Prior-Independent Auctions for Heterogeneous Bidders

We study the design of prior-independent auctions in a setting with heterogeneous bidders. In particular, we consider the setting of selling to $n$ bidders whose values are drawn from $n$ independent but not necessarily identical distributions. We work in the robust auction design regime, where we assume the seller has no knowledge of the bidders' value distributions and must design a mechanism that is prior-independent. While there have been many strong results on prior-independent auction design in the i.i.d. setting, not much is known for the heterogeneous setting, even though the latter is of significant practical importance. Unfortunately, no prior-independent mechanism can hope to always guarantee any approximation to Myerson's revenue in the heterogeneous setting; similarly, no prior-independent mechanism can consistently do better than the second-price auction. In light of this, we design a family of (parametrized) randomized auctions which approximates at least one of these benchmarks: For heterogeneous bidders with regular value distributions, our mechanisms either achieve a good approximation of the expected revenue of an optimal mechanism (which knows the bidders' distributions) or exceeds that of the second-price auction by a certain multiplicative factor. The factor in the latter case naturally trades off with the approximation ratio of the former case. We show that our mechanism is optimal for such a trade-off between the two cases by establishing a matching lower bound. Our result extends to selling $k$ identical items to heterogeneous bidders with an additional $O\big(\ln^2 k\big)$-factor in our trade-off between the two cases

Optimal (and Benchmark-Optimal) Competition Complexity for Additive Buyers over Independent Items

The Competition Complexity of an auction setting refers to the number of additional bidders necessary in order for the (deterministic, prior-independent, dominant strategy truthful) Vickrey-Clarke-Groves mechanism to achieve greater revenue than the (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. We prove that the competition complexity of $n$ bidders with additive valuations over $m$ independent items is at most $n(\ln(1+m/n)+2)$, and also at most $9\sqrt{nm}$. When $n \leq m$, the first bound is optimal up to constant factors, even when the items are i.i.d. and regular. When $n \geq m$, the second bound is optimal for the benchmark introduced in [EFFTW17a] up to constant factors, even when the items are i.i.d. and regular. We further show that, while the Eden et al. benchmark is not necessarily tight in the $n \geq m$ regime, the competition complexity of $n$ bidders with additive valuations over even $2$ i.i.d. regular items is indeed $\omega(1)$. Our main technical contribution is a reduction from analyzing the Eden et al. benchmark to proving stochastic dominance of certain random variables

The Power of Two-sided Recruitment in Two-sided Markets

We consider the problem of maximizing the gains from trade (GFT) in two-sided markets. The seminal impossibility result by Myerson shows that even for bilateral trade, there is no individually rational (IR), Bayesian incentive compatible (BIC) and budget balanced (BB) mechanism that can achieve the full GFT. Moreover, the optimal BIC, IR and BB mechanism that maximizes the GFT is known to be complex and heavily depends on the prior. In this paper, we pursue a Bulow-Klemperer-style question, i.e. does augmentation allow for prior-independent mechanisms to beat the optimal mechanism? Our main result shows that in the double auction setting with $m$ i.i.d. buyers and $n$ i.i.d. sellers, by augmenting $O(1)$ buyers and sellers to the market, the GFT of a simple, dominant strategy incentive compatible (DSIC), and prior-independent mechanism in the augmented market is least the optimal in the original market, when the buyers' distribution first-order stochastically dominates the sellers' distribution. Furthermore, we consider general distributions without the stochastic dominance assumption. Existing hardness result by Babaioff et al. shows that no fixed finite number of agents is sufficient for all distributions. In the paper we provide a parameterized result, showing that $O(log(m/rn)/r)$ agents suffice, where $r$ is the probability that the buyer's value for the item exceeds the seller's value