This is part of a sequence on decision analysis; the first post is a primer on Uncertainty.

Decision analysis has two main parts: abstracting a real situation to math, and then cranking through the math to get an answer. We started by talking a bit about how probabilities work, and I'll finish up the inner math in this post. We're working from the inside out because it's easier to understand the shell once you understand the kernel. I'll provide an example of prospects and deals to demonstrate the math, but first we should talk about axioms. In order to be comfortable with using this method, there are five axioms1 you have to agree with, and if you agree with those axioms, then this method flows naturally. They are: Probability, Order, Equivalence, Substitution, and Choice.

Probability

You must be willing to assign a probability to quantify any uncertainty important to your decision. You must have consistent probabilities.

Order

You must be willing to order outcomes without any cycles. This can be called transitivity of preferences: if you prefer A to B, and B to C, you must prefer A to C.

Equivalence

If you prefer A to B to C, then there must exist a p where you are indifferent between a deal where you receive B with certainty and a deal where you receive A with probability p and C otherwise.

Substitution

You must be willing to substitute an uncertain deal for a certain deal or vice versa if you are indifferent between them by the previous rule. Also called "do you really mean it?"

Choice

If you have a choice between two deals, both of which offer A or C, and you prefer A to C, then you must pick the deal with the higher probability of A.

These five axioms correspond to five actions you'll take in solving a decision problem. You assign probabilities, then you order outcomes, then you determine equivalence so you can substitute complicated deals for simple deals, until you're finally left with one obvious choice.

You might be uncomfortable with some of these axioms. You might say that your preferences genuinely cycle, or you're not willing to assign numbers to uncertain events, or you want there to be an additional value for certainty beyond the prospects involved. I can only respond that these axioms are prescriptive, not descriptive: you will be better off if you behave this way, but you must choose to.

Let's look at an example:

My Little Decision

Suppose I en

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Decision Analysis