Stochastic Dominance

Understand First-Order Stochastic Dominance by comparing distribution pairs and their cumulative distribution functions.

Stochastic Dominance

Concept Overview

Stochastic dominance is a partial order on probability distributions. It is widely used in decision theory and economics to compare random variables. First-Order Stochastic Dominance (FSD) occurs when one distribution unambiguously yields higher values than another, meaning that any decision-maker with an increasing utility function would prefer it.

Formally, a distribution B first-order stochastically dominates distribution A if its cumulative distribution function (CDF) is always less than or equal to that of A: F_B(x) ≤ F_A(x) for all x, with strict inequality for at least one x. Because CDFs measure the probability of falling at or below a threshold, a lower CDF means a higher probability of larger outcomes.

Worked Example

Let's verify First-Order Stochastic Dominance for the Shift distribution pair on the support {0, 1, 2}.

Distribution A: Mass vector is (0.5, 0.5, 0). So P(A=0) = 0.5, P(A=1) = 0.5, P(A=2) = 0.
Distribution B: Mass vector is (0, 0.5, 0.5). So P(B=0) = 0, P(B=1) = 0.5, P(B=2) = 0.5.

We calculate the CDFs, F(x) = P(X ≤ x), for each value in the support:

Threshold (x)	F_A(x)	F_B(x)	Comparison
0	0.5	0	F_B(0) < F_A(0)
1	1.0 (0.5+0.5)	0.5 (0+0.5)	F_B(1) < F_A(1)
2	1.0	1.0 (0+0.5+0.5)	F_B(2) = F_A(2)

Since F_B(x) ≤ F_A(x) for all x, and the inequality is strict at x = 0 and x = 1, we conclude that B dominates A.

For every nondecreasing utility, B has expected utility at least as large as A. A constant utility can tie. The supported Linear u(x) = x and Square u(x) = x² are strictly increasing on this support and give a strict improvement for Shift. For Linear:

E[u(A)] = 0.5(0) + 0.5(1) + 0(2) = 0.5
E[u(B)] = 0(0) + 0.5(1) + 0.5(2) = 1.5

Experiment

You can use the Stochastic Dominance interactive module to visualize CDF comparisons and expected utilities.

Select the Crossing option in the Distribution pair control.
Observe that the CDF step lines cross. F_A(0) = 0.5 is greater than F_B(0) = 0, but F_A(1) = 0.5 is less than F_B(1) = 1.0. Thus, neither dominates.
Adjust the Threshold x control to 1.0. Read F_A(1) = 1/2 and F_B(1) = 1 in the visible readouts and complete CDF table.
Change the Utility control from Linear to Square. Read E[u(A)] = 2 and E[u(B)] = 1. Although these expected utilities change, this cannot establish dominance on its own since the full CDF check showed they cross.
Select Distribution pair = Equal, keep Threshold x = 1, and choose Utility = Square. Both means of the utility are (0² + 1² + 2²)/3 = 5/3 ≈ 1.666667. The CDFs remain equal; changing the utility does not change the distributions.

Limitations and Boundaries

This experiment compares three fixed pairs on a finite support. It checks the entire CDF rather than inferring a universal ordering from one queried threshold or one selected utility. Six-decimal displays approximate the exact fractions.

Finite Support: The distributions are evaluated over a fixed discrete support {0, 1, 2}. Real-world applications often involve continuous distributions where checking dominance requires integrating the density functions.
Right-Continuous CDFs: The evaluated CDFs are mathematically complete right-continuous step functions. This means the value at any step threshold includes the probability mass of that threshold.
Precision and Rounding: Visual representations and some numeric displays round values (e.g., to six decimal places). The true dominance check retains unrounded arithmetic internally to distinguish exact equality from minor differences.
Higher-Order Dominance: This module only checks First-Order Stochastic Dominance. Even if CDFs cross, a distribution might still Second-Order dominate if the area under its CDF is smaller, which is relevant for risk-averse decision-makers.

References

The following reference describes the mathematical definition of Stochastic Dominance and details its properties in terms of cumulative distribution functions.

MIT 14.124: Microeconomic Theory IV - Decision Making Under Uncertainty
These lecture slides detail Stochastic Dominance, explaining the definition of First-Order Stochastic Dominance through cumulative distribution functions and its relation to expected utility.

Stochastic Dominance