Optimization and Games for Controllable Markov Chains by Julio B. Clempner & Alexander Poznyak

Author:Julio B. Clempner & Alexander Poznyak
Language: eng
Format: epub
ISBN: 9783031435751
Publisher: Springer Nature Switzerland

(8.5.3)

The process continues until for each l. At the end, we obtain the estimated transition matrices , the estimated values of . The resulting strategy profile is a Bayesian-Nash equilibrium. The resulting mechanism is incentive-compatible because it is in equilibrium and satisfies Eq. (8.3.1).

8.6 Nash Equilibrium as a Solution of a Max-Min Problem

We consider the Nash equilibrium problem [21, 22]. To this end, we first recall the definition of the (standard) Nash equilibrium problem.

We consider a game played by a set players indexed by . Each player control the variable . Let us consider a game whose strategies are denoted by where is a convex and compact set, and . Let , the vector formed by all these decision variables, be the joint strategy of the players and be a strategy of the rest of the players adjoint to . To emphasize the l th player’s variables within the vector v, we sometimes write where where subsumes all the other players’ variables. We consider a Nash equilibrium problem with n players and denote by the vector representing the v-th player’s strategy where and , such that .

Let us consider be the l-th player’s reward function (cost function). We assume that these reward functions are at least continuous, and we further assume that the functions are concave in the variable . The players try to reach one of the non-cooperative equilibrium, that is, they try to find a strategy satisfying for any and any that

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