The Harsanyi value for nontransferable utility games with restricted cooperation

A nontransferable utility (NTU) game assigns a set of feasible pay-off vectors to each coalition. In this article, we study NTU games in situations in which there are restrictions on coalition formation. These restrictions will be modelled through interior structures, which extend some of the structures considered in the literature on transferable utility games for modelling restricted cooperation, such as permission structures or antimatroids. The Harsanyi value for NTU games is extended to the set of NTU games with interior structure.     

Computing solutions for matching games

A matching game is a cooperative game (N, v) defined on a graph G = (N, E) with an edge weighting w: E? \mathbb R₊{w: E\to {\mathbb R}_+}. The player set is N and the value of a coalition S í N{S \subseteq N} is defined as the maximum weight of a matching in the subgraph induced by S. First we present an O(nm + n ² log n) algorithm that tests if the core of a matching game defined on a weighted graph with n vertices and m edges is nonempty and that computes a core member if the core is nonempty. This algorithm improves previous work based on the ellipsoid method and can also be used to compute stable solutions for instances of the stable roommates problem with payments. Second we show that the nucleolus of an n-player matching game with a nonempty core can be computed in O(n ⁴) time. This generalizes the corresponding result of Solymosi and Raghavan for assignment games. Third we prove that is NP-hard to determine an imputation with minimum number of blocking pairs, even for matching games with unit edge weights, whereas the problem of determining an imputation with minimum total blocking value is shown to be polynomial-time solvable for general matching games.     

Rooted-tree solutions for tree games

In this paper, we study cooperative games with limited cooperation possibilities, represented by a tree on the set of agents. Agents in the game can cooperate if they are connected in the tree. We introduce natural extensions of the average (rooted)-tree solution (see [Herings, P., van der Laan, G., Talman, D., 2008. The average tree solution for cycle free games. Games and Economic Behavior 62, 77–92]): the marginalist tree solutions and the random tree solutions. We provide an axiomatic characterization of each of these sets of solutions. By the way, we obtain a new characterization of the average tree solution.     

Minimum norm solutions for cooperative games

A solution f for cooperative games is a minimum norm solution, if the space of games has a norm such that f(v) minimizes the distance (induced by the norm) between the game v and the set of additive games. We show that each linear solution having the inessential game property is a minimum norm solution. Conversely, if the space of games has a norm, then the minimum norm solution w.r.t. this norm is linear and has the inessential game property. Both claims remain valid also if solutions are required to be efficient. A minimum norm solution, the least square solution, is given an axiomatic characterization.      

On consistent solutions for strategic games

Nash equilibria for strategic games were characterized by Peleg and Tijs (1996) as those solutions satisfying the properties of consistency, converse consistency and one-person rationality.  There are other solutions, like the ɛ-Nash equilibria, which enjoy nice properties and appear to be interesting substitutes for Nash equilibria when their existence cannot be guaranteed. They can be characterized using an appropriate substitute of one-person rationality. More generally, we introduce the class of “personalized” Nash equilibria and we prove that it contains all of the solutions characterized by consistency and converse consistency. Received January 1996/Final version December 1996     

Distributive solutions for absolutely stable games

Every absolutely stable game has von Neumann-Morgenstern stable set solutions. (Simple games and [n, n?1]-games are included in the class of absolutely stable games.) The character of these solutions suggests that the distributive aspect of purely discriminatory solutions is of as much conceptual importance as the discriminatory aspect.     

Mixed strategy solutions for quadratic games

Mixed strategy solutions are given for two-person, zero-sum games with payoff functions consisting of quadratic, bilinear, and linear terms, and strategy spaces consisting of closed balls in a Hilbert space. The results are applied to linear-quadratic differential games with no information, and with quadratic integral constraints on the control functions.     

