Associated Consistency and the Aumann–Drèze Value

Abstract

This paper presents an axiomatic characterization of the Aumann–Drèze value [Aumann, R. J. and Drèze, J. H. [1974] Cooperative games with coalition structures, Int. J. Game Theory 3, 217–237] for cooperative games with coalition structures. We build an associated game that extends the original associated game presented by Hamiache [[2001] Associated consistency and Shapley value, Int. J. Game Theory 30, 279–289] to cooperative games with coalition structures. We use a similar approach to the one used in Hamiache and Navarro [[2020] Associated consistency, value and graphs, Int. J. Game Theory 49, 227–249]. This new associated game is expressed through a matrix form. We show that the series of successive associated games is convergent and that its limit is an inessential game. This allows us to propose a characterization of the Aumann–Drèze value that relies on associated consistency, inessential game and continuity axioms. Hence, this paper strengthens the results of Hamiache [[2001] Associated consistency and Shapley value, Int. J. Game Theory 30, 279–289] and Hamiache and Navarro [[2020] Associated consistency, value and graphs, Int. J. Game Theory 49, 227–249] considering that if these axioms are viewed as desirable, we are now able to provide a unique value for three different types of problems: the Shapley value on standard games [Hamiache, G. [2001] Associated consistency and Shapley value, Int. J. Game Theory 30, 279–289], the Hamiache–Navarro value on games with graphs [Hamiache, G. and Navarro, F. [2020] Associated consistency, value and graphs, Int. J. Game Theory 49, 227–249] and the Aumann–Drèze value for games with coalition structures.