Lecture Notes on Game Theory

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1 Lecture Notes on Game Theory Levent Koçkesen 1 Bayesian Games So far we have assumed that all players had perfect information regarding the elements of a game. These are called games with complete information. A game with incomplete information, on the other hand, tries to model situations in which some players have private information before the game begins. The initial private information is called the type of the player. For example, types could be the privately observed costs in an oligopoly game, or privately known valuations of an object in an auction, etc. 1.1 Preliminaries A Bayesian game is a strategic form game with incomplete information. It consists of: ² a set of players, N = f1;:::;ng ; and for each i 2 N ² an action set, A i ; (A = i2n A i ) ² atypeset, i ; ( = i2n i ) ² a probability function, ² a payo function, p i : i!4( i ) u i : A! R: The function p i summarizes what player i believes about the types of the other players given her type. So, p i (µ i jµ i ) is the conditional probability assigned to the type pro le µ i 2 i : Similarly, u i (ajµ) is the payo of player i when the action pro le is a and the type pro le is µ: In summary, we can de ne a Bayesian game with the tuple G B =(N;(A i ) ; ( i ) ; (p i ) ; (u i )) : We call a Bayesian game nite if N; A i and i are all nite, for all i 2 N: A pure strategy for player i in a Bayesian game is a function which maps player i s type into her action set a i : i! A i : 1

2 In many situations it is easier to work with in nite type spaces. In this case the probability distributions p i must be de ned on all measurable subsets of i, instead of just individual elements of i : We say that beliefs (p i ) i2n in a Bayesian game are consistent ifthereissomecommon prior distribution over the set of type pro les such that each player s beliefs given her type are just the conditional probability distribution that can be computed from the prior distribution by Bayes s rule. In the nite case, beliefs are consistent if there exists some probability distribution P in 4 ( ) such that, p i (µ i jµ i )= P (µ) P s i 2 i P (s i; µ i ) for all µ 2 and for all i 2 N: A mixed strategy for player i is i : i!4(a i ) so that i (a i jµ i ) is the probability assigned by i to action a i by type µ i of player i: 1.2 Bayesian Equilibrium De nition 1 A Bayesian equilibrium of a Bayesian game (N;(A i ) ; ( i ) ; (p i ) ; (u i )) is a mixed strategy pro le =( i ) i2n ; such that for every player i 2 N and every type µ i 2 i ; we have 0 1 X i (:jµ i ) 2 arg max p i (µ i jµ i ) Y j (a j jµ j ) A (a i ) u i (ajµ) : 24(A i ) µ i 2 i a2a j2nnfig Remark 1 Type, in general, can be any private information that is relevant to the player s decision making, such as the payo function, player s beliefs about other players payo functions, her beliefs about what other players believe her beliefs are, and so on. Remark 2 Notice that, in the de nition of a Bayesian equilibrium we need to specify strategies for each type of a player, even if in the actual game that is played all but one of these types are non-existent. This is because, given a player s imperfect information, analysis of that player s decision problem requires us to consider what each type of the other players would do, if they were to play the game. Example 2 (Battle of Sexes with incomplete information). Suppose player 2 has perfect information and two types l and h: Type l loves going out with player 1 whereas type h hates it. Player 1 has only one type and does not know which type is player 2. Her beliefs place 2

3 probability 1=2 on each type. The following tables give the payo s to each action and type pro le: B S B S B 2,1 0,0 S 0,0 1,2 type l We can represent this situation as a Bayesian game: ² N = f1; 2g ² A 1 = A 2 = fb;sg ² 1 = fxg ; 2 = fl; hg ² p 1 (ljx) =p 1 (hjx) =1=2; p 2 (xjl) =p 2 (xjh) =1: ² u 1 ;u 2 are given in the tables above. B 2,0 0,2 S 0,1 1,0 type h Let us nd the Bayesian equilibria of this game by analyzing the decision problem of each player of each type: Player 2 of type l : Given player 1 s strategy 1 ; his expected payo to ² action B is 1 (Bjx) ; ² action S is 2(1 1 (Bjx)) so that his best response is to play B if 1 (Bjx) > 2=3 andtoplays if 1 (Bjx) < 2=3: Player 2 of type h : Given player 1 s strategy 1 ; his expected payo to ² action B is (1 1 (Bjx)) ; ² action S is 2 1 (Bjx) so that his best response is to play B if 1 (Bjx) < 1=3 andtoplays if 1 (Bjx) > 1=3: Player 1: Given player 2 s strategy 2 (:jl) and 2 (:jh) ; her expected payo to ² action B is (Bjl)(2) (Bjh)(2)= 2 (Bjl)+ 2 (Bjh) ; ² action S is 1 2 (1 2 (Bjl)) (1) (1 2 (Bjh)) (1) = 1 2 (Bjl)+ 2 (Bjh) : 2 3

