# A Complementarity Problem Formulation for Chance-constraine Games

### Vikas Vikram Singh, Oualid Jouini, Abdel Lisser

#### Abstract

We consider a two player bimatrix game where the entries of each player’s payoff matrix are independent random variables following a certain distribution. We formulate this as a chance-constrained game by considering that the payoff of each player is defined by using a chance-constraint. We consider the case of normal and Cauchy distributions. We show that a Nash equilibrium of the chance-constrained game corresponding to normal distribution can be obtained by solving an equivalent nonlinear complementarity problem. Further if the entries of the payoff matrices are also identically distributed with non-negative mean, we show that a strategy pair, where each player’s strategy is the uniform distribution on his action set, is a Nash equilibrium of the chance-constrained game. We show that a Nash equilibrium of the chance-constrained game corresponding to Cauchy distribution can be obtained by solving an equivalent linear complementarity problem.

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#### Paper Citation

#### in Harvard Style

Singh V., Jouini O. and Lisser A. (2016). **A Complementarity Problem Formulation for Chance-constraine Games** . In *Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-171-7, pages 58-67. DOI: 10.5220/0005754800580067

#### in Bibtex Style

@conference{icores16,

author={Vikas Vikram Singh and Oualid Jouini and Abdel Lisser},

title={A Complementarity Problem Formulation for Chance-constraine Games},

booktitle={Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,},

year={2016},

pages={58-67},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005754800580067},

isbn={978-989-758-171-7},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,

TI - A Complementarity Problem Formulation for Chance-constraine Games

SN - 978-989-758-171-7

AU - Singh V.

AU - Jouini O.

AU - Lisser A.

PY - 2016

SP - 58

EP - 67

DO - 10.5220/0005754800580067