# Asymptotic value in frequency-dependent games with separable payoffs: a differential approach

## TL;DR

This paper investigates the long-term value of frequency-dependent zero-sum games with separable payoffs using a differential approach, establishing the existence and equivalence of asymptotic and continuous-time game values.

## Contribution

It introduces a differential game framework for frequency-dependent payoffs and proves the existence and equality of asymptotic and continuous-time game values.

## Key findings

- Asymptotic value exists in both n-stage and discounted games.
- The differential game has a well-defined value despite irregularities.
- Asymptotic and continuous-time game values coincide.

## Abstract

We study the asymptotic value of a frequency-dependent zero-sum game with separable payoff following a differential approach. The stage payoffs in such games depend on the current actions and on a linear function of the frequency of actions played so far. We associate to the repeated game, in a natural way, a differential game and although the latter presents an irregularity at the origin, we prove that it has a value. We conclude, using appropriate approximations, that the asymptotic value of the original game exists in both the n-stage and the {\lambda}-discounted games and that it coincides with the value of the continuous time game.