Monotonicity of concentration in the number of agents

Establish that for symmetric agents under the minimum token selection rule and service availability density d ≥ 2, the long-run concentration probability p_{n,M} = P_{π}(|s_1| ≤ M) is nonincreasing in n. Formally, prove that p_{n+1,M} ≤ p_{n,M} for all integers n ≥ 2 and all M ∈ Z+, where π denotes the stationary distribution and s_1 is the token count of a representative agent.

Background

The authors define p_{n,M} as the stationary probability that an agent’s token count lies within ±M when there are n symmetric agents. Simulations suggest that as n increases, p_{n,M} decreases, indicating that the token distribution becomes less concentrated around zero.

Formalizing and proving this monotonicity would clarify how market size impacts stability and concentration, and it would strengthen the analytical understanding of token distributions beyond the current bounds obtained for n = 2 and asymptotically as n → ∞.