The priority value for cooperative games with a priority structure

Abstract

The views expressed are those of the authors and do not necessarily reflect those of CRESE.

Axiomatic characterizations of the family of Weighted priority values

“…Note that an ordered partition over the agent set is a special case of a poset: it corresponds to a specific ranked poset where each agent with the same rank dominates, and so has priority over, each other agent with a lower rank, and agents with the same rank are incomparable with respect to the poset. As Kalai and Samet (1987), Béal et al (2022) consider that the priority structure does not influence the coalition formation process but the distribution of payoffs. They propose the Priority value which shares the Harsanyi dividend of a coalition in a game equally to its priority members.…”

“…There are many axiomatic foundations for the Shapley value, the Positively weighted Shapley values, and the Weighted Shapley values, and Béal et al (2022) propose two axiomatic characterizations of the Priority value. Recently, Casajus and Yokote (2017) provide a characterization of the Shapley value by three properties: Efficiency, Null agent, and Weak differential marginality.…”

“…If the ordered partition is the coarsest one, then the Weighted Shapley values coincide with the Positively weighted Shapley values. Béal et al (2022) define a nonsymmetric version of the Shapley value by endowing a game with a priority structure represented by a partially ordered set (a poset henceforth) over the agent set. The poset reflects the asymmetries that may exist between agents, for example, different needs, unequal merits, beyond the parameters of the game.…”

“…To complete the characterization of Weighted priority values we use two other axioms. The first one is borrowed from Béal et al (2022). The Priority agent out axiom indicates that removing an agent does not affect the payoffs of the agents over which it has 1 Differential marginality is itself a weak version of Strong differential monotonicity introduced by Casajus and Huettner (2013).…”

Solutions for cooperative games with restricted coalition formation and almost core allocations

“…Faigle and Kern [48] considered restricted structures modelling precedence constraints where the set of players is (partially) ordered by some precedence relations. Algaba et al [3] developed the so-called antimatroid which generalizes the concept of permission structures, and Béal et al [18] developed priority structures which are mathematically identical to the acyclic conjunctive permission structures in [55] and the structure of precedence constraints. Furthermore, several other combinatorial structures are incorporated into general TU-games, such as union stable systems [2], convex geometries [23,24], augmenting systems [25], regular set systems [78], union closed systems [113], accessible union stable network structures [4], voting structures [1], intersection closed system [19], etc.…”