The Lovász Extension of Market Games
The multilinear extension of a cooperative game was introduced by Owen in 1972. In this contribution we study the Lovász extension for cooperative games by using the marginal worth vectors and the dividends. First, we prove a formula for the marginal worth vectors with respect to compatible orderings. Next, we consider the direct market generated by a game. This model of utility function, proposed by Shapley and Shubik in 1969, is the concave biconjugate extension of the game. Then we obtain the following characterization: The utility function of a market game is the Lovász extension of the game if and only if the market game is supermodular. Finally, we present some preliminary problems about the relationship between cooperative games and combinatorial optimization.
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- Driessen, T. S. H. and Rafels, C. (1999), Characterization of k-convex games, Optimization 46, 403–431.Google Scholar
- Einy, E. and Wettstein, D. (1996), Equivalence between bargaining sets and the core in simple games, International Journal of Game Theory 25, 65–71.Google Scholar
- Fujishige, S. (1984), Theory of submodular programs:a Fenchel-type min–max theorem and subgradients of submodular functions, Mathematical Programming 29, 142–155.Google Scholar
- Fujishige, S. (1991), Submodular Functions and Optimization. Amsterdam: North-Holland.Google Scholar
- Ichiishi, T. (1981), Supermodularity:applications to convex games and to the greedy algorithm for LP, Journal of Economic Theory 25, 283–286.Google Scholar
- Kannai, Y. (1992), The core and balancedness in Aumann, R. J. and Hart, S. (eds), Handbook of Game Theory, Vol. I, Amsterdam: North-Holland, 355–395.Google Scholar
- Lovász, L. (1983), Submodular functions and convexity, in Bachem, A. Gröstschel, M. and Korte, B. (eds.), Mathematical Programming:The State of the Art, Berlin: Springer-Verlag, 235–257.Google Scholar
- Martínez-Legaz, J. E. (1996), Dual representation of cooperative games based on Fenchel-Moreau conjugation, Optimization 36, 291–319.Google Scholar
- Murota, K. (1998), Discrete convex analysis, Mathematical Programming 83, 313–371.Google Scholar
- Owen, G. (1972), Multilinear extension of games, Management Science 18, 64–79.Google Scholar
- Shapley, L. S. (1971), Cores of convex games, International Journal of Game Theory 1, 11–26.Google Scholar
- Shapley, L. S. and Shubik, M. (1969), On market games, Journal of Eco-nomic Theory 1, 9–25.Google Scholar
- Weber, R. J. (1988), Probabilistic values for games, in Roth, A. E. (ed.), The Shapley Value, Cambridge: Cambridge:University Press, 101–119.Google Scholar