Dependence of the minimum admission price on the decay parameter δ

Determine how the minimum admission price p_map depends on the decay parameter δ in the monopolist pricing dynamics with quasi‑patient users, where δ denotes the fraction of pending transactions that persist to the next round and p_map is the smallest price such that all transactions paying at least p_map are eventually included under the dynamics described by the model (daily demand Q, supply s, total demand D_t, and pent‑up demand Z_t).

Background

The paper introduces a monopolist pricing model for blockchains with quasi‑patient users, parameterized by δ ∈ [0,1], the fraction of pending transactions that remain in the mempool across rounds. The authors analyze price dynamics and provide upper and lower bounds on the minimum admission price—defined as the smallest price ensuring eventual inclusion of all transactions paying at least that price—under various demand functions and δ values.

While the work establishes bounds and qualitative behaviors (e.g., transitions between monopolist and minimum admission prices, dependence on the structure of the demand function for δ < 1), the precise characterization of the minimum admission price p_map as a function of δ is left unresolved. The authors note empirical irregularities in p_map for a linear demand (Q(p)=1−p), suggesting the need for a rigorous analysis of p_map(δ).