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1

Common-Knowledge and Bayesian Equilibrium in Network Game

Matsuhisa T.

Mathematica Applicanda

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2018

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Vol. 46, no. 2

211--243

EN

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).

PL

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.

2

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

Ramsey D. M.

Mathematica Applicanda

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2018

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Vol. 46, no. 2

171--195

EN

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.

PL

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.

3

A secretary problem with missing observations

Ramsey D. M.

Mathematica Applicanda

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2016

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Vol. 44, No. 1

149--165

EN

Two versions of a best choice problem in which an employer views a sequence of N applicants are considered. The employer can hire at most one applicant. Each applicant is available for interview (and, equivalently, for employment) with some probability p. The available applicants are interviewed in the order that they are observed and the availability of the i-th applicant is ascertained before the employer can observe the (i + 1)-th applicant. The employer can rank an available applicant with respect to previously interviewed applicants. The employer has no information on the value of applicants who are unavailable for interview. Applicants appear in a random order. An employer can only offer a position to an applicant directly after the interview. If an available applicant is offered the position, then he will be hired. In the first version of the problem, the goal of the employer is to obtain the best of all the applicants. The form of the optimal strategy is derived. In the second version of the problem, the goal of the employer to obtain the best of the available applicants. It is proposed that the optimal strategy for this second version is of the same form as the form of the optimal strategy for the first version. Examples and the results of numerical calculations are given.

PL

Artykuł rozważa dwie wersje problemu wyboru najlepszego obiektu, w których pracodawca obserwuje sekwencyjnie N pracobiorców. Każdy pracobiorca jest wolny z prawdopodobieństwem p, a gdy nie jest wolny pracodawca nie może ani prowadzić rozmowy z nim, ani go zatrudnić. Rozmowa o pracy z i-tym pracobiorcą (o ile ma miejsce) jest przeprowadzana zanim pojawi się (i+1)-szy pracobiorca. Pracodawca może tylko porównać pracobiorców, czyli sporządzić ranking tych, z którymi już prowadził rozmowę. Nie posiada on żadnych informacji dotyczących wartości pracobiorców, którzy nie są wolni. Pracobiorcy pojawiają się w losowej kolejności. Pracodawca może zatrudnić tylko jednego pracobiorcę i to tylko zaraz po rozmowie z nim. Wolny pracobiorca zawsze przyjmuje ofertę pracy. Według pierwszego modelu, celem pracodawcy jest zatrudnienie najlepszego z wszystkich pracobiorców. Wyznaczono postać optymalnej strategii. Według drugiego modelu, celem pracodawcy jest zatrudnienie najlepszego z wolnych pracobiorców. Podane jest uzasadnienie, iż postać optymalnej strategii jest taka sama jak przy pierwszym modelu. Rozważono parę przykładów i podano wyniki obliczeń numerycznych.