Value-Free Reductions

We introduce the value-free (v-f ) reductions, which are 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 for them, and we characterize several v-f reductions (among which the value-free version of the reduced games propose by Hart and Mas-Colell, 1989, and Oishi et al., 2016). Unlike reduced games, which were introduced to characterize values in terms of consistency properties, v-f reductions are not defined in reference to values. However, a "path-independent" v-f reduction induces a value. We characterize v-f reductions that induce the Shapley value, the stand-alone value, and the Banzhaf value. Moreover, we can connect our approach to the literature on consistency because any value induced by a path-independent v-f reduction is consistent with that reduction.