Alternative version of Shapley’s theorem on closed coverings of a simplex

Ichiishi, Tatsuro

doi:10.1090/s0002-9939-1988-0964853-5

Cited by 22 publications

(18 citation statements)

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“…The following intersection theorem on the unit simplex is due to Ichiishi [6] and can be considered as the dual of the Shapley Lemma.…”

Section: Qedmentioning

confidence: 99%

“…The reformulation of this theorem given by Fan [1] and Freidenfelds [2] can also be found in Scarf [14] and is known as Scarf's lemma. Further generalizations of intersection theorems on the unit simplex can be found in Scarf (13], Shapley [15], Gale [5], Freund [4], Ichiishi [6] and Joosten and Talman [S]. bToreover, generalizations of intersection theorems to thf, cube or simplotope are stated in Freund [4], van der Laan, Talman and Van der llcyden [I I], van dcr l,aan and Talman [12] and Talman [17].…”

Section: Introductionmentioning

confidence: 99%

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Intersection theorems on polytopes

1999

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Intersection theorems on polytopesvan der Laan, G.; Talman, Dolf; Yang, Z. Publication date: 1994 Link to publication Citation for published version (APA):van der Laan, G., Talman, A. J. J., & Yang, Z. (1994). Intersection theorems on polytopes. (CentER Discussion Paper; Vol. 1994-20). Unknown Publisher. General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.-Users may download and print one copy of any publication from the public portal for the purpose of private study or research -You may not further distribute the material or use it for any profit-making activity or commercial gain -You may freely distribute the URL identifying the publication in the public portal Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediately and investigate your claim. AbstractIntersection theorcros are used to prove the existence of solutions to mathematical programming and game theoretic problems. Well-known intersection theorems are thc theorerns of Sperner, Knaster-Kuratowski-Mazurkiewicz (KKM), Scarf, Shapley, Ichiishi and Gale on the unit simplex. Recently the intersection result of KKM has been generalized by Ichiishi and Idzik to closed coverings of a compact convex polyhedron, called a pulytopc. In this paper we formulate a general intersection theorem on the polytope. To do so, we need to generalize the concept of balancedness as is used by Shapley and Ichiishi. The theorem implies most of the results stated above as special cases. First, we show that the theorems of KKM, Sperner, Scarf, Shapley and Ichiishi on the unit simplex and also some theorems of Ichiishi and Idzik on a polytope all satisfy t.he conditions of our theorem on the polytope. Secondly, the general thcurem allows us to formulate the analogs of these theorems on the polytope.

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“…The following intersection theorem on the unit simplex is due to Ichiishi [6] and can be considered as the dual of the Shapley Lemma.…”

Section: Qedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Intersection theorems on polytopes

1999

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“…In order to prove the intersection theorems, a non-constructive proof of the existence of a connected set of constrained equilibria containing both trivial constrained equilibria is given using Browder's fixed point theorem. The intersection theorems of Chapter 7 generalize the well-known results on the unit simplex given in Knaster, Kuratowski, and Mazurkiewicz (1929) (KKM Lemma), in Sperner (1928) and Scarf (1967) (Sperner Lemma), in Shapley (1973) (KKMS Lemma), and in Ichiishi (1988) (Ichiishi Lemma). Moreover, the results can be used to sharpen the usual formulation of the Sperner Lemma on the unit cube.…”

Section: Introductionmentioning

confidence: 52%

“…The KKM Lemma and the Sperner Lemma can be used to prove Brouwer's fixed point theorem and also to show the existence of a Walrasian equilibrium of an economy. Both the KKMS Lemma and the Ichiishi Lemma are very useful when showing the non-emptiness of the core of a cooperative game, see Shapley (1973), Ichiishi (1988), and Shapley and Vohra (1991). The Gale Lemma is used in Gale (1984) to show the existence of a Walrasian equilibrium in an economy with indivisible commodities.…”

Section: Introductionmentioning

confidence: 99%

Static and Dynamic Aspects of General Disequilibrium Theory

Herings

1996

Theory and Decision Library

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“…It will be shown that a socially stable game has a non-empty socially stable core. To do so, we formulate an intersection theorem on the unit simplex that generalizes the well-known intersection theorem used by Shapley (1973) (see also Herings, 1997;Ichiishi, 1988;van der Laan et al, 1999). Since socially stable games have a non-empty socially stable core, they also have a non-empty core.…”

Section: Introductionmentioning

confidence: 99%

Socially Structured Games

2006

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ABSTRACT. We generalize the concept of a cooperative non-transferable utility game by introducing a socially structured game. In a socially structured game every coalition of players can organize themselves according to one or more internal organizations to generate payoffs. Each admissible internal organization on a coalition yields a set of payoffs attainable by the members of this coalition. The strengths of the players within an internal organization depend on the structure of the internal organization and are represented by an exogenously given power vector. More powerful players have the power to take away payoffs of the less powerful players as long as those latter players are not able to guarantee their payoffs by forming a different internal organization within some coalition in which they have more power.We introduce the socially stable core as a solution concept that contains those payoffs that are both stable in an economic sense, i.e., belong to the core of the underlying cooperative game, and stable in a social sense, i.e., payoffs are sustained by a collection of internal organizations of coalitions for which power is distributed over all players in a balanced way. The socially stable core is a subset and therefore a refinement of the core. We show by means of examples that in many cases the socially stable core is a very small subset of the core.We will state conditions for which the socially stable core is non-empty. In order to derive this result, we formulate a new intersection theorem that generalizes the KKMS intersection theorem. We also discuss the relationship between social stability and the wellknown concept of balancedness for NTU-games, a sufficient condition for non-emptiness of the core. In particular we give an example of a socially structured game that satisfies social stability and therefore has a non-empty core, but whose induced NTU-game does not satisfy balancedness in the general sense of Billera.KEY WORDS: balancedness, core, non-transferable utility game, social stability.JEL CLASSIFICATION: C71.

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Alternative version of Shapley’s theorem on closed coverings of a simplex

Cited by 22 publications

References 14 publications

Intersection theorems on polytopes

Intersection theorems on polytopes

Static and Dynamic Aspects of General Disequilibrium Theory

Socially Structured Games

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