Coalition Resilient Outcomes in Max k-Cut Games

Abstract

We investigate strong Nash equilibria in the max k-cut game, where we are given an undirected edge-weighted graph together with a set {1, . . . , k} of k colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player v consists of the k colors. When players select a color they induce a k-coloring or simply a coloring. Given a coloring, the utility (or payoff ) of a player u is the sum of the weights of the edges {u, v} incident to u, such that the color chosen by u is different from the one chosen by v. Such games form some of the basic payoff structures in game theory, model lots of real-world scenarios with selfish agents and extend or are related to several fundamental classes of games. Very little is known about the existence of strong equilibria in max k-cut games. In this paper we make some steps forward in the comprehension of it. We first show that improving deviations performed by minimal coalitions can cycle, and thus answering negatively the open problem proposed in [13]. Next, we turn our attention to unweighted graphs. We first show that any optimal coloring is a 5-SE in this case. Then, we introduce x-local strong equilibria, namely colorings that are resilient to deviations by coalitions such that the maximum distance between every pair of nodes in the coalition is at most x. We prove that 1-local strong equilibria always exist. Finally, we show the existence of strong Nash equilibria in several interesting specific scenarios.

Optimal Coloring Strategies for the Max k-Cut Game

“…In [13], the existence of a 5-SE in unweighted, undirected graphs is shown. In this subsection, we extend this result by proving the existence of a 7-SE.…”

“…This observation reinforces the idea that certain properties related to deviations are preserved across all permutations within the set of optimal colorings. Lastly, we define G(C) as the restriction of the graph G to the members of the coalition C, i.e., the set of vertices of G(C) is C. Moreover, we say that H is an isolated component of the graph G if G(H) is a connected subgraph of G and, for each vertex i ∈ V(H), and for each vertex j / ∈ V(H), {i, j} / ∈ E. Now, we report an important result of [13] that we will use alongside the pigeon-hole principle.…”

“…The objective is to minimize the number of colors while satisfying the condition that the adjacent vertices have distinct colors. The pivotal outcome in [13] reveals that optimal colorings are 5-SE in undirected unweighted graphs. By building upon the findings of [13,14], our paper employs an innovative combinatorial approach, illustrating that optimal colorings extend to 7-SE.…”

“…The pivotal outcome in [13] reveals that optimal colorings are 5-SE in undirected unweighted graphs. By building upon the findings of [13,14], our paper employs an innovative combinatorial approach, illustrating that optimal colorings extend to 7-SE.…”

SIC Error Towards the Packet Error Rate: Optimal Partitioning in OTFS-NOMA Systems

“…D. Max K−cut games [28], [29] Take into account an undirected graph 𝒢 = 𝒱, ℰ, 𝓌 , where |𝒱| = 𝒷 1 , |ℰ| = 𝒷 2 , while we let 𝓌 : ℰ → R + be the edge-weight-map function which technically maps every edge to a positive weigth. Let the term…”