Cooperation in dividing the cake

Abstract

This paper defines models of cooperation among players partitioning a completely divisible good (such as a cake or a piece of land). The novelty of our approach lies in the players' ability to form coalitions before the actual division of the good with the aim to maximize the average utility of the coalition. A social welfare function which takes into account coalitions drives the division. In addition, we derive a cooperative game which measures the performance of each coalition. This game is compared with the game in which players start cooperating only after the good has been portioned and has been allocated among the players. We show that a modified version of the game played before the division outperforms the game played after the division.

Finding maxmin allocations in cooperative and competitive fair division

“…Dall'Aglio et al [8] proposed a strategic model of interaction, where players, before the division takes place, gather into mutually disjoint coalitions. Within each coalition, players pursue an efficient allocation of their collective share of the cake.…”

“…This is certainly the most intuitive setting. Although very natural, this proposal suffers from a serious drawback, since players participating in the game η(·, w) may be better off waiting to seek for cooperation well after the cake has been divided (see [8]);…”

“…Brams and Taylor [6] discuss the issue of the manipulability of the preferences: in most cases children may benefit from declaring false preferences. A different approach takes into account the possibility for the children to form coalitions after (Legut [15] and Legut et al [16]) or before (Dall'Aglio et al [8]) the division of the cake. In both cases coalitional games are defined and analysed.…”

“…In the case of early cooperation among children, the game is based on a maxmin allocation problem among coalitions, each one having a joint utility function and a weight. The first properties of the game are studied in [8]. It turns out that the analysis of the game is made harder by the difficulties in computing the characteristic function, i.e., the value associated to each coalition.…”

“…In Section 2 we describe the maxmin fair division problem with coalitions through the strategic model of interaction among players in [8] and the geometrical setting employed in [1], [2] and [7]. In Section 3 we present the upper and the lower bounds for the objective value.…”