What if We Only Have Approximate Stochastic Dominance?

In many practical situations, we need to select one of the two alternatives, and we do not know the exact form of the user’s utility function – e.g., we only know that it is increasing. In this case, stochastic dominance result says that if the cumulative distribution function (cdf) corresponding to the first alternative is always smaller than or equal than the cdf corresponding to the second alternative, then the first alternative is better. This criterion works well in many practical situations, but often, we have situations when for most points, the first cdf is smaller but at some points, the first cdf is larger. In this paper, we show that in such situations of approximate stochastic dominance, we can also conclude that the first alternative is better – provided that the set of points x at which the first cdf is larger is sufficiently small. 1 Stochastic Dominance: Reminder and Formulation of the Problem In finance, we need to make decisions under uncertainty. In financial decision making, we need to select one of the possible decisions: e.g., whether we sell or buy a given financial instrument (share, option, etc.). Ideally, we should select a decision which leaves us with the largest monetary value x. However, in practice, we cannot predict exactly the monetary consequences of each action: because of the changing external circumstances, in similar situations the same decision can lead to gains and to losses. Thus, we need to make a decision in a situation when we do not know the exact consequences of each action. In finance, we usually have probabilistic uncertainty. Numerous financial transactions are occurring every moment. For the past transactions, we know the monetary consequences of different decisions. By analyzing these past transactions, we can estimate, for each decision, the frequencies with which this decision leads to different monetary outcomes x. When the sample size is large – and for financial transactions it is large – the corresponding frequencies become very close to the actual probabilities. Thus, in fact, we can estimate the probabilities of different values x. 2 Vladik Kreinovich, Hung T. Nguyen, and Songsak Sriboonchitta Comment. Strictly speaking, this is not always true: we may have new circumstances, we can have a new financial instrument for which we do not have many records of its use – but in most situations, knowledge of the probabilities is a reasonable assumption. How to describe the corresponding probabilities. As usual, the corresponding probabilities can be described either by the probability density function f(x) or by the cumulative distribution function F (t) def = Prob(x ≤ t). If we know the probability density function f(x), then we can reconstruct the cumulative distribution function as F (t) = ∫ t −∞ f(x) dx. Vice versa, if we know the cumulative distribution function F (t), we can reconstruct the probability density function as its derivative f(x) = F ′(x). How to make decisions under probabilistic uncertainty: a theoretical recommendation. Let us assume that we have several possible decisions whose outcomes are characterized by the probability density functions f1(x), f2(x), . . . According to the traditional decision making theory (see, e.g., [3, 5–7]), the decisions of a rational person can be characterized by a function u(x) called utility function such that this person always selects a decision with the largest value of expected utility ∫ fi(x) · u(x) dx. A decision corresponding to the probability distribution function f1(x) is preferable to the decision corresponding to the probability distribution function f2(x) if ∫ f1(x) · u(x) dx > ∫ f2(x) · u(x) dx, i.e., equivalently, if ∫ ∆f(x) · u(x) dx > 0, where we denoted ∆f(x) def = f1(x)− f2(x). Comment. It is usually assumed that small changes in x lead to small changes in utility, i.e., in formal terms, that the function u(x) is differentiable. From a theoretical recommendation to practical decision. Theoretically, we can determine the utility function of the decision maker. However, since such a determination is very time-consuming, it is rarely done in real financial situations. As a result, in practice, we only have a partial information about the utility function. One thing we know for sure if that the larger the monetary gain x, the better the resulting situation; in other words, we know that the utility u(x) grows with x, i.e., the utility function u(x) is increasing. Often, this is the only information that we have about the utility function. How can we make a decision in such a situation? How to make decisions when we only know that utility function is increasing: analysis of the problem. When is the integral ∫ ∆f(x) · u(x) dx positive? What If We Only Have Approximate Stochastic Dominance? 3 To answer this question, let us first note that while theoretically, we have gains and losses which can be arbitrarily large, in reality, both gains and losses are bounded by some value T . In other words, fi(x) = 0 for x ≤ −T and for x ≥ T and thus, Fi(−T ) = Probi(x ≤ −T ) = 0 and Fi(T ) = Probi(x ≤ T ) = 1. In this case, ∫ ∆f(x) · u(x) dx = ∫ T