The Attractor of the Replicator Dynamic in Zero-Sum Games

Oliver Biggar, Iman Shames
Proceedings of The 35th International Conference on Algorithmic Learning Theory, PMLR 237:161-178, 2024.

Abstract

In this paper we characterise the long-run behaviour of the replicator dynamic in zero-sum games (symmetric or non-symmetric). Specifically, we prove that every zero-sum game possesses a unique global replicator attractor, which we then characterise. Most surprisingly, this attractor depends only on each player’s preference order over their own strategies and not on the cardinal payoff values, defined by a finite directed graph we call the game’s preference graph. When the game is symmetric, this graph is a tournament whose nodes are strategies; when the game is not symmetric, this graph is the game’s response graph. We discuss the consequences of our results on chain recurrence and Nash equilibria.

Cite this Paper

BibTeX
@InProceedings{pmlr-v237-biggar24a, title = {The Attractor of the Replicator Dynamic in Zero-Sum Games}, author = {Biggar, Oliver and Shames, Iman}, booktitle = {Proceedings of The 35th International Conference on Algorithmic Learning Theory}, pages = {161--178}, year = {2024}, editor = {Vernade, Claire and Hsu, Daniel}, volume = {237}, series = {Proceedings of Machine Learning Research}, month = {25--28 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v237/biggar24a/biggar24a.pdf}, url = {https://proceedings.mlr.press/v237/biggar24a.html}, abstract = {In this paper we characterise the long-run behaviour of the replicator dynamic in zero-sum games (symmetric or non-symmetric). Specifically, we prove that every zero-sum game possesses a unique global replicator attractor, which we then characterise. Most surprisingly, this attractor depends only on each player’s preference order over their own strategies and not on the cardinal payoff values, defined by a finite directed graph we call the game’s preference graph. When the game is symmetric, this graph is a tournament whose nodes are strategies; when the game is not symmetric, this graph is the game’s response graph. We discuss the consequences of our results on chain recurrence and Nash equilibria.} }
Endnote
%0 Conference Paper %T The Attractor of the Replicator Dynamic in Zero-Sum Games %A Oliver Biggar %A Iman Shames %B Proceedings of The 35th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Claire Vernade %E Daniel Hsu %F pmlr-v237-biggar24a %I PMLR %P 161--178 %U https://proceedings.mlr.press/v237/biggar24a.html %V 237 %X In this paper we characterise the long-run behaviour of the replicator dynamic in zero-sum games (symmetric or non-symmetric). Specifically, we prove that every zero-sum game possesses a unique global replicator attractor, which we then characterise. Most surprisingly, this attractor depends only on each player’s preference order over their own strategies and not on the cardinal payoff values, defined by a finite directed graph we call the game’s preference graph. When the game is symmetric, this graph is a tournament whose nodes are strategies; when the game is not symmetric, this graph is the game’s response graph. We discuss the consequences of our results on chain recurrence and Nash equilibria.
APA
Biggar, O. & Shames, I.. (2024). The Attractor of the Replicator Dynamic in Zero-Sum Games. Proceedings of The 35th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 237:161-178 Available from https://proceedings.mlr.press/v237/biggar24a.html.

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