Learning and Collusion in Multi-unit Auctions

TL;DR

This paper studies repeated multi-unit auctions with uniform pricing, developing algorithms for offline bid optimization and online learning, analyzing equilibrium stability, and highlighting differences in collusion susceptibility between two pricing formats.

Contribution

It introduces polynomial-time algorithms for bid optimization, designs online learning strategies with sublinear regret, and compares the collusion resistance of different auction formats.

Findings

The $K$-th price auction format resists collusion, unlike the $(K+1)$-st price format.

Efficient online algorithms with sublinear regret are developed for bidding strategies.

The offline bid optimization problem is solvable via maximum-weight path in a DAG.

Abstract

We consider repeated multi-unit auctions with uniform pricing, which are widely used in practice for allocating goods such as carbon licenses. In each round, $K$ identical units of a good are sold to a group of buyers that have valuations with diminishing marginal returns. The buyers submit bids for the units, and then a price $p$ is set per unit so that all the units are sold. We consider two variants of the auction, where the price is set to the $K$-th highest bid and $(K+1)$-st highest bid, respectively. We analyze the properties of this auction in both the offline and online settings. In the offline setting, we consider the problem that one player $i$ is facing: given access to a data set that contains the bids submitted by competitors in past auctions, find a bid vector that maximizes player $i$'s cumulative utility on the data set. We design a polynomial time algorithm for this problem, by showing it is equivalent to finding a maximum-weight path on a carefully constructed directed acyclic graph. In the online setting, the players run learning algorithms to update their bids as they participate in the auction over time. Based on our offline algorithm, we design efficient online learning algorithms for bidding. The algorithms have sublinear regret, under both full information and bandit feedback structures. We complement our online learning algorithms with regret lower bounds. Finally, we analyze the quality of the equilibria in the worst case through the lens of the core solution concept in the game among the bidders. We show that the $(K+1)$-st price format is susceptible to collusion among the bidders; meanwhile, the $K$-th price format does not have this issue.