Bargaining and Value

Abstract

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Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

“…In the literature various implementations of the Shapley value can be found, see e.g. Gul (1989), Hart and Mas-Colell (1996) and Pérez-Castrillo and Wettstein (2001). In this section we adapt the last mechanism and obtain the following mechanism that implements any egalitarian Shapley value as the unique subgame perfect equilibrium outcome.…”

“…Actually, only few works address the issue of implementing mixed solutions like the egalitarian Shapley values. Hart and Mas-Colell (1996) discussed a generalized version of their multilateral bargaining game and found that, for some θ ∈ (0, 1), if the proposer drops out with probability (1−ρ)θ, whereas each responder drops out with equal probability (1 − ρ)(1 − θ)/(s − 1), 12 then the game implements the corresponding egalitarian Shapley value with respect to θ in stationary subgame perfect equilibria. However, we like to point out that this result only obtains the egalitarian Shapley value as the equilibrium outcome in expected terms rather than in actual terms, as the game requires an ex ante probability distribution for players to be proposer.…”

Implementation and axiomatization of discounted Shapley values

“…Gul (1989), Hart and Mas-Colell (1996) and Pérez-Castrillo and Wettstein (2001). A cooperative game solution is (strategically) implemented if for every cooperative game one defines an associated strategic or extensive form game (a mechanism) such that applying some strategic solution, for example the subgame perfect equilibrium, always yields payoff outcomes according to this cooperative solution.…”

Cooperative Games in Strategic Form

“…One may conjecture, that, as in Hart and Mas-Colell (1996a), as ρ approaches 1 the SP equilibrium payoffs approach a Maschler-Owen (1992) consistent NTU value of (N, V G,σ ). It is not difficult to see that this is indeed the case if, for every S, the limit of the SP equilibrium payoffs of S lies in a smooth piece of the efficient boundary of 8 V G,σ (S).…”

“…There is another source of inspiration for the PC approach: the bargaining procedure formulated by ourselves (Hart and Mas-Colell, 1996a) for the context where the underlying reality is a game in coalitional form. The sequential nature of the announcements and proposals we take from there, along with the idea that a rejected proposer becomes passive for the rest of the game (with some probability).…”