Advertisement

Uncertainty in Games: Using Probability-Distributions as Payoffs

  • Stefan Rass
  • Sandra König
  • Stefan Schauer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9406)

Abstract

Many decision problems ask for optimal behaviour in (often competitive) situations, where optimality is understood as maximal revenue. The axiomatic approach of von Neumann and Morgenstern establishes the existence of suitable revenue functions, assuming an ordered revenue space. A prominent materialization of this is game theory, where utility functions map actions of several players onto comparable payoffs, typically real numbers. Inspired by an application of that theory to risk management in utility networks, we observed that the usual game-theoretic models are inapplicable due to intrinsic randomness of the effects that an action has. This uncertainty comes from physical and environmental factors that affect the game-play outside of any players influence. To tackle such scenarios, we introduce games in which the payoffs are entire probability distributions (rather than numbers). Towards a sound decision theory, we define a total ordering on a restricted subset of probability distribution functions, and demonstrate how optimal decisions and even basic game theory can be (re)established over abstract revenue spaces of probability distributions. Our results belong to the category of risk control, and are applicable to contemporary security risk management, where decisions must be made under uncertainty and the effects of management actions are almost never deterministic.

Keywords

Risk Management Mixed Strategy Payoff Distribution Utility Network Matrix Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgment

This work was supported by the European Commission’s Project No. 608090, HyRiM (Hybrid Risk Management for Utility Networks) under the 7th Framework Programme (FP7-SEC-2013-1).

References

  1. 1.
    Fudenberg, D., Tirole, J.: Game Theory. MIT Press, London (1991)zbMATHGoogle Scholar
  2. 2.
    Glicksberg, I.L.: A further generalization of the Kakutani fixed point theorem, with application to nash equilibrium points. Proc. Am. Math. Soc. 3, 170–174 (1952)MathSciNetzbMATHGoogle Scholar
  3. 3.
    von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  4. 4.
    Rass, S.: On Game-Theoretic Risk Management (Part One) - Towards a Theory of Games with Payoffs that are Probability-Distributions. ArXiv e-prints, June 2015Google Scholar
  5. 5.
    Robert, C.P.: The Bayesian Choice. Springer, New York (2001)Google Scholar
  6. 6.
    Robinson, A.: Nonstandard Analysis. Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam (1966)Google Scholar
  7. 7.
    Stoyan, D., Müller, A.: Comparison Methods for Stochastic Models and Risks. Wiley, Chichester (2002)zbMATHGoogle Scholar
  8. 8.
    Szekli, R.: Stochastic Ordering and Dependence in Applied Probability. Lecture Notes in Statistics, vol. 97. Springer, Heidelberg (1995)zbMATHGoogle Scholar
  9. 9.
    Wand, M.P., Jones, M.C.: Kernel Smoothing. Chapman & Hall/CRC, London (1995)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute of Applied Informatics, System Security GroupUniversität KlagenfurtKlagenfurtAustria
  2. 2.Digital Safety & Security DepartmentAustrian Institute of Technology GmbHKlagenfurtAustria

Personalised recommendations