# Stochastic dominance

For other uses, see Dominance.

Stochastic dominance[1][2] is a form of stochastic ordering. The term is used in decision theory and decision analysis to refer to situations where one gamble (a probability distribution over possible outcomes, also known as prospects) can be ranked as superior to another gamble. It is based on preferences regarding outcomes. A preference might be a simple ranking of outcomes from favorite to least favored, or it might also employ a value measure (i.e., a number associated with each outcome that allows comparison of multiples of one outcome with another, such as two instances of winning a dollar vs. one instance of winning two dollars.) Only limited knowledge of preferences is required for determining dominance. Risk aversion is a factor only in second order stochastic dominance.

Stochastic dominance does not necessarily give a total order: for some pairs of gambles, neither one stochastically dominates the other, yet they cannot be said to be equal.

A related concept not included under stochastic dominance is deterministic dominance, which occurs when the least preferable outcome of gamble A is more valuable than the most highly preferred outcome of gamble B.

## Statewise dominance

The simplest case of stochastic dominance is statewise dominance (also known as state-by-state dominance), defined as follows: gamble A is statewise dominant over gamble B if A gives a better outcome than B in every possible future state (more precisely, at least as good an outcome in every state, with strict inequality in at least one state). For example, if a dollar is added to one or more prizes in a lottery, the new lottery statewise dominates the old one. Similarly, if a risk insurance policy has a lower premium and a better coverage than another policy, then with or without damage, the outcome is better. Anyone who prefers more to less (in the standard terminology, anyone who has monotonically increasing preferences) will always prefer a statewise dominant gamble.

## First-order stochastic dominance

Statewise dominance is a special case of the canonical first-order stochastic dominance, defined as follows: Gamble A has first-order stochastic dominance over gamble B if for any good outcome x, A gives at least as high a probability of receiving at least x as does B, and for some x, A gives a higher probability of receiving at least x. In notation form, $P [A \ge x]\ge P [B \ge x]$ for all x, and for some x, $P[A \ge x]>P[B \ge x]$. In terms of the cumulative distribution functions of the two gambles, A dominating B means that $F_A(x) \le F_B(x)$ for all x, with strict inequality at some x. For example, consider a die-toss where 1 through 3 wins $1 and 4 through 6 wins$2 in gamble B. This is dominated by a gamble C that yields $3 for 1 through 3 and$1 for 4 through 6, and it is also dominated by a gamble A that gives $1 for 1 and 2 and$2 for 3 through 6. Gamble A has statewise dominance over B, but gamble C has first-order stochastic dominance over B without statewise dominance. This is because, in states 4 to 6, gamble C has a worse outcome than B, however $P [C \ge x] = P [B \ge x]$ for all $x \le 2$ and $P [C \ge x]> P [B \ge x]$ for all $2 < x \le 3$ . Further, although when A dominates B, the expected value of the payoff under A will be greater than the expected value of the payoff under B, this is not a sufficient condition for dominance, and so one cannot order lotteries with regard to stochastic dominance simply by comparing the means of their probability distributions.

Every expected utility maximizer with an increasing utility function will prefer gamble A over gamble B if A first-order stochastically dominates B.

First-order stochastic dominance can also be expressed as follows: If and only if A first-order stochastically dominates B, there exists some gamble $y$ such that $x_B \overset {d}{=} (x_A+y)$ where $y\le 0$ in all possible states (and strictly negative in at least one state); here $\overset{d}{=}$ means "is equal in distribution to" (that is, "has the same distribution as"). Thus, we can go from the graphed density function of A to that of B by, roughly speaking, pushing some of the probability mass to the left.

## Second-order stochastic dominance

The other commonly used type of stochastic dominance is second-order stochastic dominance. Roughly speaking, for two gambles A and B, gamble A has second-order stochastic dominance over gamble B if the former is more predictable (i.e. involves less risk) and has at least as high a mean. All risk-averse expected-utility maximizers (that is, those with increasing and concave utility functions) prefer a second-order stochastically dominant gamble to a dominated gamble.

