Lecture Notes on Game Theory

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1 Lecture Notes on Game Theory Levent Koçkesen Strategic Form Games In this part we will analyze games in which the players choose their actions simultaneously (or without the knowledge of other players choices). The rst three chapters will assume that players payo functions are common knowledge. Under this assumption, next chapter will de ne strategic form games and analyze Nash equilibrium and following chapter will discuss some other solution concepts, namely rationalizability, iterated elimination of dominated strategies, and correlated equilibrium. In the last chapter we relax the assumption of common knowledge of payo functions and introduce incomplete information regarding the payo functions. This will induce what is commonly called a Bayesian game. We will introduce this type of games and analyze its equilibria.. 1 Nash Equilibrium 1.1 Preliminaries De nition 1 A strategic form game consists of ² a nitesetofplayers:n = f1; 2;:::;ng ² action spaces of players, denoted A i ;i2 N ² payo functions of players: u i : A! R We refer to such a game by the tuple N;(A i ) i2n ; (u i ) i2n : Remark 1 An outcome is an action pro le (a 1 ;a 2 ;:::;a n ) ; and the outcome space is A = i2n A i : Remark 2 Payos are ordinal Remark 3 If A i are all nite, then the game is called a nite game. Remark 4 The game is common knowledge among the players 1

2 Finite strategic form games with two players can be represented as a bimatrix: L R U u 1 (U; L) ;u 2 (U; L) u 1 (U; R) ;u 2 (U; R) D u 1 (D; L) ;u 2 (D; L) u 1 (D; R) ;u 2 (D; R) where by convention player 1 chooses row action and player 2 chooses column action. Here, we have ² N = f1; 2g ² A 1 = fu; Dg ;A 2 = fl; Rg ;A= f(u; L) ; (U; R) ; (D; L) ; (D; R)g Example 1 Prisoners Dilemma N C N 5; 5 ; 6 C 6; 1; 1 Example 2 Hawk-Dove D H D 3; 3 1; 5 H 5; 1 ; Example 3 Stag Hunt S H S 2; 2 ; 1 H 1; 1; 1 Example 4 Battle of the Sexes B S B 2; 1 ; S ; 1; 2 Example 5 Matching Pennies H T H 1; 1 1; 1 T 1; 1 1; 1 2

3 1.2 Nash Equilibrium An equilibrium of a game is a pro le of strategies. In this section a strategy for player i is simply an element of A i : Later on we will discuss mixed strategies in which case we will introduce another de nition for strategies. One of the most common interpretations of Nash equilibrium is that it is a steady state in the sense that no rational player has an incentive to unilaterally deviate from it. Let x i (x 1 ;x 2 ;:::;x i1 ;x i+1 ;:::;x n ) ;A i j2nnfig A j ; and (y i ;x i ) (x 1 ;x 2 ;:::;x i1 ;y i ;x i+1 ;:::;x n ) : De nition 2 A Nash equilibrium of N;(A i ) i2n ; (u i ) i2n is an action pro le a 2 A such that for every i 2 N u i a i ;ai ui ai ;ai for all ai 2 A i : Another, and sometimes a more convenient way of de ning Nash equilibrium is via best response correspondences B i : A i A i such that B i (a i )=fa i 2 A i : u i (a) u i (a i ;a i) for all a i 2 A ig : A Nash equilibrium is an action pro le a such that a i 2 B i a i for all i 2 N: Given a strategic form game G N;(A i ) i2n ; (u i ) i2n ; the set of Nash equilibria is denoted N (G) : Example 6 Hawk-Dove Example 7 Matching Pennies Example 8 Stag Hunt Example 9 Battle of Sexes Nash equilibrium concept has been motivated in many dierent ways, mostly on an informal basis. We will now give a brief discussion of some of these motivations: Self Enforcing Agreements. Let us assume that two players debate about how they should play a given 2-person game in strategic form through preplay communication. If no binding agreement is possible between the players, then what sort of an agreement would they be able to implement, if any? Clearly, the agreement (whatever it is) should be self enforcing in the sense that no player should have a reason to deviate from her promise if 3

