# Prior-Free Dynamic Auctions with Low Regret Buyers

Advances in Neural Information Processing Systems (2019), pp. 4803-4813

## Abstract

We study the problem of how to repeatedly sell to a buyer running a no-regret, mean-based algorithm. Previous work (Braverman et al., EC '18) shows that it is possible to design effective mechanisms in such a setting that extract almost all of the economic surplus, but these mechanisms require the buyer's values each round to be drawn iid from a fixed distribution. In this paper, we do away with this assumption and consider the {\it prior-free setting} where the buyer's value each round is chosen adversarially (possibly adaptively).

We show that even in this prior-free setting, it is possible to extract a $(1-\varepsilon)$-approximation of the full economic surplus for any $\varepsilon > 0$. The menu complexity of our mechanism (the number of options offered to a buyer in any round) scales independently of the number of rounds $T$ and polynomially in $\varepsilon$. We show that this is optimal up to a polynomial factor; any mechanism achieving this approximation factor, even when values are drawn stochastically, requires menu complexity at least $\Omega(1/\varepsilon)$.

Finally, we examine what is possible when we constrain our mechanism to a natural auction format where overbidding is dominated. Braverman et al. show that even when values are drawn from a known stochastic distribution supported on $[1/H, 1]$, it is impossible in general to extract more than $O(\log\log H / \log H)$ of the economic surplus. We show how to achieve the same approximation factor in the {\it prior-independent} setting (where the distribution is unknown to the seller), and an approximation factor of $O(1 / \log H)$ in the prior-free setting.