International Game Theory Review

The aim of this work is to give a characterization of the Shapley–Shubik and the Banzhaf power indices for (3,2)-simple games. We generalize to the set of (3,2)-simple games the classical axioms for power indices on simple games: transfer, anonymity, null player property and efficiency. However, these four axioms are not enough to uniquely characterize the Shapley–Shubik index for (3,2)-simple games. Thus, we introduce a new axiom to prove the uniqueness of the extension of the Shapley–Shubik power index in this context. Moreover, we provide an analogous characterization for the Banzhaf index for (3,2)-simple games, generalizing the four axioms for simple games and adding another property.

In a generalized tournament, players may have an arbitrary number of matches against each other and the outcome of the games is measured on a cardinal scale with lower and upper bounds. An axiomatic approach is applied to the problem of ranking the competitors. Self-consistency (SC) requires assigning the same rank for players with equivalent results, while a player showing an obviously better performance than another should be ranked strictly higher. According to order preservation (OP), if two players have the same pairwise ranking in two tournaments where the same players have played the same number of matches, then their pairwise ranking is not allowed to change in the aggregated tournament. We reveal that these two properties cannot be satisfied simultaneously on this universal domain.

Power indices in simple games measure the relevance of a player through her ability in being critical, i.e. essential for a coalition to win. We introduce new indices that measure the power of a player in being decisive through the collaboration of other players. We study the behavior of these criticality indices to compare the power of different players within a single voting situation, and that of the same player with varying weight across different voting situations. In both cases we establish monotonicity results in line with those of Turnovec [1998]. Finally, we examine which properties characterizing the indices of Shapley–Shubik and Banzhaf are shared by these new indices.

The problem of cooperation in repeated and multistage games is considered. The strong equilibrium (equilibrium stable against deviations of coalitions) with payoffs which can be attained under cooperation is constructed for a wide class of such games. The new solution concept based on solutions of stage games is introduced and in some cases this solution is a subset of the core defined for repeated and multistage games in a classical way. It is also proved that this newly introduced solution concept is strongly time consistent. The strong time consistency of the solution is a very important property since in case it does not take place players in reality in some time instant in subgame on cooperative trajectory may switch from the previously selected optimal solution to any other optimal solution in the subgame and as result realize the solution which will not be optimal in the whole game.

Two-echelon models describe situations in which there are two differentiated groups of agents. Some examples of these models can be found in supply chain problems, transportation problems or two-sided markets. In this paper, we deal with two-sided transportation problems which can be used to describe a wide variety of logistic and market problems. We approach the problem from the perspective of cooperative games and study some solution concepts closely related to the game theoretical concept of core, but rather than focus specifically on the core of a transportation game, we introduce and study a new solution concept, a core catcher, which can be motivated by a kind of bounded rationality which can arise in these cooperative contexts.

We provide necessary and sufficient conditions for a non-empty core in many-to-one assignment games. When players on the “many” side (buyers) are substitutes with respect to any given player on the other side (firms), we show that non-emptiness requires an additional condition that limits the competition among the buyers. When buyers are complements with respect to any given firm, a sufficient condition for non-emptiness is that buyers also be complements with respect to all of the firms, collectively. A necessary condition is that no firm can be guaranteed a profit when the core is non-empty.