# Coherence of probabilistic constraints on Nash equilibria

## DOI:

https://doi.org/10.5753/jbcs.2022.2434## Keywords:

Nash equilibrium, Uncertain game, Probabilistic constraints, Coherence of constraints## Abstract

In this work, we first deal with the modeling of game situations that reach one of possibly many Nash equilibria. Before an instance of such a game starts, an external observer does not know, a priori, what is the exact profile of actions -- constituting an equilibrium -- that will occur; thus, he assigns subjective probabilities to players' actions. Such scenario is formalized as an observable game, which is a newly introduced structure for that purpose. Then, we study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, called the PCE-Coherence problem. We show several results concerning algorithms and complexity for PCE-Coherence when pure Nash equilibria and specific classes of games, called GNP-classes, are considered. In this context, we also study the computation of maximal and minimal probabilistic constraints on actions that preserves coherence. Finally, we study these problems when mixed Nash equilibria are allowed in GNP-classes of 2-player games.

### Downloads

## References

Aumann, R. J. (1974). Subjectivity and correlation in randomized strategies. *Journal of Mathematical Economics*, 1(1):67-96.

Bertsimas, D. and Tsitsiklis, J. (1997). *Introduction to Linear Optimization*. Athena Scientific series in optimization and neural computation. Athena Scientific.

Cheeseman, P., Kanefsky, B., and Taylor, W. M. (1991). Where the really hard problems are. In *12th IJCAI*, pages 331-337. Morgan Kaufmann.

Conitzer, V. and Sandholm, T. (2008). New complexity results about Nash equilibria. *Games and Economic Behavior*, 63(2):621-641.

Cook, S. A. (1971). The complexity of theorem-proving procedures. In *Conference Record of Third Annual ACM Symposium on Theory of Computing (STOC)*, pages 151-158. ACM.

Daskalakis, C., Goldberg, P. W., and Papadimitriou, C. H. (2009). The complexity of computing a Nash equilibrium. *SIAM Journal on Computing*, 39(1):195-259.

de Finetti, B. (1931). Sul significato soggettivo della probabilità. *Fundamenta Mathematicae*, 17(1):298-329. Translated into English as "On the Subjective Meaning of Probability'', In: P. Monari and D. Cocchi (Eds.), Probabilità e Induzione, Clueb, Bologna, 291-321, 1993.

de Finetti, B. (2017). *Theory of probability: A critical introductory treatment*. John Wiley & Sons. Translated by Antonio Machí and Adrian Smith.

Eckhoff, J. (1993). Helly, Radon, and Caratheodory type theorems. In Gruber, P. M. and Wills, J. M., editors, *Handbook of Convex Geometry*, pages 389-448. Elsevier Science Publishers.

Etessami, K. and Yannakakis, M. (2010). On the complexity of Nash equilibria and other fixed points. *SIAM Journal on Computing*, 39(6):2531-2597.

Finger, M. and De Bona, G. (2011). Probabilistic satisfiability: Logic-based algorithms and phase transition. In Walsh, T., editor, *IJCAI*, pages 528-533. IJCAI/AAAI.

Finger, M. and De Bona, G. (2015). Probabilistic satisfiability: algorithms with the presence and absence of a phase transition. *Annals of Mathematics and Artificial Intelligence*, 75(3):351-379. DOI: 10.1007/s10472-015-9466-6.

Fischer, F., Holzer, M., and Katzenbeisser, S. (2006). The influence of neighbourhood and choice on the complexity of finding pure Nash equilibria. *Information Processing Letters*, 99(6):239-245.

Gent, I. P. and Walsh, T. (1994). The SAT phase transition. In *ECAI94 - Proceedings of the Eleventh European Conference on Artificial Intelligence*, pages 105-109. John Wiley & Sons.

Georgakopoulos, G., Kavvadias, D., and Papadimitriou, C. H. (1988). Probabilistic satisfiability. *Journal of Complexity*, 4(1):1-11. DOI: 10.1016/0885-064X(88)90006-4.

Gilboa, I. and Zemel, E. (1989). Nash and correlated equilibria: Some complexity considerations. *Games and Economic Behavior*, 1(1):80-93.

Gottlob, G., Greco, G., and Scarcello, F. (2005). Pure Nash equilibria: Hard and easy games. *Journal of Artificial Intelligence Research*, 24:357-406.

Hansen, P. and Jaumard, B. (2000). Probabilistic satisfiability. In *Handbook of Defeasible Reasoning and Uncertainty Management Systems. Vol.5*, page 321. Springer Netherlands.

Kautz, H. A. and Selman, B. (1992). Planning as satisfiability. In *ECAI*, pages 359-363.

Nash, J. (1951). Non-cooperative games. *Annals of Mathematics*, 54(2):286-295.

Nash, J. F. (1950a). Equilibrium points in n-person games. *Proceedings of the National Academy of Sciences*, 36(1):48-49.

Nash, J. F. (1950b). *Non-Cooperative Games*. PhD thesis, Princeton University.

Nilsson, N. (1986). Probabilistic logic. *Artificial Intelligence*, 28(1):71-87.

Papadimitriou, C. H. (2007). The complexity of finding Nash equilibria. In Nisan, N., Roughgarden, T., Tardos, E., and Vazirani, V. V., editors, *Algorithmic game theory*, pages 29-51. Cambridge University Press.

Papadimitriou, C. H. and Roughgarden, T. (2008). Computing correlated equilibria in multi-player games. *Journal of the ACM*, 55(3):1-29. DOI: 10.1145/1379759.1379762.

Warners, J. P. (1998). A linear-time transformation of linear inequalities into conjunctive normal form. *Inf. Process. Lett.*, 68(2):63-69.

## Downloads

## Published

## How to Cite

*Journal of the Brazilian Computer Society*,

*28*(1), 38–51. https://doi.org/10.5753/jbcs.2022.2434

## Issue

## Section

## License

Copyright (c) 2022 The authors

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.