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Extra info for Game Theory: Mathematical Models of Conflict
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10 (The binary game). In this game A and B alternately choose binary digits 0 or 1. A complete play of the game consists of a countable number of such moves or choices. Thus each play consists of an infinite sequence s(1), s(2), s(3),... of digits 0 or 1 which determines a real number s = I: s(n). 2n Player A wins one pound from B if s belongs to a certain subset U of the unit interval, while B wins one pound from A if s is not in U. 7 An alternative way to visualise this game is as an 'intervals game'.
6 2 x n AND SYMMETRIC GAMES. Our primary concern with a matrix game is the set of optimal strategies for each player. The von Neumann minimax theorem tells us that these sets are non-empty. 15) discloses that each player possesses at least one optimal strategy which has at most minIm, n] non-zero components. This fact can be quite useful. In particular for a 2 x n (or m x 2) game it tells us that if we solve every 2 x 2 subgame, that is, study each 2 x 2 submatrix of the original matrix, then at least one of these solutions will correspond to a solution of the original game.
Case 1. Xo is labelled N. , A r, 0 :5 Aj :5 1, kAj = 1, denote the probabilities for the vertices ai' ... , a r to be selected. Let x be any vertex of f. If x = Xo we do not need to define aiO(x), nor do we need to define it if x is any other vertex labelled with an N. Otherwise x E f j for some j and is labelled with an i, 1 :5 i :5 n. We then define °(x) aj For any pure strategies plainly have ('t l , ... , 't n) ° = aij(x). of I' we denote the restriction of Tj to f j by Tj I fj' We r L P;Ctl, ..