4 Therefore, her best response is to play B if 2 (Bjl) + 2 (Bjh) > 2=3 and to play S if 2 (Bjl)+ 2 (Bjh) < 2=3: Let us rst check if there is a pure strategy equilibrium in which both types of player 2 play B; i.e. 2 (Bjl) = 2 (Bjh) = 1: In this case player 1 s best response is to play B as well to which playing B is not a best response for player 2 type h: Similarly check that 2 (Bjl) = 2 (Bjh) =0and 2 (Bjl) =0and 2 (Bjh) =1cannotbepartofaBayesian equilibrium. Let s check if 2 (Bjl) =1and 2 (Bjh) =0could be part of an equilibrium. In this case player 1 s best response is to play B: Player 2 type l s best response is to play B and that of type h is S: Therefore, ( 1 (Bjx) ; 2 (Bjl) ; 2 (Bjh)) = (1; 1; 0) is a Bayesian equilibrium. Clearly, there is no equilibrium in which both types of player 2 mixes. Suppose only type l mixes. Then, 1 (Bjx) =2=3; which implies that 2 (Bjl)+ 2 (Bjh) =2=3: This, in turn, implies that 2 (Bjh) = 0: Since 2 (Bjh) = 0 is a best response to 1 (Bjx) = 2=3; the following is another Bayesian equilibrium of this game ( 1 (Bjx) ; 2 (Bjl) ; 2 (Bjh)) = (2=3; 2=3; 0) : As an exercise show there is one more equilibrium given by ( 1 (Bjx) ; 2 (Bjl) ; 2 (Bjh)) = (1=3; 0; 2=3) : Example 3 (Cournot Duopoly with incomplete information) The pro t functions are given by u i = q i (µ i q i q j ) : Firm 1 has one type µ 1 =1; but rm 2 has private information about its type µ 2 : Firm 1 believes that µ 2 =3=4 with probability 1=2 and µ 2 =5=4 with probability 1=2; and this belief is common knowledge. We will look for a pure strategy equilibrium of this game. Firm 2 of type µ 2 s decision problem is to max q 2 (µ 2 q 1 q 2 ) q 2 which is solved at q 2 (µ 2 )= µ 2 q 1 : 2 Firm 1 s decision problem, on the other hand, is ½ 1 max q 1 2 q 1 (1 q 1 q 2 (3=4)) + 1 ¾ 2 q 1 (1 q 1 q 2 (5=4)) 4

5 which is solved at q 1 = 2 q 2 (3=4) q 2 (5=4) : 4 Solving yields, q 1 = ;q 2 (3=4) = 24 ;q 2 (5=4) = 5 24 : As an exercise, show that this is the unique equilibrium. Example 4 (Public Good Contribution) Two players simultaneously decide whether to contribute towards producing a public good. The good is produced if at least one of them contributes. The bene t derived from the public good is 1 and the cost to player i of contributing is c i : The payo s are depicted in the following table C N C 1 c 1 ; 1 c 2 1 c 1 ; 1 N 1; 1 c 2 0; 0 : The bene ts are common knowledge, but each player s cost is private information. However, both players believe it is common knowledge that each c i is drawn independently from a uniform distribution on [0; 2]: Player 1 s decision problem: Action C yields 1 c 1 whereas action N yields Z (Cjc 2 ) dc 2 : Thus, 1 (Cjc 1 )=1if Z 2 1 c 1 < (Cjc 2 ) dc 2 c 1 : Similarly for player 2, 2 (Cjc 2 )=1if Z 2 1 c 2 < (Cjc 2 ) dc 2 c 2: Since, at c 1 = c 1 ; player 1 is indi erent between contributing and not contributing we have Similarly, Solving yields, c 1 =1 Z c 2 0 Z (1)dc 1 2 c 2 (0)dc 2 =1 c 2 2 : 2 c 2 =1 c 1 2 : c 1 = c 2 = 2 3 : Notice that in the complete information version with costs c i 2 (2=3; 1) at least one of the players contribute in equilibrium, whereas in the incomplete information version no player contributes. Why is the outcome di erent in incomplete information version? 5

