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Optimal behavior in (competitive) situation is traditionally determined withthe help of utility functions that measure the payoff of different actions.Given an ordering on the space of revenues (payoffs), the classical axiomaticapproach of von Neumann and Morgenstern establishes the existence of suitableutility functions, and yields to game-theory as the most prominentmaterialization of a theory to determine optimal behavior. Although thisappears to be a most natural approach to risk management too, applications incritical infrastructures often violate the implicit assumption of actionsleading to deterministic consequences. In that sense, the gameplay in acritical infrastructure risk control competition is intrinsically random in thesense of actions having uncertain consequences. Mathematically, this takes usto utility functions that are probability-distribution-valued, in which case weloose the canonic (in fact every possible) ordering on the space of payoffs,and the original techniques of von Neumann and Morgenstern no longer apply.  This work introduces a new kind of game in which uncertainty applies to thepayoff functions rather than the player's actions (a setting that has beenwidely studied in the literature, yielding to celebrated notions like thetrembling hands equilibrium or the purification theorem). In detail, we showhow to fix the non-existence of a (canonic) ordering on the space ofprobability distributions by only mildly restricting the full set to a subsetthat can be totally ordered. Our vehicle to define the ordering and establishbasic game-theory is non-standard analysis and hyperreal numbers.

On Game-Theoretic Risk Management (Part One) -- Towards a Theory of Games with Payoffs that are Probability-Distributions