From hierarchies to well-foundedness

We highlight that the connection of well-foundedness and recursive definitions is more than just convenience. While the consequences of making well-foundedness a sufficient condition for the existence of hierarchies (of various complexity) have been extensively studied, we point out that (if parameters are allowed) well-foundedness is a necessary condition for the existence of hierarchies e.g. that even in an intuitionistic setting \({(\Pi_1^0-\mathsf{CA}_0)_\alpha \vdash \mathsf{wf}(\alpha)\, {\rm where}\, (\Pi_1^0-\mathsf{CA}_0)_\alpha}\) stands for the iteration of \({\Pi^0_1}\) comprehension (with parameters) along some ordinal \({\alpha}\) and \({\mathsf{wf}(\alpha)}\) stands for the well-foundedness of \({\alpha}\) .     

Achievable hierarchies in voting games with abstention

It is well known that he influence relation orders the voters the same way as the classical Banzhaf and Shapley–Shubik indices do when they are extended to the voting games with abstention (VGA) in the class of complete games. Moreover, all hierarchies for the influence relation are achievable in the class of complete VGA. The aim of this paper is twofold. Firstly, we show that all hierarchies are achievable in a subclass of weighted VGA, the class of weighted games for which a single weight is assigned to voters. Secondly, we conduct a partial study of achievable hierarchies within the subclass of H-complete games, that is, complete games under stronger versions of influence relation.     

Share functions for cooperative games with levels structure of cooperation

In a standard TU-game it is assumed that every subset of the player set N can form a coalition and earn its worth. One of the first models where restrictions in cooperation are considered is the one of games with coalition structure of Aumann and Drèze (1974). They assumed that the player set is partitioned into unions and that players can only cooperate within their own union. Owen (1977) introduced a value for games with coalition structure under the assumption that also the unions can cooperate among them. Winter (1989) extended this value to games with levels structure of cooperation, which consists of a game and a finite sequence of partitions defined on the player set, each of them being coarser than the previous one.     

Group stability of hierarchies in games with spillovers

This paper studies the stability properties of hierarchies in cooperative problems with spillovers. The analysis builds on a recent paper by Demange [Demange, G. (2004) “On Group Stability in Hierarchies and Networks”, Journal of Political Economy 112(4), 754–778.], in which hierarchical structures are shown to attain stability by limiting the ability of certain coalitions to act autonomously and object to cooperative outcomes. We show that the stability properties of hierarchies crucially depend on the interplay of spillovers with the expectations that blocking coalitions form on the reaction of outsiders. In particular, we focus on pessimistic, passive and optimistic expectations, and on the classes of negative and positive spillovers. Under negative spillovers, hierarchies are shown to guarantee stability in all cooperative problems, but fail to sustain as stable all cooperative outcomes that are instead stable within non hierarchical organizations. Under positive spillovers we obtain opposite results: hierarchies fail to always guarantee stability, but sustain as stable all cooperative outcomes that are stable in non hierarchical organizations.     

Union-wise egalitarian solutions in cooperative games with a coalition structure

4OR - The egalitarian principle has been widely adopted in designing solution concepts for cooperative games. In light of egalitarianism, we introduce two egalitarian values satisfying union...     

Dependence and independence: From linear hierarchies to nonlinear networks

Most decisions need to be free from assumptions of independence to be faithful to the complex problems in which they arise. This paper illustrates how to generate priorities for decisions involving general types of dependence of criteria on alternatives, criteria on criteria and alternatives on alternatives. It is based on the feedback system framework of the Analytic Hierarchy Process of which a hierarchy is a special case.     

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...     

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.     

From Lie algebra crossed modules to tensor hierarchies

The present paper, though inspired by the use of tensor hierarchies in theoretical physics, establishes their mathematical credentials, especially as genetically related to Lie algebra crossed modules. Gauging procedures in supergravity rely on a pairing – the embedding tensor – between a Leibniz algebra and a Lie algebra. Two such algebras, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. This paper is devoted to showing that any Lie-Leibniz triple induces a differential graded Lie algebra – its associated tensor hierarchy – whose restriction to the category of Lie algebra crossed modules is the canonical assignment associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of ‘lie-ization’ of the former. We deem the present construction of such tensor hierarchies clearer and more straightforward than previous derivations. We stress that such a construction suggests the existence of further well-defined Leibniz gauge theories.     

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.     

Nonzero-sum differential games with bargaining solutions

Nash's bargaining solution for finite games is extended to differential games with nonzero-sum integral payoffs. Sufficient conditions for the optimality of a strategy pair are established. An example is given.     

Core-like solutions for games with probabilistic choice of strategies

The α-core, the β-core, and the strong equilibria of various games are studied, and their relationships are established under the assumption that the given pure-strategy spaces are convex and compact, and the given von Neumann-Morgenstern utility functions are concave and continuous. Sufficient condition for nonemptiness of the β-core of a gamewith correlated strategies are derived as a corollary. By this corollary a version of the Aumann Proposition (that the β-core of a one-shot game with correlated strategies is precisely rhe set of strong equilibrium utility allocations of the associated repeated game), as presented by the present author elsewhere, is made non-vacuous.     

Special polynomials associated with rational solutions of some hierarchies

New special polynomials associated with rational solutions of the Painlevé hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey–Dodd–Gibbon, the Kaup–Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.     

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     

Harsanyi power solutions for graph-restricted games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.     

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.