The projective core of symmetric games with externalities
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The projective core of symmetric games with externalities Takaaki Abe1 · Yukihiko Funaki1 Accepted: 11 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The purpose of this paper is to study which coalition structures have stable distributions. We employ the projective core as a stability concept. Although the projective core is often defined only for the grand coalition, we define it for every coalition structure. We apply the core notion to a variety of economic models including the public goods game, the Cournot and Bertrand competition, and the common pool resource game. We use a partition function to formulate these models. We argue that symmetry is a common property of these models in terms of a partition function. We offer some general results that hold for all symmetric partition function form games and discuss their implications in the economic models. We also provide necessary and sufficient conditions for the projective core of the models to be nonempty. In addition, we show that our results hold even in the presence of small perturbations of symmetry. Keywords Core · Externalities · Oligopoly · Public goods
1 Introduction Most economic/political situations include both competition and cooperation among players. Although competition and cooperation influence each other, they are often separately analyzed in two different models: noncooperative game theory and cooperative game theory. While dividing a situation into the two different models enables us to focus on their respective specialties, it eliminates rich interactions between cooperation and competition. Thrall (1961) and Thrall and Lucas (1963) are early attempts
Takaaki Abe gratefully acknowledges the financial support from JSPS Grant-in-Aid for Research Activity Start-up (No.19K23206) and Waseda Univeristy Grant-in-Aid for Research Base Creation (2019C-486).
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Takaaki Abe [email protected] Yukihiko Funaki [email protected]
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School of Political Science and Economics, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-Ku, Tokyo 169-8050, Japan
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T. Abe, Y. Funaki
to address this problem. These authors introduced partition function form games, also known as games with externalities, to describe the competition among coalitions and cooperation within each coalition. In partition function form games, the worth of a coalition depends on both the coalition and the coalition structure of the other coalitions. This complexity causes a problem when we generalize one of the solution concepts for cooperative games to games with externalities: the core. The core is a set of payoff allocations from which no coalition has an incentive to deviate, where a deviation means that a group of players forms their own coalition and splits off from an allocation to improve the payoff of the deviating coalition. In order to determine the worth of the deviating coalition, one must also know the coalition structure that the non-deviating players form after the deviation. The literature handles this difficulty by various
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