Prior-Independent Bidding Strategies for First-Price Auctions

TL;DR

A principled analysis of prior-independent bidding strategies for first-price auctions using worst-case regret as the performance measure and a systematic and computationally-tractable procedure for deriving minimax-optimal bidding strategies.

Abstract

First-price auctions are one of the most popular mechanisms for selling goods and services, with applications ranging from display advertising to timber sales. Unlike their close cousin, the second-price auction, first-price auctions do not admit a dominant strategy. Instead, each buyer must design a bidding strategy that maps values to bids -- a task that is often challenging due to the lack of prior knowledge about competing bids. To address this challenge, we conduct a principled analysis of prior-independent bidding strategies for first-price auctions using worst-case regret as the performance measure. First, we develop a technique to evaluate the worst-case regret for (almost) any given value distribution and bidding strategy, reducing the complex task of ascertaining the worst-case competing-bid distribution to a simple line search. Next, building on our evaluation technique, we minimize worst-case regret and characterize a minimax-optimal bidding strategy for every value distribution. We achieve it by explicitly constructing a bidding strategy as a solution to an ordinary differential equation, and by proving its optimality for the intricate infinite-dimensional minimax problem underlying worst-case regret minimization. Our construction provides a systematic and computationally-tractable procedure for deriving minimax-optimal bidding strategies. When the value distribution is continuous, it yields a deterministic strategy that maps each value to a single bid. We also show that our minimax strategy significantly outperforms the uniform-bid-shading strategies advanced by prior work. Our result allows us to precisely quantify, through minimax regret, the performance loss due to a lack of knowledge about competing bids. We leverage this to analyze the impact of the value distribution on the performance loss, and find that it decreases as the buyer's values become more dispersed.

Figures (4)

• Figure 1: ODE which constructs a minimax-optimal bidding strategy for the uniform value distribution. If one starts at (0,0) and moves with a slope equal to the ratio of the dashed line to dotten line, then the resulting curve will trace out the CDF of bids under a minimax-optimal bidding strategy. The strategy is then to simply bid the corresponding quantile for every value, i.e., bid $b$ for value $v$ if and only if the quantiles of $b$ and $v$ are equal under the bid and value CDFs respectively.
• Figure 2: Summary of our insights. For every $a$, the blue curve represents the largest worst-case regret incurred by our minimax-optimal bidding strategy across all value distributions with a density bounded above $\frac{1}{1-a}$. The black curve corresponds to the worst-case regret of the bidding strategy from kasberger2023robust which bids $0.5 \cdot v$ for every value $v$, when the value distribution is a uniform on $[a,1]$.
• Figure 3: Performance of uniform-bid-shading strategies for Beta$(\rho,\rho)$ value distributions. In (a), we report the worst-case regret of different uniform-bid-shading strategies $s_{\alpha}$ as a function of $\rho$, when the value distributions is of the form Beta$(\rho,\rho)$. We also report the worst-case regret for the uniform-bid-shade strategy which uses the best $\alpha$ for each $\rho$ (see violet curve). In (b), we plot the best choice of $\alpha$ for each value of $\rho$.
• Figure 4: Impact of the value distribution on performance. We report the worst-case regret of various bidding strategies as a function of $a$, when the value distributions is of the form $\mathop{\mathrm{Unif}}\nolimits(a,1)$. The black curve corresponds to the uniform-bid-shading strategy with $\alpha = 0.5$, the violet curve to the strategy which uses the optimal uniform-bid-shading strategy for each $a$, and the blue curve to the minimax-optimal strategy.

Theorems & Definitions (37)

• Theorem 1
• Corollary 1
• Example 1
• Definition 1
• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4