On the Convergence of Learning Algorithms in Bayesian Auction Games

TL;DR

The paper reframes Bayes--Nash equilibrium in Bayesian auctions with continuous types as infinite-dimensional variational inequalities and analyzes convergence of learning dynamics via the Gateaux derivative. It demonstrates that both monotonicity and Minty-type conditions fail for the first- and second-price auctions, though the second-price case admits a Minty-type solution while the first-price does not; nevertheless, the Bayes--Nash equilibrium is unique within the class of uniformly increasing bid functions, implying gradient-based learning will converge if it reaches the interior solution. Through analytic and numerical work, it shows that gradient flows converge to the equilibrium despite Minty-condition violations, and provides explicit forms for the equilibria: $eta^*(x)=x$ for second-price and $eta^*(x)=rac{1}{G(x)}rac{d}{dx}igl( extstylerac{1}{2}G(x)x^2igr)$ (i.e., $eta^*(x)=rac{x}{2}$ when $F$ is uniform and $n=2$) for first-price, with the latter not admitting an interior Minty solution. The results offer explanation for observed convergence in diverse auction models and underscore the role of VI structure over monotonicity alone in guiding learning dynamics in auctions with continuous types. The work suggests that convergence guarantees may rely on the existence and uniqueness of interior VI solutions rather than monotonicity, with practical implications for autonomous bidding agents and numerical solvers in auction models.

Abstract

Equilibrium problems in Bayesian auction games can be described as systems of differential equations. Depending on the model assumptions, these equations might be such that we do not have a rigorous mathematical solution theory. The lack of analytical or numerical techniques with guaranteed convergence for the equilibrium problem has plagued the field and limited equilibrium analysis to rather simple auction models such as single-object auctions. Recent advances in equilibrium learning led to algorithms that find equilibrium under a wide variety of model assumptions. We analyze first- and second-price auctions where simple learning algorithms converge to an equilibrium. The equilibrium problem in auctions is equivalent to solving an infinite-dimensional variational inequality (VI). Monotonicity and the Minty condition are the central sufficient conditions for learning algorithms to converge to an equilibrium in such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity holds for the respective VIs. The second-price auction's equilibrium is a Minty-type solution, but the first-price auction is not. However, the Bayes--Nash equilibrium is the unique solution to the VI within the class of uniformly increasing bid functions, which ensures that gradient-based algorithms attain the equilibrium in case of convergence, as also observed in numerical experiments.

Figures (1)

• Figure 1: Gradients for piecewise linear bid functions with two pieces.

Theorems & Definitions (14)

• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Proposition 4.1
• proof
• Lemma 5.1
• proof
• Remark 5.1
• Lemma 5.2