## Abstract

The paper deals with systems composed of a large number of N interacting objects (e.g., agents, particles) controlled by two players defining a stochastic zero-sum game. The objects can be classified according to a finite set of classes or categories over which they move randomly. Because N is too large, the game problem is studied following a mean field approach. That is, a zero-sum game model GM_{N}, where the states are the proportions of objects in each class, is introduced. Then, letting N→ ∞ (the mean field limit) we obtain a new game model GM, independent on N, which is easier to analyze than GM_{N}. Considering a discounted optimality criterion, our objective is to prove that an optimal pair of strategies in GM is an approximate optimal pair as N→ ∞ in the original game model GM_{N}.

Original language | English |
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Pages (from-to) | 512-537 |

Number of pages | 26 |

Journal | Dynamic Games and Applications |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2021 |

### Bibliographical note

Funding Information:Work was partially supported by Consejo Nacional de Ciencia y Tecnolog?a (CONACYT-M?xico) under grant Ciencia Frontera 2019-87787.

Funding Information:

Work was partially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT-México) under grant Ciencia Frontera 2019-87787.

Publisher Copyright:

© 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.

## Keywords

- Discounted criterion
- Mean field theory
- Systems of interacting objects
- Zero-sum games