4 she believes that the other player will keep his end of the bargain. A Nash equilibrium is an outcome that would correspond to a self enforcing agreement in this sense. Once it is reached, no individual has an incentive to deviate from it unilaterally. Social Conventions. Consider a strategic interaction played between two players, where player 1 is randomly picked from a population and player 2 is randomly picked from another population. For example, the situation could be a bargaining game between a randomly picked buyer and a randomly picked seller. Now imagine that this situation is repeated over time, each iteration being played between two randomly selected players. If this process settles down to an action pro le, that is if time after time the action choices of players in the role of player 1 and those in the role of player 2 are always the same, then we may regard this outcome as a convention. Even if players start with arbitrary actions, as long as they remember how the actions of the previous players fared in the past, choose those actions that are better, and experiment with some small probability, then any social convention must correspond to a Nash equilibrium. If an outcome is not a Nash equilibrium, then at least one of the players is not best responding, and sooner or later a player in that role will happen to land on a better action which will then be adopted by the players afterwards. Put dierently, an outcome which is not a Nash equilibrium lacks a certain sense of stability, and thus if a convention were to develop about how to play a given game through time, we would expect this convention to correspond to a Nash equilibrium of the game. Focal Points. Focal points are outcomes which are distinguished from others on the basis of some characteristics which are not included in the formalism of the model. Those characteristics may distinguish an outcome as a result of some psychological or social process and may even seem trivial, such as the names of the actions. Focal points may also arise due to the optimality of the actions, and Nash equilibrium is considered focal on this basis. Learned Behavior. Consider two players playing the same game repeatedly. Also suppose that each player simply best responds to the action choice of the other player in the previous interaction. It is not hard to imagine that over time their play may settle on an outcome. If this happens, then it has to be a Nash equilibrium outcome. There are, however, two problems with this interpretation: (1) the play may never settle down, (2) the repeated game is dierent from the strategic form game that is played in each period and hence it cannot be used to justify its equilibrium. So, whichever of the above parables one may want to entertain, they all seem to suggest that, if a reasonable outcome of a game in strategic form exists, it must possess the property of being a Nash equilibrium. In other words, being a Nash equilibrium is a necessary condition for being a reasonable outcome: But notice that this is a one-way statement; it would not be reasonable to claim that any Nash equilibrium of a given game corresponds to an outcome that is likely to be observed when the game is actually played. 4

5 1.3 Existence of Nash equilibrium Our treatment is very standard and is based on a standard xed point theorem which we shall state without proof. Before giving this theorem, we need to recall that a correspondence on a set A is any mapping that associates with each a 2 A asubset(a) of A: We write :A A to indicate that is a correspondence. (Of course, if (a) is a singleton for each a 2 A; then can actually be thought of as a function that maps A to A:) We say that is nonempty and convex-valued if (a) is nonempty and convex for each a 2 A: On the other hand, a correspondence :A A is said to have a closed graph if, for all a 2 A; a m! a; b m 2 (a m ) and b m! b imply that b 2 (a): We are now ready to state Theorem 1 (Kakutani s Fixed Point Theorem): Let A ½ R n be a compact and convex set. If :A A is a nonempty and convex-valued correspondence with a closed graph, then there exists an a 2 A such that a 2 (a): Proof. See Berge (1963, p ). The following theorem, which is of fundamental importance in game theory, is usually referred to as Nash s existence theorem. This result is one of the main reasons why the notion of Nash equilibrium is such a widely used concept. Theorem 2 (Nash): Let G N;(A i ) i2n ; (u i ) i2n be a strategic form game such that ² A i is a nonempty, convex and compact subset of R m i ; ² u i is continuous on A and quasi-concave on A i ;i=1; :::; N: Then N(G) 6= ;: Proof. De ne the correspondence B : A A by B(a) (B 1 (a 1 ); :::; B n (a n )): (Recall that B i is called the best response correspondence of i:) Noticethatifa 2 B(a); then a i 2 B i (a i ) for all i; and hence a 2 N (G) : Consequently, if we can show that Kakutani s theorem applies to B; the proof will be complete. First notice that A is easily checked to be compact and convex since each A i has these two properties. (Verify.) Moreover, B is nonempty-valued since, by Weierstrass theorem, each B i is nonempty-valued. To see that B is convex-valued, consider any x; y 2 B(a): Then, for all i 2 N; u i (x i ;a i )=u i (y i ;a i ) u i (a i ;a i ) for all a i 2 A i ; and thus by using the quasi-concavity of u i on A i ; we nd u i ( x i +(1 )y i ;a i ) u i (x i ;a i ) u i (a i ;a i ) 5

