What is the Bernoulli Utility Function?
Simply put that, a Bernoulli Utility Function is a kind of utility function that model a risk-taking behavior such that,
- If someone has more wealth, she will be much comfortable to take more risks, if the rewards are high. (i.e. a rich gambler)
- But, if someone has less wealth, she will be more concerned about the worse case, and therefore, she will think twice before taking a risk of losing, even though, the reward can be high.
That makes sense, right?
And, that is the idea of the Bernoulli Utility function.
Say, if you have a good amount of money saved in your bank, you can feel safer to invest in a business where the worst-case outcome of that business will not make you bankrupt. But if you do not have much money saved in your bank account, then you would better keep that money and won't gamble with your last asset. As we can see in the following picture, someone with a sack of money is taking the risk of walking on a line over the fire. But, with little money, someone is running away from that path.
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. To be more specific in terms of math, he proposes that marginal utility is inversely proportional to wealth. If total wealth is expressed as W, and utility function is U(W), then
Here, someone's Utility Function is denoted as U(W) and marginal utility is the first derivative of the Utility function U(W). 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.
What does it mean by scaling parameters?
Say, you want your utility function such that, for a given scenario, the maximum possible payoff should give U(maximum payoff) = 1. and the minimum payoff should be U(minimum payoff) = 0. Using some parameters, you can adjust the utility function in that way. These parameters are called scaling parameters.
You can determine the value of "a" and "b" like this.
Set any value to W, i.e. 100, and ask yourself, what is your utility value for that wealth? You get a number. (i.e. 0.1 Utils) Put that number to the above equation.
Then, set another value to W, i.e. 1000 or whatever you like, then ask yourself again, what is your Utility value for such high wealth. You will get another number. (i.e. 0.9). Then, you will get 2 equations where the variables are just "a" and "b". When you have 2 equations with 2 variables, using linear algebra, you can solve the value for those variables, right?. So, you will get "a" and "b" accordingly.
Solving these 2 linear equations, we get,
Our Decision Analysis Software (Decision Tree Software or Rational Will) can calculate that parameter based on the Minimum and Maximum possible values in the decision context, which is collected from the user.
Another form of the equation
As the W represents the total wealth, if your payoff is a variable denoted by "x" and if you have net wealth "S", then your total wealth W would be equal to x + S, right? Therefore, the Bernoulli utility function can be rewritten as
Where "S" represents the money in the savings 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 savings account, he can be less risk-averse. But if someone has a very limited amount of money in his savings account, he will fear more about losing money as he/she cannot afford to lose money.
If we plot a Bernoulli Utility Function for various wealth, this idea will be very clear. Let's do that. Say, in a risky investment, someone can gain from 0$ to max 400$. Assume that she has just 10$ in her savings account. Also, assume that we have evaluated her utility function is:
If you are confused about how these numbers came to this equation, don't worry. Just think that, based on various questionnaires. we evaluated her scaling parameters as a = 33.1 and b=-99.18.
So, when S = 10$, we get the following plot of the above utility function.
If you are familiar with various utility function plots, then you can recognize that such a plot represents a utility function of a risk-averse person. (Here, the person has just 10$, which is a very low amount, therefore, she is more risk-averse).
But, if you increase the value of net wealth to a high number like S = 1000. Then, the utility function plot looks like this:
Now, notice, that, this plot clearly shows that the person is a Risk Neutral. A straight line is generally a utility function of a risk-neutral person.
Marginal Utility function
Marginal Utility, basically, means, if someone gains a very little amount of reward or payoff, how much the utility will change with respect to that little payoff. In the mathematical terms, it is the first-order derivative of the Utility Function U(x). Therefore, for a Bernoulli utility function, the marginal utility function is:
According to behavioral economics, the mathematical expression of the absolute risk aversion for any utility function is defined as:
Applying the above operation on the Bernoulli utility function, we get the absolute risk aversion as:
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 defined as:
Applying the above formula, we can get the relative risk aversion for a Bernoulli utility function as
The constraint of a Bernoulli Utility Function
Even though the Bernoulli Utility function can model realistic behavior very well, yet there is a minor detail that needs to be remembered when using such an equation.
What is the natural logarithm of zero?
