Existence of Multilateral Nash equilibria for families of games
Matija Blagojevic, Christof Schütte
TL;DR
The paper develops a comprehensive theory of multilateral (k-lateral) Nash equilibria for N-player games and extends it to families of games parameterized by a base space. It introduces multilateral best-reply correspondences and a multilateral Nikaido--Isoda framework, deriving existence criteria that rely on topological and geometric data, including Euler and Stiefel–Whitney classes and the Kneser graph clique-covering number $\xi(N,k)$. Through fiber-bundle formulations and cohomological obstructions, it proves that for many families there exists a parameter value yielding a $k$-lateral NE, and it provides a spectrum of examples—from simple geometric constructions to finite-player games and macroeconomic models—that illustrate the rarity and stability of higher-lateral equilibria. The results connect classical game theory with modern topology and geometry, offering novel tools for stability analysis in economics and beyond, and revealing deep structural insights via the clique structure of Kneser graphs. Moreover, the framework demonstrates how multilateral equilibria can be guaranteed under broad, often minimal, assumptions on the constituent games and their parameterization.
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.