Learning to Bid in FCR Markets: A Best-of-Both-Worlds Approach

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.

We study the problem of learning to bid when the bidder's value is dynamic, i.e., when the current value depends on past outcomes. Specifically, we consider a bidder participating in repeated second-price auctions whose value depends on the time elapsed since their last successful bid, with auctions arriving in continuous time and only aggregated feedback revealed at the end of the horizon. Such a bidder must (1) balance the immediate benefit of winning the current auction against its impact on future values and (2) learn unknown environmental parameters. We derive regret bounds for a class of learning methods that combine plug-in estimators with a differential-equation characterization of the optimal policy, and show that a specific confidence bound algorithm learns the optimal policy with a near optimal regret of $\widetilde{O}(\log N)$ for piecewise linear primitives, and $\widetilde{O}(N^{1/3})$ for general, smooth primitives, achieving these regrets without explicit randomization. These theoretical results are supported by numerical experiments.

The growing share of Renewable Energy Sources (RES) in modern power systems increases both grid imbalances and frequency deviations, reinforcing the need for ancillary services such as Frequency Containment Reserve (FCR) and passive balancing. Battery Energy Storage Systems (BESS) are well-suited for these services, but prior research typically relies on uniform FCR bids that remain constant throughout the control period. Such static bids fail to fully exploit BESS flexibility, as they do not balance the trade-off between reserving energy for FCR delivery and using it for imbalance arbitrage, limiting the achievable value in value-stacking settings. To address this limitation, we propose a two-stage control framework for the European context that introduces non-uniform FCR bids. In the first stage, we derive a time-varying bid sequence using data-driven Monte Carlo (MC) optimization. In the second stage, a Deep Reinforcement Learning (DRL) agent leverages the residual flexibility for real-time imbalance trading while proactively managing the State of Energy (SoE) to ensure compliance with FCR requirements. The framework is presented as a proof of concept, highlighting the potential benefits of time-varying bidding strategies. By incorporating daily cycle budgets and time-varying reserve commitments, our approach achieves a 7.56% profit increase compared to uniform baselines. These results show that non-uniform bidding can unlock additional value by more effectively aligning reserve obligations with rapidly changing imbalance opportunities.