Abstract
We argue that the existing regret matchings for Nash equilibrium approximation conduct “jumpy” strategy updating when the probabilities of future plays are set to be proportional to positive regret measures. We propose a geometrical regret matching that features “smooth” strategy updating. Our approach is simple, intuitive, and natural. The analytical and numerical results show that “smoothly” suppressing “unprofitable” pure strategies is sufficient for the game to evolve toward Nash equilibrium, suggesting that, in reality, the tendency for equilibrium could be pervasive and irresistible. Technically, iterative regret matching gives rise to a sequence of adjusted mixed strategies for us to examine its approximation to the true equilibrium point. The sequence can be studied in the metric space and visualized nicely as a clear path toward an equilibrium point. Our theory has limitations in optimizing the approximation accuracy.
Highlights
The players keep tracking the regrets on the past plays and making the future plays with probabilities proportional to positive regret measures
Iterative regret matching can be seen as continuing updating of mixed strategy with regret information: the mixed strategy to update is a statistical structure of the whole past plays, and the mixed strategy updated will determine the probabilities of plays in the immediate future
To approximate a Nash equilibrium of the noncooperative game, we propose and test a regret matching with geometrical flavor
Summary
In 2000, Hart and Mas-Colell proposed an iterative algorithm called regret matching to approximate a correlated equilibrium. The players keep tracking the regrets on the past plays and making the future plays with probabilities proportional to positive regret measures. The players keep tracking the regrets on the past plays and making the future plays with probabilities proportional to positive regret measures. This algorithm is natural in that the players do not have to ply about their opponents’ payoff functions, as opposed to the non-adaptive variety, e.g., the celebrated Lemke–Howson algorithm, which takes two players’ payoff matrices as input and pinpoints equilibrium points as output.
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