Critical threshold for phase transition in cost-function monotonicity

Establish the existence of a critical threshold t* for the incentive threshold parameter t in the individual-based institutional incentive schemes (reward, punishment, and hybrid) for well-mixed finite populations interacting via cooperation dilemmas, such that the expected institutional cost E(θ) is always non-decreasing in the per-capita incentive θ for all selection intensities β when t ≤ t*, whereas for t* < t ≤ N−1 the cost E(θ) is non-decreasing when β is sufficiently small but ceases to be monotonic when β is sufficiently large. Determine t* (e.g., as a function of the population size N) and provide a rigorous proof of this phase transition behavior for general N.

Background

The paper analyzes the expected total cost E(θ) of individual-based institutional incentives in finite, well-mixed populations. For t=1 (reward-only), the authors prove E(θ) is always non-decreasing in θ for all β. Prior analytical results for the full-invest case t=N−1 show that E(θ) can become non-monotonic for sufficiently large β, while remaining non-decreasing for small β, indicating qualitatively different behavior across t.

Based on these findings, the authors posit a phase transition as t varies: there should exist a critical threshold t* that separates regimes of guaranteed monotonicity from those where monotonicity breaks for large β. They note numerical evidence suggests t*=N−1 for small N but state that a general proof and characterization of t* remain elusive.