# Learning with Delayed Payoffs in Population Games using Kullback-Leibler Divergence Regularization

arXiv: https://arxiv.org/abs/2306.07535 · HTML: https://mlsanity.com/paper/2306.07535/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jun 2023.

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

- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)

## Abstract (arXiv)

We study a multi-agent decision problem in large population games. Agents from multiple populations select strategies for repeated interactions with one another. At each stage of these interactions, agents use their decision-making model to revise their strategy selections based on payoffs determined by an underlying game. Their goal is to learn the strategies that correspond to the Nash equilibrium of the game. However, when games are subject to time delays, conventional decision-making models from the population game literature may result in oscillations in the strategy revision process or convergence to an equilibrium other than the Nash. To address this problem, we propose the Kullback-Leibler Divergence Regularized Learning (KLD-RL) model, along with an algorithm that iteratively updates the model's regularization parameter across a network of communicating agents. Using passivity-based convergence analysis techniques, we show that the KLD-RL model achieves convergence to the Nash equilibrium without oscillations, even for a class of population games that are subject to time delays. We demonstrate our main results numerically on a two-population congestion game and a two-population zero-sum game.