Now we introduce the vectors u and v whose components are u j y j x T Ay and v

# Now we introduce the vectors u and v whose components

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Now we introduce the vectors u and v , whose components are u j = y j x T Ay , and v i = x i x T By , respectively. We can use this substitution to derive the inequalities i b i,j v i 1 j j a i,j u j 1 i, where u j > 0 implies that i b i,j v i = 1, and v i > 0 implies that j a i,j u j = 1. We can now add two vectors of slack variables , r and t , and transform these inequalities into equalities, i.e., { B T v + t = 1 n , t 0 Au + r = 1 m , r 0 , where 1 k denotes the k -vector whose components are all equal to 1. Now we have to add the constraint that characterizes the support of a NE in the mixed strategies in terms of best response. Such property translates into an orthogonality condition, i.e., r · v = t · u = 0, which expresses the fact that if r i > 0, i.e., if j a ij y j < xAy , then x i = 0, otherwise x is not a best response. The same argument shows the other orthogonality requirement. Note that the pair of stochastic vectors which constitute the NE can be recovered by normalizing u and v . This process leads to the Linear Complementarity formulation of the NE conditions (LC-NASH): Let w be the ( n + m )-vector obtained concatenating v and u , and z be the ( n + m )-vector obtained concate- nating t and r . LC-NASH : Find nonnegative ( n + m )-vectors w and z such that Hw + z = 1 ( n + m ) w T z = 0 , 2 We can add to all the entries of both payoff matrices a large positive constant without changing the nature of the game under investigation.
4-8 Lecture 4: Nash Equilibria where H = ( 0 A B T 0 ) . 4.5 Nash Equilibria for Multiplayer Games The definition of NE can be immediately extended to multi-player games. Consider a k -player game, and let u j ( s 1 , s 2 , . . . , s k ) denote the payoff to player j , when the players use strategies s 1 , . . . , s k ( s i is the strategy of player i , i = 1 , . . . , k ). A strategy profile u = ( u 1 , u 2 , . . . , u k ) can be denoted by ( u i , u i ), i ∈ { 1 , 2 , . . . , k } , where u i = ( u 1 , u 2 , . . . , u i 1 , u i +1 , . . . , u k ) . Definition 4.18 (NE for k -player games) . An NE is a strategy profile x such that no player can increase her payoff by switching her strategy unilaterally. Formally, for each player i , and for every mixed strategy y i , u i ( x ) u i ( x i , y i ) . Consider games where the payoffs are rational numbers. A major difference between two- and multi-player games is that if the number of players is at least three, then all the NE can be irrational numbers. On the contrary, all the NE in two-player games are rational. Example 4.19 (Three-player game with irrational NE) . Let a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , be the strategies available to player 1, 2, and 3, respectively. Consider the payoff matrices b 1 b 2 a 1 (0,0,1) (1,0,0) a 2 (1,1,0) (2,0,8) b 1 b 2 a 1 (2,0,9) (0,1,1) a 2 (0,1,1) (1,0,0) where the matrix on the left gives the payoffs when player 3 chooses c 1 , and the matrix on the right gives the payoffs when player 3 chooses c 2 . Each entry of the matrices is a triple, whose i -th component is the payoff to player i , i = 1 , 2 , 3 . This game has a unique NE given by ( p, 1 p ) , ( q, 1 q ) , ( r, 1 r ) , where p = 30 2 51 29 , q = 2 51 6 21 , and r = 9 51 12 .