# Distributionally Robust Games with Risk-averse Players

### Nicolas Loizou

#### Abstract

We present a new model of incomplete information games without private information in which the players use a distributionally robust optimization approach to cope with the payoff uncertainty. With some specific restrictions, we show that our “Distributionally Robust Game” constitutes a true generalization of three popular finite games. These are the Complete Information Games, Bayesian Games and Robust Games. Subsequently, we prove that the set of equilibria of an arbitrary distributionally robust game with specified ambiguity set can be computed as the component-wise projection of the solution set of a multi-linear system of equations and inequalities. For special cases of such games we show equivalence to complete information finite games (Nash Games) with the same number of players and same action spaces. Thus, when our game falls within these special cases one can simply solve the corresponding Nash Game. Finally, we demonstrate the applicability of our new model of games and highlight its importance.

#### References

- Aghassi, M. and Bertsimas, D. (2006). Robust game theory. Mathematical Programming, 107(1-2):231-273.
- Artzner, P., Delbaen, F., Eber, J.-M., and Heath, D. (2002). Coherent measures of risk1. Risk management: value at risk and beyond, page 145.
- Harsanyi, J. C. (1967,1968). Games with incomplete information played by “bayesian” players, i-iii. Management science, 14:159-182,320-334,486-502.
- Hayashi, S., Yamashita, N., and Fukushima, M. (2005). Robust nash equilibria and second-order cone complementarity problems. Journal of Nonlinear and Convex Analysis, 6(2):283.
- Löfberg, J. (2004). Yalmip: A toolbox for modeling and optimization in matlab. In Computer Aided Control Systems Design, 2004 IEEE International Symposium on, pages 284-289. IEEE.
- Loizou, N. (2015). Distributionally robust game theory. arXiv preprint arXiv:1512.03253.
- Nash, J. (1951). Non-cooperative games. Annals of mathematics, pages 286-295.
- Nash, J. F. et al. (1950). Equilibrium points in n-person games. Proceedings of the national academy of sciences, 36(1):48-49.
- Natarajan, K., Pachamanova, D., and Sim, M. (2009). Constructing risk measures from uncertainty sets. Operations Research, 57(5):1129-1141.
- Nishimura, R., Hayashi, S., and Fukushima, M. (2012). Semidefinite complementarity reformulation for robust nash equilibrium problems with euclidean uncertainty sets. Journal of Global Optimization, 53(1):107-120.
- Qu, S. and Goh, M. (2012). Distributionally robust games with an application to supply chain. Harbin Institute of Technology.
- Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2:21-42.
- Rockafellar, R. T. and Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of banking & finance, 26(7):1443-1471.
- Sion, M. et al. (1958). On general minimax theorems. Pacific J. Math, 8(1):171-176.
- Sun, H. and Xu, H. (2015). Convergence analysis for distributionally robust optimization and equilibrium problems. Mathematics of Operations Research.
- Wiesemann, W., Kuhn, D., and Sim, M. (2014). Distributionally robust convex optimization. Operations Research, 62(6):1358-1376.

#### Paper Citation

#### in Harvard Style

Loizou N. (2016). **Distributionally Robust Games with Risk-averse Players** . In *Proceedings of 5th the International Conference on Operations Research and Enterprise Systems - Volume 1: ICORES,* ISBN 978-989-758-171-7, pages 186-196. DOI: 10.5220/0005753301860196

#### in Bibtex Style

@conference{icores16,

author={Nicolas Loizou},

title={Distributionally Robust Games with Risk-averse Players},

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

year={2016},

pages={186-196},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0005753301860196},

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 - Distributionally Robust Games with Risk-averse Players

SN - 978-989-758-171-7

AU - Loizou N.

PY - 2016

SP - 186

EP - 196

DO - 10.5220/0005753301860196