Reach together: How populations win repeated games

TL;DR

ReachTogether addresses whether a uniform coalition strategy exists that guarantees a target is reached in every population size for repeated population games with absorbing (reachability) payoffs. The authors introduce an automaton-based encoding of per-population utilities and develop a finite frontier semigroup to summarize strategy effects, enabling a polynomial-space decision procedure. A key technical contribution is a Ramsey-theoretic characterization: a winning strategy exists iff there are frontiers $f$ and $g$ with $f$ initial, $g$ $\omega$-iterable, and compatible via $f\star g \to f$ and $g\star g \to g$. They place ReachTogether in $PSPACE$ by solving an exponential-size frontier graph problem in space and prove matching hardness via a unary-space DTM reduction, establishing $PSPACE$-completeness. The results bridge population-repetition models with parameterized/concurrent games and suggest further avenues for analyzing uniform winning regions and related tiling problems under finite-state abstractions.

Abstract

In repeated games, players choose actions concurrently at each step. We consider a parameterized setting of repeated games in which the players form a population of an arbitrary size. Their utility functions encode a reachability objective. The problem is whether there exists a uniform coalition strategy for the players so that they are sure to win independently of the population size. We use algebraic tools to show that the problem can be solved in polynomial space. First we exhibit a finite semigroup whose elements summarize strategies over a finite interval of population sizes. Then, we characterize the existence of winning strategies by the existence of particular elements in this semigroup. Finally, we provide a matching complexity lower bound, to conclude that repeated population games with reachability objectives are PSPACE-complete.

Figures (7)

• Figure 1: Example of a labelled deterministic automaton $\langle \calA,\lambda \rangle$ (left) for which there is a winning population strategy in $\calG(\langle \calA,\lambda \rangle)$ (right).
• Figure 2: Example of a labelled automaton which is a positive instance of ReachTogether if and only if $n_1$ and $n_2$ are co-primes.
• Figure 3: A construction to prove -hardness.
• Figure 4: Link with concurrent parameterized games BBM-fsttcs20 with a reachability objective.
• Figure 5: Schematic representation of a winning strategy decomposition.

Theorems & Definitions (37)

• Theorem 1
• Definition 2
• Definition 3
• Remark 4
• Example 5
• Example 6
• Definition 7
• Example 8
• Lemma 9
• Definition 10