6 1.3 Puri cation of Mixed Strategy Equilibria Interpreting equilibria in mixed strategies is di cult. Not only the idea of individuals ipping coins to determine their actions is counter-intuitive, but also it is di cult to provide reasons for individuals choosing the exact equilibrium probabilities when they are indi erent between actions. As a response to these di culties Harsanyi (1973) showed that mixed strategy equilibria can be interpreted as limits of Bayesian equilibria of perturbed games in which each player is almost always choosing his uniquely optimal action. Harsanyi perturbs the games by slightly perturbing the payo s, calculates the Bayesian equilibria of the games parametrized on the perturbations and then take the perturbations to zero. (A single sequence of perturbed games can be used to purify all the mixed strategy equilibria of the limit game. In this section, rather than delving into the technical details of the argument, we will demonstrate the idea using an example. Consider the following two-player game L R T 0; 0 0; 1 : B 1; 0 1; 3 In the unique Nash equilibrium of this game, player 1 plays T with probability 3=4 and player 2 plays L with probability 1=2: Now, consider the following perturbed game L R T "µ 1 ;"µ 2 "µ 1 ; 1 B 1;"µ 2 1; 3 where " 2 (0; 1) ; and types µ 1 ;µ 2 are independently, identically and uniformly distributed over [0; 1] : Let s rst calculate the Bayesian equilibria of this game. For player 1 expected payo of playing T is "µ 1 ; whereas that of B is Z 1 Z 1 2 (Ljµ 2 )(1)dµ 2 + (1 2 (Ljµ 2 )) ( 1) dµ 2 so that 0 µ 1 > 1+2R (Ljµ 2 ) dµ 2 x =) 1 (T jµ 1 )=1; " µ 1 < 1+2R (Ljµ 2 ) dµ 2 x =) 1 (T jµ 1 )=0: " Similarly for player 2, µ 2 > 3 4 R (T jµ 1 ) dµ 1 y =) 2 (Ljµ 2 )=1; " µ 2 < 3 4 R (T jµ 1 ) dµ 1 y =) 2 (Ljµ 2 )=0: " 0 6

7 At µ 1 = x player 1 is indi erent, i.e., ½Z y Z 1 ¾ "x = 1+2 (0) dµ 2 + (1) dµ 2 0 y = 1+2(1 y) ; and at µ 2 = y player 2 is indi erent, i.e., ½Z x Z 1 ¾ "x = 3 4 (0) dµ 1 + (1) dµ 1 0 x = 3 4(1 y) : Solving for x and y; we get So, the Bayesian equilibrium satis es, and x = 2+" 8+" 2 y = 4 " 8+" 2 : µ 1 > 2+" 8+" 2 =) 1 (T jµ 1 )=1; µ 1 < 2+" 8+" 2 =) 1 (T jµ 1 )=0; µ 2 > 4 " 8+" 2 =) 2 (Ljµ 2 )=1; µ 2 < 4 " 8+" 2 =) 2 (Ljµ 2 )=0: 1 (:jµ 1 ) and 2 (:jµ 2 ) can be chosen arbitrarily in the zero probability events that µ 1 = 2+" and µ 8+" 2 2 = 4 " : So, the equilibrium is almost always unique and is a pure strategy 8+" 2 equilibrium. Notice that as " goes to zero, the equilibrium converges to the unique mixed strategy equilibrium of the original game. Therefore, one can interpret randomization by players as depending on minor factors that have been omitted from the description of the game. The players almost always play pure strategies, but they condition these strategies upon un-modeled random events which have very slight e ects on the game. 1.4 Auctions Auctions are means of allocating goods such as works of art, government bonds, in which the individual who declares to pay the highest price (i.e., the highest bidder) obtains the good. 7

