Existence of Nash equilibria in general concurrent reachability games

Establish whether Nash equilibria exist for general n-player concurrent reachability games with infinite horizons and qualitative objectives.

Background

The paper distinguishes finite-horizon stochastic games, for which subgame-perfect Nash equilibria exist by backward induction, from infinite-horizon concurrent stochastic games with qualitative objectives. Although equilibrium existence is known for some restricted objective classes, such as safety objectives, the general reachability case remains unresolved. This issue is separate from the paper’s finite-horizon model-free learning results and concerns the underlying game-theoretic existence question for arbitrary numbers of players.

References

For infinite-horizon concurrent stochastic games with qualitative objectives, equilibrium existence is only known for certain low levels of the Borel hierarchy (e.g., safety objectives). For example, existence of $$ for general $n$-player concurrent reachability games remains an open question.

— Learning Rational Strategies for Temporal Objectives in Multi-Agent Systems  (2610.22287 - Adalat et al., 15 Oct 2026) in Section 3, subsection “Existence of Equilibria”