Expected Value

Expected Value

Each outcome contributes its utility multiplied by its probability to the expected value of an option. For example, a $4.00 payoff for picking a green bean is weighted by the probability of that outcome. Adding the contributions across all possible outcomes gives the option’s expected value. We do this multiplication in the sixth step, entering the expected values in a column labeled “EV” (Figure 14-6). We then add the expected values for each option and enter the sums in another column labeled “Total EV” (Figure 14-7).

[Figure 14-5: Bettor’s profit]

The referenced figures are not reproduced here, so the original tree cannot be inspected.

In the seventh step, we rank the expected values from first to last, adding yet another column (Figure 14-8). The expected values show that, if you elect to place a wager, the best decision is to buy a chance on picking a yellow bean.

In the eighth and last step, we do a sanity check. Does this analysis make sense? Do we understand the rationale? Let’s ask ourselves whether betting on a yellow bean is reasonable. Although each of the green beans is worth $4.00, only nine of the ninety beans in the jar are green. And while half the beans are red, the number of yellows is only slightly less than half and the yellows are worth more. This is the source’s rationale for the ranking; the calculation also needs to include the loss when the selected color is not drawn.

[Figure 14-6: Bettor’s profit]

[Figure 14-7: Bettor’s profit]

[Figure 14-8: Bettor’s profit]

In a utility tree, headers identify the sequence of events for each scenario.

A branch can be omitted from the arithmetic only if its contribution is zero or is already accounted for elsewhere. Keeping only matching-color branches would be wrong when the other branches carry losses, as in the example below.

After ranking the options, examine why their expected values differ. Check which payoffs and probabilities drive the result, whether the assumptions are credible, and how sensitive the ranking is to changes.

These then are the eight steps of utility-tree analysis:

Step 1: Identify the options and outcomes to be analyzed.

Step 2: Identify the perspective of the analysis.

Step 3: Construct a scenario tree for each option.

Step 4: Assign a utility value to each option-outcome combination—each branch (scenario) of the tree—by asking the Utility Question: If we select this option, and this outcome occurs, what is the utility from the perspective of…?

Step 5: Assign a probability to each outcome. Determine or estimate this probability by asking the Probability Question: If this option is selected, what is the probability this outcome will occur? The probabilities of all outcomes for a single option must add up to 1.0.

Step 6: Determine the expected values by multiplying each utility by its probability and then adding the expected values for each option.

Step 7: Determine the ranking of the alternative options.

Step 8: Perform a sanity check.

Use these steps to compare alternatives and explain the resulting ranking.