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By Wayne F.Bialas

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His nodes are partitioned among two information sets {η21 , η22 }. When Player 2 chooses his action, he only knows which information set he is in, not which node. Player 1 could analyze the game as follows: • If Player 1 chooses u1 = L then Player 2 would respond with u2 = L resulting in a payoff of (0, 1). • If Player 1 chooses u1 ∈ {M, R} then the players are really playing the 3-8 following subgame: η11 Player 1 L M η12 R η 22 L (0,1) R L (2,-1) Player 2 R (-3,-2) (0,-3) R L (-2,-1) (1,0) which can be expressed in normal form as M R L (-3,-2) (-2,-1) R (0,-3) (1,0) in which (R, R) is a Nash equilibrium strategy in pure strategies.

1 then uses the expression y¯ni i = yni i + cni i 1 + j∈Mi cij Note that the denominator is the sum (taken over ni ) of the terms in the numerator. If all of the cij vanish, we get y¯ni i = yni i . 1. Every N -person finite game in normal (strategic) form has a Nash equilibrium solution using mixed strategies. 3-3 Proof: Define ψni i and cni i , as above. Consider the expression y¯ni i = (1) yni i + cini 1 + j∈Mi cij We will try to find solutions yni i to Equation 1 such that y¯ni i = yni i ∀ ni ∈ Mi and ∀ i = 1, .

A specific vertex indicating the starting point of the game. 2. N cost functions each assigning a real number to each terminating node of the graph. The ith cost function represents the gain to Player i if that node is reached. 3. a partition of the nodes among the N players. 4. a sub-partition of the nodes assigned to Player i into information sets {ηki }. The number of branches emanating from each node of a given information 3-12 set is the same, and no node follows another node in the same information set.

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Lecture Note Set1. Game Theory by Wayne F.Bialas

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