# On the non-emptiness of the fuzzy core

## Author Info

• Nizar Allouch

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## Abstract

The seminal contribution of Debreu-Scarf [4] connects the two concepts of core and competitive equilibrium in exchange economies. In effect, their core-equilibrium equivalence result states that, when the set of economic agents is replicated, the set of core allocations of the replica economy shrinks to the set of competitive allocations. Florenzano [6] defines the fuzzy core as the set of allocations which cannot be blocked by any coalition with an arbitrary rate of participation and then shows the asymptotic limit of cores of replica economies coincides with the fuzzy core. In this note, we provide an elementary proof of the non-emptiness of the fuzzy core for an exchange economy. Unlike the classical Debreu-Scarf limit theorem [4] and its numerous extensions our result does not require any asymptotic intersection -or limit- of the set of core allocations of replica economies.

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## Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 37 (2008)
Issue (Month): 2 (June)
Pages: 203-210

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Handle: RePEc:spr:jogath:v:37:y:2008:i:2:p:203-210

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## Related research

Keywords: Fuzzy core; Payoff-dependent balancedness; Exchange economies; D51; C71;

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Find related papers by JEL classification:
• D51 - Microeconomics - - General Equilibrium and Disequilibrium - - - Exchange and Production Economies
• C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

## References

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1. Jean-Marc Bonnisseau & Vincent Iehle, 2004. "Payoffs-dependent Balancedness and Cores," Game Theory and Information 0403004, EconWPA.
2. Aliprantis, Charalambos D. & Brown, Donald J. & Burkinshaw, Owen, 1987. "Edgeworth equilibria in production economies," Journal of Economic Theory, Elsevier, vol. 43(2), pages 252-291, December.
3. Allouch,Nizar & Predtetchinski,Arkadi, 2005. "On the non-emptiness of the fuzzy core," Research Memorandum 002, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
4. Shapley, Lloyd & Vohra, Rajiv, 1991. "On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game," Economic Theory, Springer, vol. 1(1), pages 108-16, January.
5. Florenzano Monique, 1988. "Edgeworth equilibria, fuzzy core and equilibria of a production economy without ordered preferences," CEPREMAP Working Papers (Couverture Orange) 8822, CEPREMAP.
6. Donald J. Brown & Charalambos Aliprantis & Owen Burkinshaw, 1985. "Edgeworth Equilibria," Cowles Foundation Discussion Papers 756R, Cowles Foundation for Research in Economics, Yale University.
• Aliprantis, Charalambos D & Brown, Donald J & Burkinshaw, Owen, 1987. "Edgeworth Equilibria," Econometrica, Econometric Society, vol. 55(5), pages 1109-37, September.
7. Arkadi Predtetchinski, 2005. "The fuzzy core and the $(\Pi, \beta)$ -balanced core," Economic Theory, Springer, vol. 26(3), pages 717-724, October.
8. Predtetchinski, Arkadi & Jean-Jacques Herings, P., 2004. "A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game," Journal of Economic Theory, Elsevier, vol. 116(1), pages 84-92, May.
9. Predtetchinski,Arkadi, 2004. "The Fuzzy Core and the (Π, β)- Balanced Core," Research Memorandum 025, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
10. Herves-Beloso, Carlos & Moreno-Garcia, Emma & Yannelis, Nicholas C., 2005. "An equivalence theorem for a differential information economy," Journal of Mathematical Economics, Elsevier, vol. 41(7), pages 844-856, November.
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## Citations

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Cited by:
1. Nizar Allouch & Arkadi Predtetchinski, 2005. "On the Non-emptiness of the Fuzzy Core," Working Papers 531, Queen Mary, University of London, School of Economics and Finance.
2. Liu, Jiuqiang & Liu, Xiaodong, 2013. "A necessary and sufficient condition for an NTU fuzzy game to have a non-empty fuzzy core," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 150-156.

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