8 The bids may be called sequentially or may be submitted in sealed envelopes, and the price paid by the winner may be equal to the highest bid or some other price. Important questions to ask in analysis of auctions are: Is the auction e cient? If not, does there exist a more e cient auction design? Which auction design raises more revenue? The following two examples provides an introduction to the subject. Previously, we have looked at two forms of auctions, namely First Price and Second Price Auctions, in a complete information framework in which each bidder knew the valuations of every other bidder. In this section we relax the complete information assumption and revisit these two form of auctions. In particular, we will assume that each bidder knows only her own valuation, and the valuations are independently distributed random variables whose distributions are common knowledge. The following elements de ne the general form of an auction that we will analyze: ² Set of bidders, N = f1; 2;:::;ng ; and for each i 2 N ² a type set (set of possible valuations), i =[v; ¹v] ;v 0: ² an action set, A i = R + (actions are bids) ² a belief function: player i believes that her opponents valuations are independent draws from a distribution function F that is strictly increasing and continuous on [v; ¹v] : ² a payo function, which is de ned for any a 2 A; v 2 as follows 8 < v i P (a) u i (a; v) = ; if a m j a i for all j 6= i; and jfj : a j = a i gj = m : 0; if a j >a i for some j 6= i where P (a) is the price paid by the winner if the bid pro le is a: Notice that in the case of a tie the object is divided equally among all winners Second Price Auctions In this design, highest bidder wins and pays a price equal to the second highest bid. Although there are many Bayesian equilibria of second price auctions, bidding own valuation v i is weakly dominant for each player i: To see this let x be the highest of the other bids and consider bidding a 0 i <v i;v i ; and a 00 i >v i: Depending upon the value of x; the following table gives the payo s to each of these actions by i x a 0 i a 0 i <x<v i x = v i v i <x a 00 i a 00 i <x a 0 i win/tie;pay x lose lose lose lose : v i win; pay x win; pay x tie; pay v i lose lose a 00 i win; pay x win; pay x win; pay v i win/tie;pay x lose 8

9 1.4.2 First Price Auctions In rst price auctions, the highest bidder wins and pays her bid. Let us denote the bid of player with type v i by i (v i ) and look for symmetric equilibria, i.e. i (v) = (v) for all i 2 N: First, although we will not attempt to do so here, it is possible to show that strategies i (v i ) ; and hence (v) ; are strictly increasing and continuous on [v; ¹v] :(see Fudenberg and Tirole, 1991). So, let s assume that they are, and check if they are once we locate a possible equilibrium. Let G (b) be the probability that the bid, under strategy ; is at most b; i.e., G (b) = prob ( (v) b) = prob ³ v (b) 1 = F ³ 1 (b) : (1) (we know that is invertible since it is strictly increasing). The expected payo of player with type v who bids b is given by (v b) prob (highest bid is b) =(v b)(g (b)) n 1 : (2) (Because v i are independently distributed). Now, let B v ( ) be the best response of player with type v when all the other players play according to : Then, it has to satisfy the FOC: (G (B v ( ))) n 1 +(v B v ( )) (n 1) (G (B v ( ))) n 2 G 0 (B v ( )) = 0: (3) So, let ( ;:::; ) be an equilibrium. Then, Therefore, by (1), we have and, for any B v ( ) = (v) for all v 2 [v; ¹v] : G (B v ( )) = F ³ 1 ( (v)) = F (v) G 0 (b) = F 0 ³ 1 (b) ³ 1 0 (b) = F 0 ³ 1 (b) 0 ³ 1 ; (b) by the fact that is almost everywhere di erentiable (since it is strictly increasing), and by the inverse function theorem. Therefore, substituting (v) for b G 0 ( (v)) = F 0 ³ 1 ( (v)) ( (v)) = F 0 (v) 0 (v) : 9

10 Therefore, from (3). we get 0 (v)(f (v)) n 1 +(n 1) (v)(f (v)) n 2 F 0 (v) =(n 1) v (F (v)) n 2 F 0 (v) which is a di erential equation in : Integrating both sides, we get (v)(f (v)) n 1 = Z v v (n 1) x (F (x)) n 2 F 0 (x) dx = v (F (v)) n 1 Z v v (F (x)) n 1 dx: Solving for (v) ; R v v (v) =v (F (x))n 1 dx (F (v)) n 1 : One can easily show that is continuous and strictly increasing in v as we hypothesized. Furthermore, notice that (v) =v; but (v) <vfor v>v: That is, except the player with the lowest valuation, everybody bids less than her valuation. As an exercise, calculate assuming F is uniform on [0; 1]: 2 Correlated Equilibrium Suppose players can condition their actions on random outcomes. (Say, before the game they can agree, in a non-binding way, to use certain actions if they observe certain signals sent by a random signal generator). Aumann (1974, J. Math. Econ.) introduced the notion of correlated equilibrium to capture this idea. He used the following example: L R U 5; 1 0; 0 D 4; 4 1; 5 This game has two pure Nash equilibria: (U; L) and (D; R) and a mixed strategy equilibrium in which players play their two actions with equal probabilities. The mixed strategy equilibrium has the payo pro le (2:5; 2:5): Now, suppose they agree to ip a coin and if it is heads player 1 agrees to play U and player 2 agrees to play L; whereas if it is tails, player 1 agrees to play D and player 2 agrees to play R: They then play the above game. Suppose now that the coin ip results in heads. Given that player 2 plays L; it is a best response for player 1 to play U; and given that player 1playsU it is a best response for player 2 to play L: Similarly, if it is tails, then neither player has an incentive to deviate from the pre-play (non-binding!) agreement. Therefore, outcome (U; L) with probability 1=2 and (D; R) with probability 1=2 is a Nash equilibrium outcome of 10

