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Continuous Positional Payoffs

Alexander Kozachinskiy

TL;DR

It turns out that all continuous stochastically positional payoffs are multi-discounted, which gives a negative answer to a question of Gimbert (STACS 2007), who conjectured that all deterministically positional payoff are stochastic.

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.

Continuous Positional Payoffs

TL;DR

It turns out that all continuous stochastically positional payoffs are multi-discounted, which gives a negative answer to a question of Gimbert (STACS 2007), who conjectured that all deterministically positional payoff are stochastic.

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.

Paper Structure

This paper contains 20 sections, 32 theorems, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Deterministic infinite duration games on finite directed graphs
  5. Continuous payoffs
  6. MDPs
  7. Statement of the Main Result and its “Only If” Part
  8. Inductive Argument
  9. Fixed point argument
  10. The Structure of Continuous Positional Payoffs
  11. Strategy improvement argument
  12. Subexponential-time Algorithm
  13. Proof of Theorem \ref{['cont_alg']}
  14. Reducing to shift-deterministic payoffs
  15. Recursively local-global functions
  16. ...and 5 more sections

Key Result

Proposition 2.3

If $A$ is a finite set, $\varphi\colon A^\omega\to\mathbb{R}$ is a payoff and $g\colon \varphi(A^\omega) \to\mathbb{R}$ is a bounded non-decreasingThroughout the paper, we call a function $f\colon S\to\mathbb{R}, S\subseteq \mathbb{R}$ non-decreasing if for all $x, y\in S$ we have $x\le y \implies f

Figures (3)

  • Figure 1: A game graph where $\varphi$ is not positionally determined.
  • Figure 2: All nodes are owned by Max. For every $i = 1, 2, 3$, the node $v_i$ has two lassos $\mathcal{L}_i$ and $\mathcal{R}_i$ starting at it, one going to the left, and the other going to the right. We label their edges in such a way that $\mathsf{lab}(\mathcal{L}_i) = \alpha_i$ and $\mathsf{lab}(\mathcal{R}_i) = \beta_i$. This is possible because $\alpha_1, \alpha_2, \alpha_3$ and $\beta_1, \beta_2, \beta_3$ are ultimately periodic.
  • Figure 3: A graph for an MDP where $\varphi$ has no optimal positional strategy.

Theorems & Definitions (78)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • Definition 3.1
  • ...and 68 more