The Role of Transparency in Repeated First-Price Auctions with Unknown Valuations

Abstract

We study the problem of regret minimization for a single bidder in a sequence of first-price auctions where the bidder discovers the item's value only if the auction is won. Our main contribution is a complete characterization, up to logarithmic factors, of the minimax regret in terms of the auction's \emph{transparency}, which controls the amount of information on competing bids disclosed by the auctioneer at the end of each auction. Our results hold under different assumptions (stochastic, adversarial, and their smoothed variants) on the environment generating the bidder's valuations and competing bids. These minimax rates reveal how the interplay between transparency and the nature of the environment affects how fast one can learn to bid optimally in first-price auctions.

Figures (4)

• Figure 1: The utility function is generally neither Lipschitz nor continuous. If $M_t \le V_t$ (top left plot), then $\mathop{\mathrm{Util}}\nolimits_t$ is upper-semi continuous and one-sided Lipschitz; conversely, if $M_t \ge V_t$ (bottom left plot), then $\mathop{\mathrm{Util}}\nolimits_t$ is still one-sided Lipschitz---from the other side---and lower-semi continuous. Summing up the two types of utilities results in a total utility that may be neither one-sided Lipschitz nor semi-continuous (right plot, where the two utility functions of the other two plots are summed up. There, $b^\star$ is the optimal bid and $\Delta$ is the neighborhood of $b^\star$ where the total utility is "good enough").
• Figure 2: Left: The support of the base density $f$ lies inside the yellow and green regions. The perturbation $g_{w,\varepsilon}$ of $f$ occurs inside the green region, where the four rectangles $R^1_{w,\varepsilon},\dots, R^4_{w,\varepsilon}$ (in red and blue) lie. Right: The corresponding qualitative plots of $b \mapsto \mathbb{E}[\mathop{\mathrm{Util}}\nolimits_t(b)]$ (black, dotted) and $p \mapsto \mathbb{E}^{w,\varepsilon}[\mathop{\mathrm{Util}}\nolimits_t(b)]$ (red, solid).
• Figure 3: The expected utility function for three different distributions: $\mathbb{P}^0$ in purple, $\mathbb{P}^+$ in orange, and $\mathbb{P}^+$ in green.
• Figure 4: A representation of the map $\iota$ through which the bids in the first-price auction problem are related to the $K$-arms of the bandit problem. The interval $[0,1]$ is partitioned in $K$ disjoint intervals, the first and the last one of length $1/4+2\varepsilon$, and all the ones in between of length $2\varepsilon$. $\iota$ maps each bid to the index of the interval to which it belongs.

Theorems & Definitions (44)

• Definition 1: HaghtalabRS21
• Theorem 1
• proof
• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Theorem 2
• proof
• Theorem 3