This paper develops tools for welfare and revenue analyses of Bayes-Nash equilibria in asymmetric auctions with single-dimensional agents. We employ these tools to derive approximation results for social welfare and revenue. Our approach separates the smoothness framework of [e.g., Syrgkanis and Tardos, 2013] into two distinct parts, isolating the analysis common to any auction from the analysis specific to a given auction. The first part relates a bidder's contribution to welfare in equilibrium to their contribution to welfare in the optimal auction using the price the bidder faces for additional allocation. Intuitively, either an agent's utility and hence contribution to welfare is high, or the price she has to pay for additional allocation is high relative to her value. We call this condition value covering; it holds in every Bayes-Nash equilibrium of any auction. The second part, revenue covering, relates the prices bidders face for additional allocation to the revenue of the auction, using an auction's rules and feasibility constraints. Combining the two parts gives approximation results to the optimal welfare, and, under the right conditions, the optimal revenue. In mechanisms with reserve prices, our welfare results show approximation with respect to the optimal mechanism with the same reserves.As a centerpiece result, we analyze the single-item first-price auction with individual monopoly reserves (the price that a monopolist would post to sell to that agent alone; these reserves are generally distinct for agents with values drawn from distinct distributions). When each distribution satisfies the regularity condition of Myerson [1981], the auction's revenue is at least a 2e/(e − 1) ≈ 3.16 approximation to the revenue of the optimal auction. We also give bounds for matroid auctions with first-price or all-pay semantics, the generalized first-price position auction, and pay-your-bid auctions for single-minded combinatorial auctions. Finally, we give an extension theorem for simultaneous composition, i.e., when multiple auctions are run simultaneously, with single-valued, unit-demand agents. * We thank Vasilis Syrgkanis for comments on a prior version of this paper for which simultaneous composition did not hold, suggesting the study of simultaneous composition and for perspective on price-of-anarchy methodology.

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This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents' values are independently distributed. The previous best bound was 1 − 1/e ≈ .63, derived in Syrgkanis and Tardos (2013) using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example (from Hartline et al. (2014)), the first price auction achieves a ≈ .869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1 − 1/e ≈ .63 bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents' value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.

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Rotating savings and credit associations (roscas) are informal financial organizations common in settings where communities have reduced access to formal financial institutions. In a rosca, a fixed group of participants regularly contribute sums of money to a pot. This pot is then allocated periodically using lottery, aftermarket, or auction mechanisms. Roscas are empirically well-studied in economics. They are, however, challenging to study theoretically due to their dynamic nature. Typical economic analyses of roscas stop at coarse ordinal welfare comparisons to other credit allocation mechanisms, leaving much of roscas' ubiquity unexplained. In this work, we take an algorithmic perspective on the study of roscas. Building on techniques from the price of anarchy literature, we present worst-case welfare approximation guarantees. We further experimentally compare the welfare of outcomes as key features of the environment vary. These cardinal welfare analyses further rationalize the prevalence of roscas. We conclude by discussing several other promising avenues.