Abstract
How can players reach a Nash equilibrium? I offer one possible explanation in terms of a low-rationality learning method called probe and adjust by proving that it converges to strict Nash equilibria in an important class of games. This demonstrates that decidedly limited learning methods can support Nash equilibrium play.
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Notes
We call it actions instead of strategies since, strictly speaking, we consider infinitely repeated games where strategies specify choices at each information set of a player.
This is determined entirely by the state (a, a p) since a is the current state and the action profile of the previous state can be reconstructed from a and a p.
The notation \(\xrightarrow{O(\varepsilon^k)}\) indicates the order of the transition probability in question.
On a better reply path, u i (a k+1) > u i (a k) for exactly one player.
This follows since the stochastic potential of (A, A) and (B, B) is the same under the rule of Marden et al.
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Acknowledgments
I would like to thank the KLI for their hospitality and for hosting the workshop. This material is based upon work supported by the National Science Foundation under Grant No. EF 1038456. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
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Huttegger, S.M. Probe and Adjust. Biol Theory 8, 195–200 (2013). https://doi.org/10.1007/s13752-013-0114-2
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DOI: https://doi.org/10.1007/s13752-013-0114-2