Continuous Positional Payoffs

TL;DR

This paper investigates continuous payoffs in two-player antagonistic games on finite graphs, showing that positional determinacy aligns with prefix-monotonicity and providing multiple proof techniques and algorithmic insights.

Contribution

It establishes the equivalence between continuous payoffs' positional determinacy and prefix-monotonicity, using three proof methods and exploring algorithmic implications.

Findings

Positional determinacy for continuous payoffs equals prefix-monotonicity.

Three proof techniques demonstrate the main equivalence.

Algorithmic insights into continuous positionally determined payoffs.

Abstract

What payoffs are positionally determined for deterministic two-player antagonistic games on finite directed graphs? In this paper we study this question for payoffs that are continuous. The main reason why continuous positionally determined payoffs are interesting is that they include the multi-discounted payoffs. We show that for continuous payoffs, positional determinacy is equivalent to a simple property called prefix-monotonicity. We provide three proofs of it, using three major techniques of establishing positional determinacy -- inductive technique, fixed point technique and strategy improvement technique. A combination of these approaches provides us with better understanding of the structure of continuous positionally determined payoffs as well as with some algorithmic results.