Uncertainty in Games: Using Probability-Distributions as Payoffs
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.
KeywordsRisk Management Mixed Strategy Payoff Distribution Utility Network Matrix Game
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).
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