Allocation Rules for Games with Optimistic Aspirations

A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions, and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. In this paper, in addition to presenting this model and justifying its relevance, we introduce allocation rules and extend the properties of efficiency, additivity, symmetry, and null player property to this setting. We demonstrate that these four properties are insufficient to find a unique allocation rule and define three properties involving null players and nullifying players that allow the identification of unique allocation rules. The allocation rules we identify are the Midpoint Shapley Value and the Equal Division Rule. 1. Introduction In this paper we introduce games with optimistic aspirations, and we identify two allocation rules for such games—the Midpoint Shapley Value and the Equal Division Rule. A game with optimistic aspirations specifies two values for each coalition of players: the first value is the worth that the players in the coalition can guarantee for themselves in the event that they coordinate their actions (where the word guarantee implies a very conservative attitude), and the second value is the amount that the players in the coalition aspire to get under reasonable but very optimistic assumptions about the demands of the players who are not included in the coalition. The two allocation rules that we define on the class of games with optimistic aspirations in this paper, the Midpoint Shapley Value and the Equal Division Rule, are found by extending the axioms that were used in Shapley [1] to define the Shapley Value and augmenting them with stronger versions of the null player property—the strong null player property, the nullifying player property, and the destroyer player property. The strong null player property and the destroyer player property lead to the Midpoint Shapley Value, while the nullifying player property leads to the Equal Division Rule. This paper contributes to the field of cooperative game theory. Games with optimistic aspirations are inspired much in the same way in which von Neumann and Morgenstern [2] already introduced cooperative games, namely, as descriptions of situations that are devoid of a specific structure of negotiations but that capture the potential of coalitions of