Game Refinement Relations and Metrics

TL;DR

This paper introduces new equivalences and metrics for two-player probabilistic games on finite states, characterizing probabilistic winning conditions and extending classical bisimulation concepts to game structures.

Contribution

It develops novel equivalences and metrics for two-player games that capture probabilistic winning probabilities and generalize bisimulation to game settings.

Findings

Characterizes differences in winning probabilities using new metrics.

Provides a framework for probabilistic equivalences in game structures.

Extends classical bisimulation to two-player game models.

Abstract

We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal, for example, reach a target state, the question of winning is thus a probabilistic one: what is the maximal probability of winning from a given state? On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state similarity. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu-calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation.