The law of supply and demand asserts that in a perfectly competitive market, the price of a good adjusts to a market clearing price. In a market clearing price p the number of sellers willing to sell the good at p equals the number of sellers willing to buy the good at price p . In this work, we provide a mathematical foundation on the law of supply and demand through the lens of online learning. Specifically, we demonstrate that if each seller employs a no-swap regret algorithm to set their individual selling price--aiming to maximize its individual revenue--the collective pricing dynamics converge to the market-clearing price p . Our findings offer a novel perspective on the law of supply and demand, framing it as the emergent outcome of an adaptive learning processes among sellers.

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.

--With the large number of prosumers deploying distributed energy resources (DERs), integrating these prosumers into a transactive energy market (TEM) is a trend for the future smart grid. A community-based double auction market is considered a promising TEM that can encourage prosumers to participate and maximize social welfare. However, the traditional TEM is challenging to model explicitly due to the random bidding behavior of prosumers and uncertainties caused by the energy operation of DERs. Furthermore, although reinforcement learning algorithms provide a model-free solution to optimize prosumers' bidding strategies, their use in TEM is still challenging due to their scalability, stability, and privacy protection limitations. T o address the above challenges, in this study, we design a double auction-based TEM with multiple DERs-equipped prosumers to transparently and efficiently manage energy transactions. We also propose a deep reinforcement learning (DRL) model with distributed learning and execution to ensure the scalability and privacy of the market environment. Simulation results show that (1) the designed TEM and DRL model are robust; (2) the proposed DRL model effectively balances the energy payment and comfort satisfaction for prosumers and outperforms the state-of-the-art methods in optimizing the bidding strategies. ITH the extensive deployment of energy storage systems, solar photovoltaics (PVs), smart home appliances, and information technology, passive consumers in the traditional electricity market are gradually converted to active prosumers (producers + consumers) with distributed energy resources (DERs), who can monitor and control energy generation, consumption, storage, and transaction to achieve specific goals, such as balancing energy costs and user comfort levels [1]-[3]. However, the bi-directional energy and information flow, as well as the variability of distributed renewable energy, raises great challenges in the operation of power systems in a flexible and economically efficient way [4]. Liu are with the Department of Computer Science and Engineering, Santa Clara University, Santa Clara, CA, USA (e-mail: jun3525114@gmail.com, Li, M. Ghafouri, and J. Y an are with Concordia Institute for Information Systems Engineering, Concordia University, Montreal, QC, Canada (e-mail: {yuanliang.li, L. Hou is with Beijing University of Posts and Telecommunications, Beijing, China (e-mail: luyang.hou@bupt.edu.cn) Zhang is with the College of Information Engineering, Shenzhen University, Shenzhen, China (e-mail: zhangp@szu.edu.cn)

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.