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

TL;DR

This paper introduces nested replicator dynamics, a learning model in games with similarity-based strategy updates, which retains key long-term properties of traditional replicator dynamics despite non-standard assumptions.

Contribution

It develops a novel nested replicator dynamics model incorporating similarity-based strategy revision and links it to nested logit choice, extending existing online learning interpretations.

Findings

Retains main long-run rationality properties of replicator dynamics

Can be viewed as a stimulus-response model with nested logit choice

Generalizes the relation between replicator dynamics and exponential weights algorithm

Abstract

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.