# From Darwin to Poincar'e and von Neumann: Recurrence and Cycles in Evolutionary and Algorithmic Game Theory

arXiv: https://arxiv.org/abs/1910.01334 · HTML: https://mlsanity.com/paper/1910.01334/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Oct 2019.

## Built on by (most cited first, 5 of 5)

- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [Online Learning in Periodic Zero-Sum Games](https://mlsanity.com/paper/2111.03377/) (2021)
- [Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics](https://mlsanity.com/paper/2203.14129/) (2022)
- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [Evolutionary Dynamics and Phi-Regret Minimization in Games](https://mlsanity.com/paper/2106.14668/) (2021)

## Abstract (arXiv)

Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic citezeeman1980population, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincar'e recurrence, i.e., almost all orbits come arbitrary close to their initial conditions infinitely often. If no interior equilibria exist, then all interior initial conditions converge to the boundary. Specifically, the strategies that are not in the support of any equilibrium vanish in the limit of all orbits. All recurrence results furthermore extend to a class of games that generalize both graphical polymatrix games as well as evolutionary games, establishing a unifying link between evolutionary and algorithmic game theory. We show that two degrees of freedom, as in Rock-Paper-Scissors, is sufficient to prove periodicity.