Topcs on the Border of Economcs and Computaton December 11, 2005 Lecturer: Noam Nsan Lecture 7 Scrbe: Yoram Bachrach 1 Nash s Theorem We begn by provng Nash s Theorem about the exstance of a mxed strategy equlbrum n every fnte game. The proof reles on one of the fxed pont theorems - Brouwer s fxed pont theorem. Theorem 1 Brouwer s fxed pont theorem: If C s a compact (closed and bounded) convex set, and f : C C s a contnuous functon, then there exsts c C for whch f(c) = c. We now use Brouwer s fxed pont theorem to prove Nash s theorem. Theorem 2 Nash s theorem: Every fnte game has a mxed strategy equlbrum. Proof Idea: Let u (s 1,..., s n ) denote player s utlty when the players choose strateges s 1,..., s n. We denote player s strategc choce x and the other players choces x. We defne the best response of player to x, BR (x ), as the x for whch for all other x we have u (x, x ) > u (x, x ). We have defned x 1,...x n to be a Nash equlbrum f each of the x s s the best response to x. Let (S ) be the mxed extenson of player s strateges. We defne a transformaton from (S 1 ) (S 2 )... (S n ) onto tself, that s the best response transformaton: (x1,..., x n ) BR(x 1 ),..., BR(x n ). The fxed pont of ths transformaton s a Nash equlbrum. The set s compact and convex. Had ths transform been a functon we could use Brouwer s theorem to show that a fxed pont (Nash equlbrum) exsts. However, there may be more than one best response, and the transform s now contnous. Thus, we would denote ˆx a certan adjustment of x toward the best response. If no adjustment s requred, we already have a best response. Proof: We defne a contnuous functon from (S 1 ) (S 2 )... (S n ) onto tself: (x 1,..., x n ) ˆx 1,..., xˆ n. We denote x j the weght of pure strategy j n the dstrbuton x. We denote c j = cj (x 1,..., x n ) = u (j, x ) u (x, x ) the payoff of player when swtchng from x to pure strategy j. c j s contnuous for all, j, snce u s lnear n x. At the equlbrum pont, every c j < 0. 7-1
We d lke to defne ˆx j = xj + cj, but we have to normalze to get a probablty dstrbuton. We denote c j+ = max(0, c j ). cj+ s also contnuous as the maxmum of two contnuous functons. We note that for every, cj+ x j = 0 as cj = u (j, x ) u (x, x ) and we are summng ths over all js. We defne ˆx j = xj +cj+ 1+ j c j+. We note that for every, j ˆxj = 1. As noted before, the c j s are contnuos, the u s are contnuous as a lnear functon. Also, the maxmum, sums and dvsons of contnuous functons are also contnuous. Thus, The functon f s contnuous. We now use Brouwer s fxed pont theorem to show that f has a fxed pont. In such a pont, for every, x = ˆx. We show that f the games s n a Nash equlbrum, we have c j+ there exsts some c j+ = 0 for every, j. If the game s n a Nash equlbrum and yet > 0, then x j = ˆx j > 0. Snce j xj cj = 0 there must exst some j that c j < 0, so c j + = 0. Then x j = ˆx j = xj j cj+. But ths s a contradcton, snce 1 + j cj+ > 1 and we get that x j s equat to tself dvded by a number greater than 1. 2 Computatonal hardness Note that the proof n the prevous secton only shows a Nash equlbrum n mxed strateges exsts, but does not show how to fnd that equlbrum. The proof of Brouwer s fxed pont theorem contans a party argument. The Nash equlbrum problem wth two players s complete n PPAD (polnomal party argument drected). Fndng any Nash equlbrum s a PPAD-complete. equlbrum to fnd makes the problem NP-complete. Almost any condton on the Nash Theorem 3 Fndng the Nash equlbrum that maxmzes the socal welfare (u 1 + u 2 ) s NP-complete. In ths problem we are gven the utlty matrxes of players 1 and 2, and as an output we need to gve the mxed strateges whch are the Nash equlbrum that maxmzes socal welfare. To proof the theorem we use a reducton from B-Clque. The B-Clque problem: we are gven a bparte graph and a number k. We need to decde whether there exst 2 sets of k vertces (one n each sde) that hold between them all the edges n the graph. Sketch of Proof: To reduce a B-Clque G =< V, E >, k problem to a maxmal socal wellfare nash equlbrum problem, we buld the payment matrxes. Let L be the set of the 7-2
