We show that fictitious play can converge at arbitrarily slow polynomial rates in two-player zero-sum games. For every integer $k \ge 2$, we construct a payoff matrix with $(k+1)^2 - 5$ actions per player for which the duality gap of the empirical strategies decays as $Θ(t^{-1/k})$ after $t$ steps. The family starts from the standard rock-paper-scissors matrix, with each higher-order game constructed recursively from the preceding one. After a prescribed common initial action, every subsequent best response under fictitious play is unique. For $k \ge 3$, these games give counterexamples to Karlin's conjectured $O(t^{-1/2})$ convergence rate, and they extend the recent $Θ(t^{-1/3})$ construction of Wang (2025) to arbitrarily slow polynomial rates.
Anas Barakat, John Lazarsfeld, Georgios Piliouras, Antonios Varvitsiotis
原文摘要
We study online learning in linear dynamical games where multiple strategic agents act on a shared state evolving according to a linear dynamical system subject to adversarial disturbances. This setting lies beyond both single-agent nonstochastic online control and classical linear-quadratic games, which typically focus on quadratic objectives and noiseless or stochastic dynamics. Each agent seeks to minimize its own sequence of convex losses under full-state observability. Following the stabilizing-baseline paradigm in nonstochastic online control, we assume access to linear controllers that stabilize the noiseless system and focus on online adaptation by learning disturbance-action corrections. This extends adversarial online control to strategic multi-agent shared-state systems, where each agent's actions shape the state trajectory and hence the realized losses of all learners. Under state-only and aggregate-input feedback models, we analyze agents running online gradient descent with memory to update their own disturbance-action policies. We prove per-agent regret bounds that are sublinear and near-optimal in the time horizon, with population dependence that varies under the specified feedback, opponent policy, and stability assumptions. In the common-interest case, where all agents have identical cost functions, we show that the induced online problem forms a time-varying potential game and derive equilibrium tracking guarantees. Together, our results provide a theoretical framework for adversarial online learning in stabilized linear dynamical games, connecting online control with learning in games.