Existence theorems of generalized K-lateral support equilibria via local cooperative strategies for multi-player pure strategy games in tensor form

Abstract

In this paper, we introduce a novel equilibrium framework in pure strategy games called $K$-lateral support equilibrium ($K$-LSE), encompassing unilateral support equilibrium and Berge equilibrium as special cases. This framework applies to a broad class of $n$-person pure strategy games, with the values of $K$ varying from 1 to $n-1$. Two fundamental theorems provide necessary and sufficient conditions for the existence of strategy profiles corresponding to pure $K$-LSEs. When the strategy set of each player has finite elements, we develop a tensor representation of $K$-LSEs and design an algorithm for computing $K$-LSEs in $n$-player pure strategy games. This algorithm provides a useful method with tensor representation for addressing equilibrium problems in complex high-dimensional game scenarios. The effectiveness of this algorithm is demonstrated through numerical examples involving a four-player game, where the algorithm successfully identifies all unilateral, bilateral and Berge equilibria. Additionally, the relationships of different $K$-LSEs are also established and discussed.