# 5.2 Expected Utility: Theory & Formula

## A Big Decision

Justin is a hotshot salesman for a technology company. Business is booming, and he has been approached by other companies about changing jobs. He has two interesting offers on the table. His wife, Maria, tells him to go with the job that offers the most money. But for Justin, it’s not that simple.

The first job offer is with a company that makes robots. The technology is very advanced, so they pay their salespeople a salary. They will start Justin off at $6,000 per month, which is more than he makes now. The other company will pay $2,000 per month in salary, but they have a bonus system: If Justin sells his quota of software systems, the pay goes up to $10,000 a month. ‘Wow!’ says Maria. ‘We could take some amazing vacations with that kind of money!’

Now Justin and Maria have a decision to make. Justin learned a lot about statistics and math when he was in college, but he needs Maria’s help to do the calculations he is thinking about. He wants to take an analytical approach to making this decision.

## Expected Value and Expected Utility

Justin thinks there is a 50% chance of making the bonus each month. He wants to approach this in terms of the **expected value** of what he will make. He thinks he will make $2,000 a month half the time, and $10,000 the other half from the software job. They can be weighted equally and he can just take an average. $2,000 + $10,000 = $12,000. Divide that by two and his expected income is $6,000, which is exactly the same as his expected income from the robot company.

Maria thinks something is missing, though. If we just measure this by dollars, we are missing the part where dollars are not all valued the same. The first dollars that Justin makes go toward paying the rent and buying food, making them extremely valuable dollars.

After that, the principle of **diminishing marginal utility** sets in. Additional dollars will be used for things like ball games and vacations. Vacations may be fun, but they aren’t as important as paying the rent and eating! So the value of additional dollars earned diminishes.

## Expected Utility Theory

Justin and Maria can bring in the concept of expected utility to better solve their dilemma. When facing a decision with uncertainty, **expected utility theory** states they should choose the alternative that offers the most utility. Using the utility, or satisfaction they will receive, instead of just dollars will allow a more accurate decision.

To determine this, Justin and Maria can take the pay amounts from these jobs and decide what the different amounts are worth to them, then apply the formula to get the expected utility from each job. Utility is a subjective concept: everyone can place a different value on how much satisfaction, or utility, a given outcome will provide. So Justin and Maria can tailor this to their own situation.

That first $2,000 is very valuable, Maria says, because it goes to pay the rent and buy food. We will assign it 20 units of utility. The additional dollars up to $6,000 are worth quite a bit less, so we will assign $6,000 a total of 45 units. Finally, the additional dollars from $6,000 to $10,000 are worth even less, so we will make $10,000 worth 60 units of utility.

## Expected Utility Formula and Calculation

Justin wants to plug their ideas into the **expected utility formula** now and see which job will maximize their utility. The formula for the expected utility from choice (C) is:

EU(C) = (PA * UA) + (PB * UB) ……. (PZ * UZ)

PA is the probability of outcome A and UA is the utility from outcome A. It’s the same for PB, etc. This can be used for as many outcomes as desired, so long as the total of the probabilities (P) equals 1.0, or 100%.

The expected utility from the robotics job is 45 units of utility, since there is a 100% chance Justin will make his salary. For the software job, it’s the 50% probability of making only $2,000, which has 20 units of utility, plus the 50% probability that he does make the bonus and get $10,000, which has 60 units of utility.

Let’s plug those into our formula:

(0.50 * 20) + (0.50 * 60) = 10 + 30 = 40 units of utility

So in order to maximize their utility, Justin will be learning about robotics!

## Lesson Summary

**Expected utility theory** states that under conditions of uncertainty, the correct choice between alternatives is the one that maximizes utility. It is different from **expected value**, which uses absolutes to measure outcomes. Using the concept of utility instead allows for subjective values to be used for the satisfaction that will be derived from each of those outcomes. It also allows the concept of **diminishing marginal utility** to be included.

The **expected utility formula** is used to calculate the expected utility for an alternative choice. The expected utility of alternative C is:

EU(C) = (PA * UA) + (PB * UB) ……. (PZ * UZ)

PA is the probability of outcome A and UA is the utility from outcome A, etc. This can be used for as many outcomes as desired, so long as the total of the probabilities (P) equals 1.0, or 100%.