A strategic approach for the discounted Shapley values

Abstract

The family of discounted Shapley values is analyzed for cooperative games in coalitional form. We consider the bargaining protocol of the alternating random proposer introduced in Hart and Mas-Colell (Econometrica 64:357-380, 1996). We demonstrate that the discounted Shapley values arise as the expected payoffs associated with the bargaining equilibria when a time discount factor is considered. In a second model, we replace the time cost with the probability that the game ends without agreements. This model also implements these values in transferable utility games, moreover, the model implements the α-consistent values in the nontransferable utility setting.

Coalition Formation and Bargaining Protocols: A Review of the Literature

“…Note that the remaining players are free to negotiate in the next round if the coalition S called by the proposer forms, assuming and . The result obtained by introducing a discount factor is the same as the one of Calvo and Gutiérrez‐López (), but the greater flexibility of this model is paid by a restriction, stated in theorem 2 of Kawamori (), in terms of the values of the coalitions for the result to hold.…”

“…In van den Brink and Funaki (), instead, it can be found a support for the DSV by simply adding a common discount factor δ for all players, while the model remains equal in all other aspects to the one of Pérez‐Castrillo and Wettstein (). Calvo and Gutiérrez‐López () show that the simple introduction of the discount factor into the model of Hart and Mas‐Colell () leads to the DSV in SSPE. In particular, if the introduced discounting factor is represented by ρ, the δ in the formula of the DSV given before will take the following value: , where α and are defined as previously.…”

Hart–Mas-Colell implementation of the discounted Shapley value

“…Calvo and Gutiérrez-López (2016) introduced time discounting into the model of Hart and Mas-Colell (1996) and showed coincidence of the discounted Shapley value and the equilibrium payoff tuple of the model. The main difference between Calvo and Gutiérrez-López (2016) and the present paper is as follows.…”

“…Calvo and Gutiérrez-López (2016) introduced time discounting into the model of Hart and Mas-Colell (1996) and showed coincidence of the discounted Shapley value and the equilibrium payoff tuple of the model. The main difference between Calvo and Gutiérrez-López (2016) and the present paper is as follows. In Calvo and Gutiérrez-López (2016), each proposer proposes only an allocation for the full coalition, i.e., full-coalition formation is assumed; in the present paper, each proposer proposes a coalition as well as an allocation for the coalition, and a condition for full-coalition formation is derived.…”

“…In Calvo and Gutiérrez-López (2016), each proposer proposes only an allocation for the full coalition, i.e., full-coalition formation is assumed; in the present paper, each proposer proposes a coalition as well as an allocation for the coalition, and a condition for full-coalition formation is derived. In Calvo and Gutiérrez-López (2016), even the nontransferable utility case is considered; in the present paper, only the transferable utility case is considered.…”