Decision Trees & Expected Value

A decision tree maps the options in an uncertain choice, their possible outcomes, and the chances of those outcomes. Expected value is the average result weighted by those chances: for example, a hypothetical fair coin flip paying 1,200 dollars for heads and nothing for tails has an expected monetary value of 600 dollars. Try this: Sketch one uncertain choice, list the possible outcomes and their chances, and compare the expected monetary payoffs of the options.

At decision nodes, branches represent available actions. At chance nodes, branches represent outcomes that cannot happen together and cover all possibilities. Their chances, given earlier events, must add up to one. End points record payoffs: the money an outcome pays or costs. 2

Evaluate the tree backward, averaging at chance nodes and keeping the highest value at decision nodes when maximizing expected payoff. For the hypothetical coin flip, half of 1,200 dollars plus half of zero dollars gives an expected monetary payoff of 600 dollars, compared with 500 dollars guaranteed. Assuming equal costs and payment timing, the coin flip has the higher expected monetary payoff. Its actual payout is still either 1,200 dollars or nothing. 2

A higher expected monetary payoff does not establish which option you should prefer. If you need 500 dollars for an essential bill, you might prefer the guarantee because receiving nothing would have serious consequences. Bernoulli distinguishes monetary amounts from utility: their value to someone in particular circumstances. 1

Probability estimates may be subjective. Try changing uncertain estimates to see whether the ranking changes; this is sensitivity analysis. The comparison remains conditional on the outcomes and payoffs included in the tree. 2

• Bernoulli distinguishes the monetary price of a gain from the utility it provides. [1]
• Bernoulli explains that a gain's utility depends on the financial circumstances of the person receiving it. [1]
• The cited English translation appeared in 1954; the Latin original appeared in 1738. [1]
• MIT's lecture teaches backward evaluation: average values at event nodes and retain the highest value at choice nodes when maximizing payoff. [2]
• The MIT lecture illustrates sensitivity analysis by varying inputs and examining whether decisions change. [2]

An expected payoff is not a guaranteed payout. Results depend on the modeled outcomes, chances and payoffs. 2 Expected money alone does not capture how someone values risk in their circumstances; evaluating utility requires additional assumptions about those preferences. 1