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.

A Game Theory Proof of Optimal Colorings Resilience to Strong Deviations

“…In our work, we extend the main results of [10,12] since we show important properties of minimal subsets and strong deviation. Moreover, we prove that there does not exist any subsets of nodes able to increase their own payoffs simultaneously.…”

“…The most important existing result was provided by the authors of [10,11], who showed that on undirected unweighted graphs, optimal colorings are 5-strong equilibria (5-SE), i.e., colorings in which no coalition of at most five vertices can profitably deviate. These results were extended to 7-SE by the same authors of this work [12].…”

Optimal Coloring Strategies for the Max k-Cut Game

“…We are now ready to prove the main result of the section. Note that this result was presented at the conference mentioned in the introduction [15]. Theorem 3.…”

“…Therefore, G(C) must be an isolated component of G. Now, we prove that for all the cases where a minimal strongly deviating coalition, C, has, at most, 7 vertices, the cut value increases. This result was presented at the conference mentioned in the introduction [15]. Theorem 2.…”

“…Subsequently, a subset of the authors presented an early version of the results, providing a strategic approach to problem resolution at the "GASCom on random generation of combinatorial structures, 12th edition" conference (GASCom2022). In [15], we presented a conceptual outline of our innovative problem-solving approach; however, it lacked the essential lemmas, propositions, and theorems crucial for a formal proof. Given the substantial interest generated by this work, we decided to revisit and expand upon it, presenting our approach with clarity and precision.…”