Existence Of Strong Equilibrium Research Articles

Optimal Coloring Strategies for the Max k-Cut Game

We explore strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors. Here, the vertices represent players, and the edges denote their relationships. Each player, v, selects a color as its strategy, and its payoff (or utility) is determined by the number of neighbors of v who have chosen a different color. Limited findings exist on the existence of strong equilibria in max k-cut games. In this paper, we make advancements in understanding the characteristics of strong equilibria. Specifically, our primary result demonstrates that optimal solutions are seven-robust equilibria. This implies that for a coalition of vertices to deviate and shift the system to a different configuration, i.e., a different coloring, a number of coalition vertices greater than seven is necessary. Then, we establish some properties of the minimal subsets concerning a robust deviation, revealing that each vertex within these subsets will deviate toward the color of one of its neighbors.

Optimal colorings of Max k-Cut game

Abstract We investigate strong Nash equilibria in the max k-cut game on an undirected and unweighted graph with a set of k colors, where vertices represent players and the edges indicate their relations. Each player v chooses one of the available colors as its own strategy, and its payoff (or utility) is the number of neighbors of v that has chosen a different color. Such games are significant since they model loads of real-worlds scenario with selfish agents and, moreover, they are related to fundamental classes of games. Few results are known concerning the existence of strong equilibria in max k-cut games in this direction. In this paper we make some progress in the understanding of the properties of strong equilibria. In particular, our main result is to show that optimal solutions are 7-strong equilibria. This means that in order a coalition of nodes is able to deviate and drive the system towards a different configuration, i.e. a different coloring, a number of nodes of the coalition strictly larger than 7 is necessary. We also conjecture that, in a generic graph with n nodes, any optimal coloring is also an n-strong equilibrium.

The Strong Consistency of Neutral and Monotonic Binary Social Decision Rules

The purpose of this paper is to investigate the strong consistency of neutral and monotonic binary social decision rules. Individuals are assumed to satisfy von Neumann-Morgenstern axioms of individual rationality. The main result of the paper shows that there does not exist any neutral and monotonic non-null non-dictatorial binary social decision rule which is strongly consistent. The relationship between restricted preferences and the existence of strong equilibria is also investigated. It is shown that for every non-dictatorial social decision function satisfying the conditions of independence of irrelevant alternatives, neutrality, monotonicity and weak Pareto-criterion there exists a profile of individual orderings satisfying value-restriction corresponding to which there is no strong equilibrium.

Existence of strong equilibrium in discontinuous games

In this paper, we give necessary and sufficient conditions for the existence of strong equilibria in games with non-ordered preferences. Our result does not require continuous preferences and, in the case of normal form games, (quasi-)concavity of payoff functions is not needed. The result has been obtained by using the Ky Fan minimax inequality. Examples illustrate our conditions in some economic situations and compare the result with the previous literature.

Strong equilibria in games with the lexicographical improvement property

We study a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the lexicographical improvement property (LIP) and show that, in finite games, it is equivalent to the existence of a generalized strong potential function. We use this characterization to derive existence, efficiency and fairness properties of strong equilibria (SE). As our main result, we show that an important class of games that we call bottleneck congestion games has the LIP and thus the above mentioned properties. For infinite games, the LIP does neither imply the existence of a generalized strong potential nor the existence of SE. We therefore introduce the slightly more general concept of the pairwise LIP and prove that whenever the pairwise LIP is satisfied for a continuous function, then there exists a SE. As a consequence, we show that splittable bottleneck congestion games with continuous facility cost functions possess a SE.