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# On Concavification and Convex Games

## Author

Listed:
• Yaron Azrieli

(Tel Aviv University)

• Ehud Lehrer

(Tel Aviv University)

## Abstract

We propose a new geometric approach for the analysis of cooperative games. A cooperative game is viewed as a real valued function $u$ defined on a finite set of points in the unit simplex. We define the \emph{concavification} of $u$ on the simplex as the minimal concave function on the simplex which is greater than or equal to $u$. The concavification of $u$ induces a game which is the \emph{totally balanced cover} of the game. The concavification of $u$ is used to characterize well-known classes of games, such as balanced, totally balanced, exact and convex games. As a consequence of the analysis it turns out that a game is convex if and only if each one of its sub-games is exact.

## Suggested Citation

• Yaron Azrieli & Ehud Lehrer, 2004. "On Concavification and Convex Games," Game Theory and Information 0408002, University Library of Munich, Germany.
• Handle: RePEc:wpa:wuwpga:0408002
Note: Type of Document - pdf; pages: 13
as

File URL: https://econwpa.ub.uni-muenchen.de/econ-wp/game/papers/0408/0408002.pdf

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
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Cited by:

1. Péter Csóka & P. Herings & László Kóczy, 2011. "Balancedness conditions for exact games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 74(1), pages 41-52, August.
2. Bachmeier, BE & Nerlich, AG & Boukamp, P & Lichtinghagen, R & Tschesche, Harald & Fritz, H & Fink, E, 2010. "Human keratinocyte cell lines differ in the expression of the collagenolytic matrix metalloproteinases-1,-8, and-13 and of TIMP-1," Center for Mathematical Economics Working Papers 381, Center for Mathematical Economics, Bielefeld University.
3. Yaron Azrieli & Ehud Lehrer, 2005. "Cooperative investment games or population games," Game Theory and Information 0503007, University Library of Munich, Germany.
4. Rodica Branzei & Dinko Dimitrov & Stef Tijs, 2008. "Convex Games Versus Clan Games," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 10(04), pages 363-372.
5. repec:spr:compst:v:74:y:2011:i:1:p:41-52 is not listed on IDEAS
6. Ehud Lehrer, 2009. "A new integral for capacities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(1), pages 157-176, April.
7. Branzei, Rodica & Dimitrov, Dinko & Tijs, Stef, 2011. "Convex games, clan games, and their marginal games," Center for Mathematical Economics Working Papers 368, Center for Mathematical Economics, Bielefeld University.

### Keywords

concavification; convex games; core; totally balanced; exact games;

### JEL classification:

• C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
• D8 - Microeconomics - - Information, Knowledge, and Uncertainty

### NEP fields

This paper has been announced in the following NEP Reports:

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