In ZK-Rollups, provers spend significant computational resources to generate validity proofs. Their costs should be compensated properly, so a sustainable prover market can form over time. Existing transaction fee mechanisms (TFMs) such as EIP-1559, however, do not work in this setting, as EIP-1559 only generates negligible revenue because of burning, while provers often create or purchase specialized hardware in hopes of creating long-term revenue from proving, somewhat reminiscent of proof-of-work miners in the case of chains like Bitcoin. In this paper, we explore the design of transaction fee mechanisms for prover markets. The desiderata for such mechanisms include efficiency (social welfare is maximized), incentive compatibility (dominant bidding strategy exists), collusion resistance (no profitable collusion among provers exists), and off-chain agreement proofness (no profitable collusion between users and provers exists). This paper presents a prover market mechanism called Proo{\phi} (pronounced proo-fee), and show the conditions under which it can satisfy these desired properties. In our mechanism, the users bid their fee for transaction inclusion, and the user transactions are selected through a first-price auction; the provers bid their proof generation capacity and cost, and the lowest-cost provers are selected. We add an upper bound of the total capacity of the included transactions each round to forestall off-chain agreements. Proo{\phi} is Bayesian incentive compatible for users and off-chain agreement proof, and is also incentive-compatible for provers, when assuming the latter cannot employ Sybil attacks. We present a preliminary analysis of the Proo{\phi} mechanism and its limitations, and raise open questions about the feasibility of an ideal mechanism for prover markets. This work is in progress, and this manuscript will be updated.
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