Marginal Contributions and Externalities in the Value

Abstract

For games in partition function form, we explore the implications of distinguishing between the concepts of intrinsic marginal contributions and externalities. If one requires efficiency for the grand coalition, we provide several results concerning extensions of the Shapley value. Using the axioms of efficiency, anonymity, marginality and monotonicity, we provide upper and lower bounds to players' payoffs when affected by external effects, and a characterization of an ``externality-free'' value. If the grand coalition does not form, we characterize a payoff configuration on the basis of the principle of balanced contributions. We also analyze a game of coalition formation that yields sharp predictions Keywords:. externalities, marginal contributions, Shapley value, Pigouvian transfers, coalition formation JEL classification:. C7, D62 Abstract For games in partition function form, we explore the implications of distinguishing between the concepts of intrinsic marginal contributions and externalities. If one requires efficiency for the grand coalition, we provide several results concerning extensions of the Shapley value. Using the axioms of efficiency, anonymity, marginality and monotonicity, we provide upper and lower bounds to players' payoffs when affected by external effects, and a characterization of an "externality-free" value. If the grand coalition does not form, we characterize a payoff configuration on the basis of the principle of balanced contributions. We also analyze a game of coalition formation that yields sharp predictions. JEL classification numbers: C7, D62

The extended and generalized Shapley value: Simultaneous consideration of coalitional externalities and coalitional structure

“…In addition to the last two of these, [1,3,8,11,12] propose extensions of the Shapley Value to games in partition function form.…”

“…Bolger [1], Potter [12], Pham Do and Norde [11], Maskin [7], Macho-Stadler, Pérez-Castrillo and Wettstein [6] and de Clippel and Serrano [3]. It should be noted that other solution concepts in cooperative game theory identify outcomes that are reasonable in a di¤erent sense, that might be roughly described as ex post plausibility: i.e.…”

Cooperative Games in Strategic Form

“…It has led, on the one hand, to the development of extensions of the notion of coalitional form (perhaps the most well known are the "games in partition form" of Thrall and Lucas, 1963;see Myerson, 1977, Maskin, 2003, de Clippel and Serrano, 2005, Macho-Stadler, Perez-Castrillo and Wettstein, 2007, for more recent work) and on the other to the consideration of particular situations where the classical form could be justified (for example the c-games of Shapley-Shubik, see Shubik, 1983, p.130).…”

Power Indices and Minimal Winning Coalitions in Simple Games with Externalities

“…For instance, Hafalir (2007) has generalized the concepts of superadditivity and convexity. However, most of the recent contributions deal with generalizations of the Shapley value (Macho-Stadler et al, 2007;de Clippel and Serrano, 2008;Dutta et al, 2010). InÁlvarez-Mozos and Tejada (2015) the Banzhaf value is generalized to games in partition function form.…”