Prior-free auctions are robust auctions that assume no distribution over bidders' valuations and provide worst-case (input-by-input) approximation guarantees.In contrast to previous work on this topic, we pursue good prior-free auctions with non-identical bidders.Prior-free auctions can approximate meaningful benchmarks for non-identical bidders only when sufficient qualitative information about the bidder asymmetry is publicly known.We consider digital goods auctions where there is a total ordering of the bidders that is known to the seller, where earlier bidders are in some sense thought to have higher valuations.We use the framework of Hartline and Roughgarden (STOC '08) to define an appropriate revenue benchmark: the maximum revenue that can be obtained from a bid vector using prices that are nonincreasing in the bidder ordering and bounded above by the second-highest bid.This monotone-price benchmark is always as large as the well-known fixed-price benchmark F (2) , so designing prior-free auctions with good approximation guarantees is only harder.By design, an auction that approximates the monotone-price benchmark satisfies a very strong guarantee: it is, in particular, simultaneously near-optimal for essentially every Bayesian environment in which bidders' valuation distributions have nonincreasing monopoly prices, or in which the distribution of each bidder stochastically dominates that of the next.Even when there is no distribution over bidders' valuations, such an auction still provides a quantifiable input-by-input performance guarantee.In this paper, we design a simple O(1)-competitive prior-free auction for digital goods with ordered bidders.We also extend the monotone-price benchmark and our O(1)-competitive prior-free auction to multi-unit settings with limited supply.

Network Extenality and Mechanism Design

We study revenue maximization in settings where agents’ valuations exhibit positive network externalities. In our model, items have unlimited supply, and agents are unit demand. In a departure from previous literature, we assume agents have value based externalities, meaning that their valuation depends not only on their own signal, but also on the signals of other agents in their neighborhood who win the item. We give a complete characterization of ex-post incentive compatible and individually rational auctions in this setting. Using this characterization, we show that the optimal auction is in fact deterministic, and can be computed in polynomial time when the agents’ signals are independent. We further show a constant factor approximation when the signals of agents are correlated, and an optimal mechanism in this case for a constant number of bidders.

A central problem in Microeconomics is to design auctions with good revenue properties. In this setting, the bidders' valuations for the items are private knowledge, but they are drawn from publicly known prior distributions. The goal is to find a truthful auction (no bidder can gain in utility by misreporting her valuation) that maximizes the expected revenue. Naturally, the optimal-auction is sensitive to the prior distributions. An intriguing question is to design a truthful auction that is oblivious to these priors, and yet manages to get a constant factor of the optimal revenue. Such auctions are called prior-free. Goldberg et al. presented a constant-approximate prior-free auction when there are identical copies of an item available in unlimited supply, bidders are unit-demand, and their valuations are drawn from i.i.d. distributions. The recent work of Leonardi et al. [STOC 2012] generalized this problem to non i.i.d. bidders, assuming that the auctioneer knows the ordering of their reserve prices. Leonardi et al. proposed a prior-free auction that achieves a $O(\log^* n)$ approximation. We improve upon this result, by giving the first prior-free auction with constant approximation guarantee.

We consider the problem of a monopolist seller who wants to sell some items to a set of buyers. The buyers are strategic, unit-demand, and connected by a social network. Furthermore, the utility of a buyer is a decreasing function of the number of neighbors who do not own the item. In other words, they exhibit negative externalities, deriving utility from being unique in their purchases. In this model, any fixed setting of the price induces a sub-game on the buyers. We show that it is an exact potential game which admits multiple pure Nash Equilibria. A natural problem is to compute those pure Nash equilibria that raise the most and least revenue for the seller. These correspond respectively to the most optimistic and most pessimistic revenues that can be raised. We show that the revenues of both the best and worst equilibria are hard to approximate within sub-polynomial factors. Given this hardness, we consider a relaxed notion of pricing, where the price for the same item can vary within a constant factor for different buyers. We show a 4-approximation to the pessimistic revenue when the prices are relaxed by a factor of 4. The interesting aspect of this algorithm is that it uses a linear programming relaxation that only encodes part of the strategic behavior of the buyers in its constraints, and rounds this relaxation to obtain a starting configuration for performing relaxed Nash dynamics. Finally, for the maximum revenue Nash equilibrium, we show a 2-approximation for bipartite graphs (without price relaxation), and complement this result by showing that the problem is NP-Hard even on trees.