By O.J. Vrieze
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Extra info for Stochastic games with finite state and action spaces
Any combination of optimal strategies also represents a Nash equilibrium. 2 Suppose that (x1 , y1 ) and (x2 , y2 ) are Nash equilibria in a zero-sum game. Then (x1 , y2 ) and (x1 , y2 ) is also a Nash equilibrium. 5) H(x, y2 ) ≤ H(x2 , y2 ) ≤ H(x2 , y). 6). This generates a chain of inequalities with the same quantity H(x2 , y1 ) in their left- and right-hand sides. 6) appear strict equalities. And (x1 , y2 ) becomes a Nash equilibrium, since for any (x, y): H(x, y2 ) ≤ H(x2 , y2 ) = H(x1 , y2 ) = H(x1 , y1 ) ≤ H(x1 , y).
X > y), both sides go to an arbitration court. The latter must support a certain player. There exist various arbitration procedures, namely, final-offer arbitration, conventional arbitration, bonus/penalty arbitration, as well as their combinations. ZERO-SUM GAMES 43 We begin analysis with final-offer arbitration. Without a conflict (if x ≤ y), this procedure leads to successful raise negotiation in the interval between x and y. For the sake of definiteness, suppose that the negotiated raise makes up (x + y)∕2.
His opponent performs similar actions. If the forces of a player exceed those of the opponent at a given pass, then his payoff equals unity (and vanishes otherwise). Furthermore, at a certain pass Colonel Blotto’s opponent has already concentrated additional forces of size 1∕2. 1 The Colonel Blotto game. 2 The payoff function of player I. Therefore, we face a constant-sum game Γ =< I, II, X, Y, H >, where X = [0, 1], Y = [0, 1] indicate the strategy sets of players I and II. Suppose that Colonel Blotto and his opponent have allocated their forces (x, 1 − x) and (y, 1 − y) between the passes.
Stochastic games with finite state and action spaces by O.J. Vrieze