### Abstract

Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness may be made, when player objectives are given by expectations of either smooth or nonsmooth functions. In general, a direct application of deterministic results is difficult since the expectation operation results in a less tractable nonlinear function. Our interest is in developing an analytical framework that only requires the analysis of the integrands of the expectations. Accordingly, in both the smooth and nonsmooth settings, we show that if an appropriate coercivity result holds in an almost-sure fashion, then the existence of an equilibrium to the original stochastic Nash game may be claimed. In the smooth setting, a corresponding sufficiency condition for uniqueness is also provided. We illustrate the utility of our framework by examining a class of stochastic Nash-Cournot games in which nonsmoothness arises from the use of a risk measure.

Original language | English (US) |
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Title of host publication | Proceedings of the 2010 American Control Conference, ACC 2010 |

Pages | 5632-5637 |

Number of pages | 6 |

State | Published - Oct 15 2010 |

Event | 2010 American Control Conference, ACC 2010 - Baltimore, MD, United States Duration: Jun 30 2010 → Jul 2 2010 |

### Other

Other | 2010 American Control Conference, ACC 2010 |
---|---|

Country | United States |

City | Baltimore, MD |

Period | 6/30/10 → 7/2/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering

### Cite this

*Proceedings of the 2010 American Control Conference, ACC 2010*(pp. 5632-5637). [5531036]

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*Proceedings of the 2010 American Control Conference, ACC 2010.*, 5531036, pp. 5632-5637, 2010 American Control Conference, ACC 2010, Baltimore, MD, United States, 6/30/10.

**On the characterization of solution sets of smooth and nonsmooth stochastic nash games.** / Ravat, Uma; Shanbhag, Vinayak V.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - On the characterization of solution sets of smooth and nonsmooth stochastic nash games

AU - Ravat, Uma

AU - Shanbhag, Vinayak V.

PY - 2010/10/15

Y1 - 2010/10/15

N2 - Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness may be made, when player objectives are given by expectations of either smooth or nonsmooth functions. In general, a direct application of deterministic results is difficult since the expectation operation results in a less tractable nonlinear function. Our interest is in developing an analytical framework that only requires the analysis of the integrands of the expectations. Accordingly, in both the smooth and nonsmooth settings, we show that if an appropriate coercivity result holds in an almost-sure fashion, then the existence of an equilibrium to the original stochastic Nash game may be claimed. In the smooth setting, a corresponding sufficiency condition for uniqueness is also provided. We illustrate the utility of our framework by examining a class of stochastic Nash-Cournot games in which nonsmoothness arises from the use of a risk measure.

AB - Variational analysis provides an avenue for characterizing solution sets of deterministic Nash games over continuous strategy sets. We examine whether similar statements, particularly pertaining to existence and uniqueness may be made, when player objectives are given by expectations of either smooth or nonsmooth functions. In general, a direct application of deterministic results is difficult since the expectation operation results in a less tractable nonlinear function. Our interest is in developing an analytical framework that only requires the analysis of the integrands of the expectations. Accordingly, in both the smooth and nonsmooth settings, we show that if an appropriate coercivity result holds in an almost-sure fashion, then the existence of an equilibrium to the original stochastic Nash game may be claimed. In the smooth setting, a corresponding sufficiency condition for uniqueness is also provided. We illustrate the utility of our framework by examining a class of stochastic Nash-Cournot games in which nonsmoothness arises from the use of a risk measure.

UR - http://www.scopus.com/inward/record.url?scp=77957761094&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957761094&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:77957761094

SN - 9781424474264

SP - 5632

EP - 5637

BT - Proceedings of the 2010 American Control Conference, ACC 2010

ER -