### Abstract

In this paper, we consider linear-quadratic risk-sensitive mean field games (LQRSMFGs). Each agent strives to minimize an exponentiated integral quadratic cost or risk-sensitive cost function, which is coupled with other agents via a mean field term. By invoking the Nash certainty equivalence principle, we first obtain a robust decentralized control law for each agent to construct a mean field system. We then provide appropriate conditions under which the mean field system admits a unique deterministic function that approximates the mean field term with arbitrarily small error when the number of agents, say N, goes to infinity. We also show the closed-loop system stability, and prove that the set of N robust decentralized control laws possesses an ε-Nash equilibrium property. Moreover, we show that ε can be taken to be arbitrarily close to zero as N → ∞, but our ε bound is weaker than its linear-quadratic mean field game (LQMFG) counterpart due to risk-sensitivity in the present case. Finally, we discuss two different limiting cases, and show that one of these is equivalent to the corresponding LQMFG.

Original language | English (US) |
---|---|

Article number | 7039801 |

Pages (from-to) | 2691-2696 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

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### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*2015-February*(February), 2691-2696. [7039801]. https://doi.org/10.1109/CDC.2014.7039801

**Linear-quadratic risk-sensitive mean field games.** / Moon, Jun; Basar, Tamer.

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7039801, pp. 2691-2696. https://doi.org/10.1109/CDC.2014.7039801

}

TY - JOUR

T1 - Linear-quadratic risk-sensitive mean field games

AU - Moon, Jun

AU - Basar, Tamer

PY - 2014/1/1

Y1 - 2014/1/1

N2 - In this paper, we consider linear-quadratic risk-sensitive mean field games (LQRSMFGs). Each agent strives to minimize an exponentiated integral quadratic cost or risk-sensitive cost function, which is coupled with other agents via a mean field term. By invoking the Nash certainty equivalence principle, we first obtain a robust decentralized control law for each agent to construct a mean field system. We then provide appropriate conditions under which the mean field system admits a unique deterministic function that approximates the mean field term with arbitrarily small error when the number of agents, say N, goes to infinity. We also show the closed-loop system stability, and prove that the set of N robust decentralized control laws possesses an ε-Nash equilibrium property. Moreover, we show that ε can be taken to be arbitrarily close to zero as N → ∞, but our ε bound is weaker than its linear-quadratic mean field game (LQMFG) counterpart due to risk-sensitivity in the present case. Finally, we discuss two different limiting cases, and show that one of these is equivalent to the corresponding LQMFG.

AB - In this paper, we consider linear-quadratic risk-sensitive mean field games (LQRSMFGs). Each agent strives to minimize an exponentiated integral quadratic cost or risk-sensitive cost function, which is coupled with other agents via a mean field term. By invoking the Nash certainty equivalence principle, we first obtain a robust decentralized control law for each agent to construct a mean field system. We then provide appropriate conditions under which the mean field system admits a unique deterministic function that approximates the mean field term with arbitrarily small error when the number of agents, say N, goes to infinity. We also show the closed-loop system stability, and prove that the set of N robust decentralized control laws possesses an ε-Nash equilibrium property. Moreover, we show that ε can be taken to be arbitrarily close to zero as N → ∞, but our ε bound is weaker than its linear-quadratic mean field game (LQMFG) counterpart due to risk-sensitivity in the present case. Finally, we discuss two different limiting cases, and show that one of these is equivalent to the corresponding LQMFG.

UR - http://www.scopus.com/inward/record.url?scp=84988287817&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988287817&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7039801

DO - 10.1109/CDC.2014.7039801

M3 - Conference article

AN - SCOPUS:84988287817

VL - 2015-February

SP - 2691

EP - 2696

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7039801

ER -