Monotonicity of optimal-action probability under ABRA

Prove that, for any fixed noise parameter β, the probability of selecting an optimal action under the stationary distribution concentrated on an optimal recurrent class of the Approximate Best Response Algorithm increases with the rationality parameter p.

Background

The paper studies Truncated Noisy Best-Response algorithms for two-player submodular maximization games. One particular member of this family is the Approximate Best Response Algorithm (ABRA), whose mistakes-based action-selection rule is governed by a rationality parameter p and a noise parameter β.

For an optimal recurrent class, the authors define q*_p as the stationary probability of selecting an optimal action. Numerical experiments suggest that q*_p increases as p increases, and the authors conjecture that this monotonicity holds for every fixed β. Establishing the conjecture would provide a formal basis for using greater rationality to improve the frequency with which ABRA selects optimal actions.

References

Based on this observation, we make the following conjecture whose proof is the left for the future work. The probability of choosing an optimal action q*_p under stationary distribution \pi*_p concentrated on an optimal class increases with rationality parameter p for any fixed noise parameter \beta.

— Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees  (2609.11863 - Singh et al., 10 Sep 2026) in Section 3.3, subsection “Expected system objective under ABRA” (hidden subsection), Conjecture 1; revisited in Section 4.1

Based on this observation, we make the following conjecture whose proof is the left for the future work. The probability of choosing an optimal action $q*_p$ under stationary distribution $\pi*_p$ concentrated on an optimal class increases with rationality parameter $p$ for any fixed noise parameter $\beta$.

— ABRA: An algorithm which cannot converge to low-quality Nash equilibria  (2609.11889 - Singh et al., 10 Sep 2026) in Hidden subsection “Expected system objective under ABRA,” immediately preceding Conjecture (labelled \ref{conjec})