6 for all a i 2 A i and all 2 [; 1]: Since this holds for each i; we conclude that x +(1 )y 2 B(a); and hence that B is convex-valued. It remains to check that B has a closed graph. Towards this end, take any sequences a m and b m in A such that a m! a; b m! b and b m 2 B(a m ): We need to show that b 2 B(a); that is, b i 2 B i (a i ) for all i: Suppose, for contradiction, that there exists an i 2 N such that b i =2 B i (a i ): Then, there exists an x i 2 A i such that u i (x i ;a i ) u i (b i ;a i ) > : By continuity of u i ; for all ">; there exists an integer M " such that ui (b m i ;am i ) u i(b i ;a i ) <" and ui (x i ;a m i) u i (x i ;a i ) <" for all m M " : So, for any "> and any m M " ; we have u i (x i ;a m i) >u i (x i ;a i ) " = u i (b i ;a i )+ ">u i (b m i ;a m i)+ 2": Thus, choosing " = =2; we obtain u i (x i ;a m i) >u i (b m i ;a m i) for all m M =2 ; contradicting b m i 2 B i (a m i) for all m: We may then conclude that b i 2 B i (a i ) for all i 2 N and hence b 2 B(a): The proof is completed by applying Kakutani s xed point theorem. If the payos of the players depend on a parameter, then the Nash equilibrium set of the associated game would in general be conditional on the value of this parameter. Thus, we may think of the equilibrium set as a correspondence mapping the value of the parameter to a set of (equilibrium) outcomes. The following exercise shows that this correspondence has a closed graph (under reasonable assumptions). As you might imagine, this result often proves useful when one needs to examine the behavior of the equilibrium set of a particular game with respect to small changes in the descriptions of the payos of the players. Exercise 1 Let 6=? be a compact set in R k and let A i 6=? be a convex and compact subset of R m i ;i=1;:::; N: Assume that u i : A! R is a continuous function which is quasi-concave on A i and consider the game G(µ) (N;(A i ) ; (u i (:; µ))) where µ 2 : De ne the correspondence : A by (µ) N(G(µ)): Prove that is a nonempty-valued correspondence with a closed graph. Let us try to illustrate what this observation entails by a simple, one player game. Let A =[; 1]; =[1; 1] and the payo function is u (x; µ) =1+µ 2xµ and the individual chooses x 2 A; given µ 2 : The following gives the Nash equilibrium correspondence, 8 >< 1; if µ< (µ) = [; 1]; if µ = : >: ; if µ> 6

7 Now, the above result says that the Nash equilibrium of any sequence of games G (µ m ) when µ m! converges to a Nash equilibrium of the game G () : It does not say, however, that any Nash equilibrium of the limit game can be obtained by taking the limit of a sequence of games. We close this section by another useful observation concerning the existence of Nash equilibrium. A game G is called symmetric if A i = A j and u i (a) =u j (a ) for all i; j =1; :::; N and all a; a 2 A such that a is obtained from a by exchanging a i and a j : For instance, the prisoners dilemma is a symmetric game. Indeed, many interesting games of economic importance are symmetric. It is then of interest to know if a symmetric game would have a symmetric equilibrium in which all players take the same actions. The following result answers this question. Proposition 3 Let G N;(A i ) i2n ; (u i ) i2n be a symmetric strategic game such that ² A i is a nonempty, convex and compact subset of R m i ; ² u i is continuous and is quasi-concave in its ith component: Then, there exists an action pro le (a ; :::; a ) 2 A such that (a ; :::; a ) 2 N(G): Proof. Exercise: Example 1 (First Price Auction) ² N = f1; 2;:::;ng ² A i = R + ;i2n ² denote a typical element of A i by b i ; interpreted as bids. ² ( v i b i ; if i =minfj 2 N : b j 2 max fb 1 ;b 2 :::;b n gg u i (b) = ; otherwise where v i is interpreted as the valuation of player i and v 1 >v 2 >:::>v n > : Let us rst show that player 1 obtains the object. It is su cient to show that b 1 2 max fb 1 ;b 2 :::;b n g. Suppose, for contradiction that there exists j 6= 1such that j =minfk 2 N : b k 2 max fb 1 ;b 2 :::;b n gg : Now if 7

