Some theorems on the core of an n-Game without Side-Payments
01 May 1970-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 18, Iss: 3, pp 567-579
About: This article is published in Siam Journal on Applied Mathematics.The article was published on 1970-05-01. It has received 82 citations till now. The article focuses on the topics: Core (game theory).
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TL;DR: In this paper, an economic model of trading in commodities that are inherently indivisible, like houses, is investigated from a game-theoretic point of view, and the concepts of balanced game and core are developed, and a general theorem of Scarf's is applied to prove that the market in question has a nonempty core, that is, at least one outcome that no subset of traders can improve upon.
1,232 citations
TL;DR: In this paper, the authors derived versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players.
Abstract: We derive versions of known results dealing with core, equilibria and Shapley-values of cooperative games in the case of cooperative fuzzy games, i.e., games defined on fuzzy subsets of the set of n players. A fuzzy coalition is an n-vector τ = τi associating with each player i his “rate of participation” τi ∈ [0, 1] in the fuzzy coalition and the real number v· is the coalition worth, assumed to be positively homogeneous. If also v is superadditive, or equivalently concave, the fuzzy game with side payments has a nonempty core. Associated with the coalition-worth function for any ordinary game with side payments is a fuzzy extension thereof, viz., the fuzzy game whose coalition-worth function is the least positively-homogeneous superadditive function majorizing the coalition-worth function of the original game. The fuzzy extension always has a nonempty core. Moreover, if the original game has a nonempty core, it coincides with that of its fuzzy extension. Analogous results are established for games without side payments. An axiomatization of “values” of fuzzy games with side payments is also given. The results are applied to show that the set of Walras equilibria coincides with the fuzzy core of an economy.
324 citations
TL;DR: A generalization of assignment games, called partitioning games, is introduced and necessary and sufficient conditions for the nonemptiness of the cores of all games with essential coalitions π are developed.
176 citations
01 Sep 1972
TL;DR: In this article, a new proof of a basic theorem of game theory, because of Scarf, that every balanced game without side payments has a nonempty core is presented, and the main tool is a new generalization of Sperner's celebrated topological lemma concerning triangulations of the simplex.
Abstract: Publisher Summary This chapter presents a new proof of a basic theorem of game theory, because of Scarf, that states that every balanced game without side payments has a nonempty core. The main tool is a new generalization of Sperner's celebrated topological lemma concerning triangulations of the simplex, which will be of independent interest. The chapter describes balanced sets and provides the simple but very useful geometric characterization of these sets. It also describes iterated barycentric partitions.
146 citations
Cites background from "Some theorems on the core of an n-G..."
...**Scarf (1967a); see also Billera (1970, 1971). Sperner (1928); also Knaster, Kuratowski, and Mazurkiewicz (1926) ....
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...**Scarf (1967a); see also Billera (1970, 1971). Sperner (1928); also Knaster, Kuratowski, and Mazurkiewicz (1926) . *Lemke and Howson (1964) ; see also Cohen (1967) , Scarf (1967b), and Kuhn (1968, 1969)....
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TL;DR: In this article, sufficient conditions are given for large replica games without side payments to have non-empty approximate cores for all sufficiently large replications, where the conditions are superadditivity, boundedness condition, and convexity of the payoff sets.
116 citations
Cites background from "Some theorems on the core of an n-G..."
...In particular, some variations of the concept of balancedness have been studied and shown to ensure nonemptiness of the core; cf. Billera (1970, 1971)....
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