Consistency, anonymity, and the core on the domain of convex games
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Consistency, anonymity, and the core on the domain of convex games Toru Hokari1
· Yukihiko Funaki2 · Peter Sudhölter3
Received: 11 October 2017 / Accepted: 24 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that neither Peleg’s nor Tadenuma’s well-known axiomatizations of the core by non-emptiness, individual rationality, super-additivity, and max consistency or complement consistency, respectively, hold when only convex rather than balanced TU games are considered, even if anonymity is required in addition. Moreover, we show that the core and its relative interior are the only two solutions that satisfy Peleg’s axioms together with anonymity and converse max consistency on the domain of convex games. JEL Classification C71
1 Introduction The core is one of the most important solutions for cooperative games. It is important mainly because it satisfies many desirable properties. In particular, it satisfies two kinds of reduced game properties, namely, “max consistency” (Peleg 1986 Davis and Maschler 1965) and “complement consistency” (Tadenuma 1992; Moulin 1985).1 There are two well-known axiomatic characterizations of the core on the domain of 1 For these two consistency axioms we use the terminology introduced by Thomson (1996) and call them max consistency and complement consistency because each name suggests how the underlying “reduced games” are defined in each case.
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Toru Hokari [email protected] Yukihiko Funaki [email protected] Peter Sudhölter [email protected]
1
Faculty of Economics, Keio University, Tokyo, Japan
2
School of Political Science and Economics, Waseda University, Tokyo, Japan
3
Department of Business and Economics, University of Southern Denmark, Odense, Denmark
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balanced TU games based on each of these two axioms: (i) The core is the unique solution that satisfies non-emptiness, individual rationality, super-additivity, and max consistency (Peleg 1986); (ii) it is the unique solution that satisfies non-emptiness, individual rationality and complement consistency (Tadenuma 1992).2 In this note, we investigate what happens when the domain is restricted to the domain of convex TU games. Although the core satisfies Peleg’s four axioms on this domain, it is not the only one.3 It so happens that except for the core itself, all known examples of such solutions violate anonymity. So, one may conjecture that an axiomatic characterization of the core might be obtained by adding anonymity to Peleg’s four axioms. In this note, we disprove this conjecture. Moreover, we show that there exist only two solutions, the core and its relative interior, that satisfy Peleg’s four axioms together with anonymity and converse max consistency. We also consider a similar problem for complement consistency. In particular, we show that the core is not the only solution on the domain of convex games that satisfies Tadenuma’s three axioms and anonymity.
2 Definitions and results Let U be an arbitrary universe of at least three players, which is
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