Resumen de The Procedural Value
Geoffroy de Clippel
We define a single-valued solution (named the procedural value) that extends both the consistent Shapley value of Maschler and Owen (1989) and Raiffa�fs discrete bargaining solution to a large class of NTU games. Though not axiomatized, the solution is motivated via the Nash program.
We consider bargaining procedures GK that are indexed by a positive integer K, related to the size of the tree. At each stage, a player (let�fs call him i) is chosen at random within the active coalition (let�fs call it S). Player i proposes a feasible allocation for S (let�fs call it x). Every member of S then sequentially chooses wether to accept player i�fs proposal. If all the members of S accepts, then x is enforced and the game ends. If somebody rejects, then S \ {i} becomes the active coalition when S has already been active K times in the past. Otherwise, S remains active. When a player is ejected from the active coalition, he receives a payoff low enough in order to promote cooperation. This payoff could be the disagreement payoff if the former NTUgame is superadditive. The game starts with the grand coalition being active.
Each game GK admits a unique subgame perfect equilibrium outcome. The sequence of subgame perfect equilibrium outcomes converge to the procedural value, as K tends to infinity.
The extensive form games GK we consider as relevant bargaining procedures, are closely related to the ones proposed by Hart and Mas-Collel (1996) in order to support the consistent Shapley value of Maschler and Owen (1992). In their framework, the parameter K is replaced by a parameter �Ï �¸ (0, 1] that represents the probability for each player to be excluded of the active coalition and to receive the punishment payoff, when his proposition is rejected by some member of the active coalition. Their extensive form games can potentially last infinitely many periods, although this happens with probability zero. They thus have to focus on stationary subgame perfect equilibria (SSPE), as folk-like theorems hold in their model for the larger set of subgame perfect equilibria (SPE). Moreover, they obtain the weaker result that every sequence of SSPE outcomes indexed by rho converges to some consistent Shapley value, as �Ï tends to zero. They do not establish the converse. Our approach also shows the limits of the robustness of their result: a slight variation (out of the class of variations defined in their sixth section) of their bargaining procedure may yield the Shapley value in the TU-case and something different from the consistent Shapley value in the NTU-case.
