Limit-value existence for non-oriented percolation games

Establish whether the sequence of finite-horizon values converges for non-oriented percolation games when payoffs are i.i.d. across both states and actions.

Background

The paper studies percolation Markov decision processes (PMDPs), the one-player restriction of percolation games, and proves almost-sure convergence of finite-horizon values for all PMDPs under i.i.d. edge payoffs. The cited two-player percolation-game results establish convergence in the oriented case, but removing orientation remains difficult even under the stronger assumption that payoffs are i.i.d. across both states and actions.

This unresolved problem concerns the corresponding two-player non-oriented model, rather than the one-player PMDP model solved in the paper. Its resolution would extend long-horizon value-existence results beyond the oriented setting.

References

And still, with this assumption the existence of the limit value for non-oriented percolation games is an open problem.

— Percolation Markov Decision Processes  (2609.19905 - García et al., 17 Sep 2026) in Remark 2.3, immediately following Theorem 2.2 in Section 2.3

Can convergence results be established when the payoffs are i.i.d. only across space?

— Percolation Markov Decision Processes  (2609.19905 - García et al., 17 Sep 2026) in Section 8, Perspectives, Point 2