11 this extended game. (Note that there is no independent randomizations by players which can generate this outcome): The expected payo at this equilibrium is (3; 3) : Remark 3 Any payo pro le in the convex hull of the Nash equilibrium payo pro les can be obtained if players can condition their strategies on publicly observable random events. Furthermore, only those outcomes can be obtained. Exercise 2.1 Prove the above claim. However, players can do even better by using a device that sends di erent but correlated signals. To illustrate, suppose there three equally likely states: A, B, andc. Alsoletplayer1 s information partition to be ffag; fb; Cgg andthatofplayer2tobeffa; Bg; fcgg: Suppose they agree on the following: Player 1 : fag!u; fb;cg!d Player 2 : fa; Bg!L; fcg!r: It is easy to verify that this is a Nash equilibrium: if state A occurs player 1 s best response is to play U: If state B or C occurs, then player 1 does not know which one occurred. She can calculate the conditional probability that player 2 observed fa; Bg as prob (fa; BgjfB;Cg) = = = 1 2 : prob (fa; Bg\fB;Cg) prob (fb;cg) 1=3 1=3+1=3 Therefore, she assesses the probability that player 2 will play L as 1=2 to which D is a best response. Verify for player 2 yourself. Notice that: prob (U; L) = prob (fag\fa; Bg) =1=3 prob (U; R) = prob (fag\fcg) =0 prob (D;L) = prob (fb; Cg\fA; Bg) =1=3 prob (D; R) = prob (fb;cg\fcg) =1=3 which leads to an expected payo of Nash equilibrium payo s. for each player, which is outside the convex hull of Remark 4 Original Nash equilibria are still equilibria of the extended game because if you ignore your signal ignoring mine is a best response. 11

12 De nition 5 A correlated equilibrium of a strategic form game (N;(A i ) ; (u i )) consists of ² a nite probability space ( ;¼) where is a set of states, and ¼ is a probability measure on ; ² for each i; a partition P i of (thisiscalledplayeri s information partition) ² for each i; a function ¾ i :! A i with ¾ i (!) =¾ i (! 0 ) whenever! 2 P i and! 0 2 P i for some P i 2P i : (¾ i is player i s strategy) such that for every i 2 N and every function i :! A i for which i (!) = i (! 0 ) whenever! 2 P i and! 0 2 P i for some P i 2P i we have X ¼ (!) u i (¾ i (!) ;¾ i (!)) X ¼ (!) u i (¾ i (!) ; i (!)) :!2!2 Example 6 Let us formally express the correlated equilibria of the opening example: The rst equilibrium is given by ² = fh; Tg ;¼(H) =¼ (T )=1=2; ²P 1 = P 2 = ffhg ; ft gg ; ² ¾ 1 (H) =U; ¾ 1 (T )=D; ¾ 2 (H) =L; ¾ 1 (T )=R: The second equilibrium is given by ² = fa; B; Cg ;¼(A) =¼ (B) =¼ (C) =1=3; ²P 1 = ffag ; fb;cgg ; P 2 = ffa; Bg ; fcgg ; ² ¾ 1 (A) =U; ¾ 1 (B) =¾ 1 (C) =D; ¾ 2 (A) =¾ 2 (B) =L; ¾ 2 (C) =R: The following is an alternative de nition of correlated equilibrium which is sometimes more useful in calculating the correlated equilibria of particular games. 12

13 De nition 7 A correlated equilibrium is any probability distribution p over A such that, for every i 2 N we have or X p (a) u i (a i ;a i ) X p (a) u i (a i ;d i (a i )) for all d i : A i! A i ; a2a a2a X p (a) u i (a i ;a i ) X p (a) u i (a i ;a 0 i) for all a i ;a 0 i 2 A i : a i 2A i a i 2A i Common Values and Winner s Curse: the example I gave in the class. 13