Introduction to the Theory of Games by J. Szép, F. Forgó (auth.)

By J. Szép, F. Forgó (auth.)

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It is easy to see that if both players apply their equilibrium strategies, then the game arrives at the terminal vertex with pay-off vector (0,0). (The path leading from the root to this terminal vertex is marked by thick lines). Suppose now that player I has forgotten his and player II's first move by the time he is to make his second move. Thus, he does not know at which of the encircled nodes of the tree he is. (See Figure 2. ) Now player I has two alternatives: he either goes to the left or to the right.

Our proof consists of three steps. (a) Let {F~)}~t be a sequence of elements from Lk • We first show that there exists a subsequence {FfJ)} and a distribution function F10 ) ELk such that at every point of continuity of F~O), F(iJ)(u)~ F10)(u). Denote r t , r 2' . the sequence of rational numbers in the interval [a k , bk ]. Since for any 1= 1, 2, ... the sequence {F~)(r,)} is bounded, there exists a convergent subsequence {F~~I) (r,)} for each I. Furthermore there is a subsequence {F~Y)} converging for any rational number r,.

O'l(I:)n" ••• , . ) for i = 1, ... , s; k = 1, ... , m. , the strategies with superscript. are briefly denoted by •. ) If the player assigned to root I is "chance", then strategies m O't = U O't(I:) , (i= 1, ... , n) obviously form an equilibrium point in r. 1:=1 Thus we have only to consider the case when a "real" player belongs to I. Let this player be io . His strategy set Eio in game r is the following: U ... m U [E~:) u {choose the 1st edge emanating from I} k=1 u {choose the mth edge emanating from I}].

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