We have already explained what a utility function is. Let's explain the Bernoulli Utility function.
Introduction to Bernoulli Utility Function
Say, if you have a good amount of money saved in your bank, you can feel safer to take a bigger risk, but if you do not have much money saved in your bank account, then you will tend to avoid a risk. That is the idea of the Bernoulli Utility function. Bernoulli proposes that the utility function used to evaluate an option should be a function of one's wealth, and not just current income flows. Bernoulli suggests a form for the utility function in terms of a differential equation. In particular, he proposes that marginal utility is inversely proportional to wealth. If total wealth is expressed as W, and utility function is U(W), then
for some constant "a". We can solve this differential equation to find the function "U(W)". By solving the equation, we get
"a" and "b" are essentially scaling parameters. Decision Tree Maker & Analyzer software can calculate that parameter based on the Minimum and Maximum possible values in the decision context, which is collected from the user.
As the W represents the total wealth, if your payoff is a variable denoted by "x", then the utility function can be rewritten as
Where "S" represents the money in the saving account. In the decision tree software, this term is presented as "Net Wealth". Bernoulli points out that with this utility function, people will be risk-averse. If someone has a huge amount of money saved in his saving account, he can be less risk-averse. But if someone has a very limited amount of money in his saving account, he will fear more about losing money as he/she cannot afford losing money.
Marginal Utility function
For a Bernoulli utility function, the marginal utility function is
The absolute risk aversion for any utility function is
Applying the above operation on the Bernoulli utility function, we get the absolute risk aversion is
From the above absolute risk aversion function, we can easily understand that, when someone has a huge amount of money, the A(x) tends to be zero. That means he/she won't be risk averse at all. But, if someone has a very little amount of money, A(x) will be a big number, and therefore, he/she will be highly risk-averse.
The relative risk aversion formula for any utility function is
Applying the above formula, we can get the relative risk aversion for a Bernoulli utility function as
Modeling in the Decision Tree software
You can model Bernoulli Utility function using the Decision Tree software, and you can incorporate Bernoulli utility function in a decision-making process when the uncertainty is involved.
Here, we can show how to use the Bernoulli utility function.
Navigate to the Objectives page and then from the context menu choose Edit button (or double-click the objective)
Once you open the objective editor, click the button "Bernoulli Utility Function".
Once the Bernoulli Utility Function editor shows up, do one more thing. From the attribute type drop-down, select Money type so that a Net Worth box is shown.
Now, you will see the Net Wealth box. Enter the money amount you have saved in your bank. You will see the curve changed and the utility function is updated.
You can check the Marginal Utility function, Absolute Risk Aversion and Relative Risk Aversion from the radio buttons as you can see at the bottom of the panel. Here is the Marginal Utility Function for the above-generated function.