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Common-Knowledge and Bayesian Equilibrium in Network Game

Matsuhisa T.

Mathematica Applicanda

2018

Vol. 46, no. 2

211--243

In this paper we investigate equilibriums in the Bayesian routing problem of the network game introduced by Koutsoupias and Papadimitriou (1999).We treat epistemic conditions for Nash equilibrium of social costs function in the network game. It highlights the role of common-knowledge on the users' individual conjectures on the others' selections of channels in the network game. Especially two notions of equilibria are presented in the Bayesian extension of the network game; expected delay equilibrium and rational expectations equilibrium. The former equilibrium is given such as each user minimizes own expectations of delay, and the latter is given as he/she maximizes own expectations of a social costs. We show that the equilibria have the properties: If all users commonly know them, then the former equilibrium yields a Nash equilibrium in the based KP-model and the latter equilibrium yields a Nash equilibrium for social costs in the network game. Further we introduce the extended notions of price of anarchy in the Bayesian network game for rational expectations equilibriums for social costs, named the expected price of anarchy and the common-knowledge price of anarchy. We will examine the relationship among the two extended price of anarchy and the classical notion of price of anarchy introduced by Koutsoupias and Papadimitriou(1999).

Rozważane będą modele matematyczne, w których zmiana parametru jest przedmiotem badań statystycznych. Specjalizowanym narzędziem do tego celu są karty kontrolne. Celem pracy jest konstrukcja kart kontrolnych do badania zmian parametru rozkładu obserwowanej cechy w oparciu o dokładne rozkłady różnych estymatorów parametrów kontrolowanych wielkości i ich porównanie.

Communication Leading to Subgroup Nash Equilibrium for Generalized Information

Matsuhisa T.

Fundamenta Informaticae

2016

Vol. 145, nr 3

323--340

This paper treats subgroup Nash equilibriums which concept is given as an extension of Nash equilibrium of a strategic game with non-partitional information, and addresses the problem how to reach the equilibrium by communication through messages according to network among players. A subgroup Nash equilibrium of a strategic game consists of (1) a subset S of players, (2) independent mixed strategies for each member of S together with (3) the conjecture of the actions for the other players outside S provided that each member of S maximizes his/her expected payoff according to the product of all mixed strategies for S and the conjecture about other players' actions. Suppose that the players have a reflexive and transitive informationwith a common prior distribution, and that each player in a subgroup S predicts the other players' actions as the posterior of the others' actions given his/her information. He/she communicates privately his/her belief about the other players' actions through messages to the recipient in S according to the communication network in S.We show that in the pre-play communication according to the revision process of their predictions about the other players' actions, their future predictions converges to a subgroup Nash equilibrium of the game in the long run.