Existence of Multilateral Nash equilibria for families of games

TL;DR

This paper introduces multilateral Nash equilibria and families of games, extending classical game theory to include group deviations and parameterized collections of games, with a proof of equilibrium existence under minimal assumptions.

Contribution

It develops the concept of multilateral Nash equilibria and demonstrates their existence in families of games, linking game structure with combinatorial graph properties.

Findings

Higher-lateral Nash equilibria exist in classical games.

The existence criterion reflects the rarity of higher-lateral equilibria.

The clique covering number of the Kneser graph is central to the theory.

Abstract

This paper introduces two fundamentally new concepts to game theory: multilateral Nash equilibria and families of games. Starting with non-cooperative games, we show how these notions together seamlessly integrate into and naturally extend the classical theory, and simultaneously enable us to prove a powerful (multilateral) Nash equilibrium existence result with minimal assumptions on the game. Classically, a Nash equilibrium is a global strategy such that whichever player unilaterally deviates from the equilibrium, also reduces his own profit. For a k-lateral Nash equilibrium we now require that whichever group of k players collectively changes their strategies, also reduces all of the deviating players' profits. In this way, we obtain a filtration of equilibria, where the higher-lateral equilibria are less frequent. Furthermore, we derive an existence criterion for multilateral Nash equilibria and demonstrate how it reflects the increasing rarity of higher-lateral equilibria. Additionally, we show that some classical games have higher-lateral Nash equilibria, which in every case reveal the structure of these games from a new point of view. A family of games is a parameterized collection of non-cooperative games, where the parameter affects every aspect of the game. Typically, we assume that this dependence is continuous, thereby introducing a new structure. That way, we can avoid analyzing the games one at a time, and instead treat the family as a whole. This allows the parameter to take a central role in our theory, and shifts our attention from seeking a special strategy to searching for a special game with preferred strategies. Our main result proves the existence of a multilateral equilibrium in a family of games, maintaining minimalistic assumptions on the games individually. Surprisingly, the clique covering number of the Kneser graph makes a central appearance.