AI agents in Algorithmic Electricity Markets: On the Emergence of Tacit Collusion

As electricity market participants increasingly adopt learning-based agents for their bidding strategies, electricity markets are becoming algorithmic. Evidence from algorithmic markets in other domains shows that tacit collusion can arise purely through independent learning. Moreover, electricity markets are typically oligopolistic and feature repeated interaction among a small number of participants, making them structurally susceptible to non-competitive behavior. In the face of these observations, this paper investigates the hypothesis that tacit collusion may emerge in electricity markets where participants' actions are controlled by autonomous learning-based algorithms. We model strategic bidding as a repeated game with imperfect public monitoring, and model the participants' emergent behavior using multi-agent reinforcement learning. We propose a multi-dimensional set of criteria (going beyond profit comparisons against Nash equilibria) to assess whether the resulting behavior constitutes tacit collusion. Our experimental results showcase that such a danger is realistic for electricity markets: there are cases where agents do learn to sustain supra-competitive outcomes that are supportive of tacit collusion indicators, even though the agents were never instructed to collude.

Regret, equilibrium, and learning in games: A guided tour

This note aims to serve as an entry point to the literature on learning in games, a topic with significant theoretical appeal and a wide range of applications -- from machine learning and data science to economics and beyond. Our presentation is structured around two complementary viewpoints: We first consider a single agent -- the learner -- engaged in a sequential decision process in an unknown, non-stationary, and possibly adversarial environment. We then examine what happens when the environment is shaped by the decisions of several interacting agents, not necessarily aware of each other's actions or goals, and all seeking to improve their individual rewards. In this general context, we examine a family of regularized learning policies based on best-responding to the past history of play, up to a regularization penalty intended to encourage exploration and prevent over-commitment to suboptimal choices. In the single-agent setting, we present some basic regret bounds for regularized learning in adversarial multi-armed bandits; in the multi-agent setting, we describe an ergodic equilibrium convergence result for zero-sum games in the spirit of classical results on fictitious play, as well as a "folk theorem" linking strategic and dynamic notions of stability -- Nash equilibria and attracting points of regularized learning, respectively. We pay special attention to the information available to the players and, through a unified analysis framework, we study both oracle- and payoff-based (bandit) methods. Our goal is to provide a coherent and comprehensible -- albeit, by necessity, not comprehensive -- account of some recent ideas in the field, and to discuss their implications for the study of rationality.

Learning under Opponent Unawareness in Linear-Quadratic Stochastic Games

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $ε$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

Paradoxes of Game Theoretic Equilibria and Price of Anarchy

For decades, static solution concepts (Nash, Correlated, and Coarse Correlated Equilibria) and the Price of Anarchy (PoA) have formed the bedrock of algorithmic game theory, with no-regret learning proving fast convergence to such game-theoretic equilibria. We show that reducing multi-agent learning to static equilibrium and black-box regret analysis obscures underlying dynamic disequilibrium and game theoretic bounds. First, interior Nash equilibria lack $C^1$ vector field information, meaning agents cannot distinguish aligned from strictly opposing incentives. Inheriting this geometry, the worst-case pure Nash equilibria dictating robust PoA bounds manifest as topologically unstable strict saddles, and in canonical congestion games, as global repellers supported on almost everywhere strictly dominated strategies. Anchoring efficiency guarantees to these unstable states causes algebraic sensitivity; we prove that accommodating all strictly positive affine costs renders the PoA unbounded. Furthermore, projecting learning trajectories onto the discrete simplex of correlated play systematically accommodates non-rationalizable behavior. Evaluating dynamics via Coarse Correlated Equilibria or proximal refinements fails to preclude strictly dominated strategies. Moreover, optimal $O(1/T)$ swap-regret minimization does not preclude macroscopic turbulence, manifesting as chaotic limit sets even in minimal games. Finally, we examine the non-atomic limit of congestion games. Though considered highly stable with tight sub-linear $Θ(p/\ln p)$ PoA bounds (where $p$ is the polynomial degree), we prove that under discrete-time learning, the unique equilibrium destabilizes into Li-Yorke chaos and global attractors whose time-averaged inefficiency degrades exponentially as $2^p$. These results necessitate re-evaluating worst-case equilibrium frameworks for dynamically grounded metrics.

GradMAP: Gradient-Based Multi-Agent Proximal Learning for Grid-Edge Flexibility

Coordinating large populations of grid-edge devices requires learning methods that remain fully decentralised in deployment while still respecting three-phase AC distribution-network physics. This paper proposes gradient-based multi-agent proximal learning (GradMAP) to address this challenge. GradMAP trains independent neural-network policies for each agent without any parameter sharing, and each agent uses only its own local observation for online decision-making without communication. During offline training, GradMAP embeds a differentiable three-phase AC power-flow model in a primal-dual learning loop and uses implicit differentiation to propagate exact network-constraint violations to update the policy parameters. To speed up training, GradMAP reuses expensive environment gradients through a proximal surrogate within a trust region defined in the more direct policy-output (action) space, instead of the probability distribution space used in other works, such as PPO. In case studies with 1,000 agents managing batteries, heat pumps, and controllable generators on the IEEE 123-bus feeder, GradMAP learns decentralised policies that minimise three-phase AC load-flow constraint violations within 15 minutes of training on a single workstation-class NVIDIA RTX PRO 5000 Blackwell 48GB GPU. This is a 3--5x training speed-up over gradient-based self-supervised learning benchmarks and substantially better training efficiency than multi-agent reinforcement-learning benchmarks. In out-of-sample tests, GradMAP also delivers among the lowest operating cost and constraint violations.