When two companies bid to buy the same target, no one knows exactly what the target is worth. Each bidder pays for due diligence: costly, imperfect homework that sharpens its own private estimate before it bids. How much of that homework is worth buying? We build a simple computer model of the bidding contest and let it teach itself to bid well by playing against itself, the way a game engine learns chess. The economic question, how much diligence pays for itself, and the computational question, when the contest becomes too complex to solve exactly, are both controlled by a single thing: how many pieces of private information a bidder carries. Our main finding is that the right amount of diligence is modest and finite. It falls as diligence gets more expensive, and it falls further when both sides are doing their homework, because competition erodes the value of knowing more. We also test a recent claim from AI research: that simple, general self-play methods can rival the specialized, expensive algorithms usually built for games like these. Running on an ordinary laptop with no costly frontier AI, we find the simple methods are the best of the self-learning approaches, though purpose-built exact methods still win whenever the game is small enough to solve outright. The simple methods earn their keep only once the game grows too large to solve exactly, which is the regime real deals live in, and there we show they still find strong bidding strategies. The contribution is threefold: a cheap, reproducible way to study deal-making under uncertainty; a concrete, model-based answer to how much due diligence is worth buying; and evidence about when lightweight, general-purpose AI is good enough to replace specialized methods. We release all the games, code, and experiments.
We study repeated bidding in multi-unit discriminatory (pay-as-bid) auctions for a single bidder with per-round utility equal to value minus $α$ times payment, where $α\in[0,1]$ is a cost-of-capital parameter. The bidder aims to maximize cumulative utility over $T$ rounds subject to a total budget $B$. The problem is challenging even without budgets: the action space is exponential in $M$, the maximum demand of the bidder and the valuation vector (context) varies over time. Exploiting a decomposition of utility across units, we develop polynomial-time learning algorithms based on shortest paths in a directed acyclic graph, obtaining sublinear regret under both full-information and bandit feedback. In the bandit setting, the regret is independent of the number of contexts due to complete cross-learning: observing the utility of the chosen action under the realized context reveals the utility for the same action under all counterfactual contexts. With budget constraints, when the average normalized per-round budget $ρ=\frac{B}{MT}<1$, we design a coupled primal-dual algorithm in which the DAG-based procedure uses dual-adjusted edge weights for primal updates, while online gradient descent updates the dual variable, yielding $ρ$-approximate sublinear regret. Finally, we give implementations whose per-round time and space are independent of the number of contexts, enabling scalability to large or even infinite context spaces.
Learning to bid in repeated multi-unit auctions with bandit feedback poses a fundamental challenge. Existing methods often rely on rigid explore-then-exploit schedules, assume stationary adversaries, and optimize solely for bidder utility, thereby limiting adaptability and strategic robustness. To address these limitations, we introduce the A3M framework, which integrates adaptive deep reinforcement learning (DRL), explicit adversarial reasoning, and principled multi-objective reward design for online auction strategy optimization. A3M employs an actor-critic DRL backbone to dynamically balance exploration and exploitation, an opponent model for fictitious play against non-stationary adversaries, and a composite reward function to jointly maximize utility, auctioneer revenue, and fairness. We provide the first comprehensive empirical evaluation of this integrated approach against established baselines in both discriminatory and uniform price auctions. Results show that A3M reduces final regret by 30--40\% in standard settings, maintains robust performance against adversarial strategy shifts, scales favorably with the number of units $K$, and enables tunable multi-objective trade-offs. An extensive ablation study confirms the necessity of each core component. Our work establishes A3M as a powerful and flexible framework for learning in complex auction environments.
Luigi Foscari, Matilde Tullii, Vianney Perchetstat.ML cs.GT cs.LG
Shilling is the use of artificial bids to make competition appear stronger and push prices upward. We study repeated first-price auctions in which shilling affects feedback but not allocation: the learner wins or loses against the real competing bid, but after a loss observes the maximum of the real bid and an independent shill bid. Thus the manipulation changes what the learner observes and hence how it learns to bid, without changing the outcome of the current auction. We analyze regret with respect to the best bid benchmark, assuming that the shill-bid distribution is known. Even then, shilling can mask the real bid, while useful side information appears only through intermittent low-shill events. Our algorithm combines a robust interval-elimination branch, which ignores the shilled report and achieves the dynamic-pricing rate $\tilde{\mathcal{O}}(T^{2/3})$, with an optimistic branch that debiases losing-side reports and exploits the resulting suffix information when it is reliable and achieves the first-price auctions rate $\tilde{\mathcal{O}}(\sqrt{T})$. A validation and racing procedure lets the algorithm use these optimistic updates without knowing the right scale or feedback geometry in advance. We complement the upper bounds with a matching lower bound, up to logarithmic factors, in the single-active-region case. Overall, the results show that even feedback-only shilling can sharply alter the statistical difficulty of repeated bidding.