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Regular Games with Imperfect Information Are Not That Regular

Laurent Doyen, Thomas Soullard

TL;DR

The paper investigates synthesis of almost-sure randomized strategies in two-player games with imperfect information defined by indistinguishability. It introduces regular games based on rectangular morphisms and develops a quadratic-time fixpoint algorithm to decide almost-sure reachability (with Büchi extending via reductions), highlighting that reductions that succeed for pure strategies do not generally carry over to randomized strategies. Through counterexamples, it shows that a strong bijective transfer between the original and abstract games is not always possible, yet a nuanced bisimulation-based relationship can still connect certain aspects of the models. Overall, the work extends positive decidability results for almost-sure reachability/Büchi from partial-information settings to the broader regular game framework, while clarifying fundamental limitations of reductions in the randomized-imperfect-information setting.

Abstract

We consider two-player games with imperfect information and the synthesis of a randomized strategy for one player that ensures the objective is satisfied almost-surely (i.e., with probability 1), regardless of the strategy of the other player. Imperfect information is modeled by an indistinguishability relation describing the pairs of histories that the first player cannot distinguish, a generalization of the traditional model with partial observations. The game is regular if it admits a regular function whose kernel commutes with the indistinguishability relation. The synthesis of pure strategies that ensure all possible outcomes satisfy the objective is possible in regular games, by a generic reduction that holds for all objectives. While the solution for pure strategies extends to randomized strategies in the traditional model with partial observations (which is always regular), we show that a similar reduction does not exist in the more general model. Despite that, we show that in regular games with Buechi objectives the synthesis problem is decidable for randomized strategies that ensure the outcome satisfies the objective almost-surely.

Regular Games with Imperfect Information Are Not That Regular

TL;DR

The paper investigates synthesis of almost-sure randomized strategies in two-player games with imperfect information defined by indistinguishability. It introduces regular games based on rectangular morphisms and develops a quadratic-time fixpoint algorithm to decide almost-sure reachability (with Büchi extending via reductions), highlighting that reductions that succeed for pure strategies do not generally carry over to randomized strategies. Through counterexamples, it shows that a strong bijective transfer between the original and abstract games is not always possible, yet a nuanced bisimulation-based relationship can still connect certain aspects of the models. Overall, the work extends positive decidability results for almost-sure reachability/Büchi from partial-information settings to the broader regular game framework, while clarifying fundamental limitations of reductions in the randomized-imperfect-information setting.

Abstract

We consider two-player games with imperfect information and the synthesis of a randomized strategy for one player that ensures the objective is satisfied almost-surely (i.e., with probability 1), regardless of the strategy of the other player. Imperfect information is modeled by an indistinguishability relation describing the pairs of histories that the first player cannot distinguish, a generalization of the traditional model with partial observations. The game is regular if it admits a regular function whose kernel commutes with the indistinguishability relation. The synthesis of pure strategies that ensure all possible outcomes satisfy the objective is possible in regular games, by a generic reduction that holds for all objectives. While the solution for pure strategies extends to randomized strategies in the traditional model with partial observations (which is always regular), we show that a similar reduction does not exist in the more general model. Despite that, we show that in regular games with Buechi objectives the synthesis problem is decidable for randomized strategies that ensure the outcome satisfies the objective almost-surely.
Paper Structure (14 sections, 10 theorems, 10 equations, 3 figures)

This paper contains 14 sections, 10 theorems, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Basic notions
  4. Games with imperfect information
  5. Strategies and outcome
  6. Almost-Sure Reachability
  7. Regular games with imperfect information
  8. Algorithm
  9. Reductions
  10. Alternating probabilistic trace equivalence
  11. Bijection and bisimulation
  12. Counterexamples
  13. Proof of Lemma \ref{['lem:Pforall']}
  14. Proof of Lemma \ref{['lem:Pexists']}

Key Result

lemma 1

BDS23 The relation $\approx$ is an equivalence and $h([\tau]_{\sim}) = [h(\tau)]_{\approx}$ for all $\tau \in \Gamma^*$.

Figures (3)

  • Figure 1: A game with imperfect information representing matching pennies, and its information tree.
  • Figure 2: Equivalences between the game ${\cal G}$, its induced abstract game ${\cal H}$, and their information trees.
  • Figure 3: A game ${\cal G}$ and the abstract game ${\cal H}$ induced by a rectangular morphism, with randomized strategies encoded by the variables $\bar{x},\bar{t}$ (in ${\cal G}$) and $\bar{y},\bar{z}$ (in ${\cal H}$).

Theorems & Definitions (12)

  • lemma 1
  • theorem 1
  • corollary 1
  • lemma 2
  • lemma 3
  • theorem 2
  • theorem 3
  • theorem 4
  • lemma 3
  • proof
  • ...and 2 more