bayes rule PBE: Definition

23. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games A perfect Baysian EQ of a signaling game is a strategy profile σ* and posterior beliefs μ(. | a 1 ) such that (P1) ∀ Θ σ 1 *(.| Θ) ∈ argmax_ α1 u 1 ( α 1, σ 2 *, Θ) (P2) ∀ α 1 σ 2 *(.| a 1 ) ∈ argmax_ α2 ∑_ Θ μ(Θ|a 1 ) u 2 ( a 1, α 2, Θ) (B) μ(Θ|a 1 ) = p(Θ) σ 1 *( a 1 | Θ) / ∑_ Θ‘ p(Θ‘) σ 1 *( a 1 | Θ‘) Interpretation of conditions: (P1) player 1 takes into account effect of a 1 on player 2‘s actions (P2) player 2 reacts optimally given posterior beliefs about Θ (B) Application of Bayes rule PBE: Definition p( a 1 )

24. A Little Glance on (Dynamic+Incomplete)-Case + Signaling games PBE: Definition [1]

25. Summary Static games + complete information : Nash Equilibrium Dynamic games + complete information: Subgame-Perfect Equilibrium Static games + incomplete information: Bayesian Equilibrium (+ others) Dynamic games + incomplete information: Perfect Baysian EQ (+ others) Equilibrium Refinements: Summary

26. [1] D.Fudenberg, J.Tirole: Game Theory; MIT Press, 1991 Bibliography