Strong Nash Equilibria in Games with the Lexicographical Improvement Property

TL;DR

This paper introduces the Lexicographical Improvement Property (LIP) for finite strategic games, establishing its implications for the existence, computation, and properties of strong Nash equilibria, especially in bottleneck congestion games.

Contribution

It defines the LIP, shows its connection to potential functions, and applies it to finite and infinite bottleneck congestion games, including polynomial algorithms for equilibrium computation.

Findings

LIP implies existence of strong Nash equilibria in finite games.

Bottleneck congestion games possess the LIP, ensuring equilibrium properties.

Polynomial algorithms can compute SNE in certain network bottleneck games.

Abstract

We introduce 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 it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium.