Exponential tail bound for stability at any number of symmetric agents

Establish that for all integers n ≥ 2 with symmetric agents under the minimum token selection rule and service availability density d ≥ 2, the stability condition (C1) holds with an exponentially decaying tail f(M) = a^M for all M ∈ Z+, for some constant a ∈ [1/3, 1/2]. Specifically, prove that the long-run probability that a given agent’s token count exceeds ±M decays exponentially with M, and identify (or bound) the constant a uniformly over n.

Background

The paper analyzes token systems under the minimum token selection rule. It proves stability (existence of a stationary distribution) for all n ≥ 2 and d ≥ 2 and derives tail bounds: an O(1/M) bound uniformly in n (Theorem 4.1) and an exponential tail with constant a = 1/2 in the large-n limit (Theorem 4.2). For n = 2, the authors compute the stationary distribution explicitly and obtain an exponential tail with constant a = 1/3.

Motivated by these two extremal cases (n = 2 and n → ∞), the authors conjecture that exponential decay holds for every finite n ≥ 2 with a constant a lying between 1/3 and 1/2. Proving this would unify the finite- and infinite-population analyses and provide sharper, uniform concentration guarantees for the token distribution across all market sizes.