Intuitive Solutions in Game Representations: The Shapley Value Revisited

An approach to transportation network analysis via transferable utility games

We strategically separate different core outcomes. The natural counterparts of a core allocation in a strategic environment are the α-core, the β-core and the strong equilibrium, modified by assuming that utility is transferable in a strategic context as well. Given a core allocation ω of a convex transferable utility (TU) game $$v$$ , we associate a strategic coalition formation game with $$ \left( {v, \omega } \right) $$ in which ω survives, while most other core allocations are eliminated. If the TU game is strictly convex, the core allocations respected by the TU-α-core, the TU-β-core and the TU-strong equilibrium shrink to ω only in the canonical family of coalition formation games associated with $$ \left( {v, \omega } \right) $$ . A mechanism, which strategically separates core outcomes from noncore outcomes for each convex TU game according to the TU-strong equilibrium notion is reported.

We consider a sequence of transferable utility (TU) games where, at each time, the characteristic function is a random vector with realizations restricted to some set of values. The first part of the paper contributes to the definition of a robust (coalitional) TU game and the development of a distributed bargaining protocol. We prove the convergence with probability 1 of the bargaining process to a random allocation that lies in the core of the robust game under some mild conditions on the underlying communication graphs. The second part of the paper addresses the more general case where the robust game may have empty core. In this case, with the dynamic game we associate a dynamic average game by averaging over time the sequence of characteristic functions. Then, we consider an accordingly modified bargaining protocol. Assuming that the sequence of characteristic functions is ergodic and the core of the average game has a nonempty relative interior, we show that the modified bargaining protocol converges with probability 1 to a random allocation that lies in the core of the average game.

This paper presents a new way to fairly distribute welfare gains derived from trade in exchange economies. The imposed fairness condition is based on the balanced marginal contributions condition satisfied by the Shapley value in transferable utility games. The solution is defined for cardinal finite exchange economies. Then, we point out that solutions for ordinal economies can be created by mapping finite ordinal exchange economies onto cardinal economies and then applying the original solution.

The non-emptiness of the weak sequential core of a transferable utility game with uncertainty

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

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 \(\alpha \)-consistent values in the nontransferable utility setting.

A non-negative transferable utility (TU) game is average monotonic if there exists a non-negative vector according to which the relative worth is not decreasing when enlarging the coalition. We generalize this definition to the nontransferable utility (NTU) case. It is shown that an average monotonic NTU game shares several properties with an average monotonic TU game. In particular it has a special core element and there exists a population monotonic allocation scheme. We show that an NTU bankruptcy game is average monotonic with respect to the claims vector.

A “restriction” is a monotonic projection assigning to each coalition of a finite player setN a subcoalition. On the class of transferable utility games with player setN, a Shapley value is associated with each restriction by replacing, in the familiar probabilistic formula, each coalition by the subcoalition assigned to it. Alternatively, such a Shapley value can be characterized by restricted dividends. This method generalizes several other approaches known in literature. The main result is an axiomatic characterization with the property that the restriction is determined endogenously by the axioms.

Null or Zero Players: The Difference between the Shapley Value and the Egalitarian Solution

Place et conditions de réalisation de l’autopsie en cas de mort maternelle

In this paper we present cooperative games and its fuzzification. A cooperative game (or coalitional game is usually described by a set of players and a characteristics function. In a transferable utility games only one number is attached to each coalition. Cooperative games are useful in considering profit allocation among multiple decision makes who cooperate with each other.

The rapid development and deployment of vehicle technologies offer opportunities to re-think the way traffic is managed. This article capitalizes on vehicle connectivity and proposes an economic instrument and corresponding cooperative framework for allocating priority at intersections. The framework is compatible with a variety of existing intersection control approaches. Similar to free markets, our framework allows vehicles to trade their time based on their (disclosed) value of time. We design the framework based on transferable utility games, where winners (time buyers) pay losers (time sellers) in each game. We conduct simulation experiments of both isolated intersections and an arterial setting. The results show that the proposed approach benefits the majority of users when compared to other mechanisms both ones that employ an economic instrument and ones that do not. We also show that it drives travelers to estimate their value of time correctly, and it naturally dissuades travelers from attempting to cheat.

If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a subcoalition of a grand coalition may only depend on the grand coalition and the size of the subcoalition. Hence, the symmetrically weighted versions of the (pre)nucleolus and the (pre)kernel are symmetric, i.e., invariant under symmetries of a game. They may, however, violate anonymity, i.e., they may depend on the names of the players. E.g., a symmetrically weighted nucleolus may assign the classical nucleolus to one game and the per capita nucleolus to another game. We generalize Sobolev’s axiomatization of the prenucleolus and its modification for the nucleolus as well as Peleg’s axiomatization of the prekernel to the symmetrically weighted versions. Only the reduced games have to be replaced by suitably modified reduced games whose definitions may depend on the weight system. Moreover, it is shown that a solution may only satisfy the mentioned sets of modified axioms if the weight system is symmetric.

Efficiency in uncertain cooperative games