# A stability property in mean field type differential games

## TL;DR

This paper explores a stability property in deterministic mean field type differential games, providing a characterization of the value function through directional derivatives and feasible directions, advancing the understanding of feedback strategies.

## Contribution

It introduces an infinitesimal stability condition involving directional derivatives, offering a new characterization of the value function in mean field differential games.

## Key findings

- Characterization of the value function via directional derivatives
- Infinitesimal stability condition involving feasible directions
- Extension of dynamic programming principle for constant controls

## Abstract

The paper is concerned with the feedback approach to the deterministic mean field type differential games. Previously, it was shown that suboptimal strategies in the mean field type differential game can constructed based on functions of time and probability satisfying the stability condition. This property realizes the dynamic programming principle for the constant control of one player. We present the infinitesimal form of this condition involving analogs of the directional derivatives. In particular, we obtain the characterization of the value function of the deterministic mean field type differential game in the terms of directional derivatives and the set of directions feasible by virtue of the dynamics of the game.