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.

In routing games with infinitesimal players, it follows from well-known convexity arguments that equilibria exist and are unique (up to induced delays, and under weak assumptions on delay functions). In routing games with players that control large amounts of flow, uniqueness has been demonstrated only in limited cases: in 2-terminal, nearly-parallel graphs; when all players control exactly the same amount of flow; when latency functions are polynomials of degree at most three. In this work, we answer an open question posed by Cominetti, Correa, and Stier-Moses (ICALP 2006) and show that there may be multiple equilibria in atomic player routing games. We demonstrate this multiplicity via two specific examples. In addition, we show our examples are topologically minimal by giving a complete characterization of the class of network topologies for which unique equilibria exist. Our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations.

We study simple and approximately optimal auctions for agents with a particular form of risk-averse preferences. We show that, for symmetric agents, the optimal revenue (given a prior distribution over the agent preferences) can be approximated by the first-price auction (which is prior independent), and, for asymmetric agents, the optimal revenue can be approximated by an auction with simple form. These results are based on two technical methods. The first is for upper-bounding the revenue from a risk-averse agent. The second gives a payment identity for mechanisms with pay-your-bid semantics.

In routing games with infinitesimal players, it follows from well-known convexity arguments that equilibria exist and are unique. In routing games with atomic players with splittable flow, equilibria exist, but uniqueness of equilibria has been demonstrated only in limited cases: in two-terminal nearly parallel graphs, when all players control the same amount of flow, and when latency functions are polynomials of degree at most three. There are no known examples of multiple equilibria in these games. In this work, we show that in contrast to routing games with infinitesimal players, atomic splittable routing games admit multiple equilibria. We demonstrate this multiplicity via two specific examples. In addition, we show that our examples are topologically minimal by giving a complete characterization of the class of network topologies for which multiple equilibria exist. Our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations.

The first-price auction is popular in practice for its simplicity and transparency. Moreover, its potential virtues grow in complex settings where incentive compatible auctions may generate little or no revenue. Unfortunately, the first-price auction is poorly understood in theory because equilibrium is not a priori a credible predictor of bidder behavior.We take a dynamic approach to studying first-price auctions: rather than basing performance guarantees solely on static equilibria, we study the repeated setting and show that robust performance guarantees may be derived from simple axioms of bidder behavior. For example, as long as a loser raises her bid quickly, a standard first-price auction will generate at least as much revenue as a second-price auction.We generalize this dynamic technique to complex pay-your-bid auction settings: as long as losers do not wait too long to raise bids, a first-price auction will reach an envy-free state that implies a strong lower-bound on revenue; as long as winners occasionally experiment by lowering their bids, the outcome will near the boundary of this envy-free set so bidders do not overpay; and when players with the largest payoffs are the least patient, bids converge to the egalitarian equilibrium. Significantly, bidders need only know whether they are winning or losing in order to implement such behavior.Along the way, we find that the auctioneer's choice of bidding language is critical when generalizing beyond the single-item setting, and we propose a specific construction called the utility-target auction that performs well. The utility-target auction includes a bidder's final utility as an additional parameter, identifying the single dimension along which she wishes to compete. This auction is closely related to profit-target bidding in first-price and ascending proxy package auctions and gives strong revenue guarantees for a variety of complex auction environments. Of particular interest, the guaranteed existence of a pure-strategy equilibrium in the utility-target auction shows how Overture might have eliminated the cyclic behavior in their generalized first-price sponsored search auction if bidders could have placed more sophisticated bids.