8 ² b j >v 2 ; then b j >v j so that player j would rather bid zero (which means he doesn t win the auction) and get zero. ² b j v 2 ; then b j <v 1 and player 1 can bid b j ; win the auction and obtain b j v 1 > : Now that we established player 1 wins the object, let us see what values b 1 can take. Clearly, b 1 v 1 : It also has to be that b 1 v 2 ; forotherwise,player2canbidb 2 2 (b 1 ;v 2 ) and increase her payo. Thus b 1 2 [v 2 ;v 1 ] : We will nally show that b j = b 1 for some j 6= 1: If not, player 1 can increase her payo by reducing her bid. So, the only candidates for equilibrium are (b 1 ;b 2 ;:::;b n ) such that b 1 2 [v 2 ;v 1 ] ;b j b 1 for all j and b j = b 1 for some j 6= 1: It is easy to con rm that all such (b 1 ;b 2 ;:::;b n ) are Nash equilibria. De nition 3 A Strict Nash equilibrium of N;(A i ) i2n ; (u i ) i2n is an action pro le a 2 A such that for every i 2 N u i (a ) >u i ai ;a i for all ai 2 A i,witha i 6= a i : Strict Nash equilibria are robust in the sense that they remain strict equilibria when the payo functions are slightly perturbed. Unfortunately, not every game has a strict Nash equilibrium, and not every Nash equilibrium is strict. For example, in the following game L R U 1; 1 ; D ; ; 2 (U; L) is a strict Nash equilibrium, whereas (D; R) is not. u 1 (U; R) ="; (D; R) ceases to be a Nash equilibrium. Notice that for all " > ; if 1.4 Mixed Strategy Equilibrium Players strategies are no longer actions but rather probability distributions over actions. Therefore, a strategy pro le induces a probability distribution over action pro les. Since, payo functions are de ned on action pro les we will assume that preferences are de ned over lotteries on A: (A lottery in this context is de ned by (A; p) where p is a probability measure over A:) To keep the analysis simple we will mostly be dealing with nite action spaces. We also assume that preferences satisfy von Neumann-Morgenstern assumptions, so that they can be represented by the expected value of some function u i : A! R: 8

9 Denote by 4 (A i ) the set of probability distributions over A i : A mixed strategy of player i is a member of 4 (A i ) ; which we assume are independent across players. Sometimes we will refertoamemberofa i as a pure strategy. For any nite set X and ± 24(X) ;±(x) denotes the probability that ± assigns to x 2 X: The support of ± is given by the set fx 2 X : ± (x) > g, denotedsupp(±): Apro le( i ) i2n of mixed strategies induces a probability distribution over A: If A is nite, then the probability assigned to a 2 A is p (aj ) Y i (a i ) : i2n (Note the role of independence) Player i s expected payo is then given by U i ( ) X p (aj ) u i (a) a2a à Y! = X a2a j2n j (a j ) u i (a) : We call the game (N;(4 (A i )) ; (U i )) the mixed extension of game (N;(A i ) ; (u i )) : Remark 5 U i is linear in ; i.e. for any 2 [; 1] we have U i ( +(1 ) ) = U i ( )+(1 ) U i ( ): Exercise 2 Verify that the above remark is true. Remark 6 Let e(a i ) be the degenerate mixed strategy of i (i.e., a pure strategy) that attaches probability one to action a i : Then U i ( ) = X i (a i ) U i (e (a i ) ; i ) : a i 2A i Exercise 3 Verify that the above remark is true. De nition 4 A mixed strategy equilibrium is a strategy pro le 2 i2n 4 (A i ) such that for every i 2 N we have U i ( ) U i i ; i for all i 24(A i ) : Proposition 4 A strategy pro le 2 i2n 4 (A i ) is a mixed strategy equilibrium if and only if for every player i 2 N every action in the support of i is a best response to i: 9