In terms of cumulative distribution functions $F_A$ and $F_B$, A is second-order stochastically dominant over B if and only if the area under $F_A$ from minus infinity to $x$ is less than or equal to that under $F_B$ from minus infinity to $x$ for all real numbers $x$, with strict inequality at some $x$; that is, $\int_{-\infty}^x [F_B(t) - F_A(t)]dt \geq 0$ for all $x$, with strict inequality at some $x$. Equivalently, $A$ dominates $B$ in the second order if and only if $E[u(A)] \geq E[u(B)]$ for all nondecreasing and concave utility functions $u(x)$.

Second-order stochastic dominance can also be expressed as follows: If and only if A second-order stochastically dominates B, there exist some gambles $y$ and $z$ such that $x_B \overset {d}{=} (x_A + y + z)$, with $y$ always less than or equal to zero, and with $E(z|x_A+y)=0$ for all values of $x_A+y$. Here the introduction of random variable $y$ makes B first-order stochastically dominated by A (making B disliked by those with an increasing utility function), and the introduction of random variable $z$ introduces a mean-preserving spread in B which is disliked by those with concave utility. Note that if A and B have the same mean (so that the random variable $y$ degenerates to the fixed number 0), then B is a mean-preserving spread of A.

### Second-order stochastic dominance in portfolio analysis

In the application of Portfolio Theory, Second order stochastic dominance plays a role when one begins to construct a framework to analyze risk-adjusted returns. MPT (Modern Portfolio Theory) employs the CAL (Capital Allocation Line) to evaluate the expected return (Mean) on the y-axis and the standard deviation (the square root of variance) x-axis. The whole concept of risk aversion is based on second order stochastic dominance in portfolio selection; however, some of the shortfalls of MPT are a result of not evaluating third order dominance or fourth order dominance criteria i.e. the skewness or kurtosis of the distribution.

"Markowitz (1959) recognized the asymmetrical inefficiencies inherited in the traditional mean-variance approach and suggested a semi-variance measure of asset risk that focuses only on the risks below a target rate or return. Post Modern Portfolio Theory (PMPT) employs the use of Mean Lower Partial Moments as a framework for analyzing risk. "Bawa (1975) generalized the semi-variance measure of risk to reflect a less restrictive class of decreasing absolute risk-averase utility function and shows that the second order mean-LPM for a class of DARA utility functions, is a preferred approximation for the optimal third order stochastic dominance selection rule compared to the mean-variance criteria." Sing and Ong

From "Marginal conditional stochastic dominance", "In finance, marginal conditional stochastic dominance is a condition under which a portfolio can be improved in the eyes of all risk-averse investors by incrementally moving funds out of one asset (or one sub-group of the portfolio's assets) and into another.[1][2][3] Each risk-averse investor is assumed to maximize the expected value of an increasing, concave von Neumann-Morgenstern utility function. All such investors prefer portfolio B over portfolio A if the portfolio return of B is second-order stochastically dominant over that of A; roughly speaking this means that the density function of A's return can be formed from that of B's return by pushing some of the probability mass of B's return to the left (which is disliked by all increasing utility functions) and then spreading out some of the density mass (which is disliked by all concave utility functions).

If a portfolio A is marginally conditionally stochastically dominated by some incrementally different portfolio B, then it is said to be inefficient in the sense that it is not the optimal portfolio for anyone. Note that this context of portfolio optimization is not limited to situations in which mean-variance analysis applies. The presence of marginal conditional stochastic dominance is sufficient, but not necessary, for a portfolio to be inefficient. This is because marginal conditional stochastic dominance only considers incremental portfolio changes involving two sub-groups of assets — one whose holdings are decreased and one whose holdings are increased. It is possible for an inefficient portfolio to not be second-order stochastically dominated by any such one-for-one shift of funds, and yet to by dominated by a shift of funds involving three or more sub-groups of assets.[4]"

