Games played by Exponential Weights Algorithms
Maurizio d'Andrea, Fabien Gensbittel, Jérôme Renault
TL;DR
The paper investigates the last-iterate convergence of Exponential Weights with constant learning rates in finite games, showing that the induced Markov dynamics on mixed strategies typically converge to the Nash equilibria with equalizing payoffs (NEEP) and, in strong coordination games, to a strict Nash equilibrium. It proves that the probability of playing a strict NE at the next stage converges to $0$ or $1$ when a strict NE exists, and that any limit point, if it exists, must lie in the NEEP set; in strong coordination games the process converges almost surely to a diagonal strict NE. The work combines stochastic-process techniques with game-theoretic structure, providing a rigorous last-iterate analysis, complemented by simulations and a discussion of open questions and future directions. Overall, it clarifies the conditions under which constant-learning-rate EW dynamics exhibit stable long-run behavior and points to the boundaries of convergence in broader game classes.$
Abstract
This paper studies the last-iterate convergence properties of the exponential weights algorithm with constant learning rates. We consider a repeated interaction in discrete time, where each player uses an exponential weights algorithm characterized by an initial mixed action and a fixed learning rate, so that the mixed action profile $p^t$ played at stage $t$ follows an homogeneous Markov chain. At first, we show that whenever a strict Nash equilibrium exists, the probability to play a strict Nash equilibrium at the next stage converges almost surely to 0 or 1. Secondly, we show that the limit of $p^t$, whenever it exists, belongs to the set of ``Nash Equilibria with Equalizing Payoffs''. Thirdly, we show that in strong coordination games, where the payoff of a player is positive on the diagonal and 0 elsewhere, $p^t$ converges almost surely to one of the strict Nash equilibria. We conclude with open questions.