On the Uniqueness of Bayesian Coarse Correlated Equilibria in Standard First-Price and All-Pay Auctions

Abstract

We study the Bayesian coarse correlated equilibrium (BCCE) of continuous and discretised first-price and all-pay auctions under the standard symmetric independent private-values model. Our study is motivated by the question of how the canonical Bayes-Nash equilibrium (BNE) of the auction relates to the outcomes learned by buyers utilising no-regret algorithms. Numerical experiments show that in two buyer first-price auctions the Wasserstein-$2$ distance of buyers' marginal bid distributions decline as $O(1/n)$ in the discretisation size in instances where the prior distribution is concave, whereas all-pay auctions exhibit similar behaviour without prior dependence. To explain this convergence to a near-equilibrium, we study uniqueness of the BCCE of the continuous auction. Our uniqueness results translate to provable convergence of deterministic self-play to a near equilibrium outcome in these auctions. In the all-pay auction, we show that independent of the prior distribution there is a unique BCCE with symmetric, differentiable, and increasing bidding strategies, which is equivalent to the unique strict BNE. In the first-price auction, we need stronger conditions. Either the prior is strictly concave or the learning algorithm has to be restricted to strictly increasing strategies. Without such strong assumptions, no-regret algorithms can end up in low-price pooling strategies. This is important because it proves that in repeated first-price auctions such as in display ad actions, algorithmic collusion cannot be ruled out without further assumptions even if all bidders rely on no-regret algorithms.

Figures (5)

• Figure 1: A buyer's distribution of bids for Wasserstein distance maximising BCCE of first-price and all-pay auctions with two buyers, for $V_n = \{0,1/20,\ldots,1\}$ and $B_n = \beta^\mathcal{A}(V_n)$. For the first-price auction, the BCCE correlates on both buyers bidding the equilibrium strategy or both buyers bidding $0$. The form of the BCCE for the all-pay auction is less clear instead, and the maximal Wasserstein-$2$ distance BCCE does not appear to admit an obvious parametric form in $n$.
• Figure 2: Values of (\ref{['opt:Wasserstein']}) for discretised first-price auctions with two buyers. In this and all further figures, (1) dashed (solid) lines correspond to bounds for B(C)CE, and (2) dotted lines correspond to a $O(n)$ rational function fit for the BCCE bounds with a degree $1$ numerator and degree $2$ denominator.
• Figure 3: Values of (\ref{['opt:Wasserstein']}) for various discretised all-pay auctions with two buyers.
• Figure 4: Values of (\ref{['opt:concentration']}) for the uniform discrete distribution on $V_n = \{0,1/n,\ldots,1.\}$ and $B_n = V_n$, for first-price and all-pay auctions.
• Figure 5: Values of $\epsilon_i(v_i,b'_i)$ and $\rho(.5,b_i,v_{-i})$ for the first-price auction with uniform discrete prior distribution on $V_n = B_n = \{0,1/20,\ldots,1.\}$.

Theorems & Definitions (66)

• Corollary
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.5
• Example 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9