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# Stochastic Dominance

Updated on January 8, 2024
Article byJyotsna Suthar
Edited byAlfina
Reviewed byDheeraj Vaidya, CFA, FRM

## What Is Stochastic Dominance?

Stochastic dominance, in statistics, is a commonly used decision theory that ranks probability distributions based on certain factors. The main purpose of this concept is to decide whether a certain outcome is preferable over another. Based on it, the decision is made.

For eg:
Source: Stochastic Dominance (wallstreetmojo.com)

This dominance belongs to the stochastic ordering category. It tries to prioritize outcomes based on their risk factors. It helps in risk assessment and decision-making. In addition, a major application of stochastic dominance is also visible in finance. However, it may ignore certain features and variations.

### Key Takeaways

• Stochastic dominance refers to the decision-making concept that allows the user to select only those distributions based on certain parameters.
• The major tests involved in the process are statewide, first-order, second-order, third-order, and high-degree dominance tests.
• The application of this dominance is visible in gambling, the insurance market, and the financial industry. Also, financial analysts and fund managers use it frequently.
• Here, the dominant outcome gets more priority than others. However, risk, inequality, skewness, and kurtosis also matter.

### Stochastic Dominance Explained

Stochastic dominance is a form of decisive model that helps the user to rank the outcomes based on their risk rate. There are different tests and types included to use the stochastic dominance approach. Each creates a list of probability distributions having more occurrences without diverting the model. Thus, the topmost, superior outcome is chosen for the decision-making.

The major application of stochastic dominance is seen when used by financial analysts and fund managers. As there are various asset classes, it is sometimes hard to select one investment option. Thus, using stochastic dominance in finance enables them to select the right investment deal for the client’s portfolio. So, out of three options, if one has higher volatility, the fund manager will choose one with less risk. Another usage is visible among gamblers and insurers. For example, if the lottery adds an extra prize amount, the new lottery will be preferable over others. Likewise, insurance policies with lower premiums and risk will have stochastic dominance.

However, at times, there is often confusion in selecting the variables. In short, the researcher might not be sure which outcome must be applied. As a result, in such situations, the stochastic dominance approach is used. Based on the outcomes, there are two stochastic dominance tests. It includes a first-order stochastic dominance and a second-order test. With the help of them, users can easily make comparisons among the distributions. But, if there are more than two alternatives, no two dominant ones will give the supposed result. Yet at least one of them will break down the complexity. Therefore, the decision-making process turns simpler.

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### Types

Let us look at the types of stochastic dominance tests that help in better distribution selection:

#### #1 – Statewise Dominance

Also known as state-by-state dominance, it is one of the simplest forms of stochastic dominance. Under this type, each state dominates the other in some situations. For example, if gamble A provides better returns than B, the former is selected. Those individuals having an increasing preference can use this test.

P (A ≥ x) ≥ P (B ≥ x)

#### #2 – First-Order Dominance Test

As the name suggests, first-order dominance states that the first outcome with the highest probability receives more importance. It is controlled by cumulative distribution functions (CDFs). For example, event A has more chances to receive 2x returns than B, while B provides 1x yield. At the same time, event C has 2x more returns than B. So, in this case, event C has first-order dominance. However, event A will still have statewide dominance against B.

Here, P (A ≥ x) ≥ P (B ≥ x), but P (C ≥ x) ≥ P (B ≥ x)

#### #3 – Second Order Dominance Test

Here, it considers both inequality and risk aversion. For instance, if event A has more chances of probability with less risk than B, the former will become second-order dominance. If it happens so, the geometric mean of A will be more than the event B.

Here, EA (x) ≥ EB (x) states the dominance of A over B.

#### #4 – Third-Degree Dominance Test

An event must be risk-averse, inequality-averse, and lower skewness to become a third-order dominance. In addition, it considers the shape and structure of the distribution before making a decision. However, if event A passes both the inequality and risk test, the next step involves the shape of the distribution.

#### #5 – Higher Degree Dominance Test

This dominance test surpasses the third-order test. In this type, even kurtosis matters apart from inequality, risk, and skewness.

### Examples

Let us look at the examples of stochastic dominance for a better understanding of the concept:

#### Example #1

Suppose Claisy is a fund manager working in a mutual fund house for the past eight years. She tries to accumulate funds from clients and invest in appropriate asset classes. And to accomplish the task, she uses stochastic dominance. So, consider she has a client named Arthur. Since he prefers less risk and long-term returns, Claisy has listed three dummy investments. The features for each portfolio are as follows:

1. Investment #1 – High returns, high volatility
2. Investment #2 – Fewer returns, low risk involved.
3. Investment #3 – Moderate returns with minimal risk

Out of all three, Claisy put them on the stochastic dominance test. Event 1 has more probability than the second investment. But Arthur needed less risk. Therefore, the first investment turns statewide dominance. However, the last investment stays appropriate to the conditions. Thus, Claisy decides to invest in Investment #3 for Arthur.

#### Example #2

A similar example accounts for gambling methods. There are certain outcomes if a person buys eight lottery tickets hoping to win the highest reward. There can be instances where ticket one may align with a high-degree dominance test. However, the rest of them might not cover risk and inequality factors. So, a gambler will choose the one that compacts all parameters.

Although stochastic dominance has a major application in the finance industry, it can have equivalent disadvantages. So, let us look at them:

1. When did stochastic dominance become popular?

The popularity of stochastic dominance dates back to the mid-20th century after the Second World War. The first mention was by French economist, Maurice Allais, in the article, “Fondements d’une Théorie Positive des Choix comportant un Risque et Critique des Postulats et Axiomes de l‘Ecole Américaine.” However, the first paper was released in 1962 in the economics and finance literature.

2. What is the importance of stochastic dominance?

The concept of stochastic dominance has greater usage in the decision tree maker. It acts as a partial order between two random variables. Here, partial order refers to events that precede the former.

3. What is the difference between stochastic and deterministic dominance?

In deterministic dominance, there are chances of similar outcomes. However, for stochastic dominance, the inputs are run partially. Therefore, chances of producing identical results are not possible. The latter inherits a certain level of randomness within the process.

This has been a guide to What Is Stochastic Dominance. Here, we explain the concept along with its examples, types, advantages, and disadvantages. You can learn more about it from the following articles –