We study the pay-as-bid auction game, a supply function model with discriminatory pricing and asymmetric firms. In this game, strategies are non-decreasing supply functions relating pric to quantity and the exact choice of the strategy space turns out to be a crucial issue: when it includes all non-decreasing continuous functions, pure-strategy Nash equilibria often fail to exist. To overcome this, we restrict the strategy space to the set of Lipschitz-continuous functions and we prove that Nash equilibria always exist (under standard concavity assumptions) and consist of functions that are affine on their own support and have slope equal to the maximum allowed Lipschitz constant. We further show that the Nash equilibrium is unique up to the market-clearing price when the demand is affine and the asymmetric marginal production costs are homogeneous in zero. For quadratic production costs, we derive a closed-form expression and we compute the limit as the allowed Lipschitz constant grows to infinity. Our results show that in the limit the pay-as-bid auction game achieves perfect competition with efficient allocation and induces a lower market-clearing price compared to supply function models based on uniform price auctions.

Over the past decade, bidding in power markets has attracted widespread attention. Reinforcement Learning (RL) has been widely used for power market bidding as a powerful AI tool to make decisions under real-world uncertainties. However, current RL methods mostly employ low dimensional bids, which significantly diverge from the N price-power pairs commonly used in the current power markets. The N-pair bidding format is denoted as High Dimensional Bids (HDBs), which has not been fully integrated into the existing RL-based bidding methods. The loss of flexibility in current RL bidding methods could greatly limit the bidding profits and make it difficult to tackle the rising uncertainties brought by renewable energy generations. In this paper, we intend to propose a framework to fully utilize HDBs for RL-based bidding methods. First, we employ a special type of neural network called Neural Network Supply Functions (NNSFs) to generate HDBs in the form of N price-power pairs. Second, we embed the NNSF into a Markov Decision Process (MDP) to make it compatible with most existing RL methods. Finally, experiments on Energy Storage Systems (ESSs) in the PJM Real-Time (RT) power market show that the proposed bidding method with HDBs can significantly improve bidding flexibility, thereby improving the profit of the state-of-the-art RL bidding methods.

We consider a dynamic system with multiple types of customers and servers. Each type of waiting customer or server joins a separate queue, forming a bipartite graph with customer-side queues and server-side queues. The platform can match the servers and customers if their types are compatible. The matched pairs then leave the system. The platform will charge a customer a price according to their type when they arrive and will pay a server a price according to their type. The arrival rate of each queue is determined by the price according to some unknown demand or supply functions. Our goal is to design pricing and matching algorithms to maximize the profit of the platform with unknown demand and supply functions, while keeping queue lengths of both customers and servers below a predetermined threshold. This system can be used to model two-sided markets such as ride-sharing markets with passengers and drivers. The difficulties of the problem include simultaneous learning and decision making, and the tradeoff between maximizing profit and minimizing queue length. We use a longest-queue-first matching algorithm and propose a learning-based pricing algorithm, which combines gradient-free stochastic projected gradient ascent with bisection search. We prove that our proposed algorithm yields a sublinear regret $\tilde{O}(T^{5/6})$ and queue-length bound $\tilde{O}(T^{2/3})$, where $T$ is the time horizon. We further establish a tradeoff between the regret bound and the queue-length bound: $\tilde{O}(T^{1-\gamma/4})$ versus $\tilde{O}(T^{\gamma})$ for $\gamma \in (0, 2/3].$

With the growing penetration of renewable energy resource, electricity market prices have exhibited greater volatility. Therefore, it is important for Energy Storage Systems(ESSs) to leverage the multidimensional nature of energy market bids to maximize profitability. However, current learning methods cannot fully utilize the high-dimensional price-quantity bids in the energy markets. To address this challenge, we modify the common reinforcement learning(RL) process by proposing a new bid representation method called Neural Network Embedded Bids (NNEBs). NNEBs refer to market bids that are represented by monotonic neural networks with discrete outputs. To achieve effective learning of NNEBs, we first learn a neural network as a strategic mapping from the market price to ESS power output with RL. Then, we re-train the network with two training modifications to make the network output monotonic and discrete. Finally, the neural network is equivalently converted into a high-dimensional bid for bidding. We conducted experiments over real-world market datasets. Our studies show that the proposed method achieves 18% higher profit than the baseline and up to 78% profit of the optimal market bidder.

We study deregulated power markets with strategic power suppliers. In deregulated markets, each supplier submits its supply function (i.e., the amount of electricity it is willing to produce at various prices) to the independent system operator (ISO), who based on the submitted supply functions, dispatches the suppliers to clear the market with minimal total generation cost. If all suppliers reported their true marginal cost functions as supply functions, the market outcome would be efficient (i.e., the total generation cost is minimized). However, when suppliers are strategic and aim to maximize their own profits, the reported supply functions are not necessarily the true marginal cost functions, and the resulting market outcome may be inefficient. The efficiency loss depends crucially on the topology of the underlying transmission network. This paper provides an analytical upper bound of the efficiency loss due to strategic suppliers, and proves that the bound is tight under a large class of transmission networks (i.e., weakly cyclic networks). Our upper bound sheds light on how the efficiency loss depends on the transmission network topology (e.g., the degrees of nodes, the admittances and flow limits of transmission lines).

Every producer chooses as its strategy a supply function returning the quantity S(p) that it is willing to sell at a minimum unit price p. The market clears at the price at which the aggregate demand intersects the total supply and firms are paid the bid prices. We study a game theoretic model of competition among such firms and focus on its equilibria (Supply function equilibrium). The game we consider is a generalization of both models where firms can either set a fixed quantity (Cournot model) or set a fixed price (Bertrand model). Our main result is to prove existence and provide a characterization of (pure strategy) Nash equilibria in the space of K-Lipschitz supply functions.