Cooperative Games with Externalities, Marginalism and the Average approach

This paper aims to characterize classes or types of values for transferable utility games in partition function form, accounting for externalities where the worth of a coalition depends on the cooperation structure outside it. We follow an axiomatic approach, focusing on values that represent two-stage anticipations. In the first stage, the worth of each coalition is computed as a weighted average of each of its possible worths (depending on the cooperation structure outside the coalition). In the second stage, these anticipated worths for all coalitions are used to derive the value for participating in the game, treating the game as if it were one without externalities. We extend previous work by Macho-Stadler et al. (Journal of Economic Theory,135, 339-356, 2007; Games and Economic Behavior,108, 49-64, 2018) beyond Efficiency and Symmetry, providing axiomatic characterizations for values based on both exogenous and endogenous weights. Additionally, we model first-step anticipations as operators from games with externalities to games without externalities, characterizing the family of weighted average operators. We also explore the relationship of the average approach with marginalism, highlighting that marginalist values constructed from the average approach require weights satisfying a consistency condition, termed recursivity, but not necessarily a symmetry condition. Finally, we contrast these results with Grabisch and Funaki's (2012) coalition formation value, which involves averages over dynamic coalition formation processes but cannot be constructed from the average approach.