Serial Monopoly on Blockchains with Quasi-patient Users

In the face of limited block size, miners (e.g., in Bitcoin) prioritize high-bid transactions, forming a large part of their revenue. If the block size were to expand significantly, meeting all transaction demand due to infrastructure or protocol improvements, bids could drop to zero or to a minimum fee, reducing mining incentives and potentially affecting security. To address this, Lavi et al. (2022) introduced a monopolistic pricing mechanism where miners only include transactions paying a minimum price, ensuring some revenue but resulting in an unbounded loss in welfare. Nisan (2023) expands this by modeling bidders as patient, who wait indefinitely long for lower prices, causing price fluctuations even with stable demand. In order to capture users' diminishing interest in having their transactions added to the ledger over time, we consider a more realistic setting with quasi-patient users, where only a fraction δ∈[0,1] of pending transactions remains in the next round. This richer model encompasses both Lavi et al.'s impatient users (δ=0) and Nisan's patient users (δ=1) as special cases. We demonstrate that Nisan's fluctuating dynamics persist for δ close to 1, while for δ close to 0, the dynamics resemble the impatient case. For δ∈(0,1), we establish new bounds on price dynamics, revealing unexpected effects. Unlike the fully patient case, the bounds of the dynamics for δ<1 depend on the demand curve and undergo a "transition phase". For some δ, the model mirrors the fully patient setting, and for smaller δ′<δ, it stabilizes at the highest monopolist price, thus collapsing to the impatient case. We provide quantitative bounds and analytical results, showing that the bounds for δ=1 are generally not tight for δ<1, and we give guarantees on the minimum ("admission") price for transactions.