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 enter a lottery for MLP toys. I can choose from two kinds of tickets: an A ticket has a 1/3 chance of giving me a Twilight Sparkle, a 1/3 chance of giving me an Applejack, and a 1/3 chance of giving me a Pinkie Pie. A B ticket has a 1/3 chance of giving me a Rainbow Dash, a 1/3 chance of giving me a Rarity, and a 1/3 chance of giving me a Fluttershy. There are two deals for me to choose between- the A ticket and the B ticket- and six prospects, which I'll abbreviate to TS, AJ, PP, RD, R, and FS.

(Typically, decision nodes are represented as squares, and work just like uncertainty nodes, and so A would be above B with a decision node pointing to both. I've displayed them side by side because I suspect it looks better for small decisions.)

The first axiom- probability- is already taken care of for us, because our model of the world is already specified. We are rarely that lucky in the real world. The second axiom- order- is where we need to put in work. I need to come up with a preference ordering. I think about it and come up with the ordering TS > RD > R = AJ > FS > PP. Preferences are personal- beyond requiring internal consistency, we shouldn't require or expect that everyone will think Twilight Sparkle is the best pony. Preferences are also a source of uncertainty if prospects satisfy multiple different desires, as you may not be sure about your indifference tradeoffs between those desires. Even when prospects have only one measure, that is, they're all expressed in the same unit (say, dollars), you could be uncertain about your risk sensitivity, which shows up in preference probabilities but deserves a post of its own.

Now we move to axiom 3: I have an ordering, but that's not enough to solve this problem. I need a preference scoring to represent how much I prefer one prospect to another. I might prefer cake to chicken and chicken to death, but the second preference is far stronger than the first! To determine my scoring I need to imagine deals and assign indifference probabilities. There are a lot of ways to do this, but let's jump straight to the most sensible one: compare every prospect to a deal between the best and worst prospect.2

I need to assign a preference probability p such that I'm indifferent between the two deals presented: either RD with certainty, or a chance at TS (and PP if I don't get it). I think about it and settle on .9: I like RD close to how much I like TS.3 This indifference needs to be two-way: I need to be indifferent about trading a ticket for that deal for a RD, and I need to be indifferent about trading a RD for that deal.4 I repeat this process with the rest, and decide .6 for R and AJ and .3 for FS. It's useful to check and make sure that all the relationships I elicited before hold- I prefer R and AJ the same, and the ordering is all correct. I don't need to do this process for TS or PP, as p is trivially 1 or 0 in that case.

Now that I have a preference scoring, I can move to axiom 4. I start by making things more complicated- I take all of the prospects that weren't TS or PP and turn them into deals of {p TS, 1-p PP}. (Pictured is just the expansion of the right tree; try expanding the tree for A. It's much easier.)

Then, using axiom 1 again, I rearrange this tree. The A tree (not shown) and B tree now have only two prospects, and I've expressed the probabilities of those prospects in a complicated way that I know how to simplify.

And we have one last axiom to apply: choice. Deal B has a higher chance of the better prospect, and so I pick it. Note that that's the case even though my actual chance of receiving TS with deal B is 0%- this is just how I'm representing my preferences, and this computation is telling me that my probability-weighted preference for deal B is higher than my probability-weighted preference for deal A. Not only do I know that I should choose deal B, but I know how much better deal B is for me than deal A.5

This was a toy example, but the beauty of this method is that all calculations are local. That means we can apply this method to a problem of arbitrary size without changes. Once we have probabilities and preferences for the possible outcomes, we can propagate those from the back of the tree through every node (decision or uncertainty) until we know what to do everywhere. Of course, whether the method will have a runtime shorter than the age of the universe depends on the size of your problem. You could use this to decide which chess moves to play against an opponent whose strategy you can guess from the board configuration, but I don't recommend it.6 Typical real-world problems you would use this for are too large to solve with intuition but small enough that a computer (or you working carefully) can solve it exactly if you give it the right input.

Next we start the meat of decision analysis: reducing the real world to math.

1. These axioms are Ronald Howard's 5 Rules of Actional Thought.

2. Another method you might consider is comparing a prospect to its neighbors; RD in terms of TS and R, R in terms of RD and FS, FS in terms of R and PP. You could then unpack those into the preference probability

3. Assigning these probabilities is tough, especially if you aren't comfortable with probabilities. Some people find it helpful to use a probability wheel, where they can see what 60% looks like, and adjust the wheel until it matches what they feel. See also 1001 PredictionBook Nights and This is what 5% feels like.

4. In actual practice, deals often come with friction and people tend to be attached to what they have beyond the amount that they want it. It's important to make sure that you're actually coming up with an indifference value, not the worst deal you would be willing to make, and flipping the deal around and making sure you feel the same way is a good way to check.

5. If you find yourself disagreeing with the results of your analysis, double check your math and make sure you agree with all of your elicited preferences. An unintuitive answer can be a sign of an error in your inputs or your calculations, but if you don't find either make sure you're not trying to start with the bottom line.

6. There are supposedly 10120 possible games of chess, and this method would evaluate all of them. Even with computation-saving implementation tricks, you're better off with another algorithm.