# Nested replicator dynamics, nested logit choice, and similarity-based learning

arXiv: https://arxiv.org/abs/2407.17815 · HTML: https://mlsanity.com/paper/2407.17815/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 74 references that are in the map.
- First posted on arXiv 25 Jul 2024.

## Builds on (most cited first, 12 of 12)

- [Cycles in adversarial regularized learning](https://mlsanity.com/paper/1709.02738/) (2017)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Survival of the strictest: Stable and unstable equilibria under regularized learning with partial information](https://mlsanity.com/paper/2101.04667/) (2021)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [Riemannian game dynamics](https://mlsanity.com/paper/1603.09173/) (2016)
- [Multi-agent online learning in time-varying games](https://mlsanity.com/paper/1809.03066/) (2018)
- [Finite composite games: Equilibria and dynamics](https://mlsanity.com/paper/1503.07935/) (2015)
- [On the robustness of learning in games with stochastically perturbed payoff observations](https://mlsanity.com/paper/1412.6565/) (2014)
- [Learning in nonatomic games, Part I: Finite action spaces and population games](https://mlsanity.com/paper/2107.01595/) (2021)
- [Nested bandits](https://mlsanity.com/paper/2206.09348/) (2022)
- [A geometric decomposition of finite games: Convergence vs. recurrence under exponential weights](https://mlsanity.com/paper/2405.07224/) (2024)
- [Survival of dominated strategies under imitation dynamics](https://mlsanity.com/paper/2209.08416/) (2022)

## Abstract (arXiv)

We consider a model of learning and evolution in games whose action sets are endowed with a partition-based similarity structure intended to capture exogenous similarities between strategies. In this model, revising agents have a higher probability of comparing their current strategy with other strategies that they deem similar, and they switch to the observed strategy with probability proportional to its payoff excess. Because of this implicit bias toward similar strategies, the resulting dynamics - which we call the nested replicator dynamics - do not satisfy any of the standard monotonicity postulates for imitative game dynamics; nonetheless, we show that they retain the main long-run rationality properties of the replicator dynamics, albeit at quantitatively different rates. We also show that the induced dynamics can be viewed as a stimulus-response model in the spirit of Erev & Roth (1998), with choice probabilities given by the nested logit choice rule of Ben-Akiva (1973) and McFadden (1978). This result generalizes an existing relation between the replicator dynamics and the exponential weights algorithm in online learning, and provides an additional layer of interpretation to our analysis and results.