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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.

A Large Population Partnership Formation Game with Associative Preferences and Continuous Time

Ramsey D. M.

Mathematica Applicanda

2018

Vol. 46, no. 2

171--195

A model of partnership formation is considered in which there are two classes of player (called male and female). There is a continuum of players and two types of both sexes. These two types can be interpreted, e.g. as two subspecies, and each searcher prefers to pair with an individual of the same type. Players Begin searching at time zero and search until they find a mutually acceptable prospective partner or the mating season ends. When a pair is formed, both individuals leave the pool of searchers. Hence, the proportion of players still searching and the distribution of types changes over time. Prospective partners are found at a rate which is nondecreasing in the proportion of players still searching. Nash equilibria are derived which satisfy a refinement based on the following optimality criterion: each searcher accepts a prospective partner if and only if the reward that would be gained from such a partnership (given that it formed) is greater or equal to the expected reward obtained by that searcher from future search. So called "completely symmetric" versions of this game are considered, where the two types of player are equally frequent. In this class of games, there exists a unique Nash equilibrium satisfying the optimality criterion, regardless of the precise rule determining the rate at chich prospective partners are found. This equilibrium is given by a threshold time t0, such that before time t0 individuals only mate with prospective partners of the same type and from time t0 onwards each searcher accepts any prospective partner. Two examples are given. One example considers the so called "singles bar" model, according to which prospective partners are found at a constant rate. The secondo example considers the "mixing population" model, according to which the rate at which prospective partners are found is proportional to the fraction of individuals who are still searching for a partner.