International Game Theory Review

The purpose of this paper is to analyze a class of voting games in a partition approach. We consider a society in which coalitions can be formed and where a finite number of voters have to choose among a set of alternatives. A coalition is winning if it can veto any proposed alternative. In our model, the veto power of a coalition is dependent on the coalition formation of the outsiders. We show that whether or not the core is non-empty depends crucially on the expectations of each coalition regarding outsiders' behavior when it wishes to veto an alternative. On the one hand, if each coalition has pessimistic expectations, then the core is non-empty if and only if the dimension of the set of alternatives is equal to one. On the other hand, if each coalition has optimistic expectations, the non-emptiness of the core is not ensured.

We introduce a solution concept for games in normal form with undetermined parameters, coalitional ZP-equilibrium, based on the notions of Z-equilibrium of [Zhukovskii and Chikrii [1994] Linear quadratic differential games, Kiev, Naoukova Doumka] and ZS-equilibrium of [Larbani and Lebbah [1999] A concept of equilibrium for a game under uncertainity. Europ. J. Oper. Res.117, 145–156]. For each coalition structure, ZP-equilibrium ensures both the stability of the partition and equilibrium of coalitional strategies (in Pareto sense). We show that under some quasiconcavity conditions on payoff functions, the coalitional ZP-equilibrium exists in compact, convex and continuous normal form games involving undetermined parameters.

In this paper we study a family of extensions of the Shapley value for games in partition function form with n players. In particular, we provide a complete characterization for all linear, symmetric, efficient and null solutions in these environments. Finally, we relate our characterization result with other ways to extend the Shapley value in the literature.

The launch of a public project requires "enough" support from a group of 'n' players, i.e., a certain threshold has to be passed. The players may be differently important for passing the threshold; they may have different costs of support and different benefits if the project is launched. If players have only binary decision sets (participate or not, vote approvingly or not) this game is called a Binary Threshold Public Goods game (BTPG). We compare the expected equilibrium payoffs in BTPGs with the same costs and benefits but different thresholds. Applying two principles of equilibrium selection, the least and the most demanding threshold, namely "one supporting player is sufficient" (Volunteer's Dilemma) and "support by all players is necessary" (Stag Hunt game) are payoff equivalent for all players. Compared with the Stag Hunt game, all intermediate thresholds are connected with Pareto-inferior payoffs.

This paper analyzes a dynamic game for the provision of a discrete public good that will be provided only when a certain number of contributions are made. In each period players are randomly ordered and each player decides whether to contribute or not to the provision of the public good. If not enough contributions are made to provide the public good, the game goes to the next period and continue until the public good is either provided or the game ends at the final period. The paper first assumes symmetric players and shows that the exact period in which the public good will be provided is determined under subgame-perfect equilibria and that for some cost structures there will be a delay in provision of the public good. Counterintuitively, in some cases the public good is provided earlier at higher costs than at lower costs. The paper then introduces a partial public good, which requires fewer contributions to be provided but has a smaller value. We show that for some costs, introducing the partial public good eliminates the possibility of providing any kind of public good. For other cost structures, introducing the partial public good may improve social welfare. At last, asymmetric costs are introduced. The results are similar to the symmetric case.

Dynamic (or stochastic) games are, in general, considerably more complicated to analyze than repeated games. This paper shows that for every deterministic dynamic game that is linear in the state, there exists a strategically equivalent representation as a repeated game. A dynamic game is said to be linear in the state if it holds for both the state transition function as well as for the one-period payoff function that (i) they are additively separable in action profiles and states and (ii) the state variables enter linearly. Strategic equivalence refers to the observation that the two sets of subgame perfect equilibria coincide, up to a natural projection of dynamic game strategy profiles on the much smaller set of repeated game histories. Furthermore, it is shown that the strategic equivalence result still holds for certain stochastic elements in the transition function if one allows for additional signals in the repeated game or in the presence of a public correlating device.

We propose lattice-theoretical methods to analyze the existence and the order structure of Berge equilibria (in the sense of Zhukovskii) in noncooperative games. We introduce Berge-modular games, and prove that the set of Berge equilibrium turns out to be a complete lattice.