Cooperation and Fair Division

When two players are facing each other, the question of cooperation is simple: either they cooperate or they do not cooperate. When instead, there are three players or more, players can form coalitions. This enriches considerably the analysis. The concept of coalition is indeed central in the analysis of games involving more than two players. If a group of players decides to form a coalition, it means that they dissociate themselves from the set of all players (called the “grand coalition”) in order to cooperate between themselves. However, the question that is being addressed concerns the cooperation between all players and the allocation among them of the resulting “social” outcome. Hence, the role played by coalitions remains potential. Beyond the maximum gain that the players can generate altogether by cooperating, we need to know the gain that each coalition of players could obtain through the cooperation of its members, independently of the actions of the players outside the coalition. When a coalition forms, what matters is the outcome it can achieve, without specifying the organizational details. Quoting Shapley, cooperative game theory “is concerned with things like cooperation, coalition, organizational structure, commitment, trust, compromise, threatThreat, enforceability and indeed the whole legal/social/cultural environment. It deemphasizes questions of tactical optimization, the detailed spelling out of rules and the numerical calculation of outcomes and payoffs”.1 The possibility of binding and enforceable agreements is an element that further differentiates cooperative games from non-cooperative games. In Aumann's words, “A game is called cooperative if there is available a mechanism, such as a court, to enforce agreements. In a cooperative game, any feasible outcome may be achieved if the players subscribe to the appropriate agreement”.