Games played by Exponential Weights Algorithms

• URL: http://arxiv.org/abs/2407.06676v1
• Date: Tue, 9 Jul 2024 08:49:51 GMT
• Title: Games played by Exponential Weights Algorithms
• Authors: Maurizio d'Andrea, Fabien Gensbittel, Jérôme Renault,
• Abstract summary: 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. 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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