Consistent and covariant solutions for TU games

One of the important properties characterizing cooperative game solutions is consistency. This notion establishes connections between the solution vectors of a cooperative game and those of its reduced game. The last one is obtained from the initial game by removing one or more players and by giving them the payoffs according to a specific principle (e.g. a proposed payoff vector). Consistency of a solution means that the restriction of a solution payoff vector of the initial game to any coalition belongs to the solution set of the corresponding reduced game. There are several definitions of the reduced games (cf., e.g., the survey of T. Driessen [2]) based on some intuitively acceptable characteristics. In the paper some natural properties of reduced games are formulated, and general forms of the reduced games possessing some of them are given. The efficient, anonymous, covariant TU cooperative game solutions satisfying the consistency property with respect to any reduced game are described.The research was supported by the NWO grant 047-008-010 which is gratefully acknowledgedReceived: October 2001     

Monotonic stable solutions for minimum coloring games

For the class of minimum coloring games (introduced by Deng et al. Math Oper Res, 24:751–766, 1999) we investigate the existence of population monotonic allocation schemes (introduced by Sprumont Games Econ Behav 2:378–394, 1990). We show that a minimum coloring game on a graph $G$ has a population monotonic allocation scheme if and only if $G$ is $(P_4,2K_2)$ -free (or, equivalently, if its complement graph $\bar{G}$ is quasi-threshold). Moreover, we provide a procedure that for these graphs always selects an integer population monotonic allocation scheme.     

The Banzhaf power index for political games

The Banzhaf index of a voting game is a measure of a priori power of the voters. The model on which the index is based treats the voters symmetrically, i.e. the ideology, outlook, etc., of the voters influencing their voting behavior is ignored. Here we present a nonsymmetric generalization of the Banzhaf index in which the ideology of the voters affecting their voting behavior is taken into account. A model of ideologies and issues is presented. The conditions under which our model gives the Shapley-Shubik index (another index of a priori power of the voters) are given. Finally several examples are presented and some qualitative results are given for straight majority and pure bargaining games.     

Non-existence of super-additive solutions for 3-person games

The super-additive solution for 2-person Nash bargaining games (with constant threat) was defined axiomatically inPerles/Maschler [1981]. That paper contains also a study of its basic properties. In this paper we show that the axioms are incompatible even for 3-person unanimity games. This raises the problem of finding a satisfactory generalization of this solution concept to multi-person games.     

Symmetric solutions for symmetric games with large cores

International Journal of Game Theory - Symmetric solutions (symmetric stable sets) and their uniqueness are investigated for symmetric games when the cores are large enough to have intersections...     

Aggregate monotonic stable single-valued solutions for cooperative games

This article considers single-valued solutions of transferable utility cooperative games that satisfy core selection and aggregate monotonicity, defined either on the set of all games, G ^N , or on the set of essential games, E ^N (those with a non-empty imputation set). The main result is that for an arbitrary set of players, core selection and aggregate monotonicity are compatible with individual rationality, the dummy player property and symmetry for single-valued solutions defined on both G ^N and E ^N . This result solves an open question in the literature (see for example Young et?al. Water Resour Res 18:463?C475, 1982).     

Monotonic solutions of cooperative games

The principle of monotonicity for cooperative games states that if a game changes so that some player's contribution to all coalitions increases or stays the same then the player's allocation should not decrease. There is a unique symmetric and efficient solution concept that is monotonic in this most general sense — the Shapley value. Monotonicity thus provides a simple characterization of the value without resorting to the usual “additivity” and “dummy” assumptions, and lends support to the use of the value in applications where the underlying “game” is changing, e.g. in cost allocation problems.     

On solutions of Bayesian games

The theory of Bayesian games, as developped by W. Böge, is axiomatically treated. A direct access to the system of complete reflections is shown. Solutions for these games are defined and characterizations for their existence are given. Concrete situations are investigated for the case of (2,2)-games.     

微分对策中的联盟解

提出时间区间[t_0,∞)上的n人微分对策两阶段联盟解. 在第一阶段不能形成大联盟的假设是自然的,即源于这一思想. 在第一阶段以联盟作为局中人的对策中计算得到其纳什均衡,之后对每个联盟的收益按Shapley值进行分配. 一个n人微分减排模型的例子阐明了上述结果.