Discrete Nondeterminism and Nash Equilibria for Strategy-Based Games

TL;DR

This paper introduces multi strategic games, a new formalism that combines features of sequential and simultaneous decision-making, guaranteeing the existence of discrete, non-deterministic equilibria with efficient computation and practical recommendation capabilities.

Contribution

It proposes a novel discrete and static approach using multi strategic games, extending traditional strategic and sequential graph games, with a guarantee of equilibrium existence.

Findings

Existence of discrete, non-deterministic equilibria is guaranteed.

Equilibria can be computed efficiently with polynomial complexity.

Numerical example demonstrates practical effectiveness of the approach.

Abstract

Several notions of game enjoy a Nash-like notion of equilibrium without guarantee of existence. There are different ways of weakening a definition of Nash-like equilibrium in order to guarantee the existence of a weakened equilibrium. Nash's approach to the problem for strategic games is probabilistic, \textit{i.e.} continuous, and static. CP and BR approaches for CP and BR games are discrete and dynamic. This paper proposes an approach that lies between those two different approaches: a discrete and static approach. multi strategic games are introduced as a formalism that is able to express both sequential and simultaneous decision-making, which promises a good modelling power. multi strategic games are a generalisation of strategic games and sequential graph games that still enjoys a Cartesian product structure, \textit{i.e.} where agent actually choose their strategies. A pre-fixed point result allows guaranteeing existence of discrete and non deterministic equilibria. On the one hand, these equilibria can be computed with polynomial (low) complexity. On the other hand, they are effective in terms of recommendation, as shown by a numerical example.