Since ln(0) is the number that we get by solving the equation:
There is no value of x that satisfies this equation.
So, if you set Net Wealth = 0, then for a value of x, your Bernoulli Utility Function will give a value that is undefined.
So, if you set Net Wealth = 0, and if your payoff's Minimum and Maximum value is such a range where 0 can be a possible number, then our software will show error as shown below.
So, in order to avoid such a problem, we recommend setting at least 1 in the Net Wealth, or your Minimum Payoff value should be greater than 0.
Say, you have two business opportunities and you want to decide which one is best. They are Investment A and Investment B. Investment A can bring 20,000$ revenue with a probability of 0.2 and 500$ with a probability of 0.8. Investment B can bring 2000$ with a probability of 0.85 and 100$ with a probability of 0.15.
Also, assume that you have a net wealth of 100$.
If you are using the Decision Tree Analyzer software then you will be greeted with the following screen. If you are using Rational Will software, click the "Decision Tree" button from the home screen to get to this view. Click the button "Identify your Objectives".
Then you will be presented with the following screen. Select "Maximize" and enter "Revenue" as shown below.
Then click the "Proceed" button. You will be asked about the type of objective. Select "Money Type". Please remember that, in order to use a Utility function, you need to use the Number type or Money Type objective.
Then you will be asked if you want to use Interest Rate based calculation where Present Monetary Value will be calculated. For this simple example, we do not need that, so click No. Then you will be asked about the minimum, maximum payoff range from the investment. Enter Minimum = 100 and Maximum = 20000.
Click Proceed. You will be asked if you want to add another objective. To keep the demonstration simple and easy to follow, let's stick with one objective. So, click No. Then, you will be presented with the following screen. To create a utility function, we need to go to the objectives manager and edit an objective. So, click the "Objectives" hyperlink.
Then, you will be taken to the Objectives manager page. Select the objective and open the context menu from a right mouse click, or double click on the objective.
The objective editor will appear. Click the Bernoulli Utility Function button as shown below. Enter the Net Wealth value = 100$.
How Scaling parameters are calculated
You may be curious to know, in the generated utility function, from where these scaling parameters 21.69 and -114.93 come from. The scaling parameters are calculated such that, the maximum payoff will result in the highest utility value which can be 1 or 100, depending on the preference. The lowest payoff will result in the lowes utility value which can be 0, or -1 or -100, depending on the preferences. The preference can be specified from the ribbon as shown here. And the maximum and minimum payoff are specified as Minimum Value and Maximum Value, shown in the following screenshot as well.
How net wealth shapes the curve
Notice that the generated plot is a concave line which indicates high-risk aversion, based on your net wealth. Just for an experiment, change the net wealth value from 100 to 10,000, you will see the plot become almost like a straight line, which indicates, risk-neutral attitude. We learned that more wealth can make a decision maker less risk-averse and we can get a demonstration of that idea in this plot.
Please note that Bernoulli Utility Function can be used for both Numeric type and Monetary type objectives. But, if the objective is not a monetary type, then the Net Wealth box does not show up. But, you can always change from one objective type to another as shown below:
Viewing other derivatives of the generated utility function.
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.
Also, another detail about this utility function, in our decision analysis software is that, when the Goal is to Maximize some criterion, then the "Money Type" attribute can be used with Bernoulli Utility Function. If the goal is to Minimize some variable, then, a money type attribute with Bernoulli utility function won't make sense, and therefore, the software will show an error message like this.
Finally, model the Decision Tree
Click Ok in your Objective editor when you are done refining your utility function. Then, you will be taken to the Objectives manager page. Click the "Work on Decision Tree" button.
Then, click the "decision Node" button to create a decision tree with a Decision Node as the root node.
Then, create a decision tree like this. If you are not familiar with how to create the decision tree in our decision tree software, please visit the getting started page. From that page, you will know how to set a payoff to a node.
Then, set payoff for each node.
Anytime, you click the Utility value link shown on each node, the Payoff editor will show up.
Within the payoff editor, click the Utils link to open the utility function chart. You can also see a green vertical line that indicates where your utility stands in the plot based on the currently set payoff. The line moves as you change the payoff instantly.
Finally, we hope our attempt to explain the Bernoulli Utility Function on this page will be helpful.