Optimistic rollups are blockchain scaling solutions that reduce main-chain computation while maintaining high security. Rollup operators execute transactions and post state commitments to the main chain, which are assumed valid unless challenged. These challenges let users dispute invalid commitments within a set period, ensuring correctness. However, if challenges can be censored, the rollup’s security and censorship resistance – a core blockchain property requiring all valid transactions to be processed – are compromised. To strengthen transaction verification, we introduce the concept of time consensus, where participants agree on an upper bound for when they received a transaction, using shared randomness and no direct communication. We design single-shot time consensus mechanisms achieving near-optimal trade-offs between success probability and overhead, along with tight lower bounds. These mechanisms yield approximate correlated equilibria approaching exactness as failure probability vanishes - though exact equilibria are impossible. We extend this to multi-shot time consensus with feedback, achieving improved trade-offs and robustness against adversarial manipulation of shared randomness.
This paper examines the optimal design of contests with n contestants, where the designer determines both the number of admitted contestants in a shortlist and the prize structure under a fixed budget. For the maximum individual effort objective, the optimal contest is remarkably simple—a two-contestant winner-take-all contest, where only two contestants are shortlisted, and a single prize is awarded. For the total effort objective, the optimal contest admits m=kn contestants and distributes the budget equally among the top m-1 contestants. We also present a general algorithm to determine the optimal k for any ability distribution (e.g., k=0.09 for exponential distribution and k=0.15 for the uniform distribution), which has practical applications. Additionally, we establish a tight upper bound of k̅=0.3162 for any distribution. Finally, we compare total effort across different contest configurations: the optimal contest without a shortlist achieves (1) , a two-contestant winner-take-all contest yields (log n) , and the optimal contest with a shortlist reaches (n) . This demonstrates the significant advantage of incorporating a shortlist.
We study the stable matching problem in which participants have complete and weak preference lists over the other participants. We focus on a new structure called the “super-stable partition” that characterizes the super-stability of the roommate problem under weak preferences. We show that a super-stable partition always exists and can be computed in polynomial time O(n^2) . The takeaway result is that a super-stable roommate matching exists if and only if the cardinality of each super ring in the super-stable partition is even.
A player’s menu—the set of all allocations she might receive given other players’ reports—is a foundational concept in mechanism design. Menus capture the structure of strategyproofness: a mechanism is strategyproof if and only if each player always receives her favorite allocation from her menu. This leads to a number of natural questions: How can canonical mechanisms be described in terms of a player picking from her menu? How does one player picking from their menu affect other players’ allocations and menus? In this paper, we investigate such questions for the canonical assignment problem with unit demand and transferable utility: the Unit Demand VCG auction (UD). We study ascending and descending unit-demand auctions, a natural algorithmic model for calculating and describing auctions. We establish (i) a descending-price auction can calculate the outcome of UD while computing any particular bidder’s menu along the way, while (ii) no ascending-price auction can do the same. We then study complexity measures concerning menus in UD, which capture how one bidder’s pick from her menu influences the full allocation and another bidder’s menu. We show that (iii) the outcome-effect complexity is (n^2) , meaning that a bidder can affect the full allocation in a way that is intractable-to-characterize. Finally, we show (iv) the options-effect complexity is O(n) , giving an efficient representation of how one bidder can affect another’s menu. Our results for (i) and (iv) stem from a new characterization of how removing one vertex effects the max-weight matching in a bipartite graph. Perhaps interestingly, the complexity in (ii) and (iii) arises not from one bidder’s impact on the allocation of items, but their impact on the prices other bidders must pay.
The problem of designing budget-feasible mechanisms (BFMs) for submodular valuation functions has been extensively studied for more than a dozen years. In this NP-hard problem, a buyer aims to procure goods or services from multiple strategic sellers under a fixed budget, with the goal of maximizing a submodular valuation function. For monotone submodular valuations, the best-known approximation guarantee is 4, achieved by a randomized mechanism proposed seven years ago at WINE 2018. Since then, it has remained an open problem whether there exists a truthful BFM with an approximation guarantee better than 4—even without any computational or complexity constraints. In this paper, we break this long-standing approximation barrier for the first time by presenting truthful deterministic and randomized BFMs that achieve an approximation guarantee of 3.798 with linear-time complexity. Moreover, as our proposed BFMs are built upon novel unified algorithmic frameworks, they simultaneously achieve improved performance guarantees over state-of-the-art BFMs for non-monotone submodular valuations. To achieve the aforementioned advantages, our randomized BFM adopts the celebrated multi-round (descending) clock auction paradigm, but offers prices in only a single round to ensure linear-time complexity. It further introduces a novel pricing technique that integrates a potential function with random sampling, allowing sellers with sufficiently high quality to be randomly accepted as candidate winners. As a result, optimizing the approximation guarantee of our randomized BFM reduces to solving a non-trivial nonlinear optimization problem, which, fortunately, admits a closed-form solution. This solution also inspires a two-stage pricing technique that enables us to derandomize the mechanism, giving rise to our unified deterministic BFM. This deterministic mechanism matches the approximation guarantee of the randomized BFM for monotone submodular valuations, and achieves a slightly weaker but still superior approximation guarantee for non-monotone submodular valuations.
