Backward Induction Foundations of the Shapley Value

Abstract

We present a noncooperative game model of coalitional bargaining, closely based on that of Gul (1989) but solvable by backward induction. In this game, Gul's condition of “value additivity” does not suffice to ensure the existence of a subgame perfect Nash equilibrium that supports the Shapley value, but a related condition—“no positive value-externalities”—does. Multiple equilibria can arise only in the event of ties, and with a mild restriction on tie-break rules these equilibria all support the Shapley value

Balanced externalities and the Shapley value

“…So, the Shapley value, postulated as the expected outcome of rational bargaining, may be seen as progressively more compelling when applied to games that are value-additive and to games with no positive value-externalities. This conclusion corresponds exactly with findings in analyses of noncooperative bargaining models -specifically those of Gul (1989) and McQuillin and Sugden (2016) -that allow for bilateral mergers. In Gul's bargaining model, value-additivity is a necessary and sufficient condition for the existence of a stationary subgame perfect equilibrium within which players' expectations converge (as a discount factor tends to 1) to the Shapley value, and (McQuillin and Sugden show) no positive value-externalities ensures that there are no other stationary subgame perfect equilibria.…”

“…By Proposition 7, within our merge-externalities approach, the solution ϕ ESV 2 emerges as the analogue, for games in partition function form, to the solution φ Sh for games in characteristic function form. By Proposition 8, and consistent with the result in McQuillin and Sugden (2016), this solution is most defensible as the expected outcome from rational bargaining for games (such as the glove game) in which there are 'no positive value-externalities'.…”

“…It turns out that the merge-externalities approach supports the value proposed by McQuillin (2009), the distinctive feature of which is that players' expectations are independent of payoffs in the underlying game associated with partitions of cardinality greater than two. It should be emphasized that this feature emerges as a result (not as an assumption) both within the axiomatic approach of McQuillin (2009) and, for a defined sub-class of games (games that have 'no positive valueexternalities'), within the noncooperative bargaining model of McQuillin and Sugden (2016).…”

“…One approach is to model bargaining as a noncooperative game. Various authors have taken this approach, showing for specific models of the bargaining process that -at least under some conditions -the Shapley value matches utility expectations (Gul, 1989;Hart and Mas-Colell, 1996;Pérez-Castrillo and Wettstein, 2001;McQuillin and Sugden, 2016). In this paper we adopt a different strategy, by aiming to explicitly capture the notion of a player's bargaining position within an axiomatic characterization.…”

“…The balanced contributions characterization relates closely to subsequent noncooperative models (Hart and Mas-Colell, 1996;Pérez-Castrillo and Wettstein, 2001) in which there is a possibility, during bargaining, that players leave the game. Our 'merge externalities' characterization relates similarly to models (Gul, 1989;McQuillin and Sugden, 2016) in which, during bargaining, players have opportunities to form intermediate coalitions instead of forming the grand coalition in a single step.…”