# Sink equilibria and the attractors of learning in games

arXiv: https://arxiv.org/abs/2502.07975 · HTML: https://mlsanity.com/paper/2502.07975/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 47 references that are in the map.
- First posted on arXiv 11 Feb 2025.

## Builds on (most cited first, 7 of 7)

- [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)
- [The graph structure of two-player games](https://mlsanity.com/paper/2209.10182/) (2022)
- [The Replicator Dynamic, Chain Components and the Response Graph](https://mlsanity.com/paper/2209.15230/) (2022)
- [Preference graphs: a combinatorial tool for game theory](https://mlsanity.com/paper/2502.03546/) (2025)
- [The Attractor of the Replicator Dynamic in Zero-Sum Games](https://mlsanity.com/paper/2302.00253/) (2023)
- [Swim till You Sink: Computing the Limit of a Game](https://mlsanity.com/paper/2408.11146/) (2024)

## Built on by (most cited first, 1 of 1)

- [What preferences can - and cannot - predict in multi-agent online learning](https://mlsanity.com/paper/2608.13810/) (2026)

## Abstract (arXiv)

Characterizing the limit behavior -- that is, the attractors -- of learning dynamics is one of the most fundamental open questions in game theory. In recent work on this front, it was conjectured that the attractors of the replicator dynamic are in one-to-one correspondence with the sink equilibria of the game -- the sink strongly connected components of a game's preference graph -- and it was established that they do stand in at least one-to-many correspondence with them. Here, we show that the one-to-one conjecture is false. We disprove this conjecture over the course of three theorems: the first disproves a stronger form of the conjecture, while the weaker form is disproved separately in the two-player and N-player (N>2) cases. By showing how the conjecture fails, we lay out the obstacles that lie ahead for characterizing attractors of the replicator, and introduce new ideas with which to tackle them. All three counterexamples derive from an object called a local source -- a point lying within the sink equilibrium, and yet which is `locally repelling'; we prove that the absence of local sources is necessary, but not sufficient, for the one-to-one property to be true. We complement this with a sufficient condition: we introduce a local property of a sink equilibrium called pseudoconvexity, and establish that when the sink equilibria of a two-player game are pseudoconvex then they precisely define the attractors. Pseudoconvexity generalizes the previous cases -- such as zero-sum games and potential games -- where this conjecture was known to hold, and reformulates these cases in terms of a simple graph property.