Stochastic dominance is a concept used in decision-making under uncertainty, especially in portfolio optimization and finance. It provides a way to rank uncertain outcomes, or more specifically probability distributions, based on specific preference relations. In an ambiguous environment, stochastic dominance can help us make better decisions by selecting alternatives that perform well under various preference criteria.
There are mainly three orders of stochastic dominance:
1. First-order stochastic dominance (FSD): Given two cumulative distribution functions (CDFs) F(x) and G(x), F dominates G in the first-order (FSD) if and only if:
F(x) ≤ G(x) ∀x ∈ ℝ
This means the probability of F yielding a value less than or equal to x is always greater than or equal to the same probability for G for any given x. In this case, all risk-averse decision-makers would prefer F to G.
2. Second-order stochastic dominance (SSD): Given two cumulative distribution functions (CDFs) F(x) and G(x), F dominates G in the second-order (SSD) if and only if:
∫ − ∞xF(u)du ≤ ∫ − ∞xG(u)du ∀x ∈ ℝ
This means that the expected utility or average payoff of F is greater than that of G for all concave utility functions: all risk-averse decision-makers with a concave utility function would prefer F to G.
3. Third-order stochastic dominance (TSD): Given two cumulative distribution functions (CDFs) F(x) and G(x), F dominates G in the third-order (TSD) if and only if the third-order integral inequality holds for all x:
∫ − ∞x[∫ − ∞yF(u)du − ∫ − ∞yG(u)du]dy ≤ 0 ∀x ∈ ℝ
This condition implies that F has less downside risk than G, and all risk-averse decision-makers with a negative third derivative of the utility function would prefer F to G.
Now, let’s discuss the concept of stochastic dominance under ambiguity. Ambiguity arises when there is a lack of information about the true probability distribution of outcomes, which makes it hard to estimate the expectation and the probability of occurrence of a specific outcome. In such cases, decision makers usually consider multiple distributions representing the uncertainty.
The concept of stochastic dominance under ambiguity can be best explained using the Choquet integral. Given a preference functional q, if F dominates G in the first-order sense, we have:
∫ − ∞xF(u)dq(u) ≤ ∫ − ∞xG(u)dq(u)
For example, consider two assets P1 and P2 with ambiguous expected returns, represented by multiple CDFs such as (F1(x), F1′(x)) and (G1(x), G1′(x)), respectively. We can say P1 stochastically dominates P2 under ambiguity if there exists a preference functional q such that for all CDFs of P1 i.e., both F1(x) and F1′(x), and for all CDFs of P2 i.e., both G1(x) and G1′(x):
∫ − ∞xF1(u)dq(u) ≤ ∫ − ∞xG1(u)dq(u)
∫ − ∞xF1′(u)dq(u) ≤ ∫ − ∞xG1′(u)dq(u)
In portfolio optimization, stochastic dominance under ambiguity helps investors in choosing an investment that minimizes the chance of a low return and/or unfavorable outcome. Investors prefer portfolios with the highest stochastic dominance ranking as they provide better performance irrespective of the ambiguity associated with the probability distributions.
Uncertainty models like the Ellsberg Paradox or the Risk-Averse Utility Function can be used in conjunction with stochastic dominance to make better-informed decisions under ambiguity. By identifying the best-ranked ambiguous alternatives or portfolios, investors can potentially increase their utility levels or reduce overall risks in their investment decisions.