Fast computation of the leastcore and prenucleolus of cooperative games

Bonnans, Joseph Frédéric; André, Matthieu

RAIRO - Operations Research - Recherche Opérationnelle, Tome 42 (2008) no. 3, pp. 299-314.

Le calcul du leastcore et du prénucléole est une manière efficace d’allouer une ressource entre $n$ joueurs. L’inconvénient est qu’il suppose la résolution d’un programme linéaire avec $2^n-2$ contraintes. Dans cet article nous montrons comment, dans le cas de jeux de production convexes, générer des contraintes en résolvant des programmes linéaires mixtes de petite taille. L’approche est étendue aux jeux avec symétries (joueurs identiques) et aux jeux avec coalitions partiellement continues. Nous étudions aussi le calcul du prénucléole, et donnons des résultats numériques prometteurs.

The computation of leastcore and prenucleolus is an efficient way of allocating a common resource among $n$ players. It has, however, the drawback being a linear programming problem with $2^n-2$ constraints. In this paper we show how, in the case of convex production games, generate constraints by solving small size linear programming problems, with both continuous and integer variables. The approach is extended to games with symmetries (identical players), and to games with partially continuous coalitions. We also study the computation of prenucleolus, and display encouraging numerical results.

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