See modern portfolio theory and marginal conditional stochastic dominance. See "Asset Allocation in a Downside Risk Framework", Tien Foo Sing and Seow Eng Ong See "Capital Market Equilibrium in a Mean-Lower Partial Moment Framework", Vijay S. Bawa and Eric B. Lindenberg

### Sufficient conditions for second-order stochastic dominance

• First-order stochastic dominance of A over B is a sufficient condition for second-order dominance of A over B.
• If B is a mean-preserving spread of A, then A second-order stochastically dominates B.

### Necessary conditions for second-order stochastic dominance

• $E_A(x) \geq E_B(x)$ is a necessary condition for A to second-order stochastically dominate B.
• If $A$ dominates $B$ in the second order, then the geometric mean of $A$ must be greater than or equal to the geometric mean of $B$.[clarification needed]
• $\min_A(x)\geq\min_B(x)$ is a necessary condition. The condition implies that the left tail of $F_B$ must be thicker than the left tail of $F_A$.

## Third-order stochastic dominance

Let $F_A$ and $F_B$ be the cumulative distribution functions of two distinct investments $A$ and $B$. $A$ dominates $B$ in the third order if and only if

• $\int_{-\infty}^x \int_{-\infty}^z [F_B(t) - F_A(t)] \, dt \, dz \geq 0$ for all $x$,
• $E_A(x) \geq E_B(x), \,$

and there is at least one strict inequality. Equivalently, $A$ dominates $B$ in the third order if and only if $E_AU(x) \geq E_BU(x)$ for all nondecreasing, concave utility functions $U$ that are positively skewed (that is, have a positive third derivative throughout).

### Sufficient condition for third-order stochastic dominance

• Second-order stochastic dominance is a sufficient condition.

### Necessary conditions for third-order stochastic dominance

• $E_A(\log(x))\geq E_B(\log(x))$ is a necessary condition. The condition implies that the geometric mean of $A$ must be greater than or equal to the geometric mean of $B$.
• $\min_A(x)\geq\min_B(x)$ is a necessary condition. The condition implies that the left tail of $F_B$ must be thicker than the left tail of $F_A$.

## Higher-order stochastic dominance

Higher orders of stochastic dominance have also been analyzed, as have generalizations of the dual relationship between stochastic dominance orderings and classes of preference functions.

## Stochastic dominance constraints

Stochastic dominance relations may be used as constraints [3] [4] in problems of mathematical optimization, in particular stochastic programming. In a problem of maximizing a real functional $f(X)$ over random variables $X$ in a set $X_0$ we may additionally require that $X$ stochastically dominates a fixed random benchmark $B$. In these problems, utility functions play the role of Lagrange multipliers associated with stochastic dominance constraints. Under appropriate conditions, the solution of the problem is also a (possibly local) solution of the problem to maximize $f(X) + E[u(X) - u(B)]$ over $X$ in $X_0$, where $u(x)$ is a certain utility function. If the first order stochastic dominance constraint is employed, the utility function $u(x)$ is nondecreasing; if the second order stochastic dominance constraint is used, $u(x)$ is nondecreasing and concave.

## References

1. ^ Hadar, J., and Russell, W.,"Rules for Ordering Uncertain Prospects", American Economic Review 59, March 1969, 25-34.
2. ^ Bawa, Vijay S., "Optimal Rules for Ordering Uncertain Prospects," Journal of Financial Economics 2, 1975, 95-121.
3. ^ Dentcheva, D., and Ruszczyński, A., "Optimization with Stochastic Dominance Constraints," SIAM Journal on Optimization 14, 2003, 548--566.
4. ^ Dentcheva, D., and Ruszczyński, A., "Semi-Infinite Probabilistic Optimization: First Order Stochastic Dominance Constraints," Optimization 53, 2004, 583--601.