Value-Free Reductions

We introduce the value-free (v-f) reductions, operators that map a coalitional game played by a set of players to another “similar” game played by a subset of those players. We propose properties that v-f reductions may satisfy, we provide a theory of duality, and we characterize several v-f reductions (among which the value-free version of the reduced games proposed by Hart and Mas-Colell, 1989, and Oishi et al., 2016). Unlike reduced games, introduced to characterize values in terms of consistency, v-f reductions are not defined in reference to values. However, a v-f reduction induces a value. We characterize v-f reductions that induce the Shapley, the stand-alone, and the Banzhaf values. We connect our approach to the theory of implementation. Finally, our new approach is a valuable tool to provide new characterizations of values in terms of consistency. We present new characterizations of the Banzhaf and the stand-alone values.