10 Proof. Letus rstshowthatif is a mixed strategy equilibrium then every action in the support i is a best response to i: Suppose, for contradiction, there exists an i 2 N and an action a i in the support of i that is not a best response to i: Then, there exists an action a i 2 A i which gives her a strictly higher payo, i.e., U i e (a i ) ; i >Ui e (a i ) ; i : Now, consider the following mixed strategy for player i; 8 >< i (a i )+ i (a i ) ; if a i = a i i (a i)= ; if a i = a >: i i (a i) ; otherwise : We have that X U i i ; i = i (a i ) U i e (ai ) ; i a i 2A i X = i (a i) U i e (ai ) ; i + i (a i ) U i e (a i ) ; i + i (a i ) U i e (a i ) ; i a i =2fa i ;a i X g = i (a i) U i e (ai ) ; i + i (a i ) U i e (a i ) ; i + i (a i ) U i e (a i ) ; i a i =2fa i ;a i X g < i (a i) U i e (ai ) ; i + i (a i ) U i e (a i ) ; i + i (a i ) U i e (a i ) ; i a i =2fa i ;a i g = U i i ; i contradicting that is a Nash equilibrium. Now suppose that for all i 2 N; each a i 2supp( i ) is a best response to i, i:e:; U i e (ai ) ; i U i e (a i ) ; i for all ai 2supp( i ) and a i 2 A i : Fix a mixed strategy i 2 4(A i ) and let j i be the probability that is assigned to the jth action by ; i.e., j i = i a j i for all j 2f; 1;:::;m i g ; where m i is the number of elements in A i : Notice that U i e (ai ) ; i = Ui e (a i ) ; i = v for all ai ;a i 2 supp ( i ) ; for some v 2 R; and, j i v j i U i e a j i ; i for all a j i 2 A i: Therefore, we have X U i i ; i = i (a i) U i e (ai ) ; i a i 2A i X = i (a i) U i e (ai ) ; i supp( i ) 1

11 = v Xm i = j i v j=1 Xm i j=1 = X j i U i supp( i ) = U i i ; i : e a j i ; i i (a i ) U i e (ai ) ; i Since i was arbitrary, we have U i i ; i Ui i ; i for all i 24(A i ) : Since this is true for all i 2 N; is a Nash equilibrium. Remark 7 The above proposition implies that every action in the support of i same payo in a Nash equilibrium. yields the Theorem 5 Every nite strategic form game has a mixed strategy Nash equilibrium. Proof. Exercise. Example 11 Hawk-Dove. Let us nd the best response correspondence for player 1 rst. The payo to action D is 3 2 (D) +(1 2 (D))and the payo to action H is 4 2 (D) : Therefore, the best response correspondence for player 1 is 8 >< 1; 2 (D) < 1=2 B 1 ( 2 (D)) = [; 1]; 2 (D) =1=2 >: ; 2 (D) > 1=2 where B 1 denotes the probability assigned to D by 1 : Similarly, 8 >< 1; 1 (D) < 1=2 B 2 ( 1 (D)) = [; 1]; 1 (D) =1=2 >: ; 1 (D) > 1=2 : Plotting these two correspondences, we nd that the set of Nash equilibria of this game is N(HD)=f(1; ) ; (; 1) ; (1=2; 1=2)g : Exercise 4 Find all Nash equilibria of the Matching Pennies game. Exercise 5 Find all Nash equilibria of the Battle of the Sexes game. 11

12 α 2(D) 1 Nash Equilibria 1/2 1/2 1 α 1 (D) Figure 1: Nash Equilibria of Hawk-Dove Game 12