Existence of Secure Equilibrium in Multi-Player Games with Perfect Information

TL;DR

This paper proves the existence of secure equilibrium in certain multi-player perfect information games, extending the concept beyond two-player scenarios to more complex game structures with probabilistic and deterministic transitions.

Contribution

It establishes the existence of secure equilibrium in multi-player perfect information games with probabilistic and deterministic transitions, broadening the theoretical understanding of game solutions.

Findings

Secure equilibrium exists in games with probabilistic transitions, countable states, and continuous payoffs.

Secure equilibrium exists in deterministic games with arbitrary states and Borel payoffs.

Results apply to various game types studied in literature.

Abstract

Secure equilibrium is a refinement of Nash equilibrium, which provides some security to the players against deviations when a player changes his strategy to another best response strategy. The concept of secure equilibrium is specifically developed for assume-guarantee synthesis and has already been applied in this context. Yet, not much is known about its existence in games with more than two players. In this paper, we establish the existence of secure equilibrium in two classes of multi-player perfect information turn-based games: (1) in games with possibly probabilistic transitions, having countable state and finite action spaces and bounded and continuous payoff functions, and (2) in games with only deterministic transitions, having arbitrary state and action spaces and Borel payoff functions with a finite range (in particular, qualitative Borel payoff functions). We show that these results apply to several types of games studied in the literature.