Table 1: The reduced game for B-Clque L R R (1, 1) f (r, l) E (0, 0) f (r, l) / E (k, k) 0... 0 (k, k) L ( k, k) 0... 0 ( k, k) 0 vertces n the left sde of the bparte graph, and R the set of vertces n the rght sde. The payment matrxes are composed of 4 submatrxes - top left, top rght, bottom left and bottom rght. In the top left submatrx, we have a row for every vertex n L and colomn for every vertex n R. If the edge (l, r) E the payments are (1, 1). If (l, r) / E the payments are (0, 0). In the top rght submatrx, we have a row for every vertex n R and colomn for every vertex n L. The payments are always ( k, k) n the dagonal, and (0, 0) anywhere else. In the bottom left submatrx, we have a row for every vertex n R and colomn for every vertex n L. The payments are always (k, k) n the dagonal, and (0, 0) anywhere else. In the bottom rght submatrx, we have a row for every vertex n L and colomn for every vertex n R. The payments are (0, 0) everywhere n that submatrx. We show that n the game descrbed by the bult matrx there s a Nash equlbrum wth a total utlty of u 1 + u 2 2 f and only f there s a k-b-clque n the orgnal graph. In the frst drecton, we can show that when we choose the strateges that are the vertces of the b-clque, we get a utlty of exactly 2. We can also show that ths s a Nash equlbrum: for example, frst player can only loose by devatng on the left sde. In the opposte drecton, we consder a Nash equlbrum wth a utlty of at least 2. The strateges wth postve probablty defne vertces n the orgnal graph. We can show that these are a b-clque. We can show that f there s any weght n the mxed strategy n the bottom, the total utlty would be less than 2. So, f the total utlty s at least 2, all the weght s n the top left submatrx. We then note that f even one strategy s mssng, we get a total utlty whch s less than 2. In order to complete the proof we show that f there are less than k chosen strateges, the weght of at least one chosen strategy s at least 1 k, and look at what we can do on the bottom left submatrx. 3 Correlated equlbrum We start by gvng the game of Chcken as an example. Consder a head-to-head race between to players. In ths game two ndvduals are challengng each other to a contest 7-3
Table 2: The game of Chcken Chcken Dare Chcken (8,8) (1,9) Dare (9,1) (0,0) where each can ether dare or chcken out. If one s gong to Dare, t s better for the other to chcken out. But f one s gong to chcken out t s better for the other to Dare. Ths leads to an nterestng stuaton where each wants to dare, but only f the other mght chcken out. In ths game, there are three Nash equlbra. The two pure strategy Nash equlbra are (D, C) and (C, D). In order to fnd the mxed strategy equlbrum we denote by p the probablty of dare. In the equlbrum we have 8p + 1(1 p) = 9p, so p = 0.5. In ths case we have u 1 = 4.5. However, both players could get (8, 8) f they both chcken out. Now consder a trusted thrd party that draws one of three cards labeled: (C, C), (D, C), and (C, D). He draws each of the cards wth equal probablty. After drawng the card the thrd party nforms the players of the strategy assgned to them on the card but not the strategy assgned to ther opponent. We show that lstenng to the trusted thrd party s an equlbrum. Suppose a player s assgned D, he would not want to devate supposng the other player played ther assgned strategy snce he wll get 9 (the hghest payoff possble). Suppose a player s assgned C. Then the other player wll play C wth probablty 0.5 and D wth probablty 0.5. The expected utlty of Darng s 0.5 0 + 0.5 9 = 4.5. The expected utlty of chckenng out s 0.5 8 + 0.5 1 = 4.5. So, there s not ncentve not to do as the thrd party has nstructed. Snce nether player has an ncentve to devate, ths s a correlated equlbrum. Interestngly, the expected payoff for ths equlbrum s 1 3 8 + 2 3 5 = 6 whch s hgher than the expected payoff of the mxed strategy Nash equlbrum. Let us denote the the probablty the trusted thrd party chooses (C, C), (C, D), (D, C), (D, D) by a, b, c, d. The condton for a correlated equlbrum s that gven the trusted thrd party s nformaton, each player has no ncentve to devate. c Thus, we get several equatons. 9 c+d + 0 d c+d 8 c c+d + 1 d c+d and we get b a. Smlarly a we get c a. On the other sde we get 8 a+b + 1 b a+b 9 a a+b + 0 b a+b and we get b a, and smlarly b d. So, the neccessary and suffcent condtons for a correlated equlbrum n ths game are: a + b + c + d = 1, a, b, c, d 0, b d, c a, c d, b a. Snce all the condtons are lnear, t s easy to fnd the optmal probabltes for the trusted 7-4
thrd party, whch maxmze the expected socal welfare, by solvng a lnear program. In the next lesson we would gve the formal defnton of correlated equlbrum. 7-5