# The Replicator Dynamic, Chain Components and the Response Graph

arXiv: https://arxiv.org/abs/2209.15230 · HTML: https://mlsanity.com/paper/2209.15230/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 44 references that are in the map.
- First posted on arXiv 30 Sep 2022.

## Builds on (most cited first, 10 of 10)

- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [The Mechanics of n-Player Differentiable Games](https://mlsanity.com/paper/1802.05642/) (2018)
- [Game-Theoretic Multiagent Reinforcement Learning](https://mlsanity.com/paper/2011.00583/) (2020)
- [α-Rank: Multi-Agent Evaluation by Evolution](https://mlsanity.com/paper/1903.01373/) (2019)
- [No-regret learning and mixed Nash equilibria: They do not mix](https://mlsanity.com/paper/2010.09514/) (2020)
- [Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games](https://mlsanity.com/paper/1905.08396/) (2019)
- [Chaos, Extremism and Optimism: Volume Analysis of Learning in Games](https://mlsanity.com/paper/2005.13996/) (2020)
- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [Nash, Conley, and Computation: Impossibility and Incompleteness in Game Dynamics](https://mlsanity.com/paper/2203.14129/) (2022)
- [The graph structure of two-player games](https://mlsanity.com/paper/2209.10182/) (2022)

## Built on by (most cited first, 3 of 3)

- [Swim till You Sink: Computing the Limit of a Game](https://mlsanity.com/paper/2408.11146/) (2024)
- [Sink equilibria and the attractors of learning in games](https://mlsanity.com/paper/2502.07975/) (2025)
- [On algorithmically boosting fixed-point computations](https://mlsanity.com/paper/2304.04665/) (2023)

## Abstract (arXiv)

In this paper we examine the relationship between the flow of the replicator dynamic, the continuum limit of Multiplicative Weights Update, and a game's response graph. We settle an open problem establishing that under the replicator, sink chain components -- a topological notion of long-run outcome of a dynamical system -- always exist and are approximated by the sink connected components of the game's response graph. More specifically, each sink chain component contains a sink connected component of the response graph, as well as all mixed strategy profiles whose support consists of pure profiles in the same connected component, a set we call the content of the connected component. As a corollary, all profiles are chain recurrent in games with strongly connected response graphs. In any two-player game sharing a response graph with a zero-sum game, the sink chain component is unique. In two-player zero-sum and potential games the sink chain components and sink connected components are in a one-to-one correspondence, and we conjecture that this holds in all games.