 - Expected Value

# Expected Value

I use the idea of “expected value” a lot. Being comfortable with this idea is pretty essential for following a lot of what’s being discussed in the "effective altruist" and "rationalist" communities. Here’s a brief summary of how “expected value” works and why it’s often important.

## Defining expected value

Say you have a scratch-off lottery ticket with the following properties:

• There’s an 80% chance that it’s worth \$0.
• There’s a 15% chance that it’s worth \$1.
• There’s a 4.9999% chance that it’s worth \$10.
• There’s a .0001% chance that it’s worth \$100,000.

How much is this ticket worth? That is, how much would be a fair price to sell it for?

One answer would be to look at the expected value of the ticket. To compute this, you multiply the value of each possible outcome by the probability of that outcome, and add across all possible outcomes.

For this ticket, that would be:

80% * \$0 + 15% * \$1 + 4.9999% * \$10 + .0001% * \$100,000

= \$0 + \$0.15 + \$0.49999 + \$0.10

= about \$0.85 (to be precise, \$0.849999)

More abstract and general version:

For any given decision you might make (such as selling a lottery ticket), you multiply the value and probability of each possible outcome, then add across all possible outcomes to get the expected value of your decision.

• Say that there are N possible outcomes of your decision: outcome 1, outcome 2, … , outcome N.
• Call the probabilities P_1 [probability of outcome 1], P_2 [probability of outcome 2], … , P_N [probability of outcome N].
• Call the values V_1 [value of outcome 1], V_2 [value of outcome 2], … , V_N [value of outcome N].
• Then, the expected value of your decision is: P_1*V_1 + P_2*V_2 + … + P_N*V_N.

## Value = expected value?

A background assumption in many discussions (particularly in the effective altruist and rationalist communities) is that “value is expected value.” That is, X is worth whatever the expected value of X is.

• This implies that the lottery ticket from the previous section is equivalent to \$0.85 in cash - there’s basically no important difference.
• It also implies that if you donate to a charity that will definitely prevent 10 deaths, this is exactly as good as donating to a charity with a 1% chance of preventing 1000 deaths (and a 99% chance of accomplishing nothing), since 1%*1000 = 10.

Why think that value is expected value?

### A simple reason: value = expected value with enough trials

Let’s start here: if you rack up enough independent events, then (on average) value will converge toward expected value.

For example, take the lottery ticket from the previous section. If you scratch it off, you might end up with \$0, or \$1, or \$10, or \$100,000. But if you scratch off a billion tickets just like this one, and the result of each is independent of the result of the others, then you are statistically guaranteed to end up with approximately \$850 million, which is \$0.85 for each ticket you scratched off.1

Now say that you have a very large number of different lottery tickets. Each one has different probabilities, different payoffs, and different expected value, and the result of each one is independent of each of the result of the others. Here too, if you have a large enough number of tickets, the total value of the tickets is guaranteed to be approximately the same as the total expected value of the tickets. That is, if you compute the expected value of each ticket, and add those up, you have an almost-perfect estimate of how much money you would actually make if you scratched them all off. (The difference between this paragraph and the last one is that you don’t need the tickets to be the same as each other - your total value equals your total expected value across all the tickets, even if they’re all different tickets.)

A couple of consequences of this:

• You could think of all the financial decisions you make in your life as a large number of different, independent scratch-off lottery tickets. Each financial decision has some probability of leaving you with more money, some probability of leaving you with less, etc. If you make each of those decisions in a way that maximizes your expected money, then over the course of your life, you’ll end up with the maximum possible amount of actual money - as long as the total number of independent decisions is large enough.
• When you’re out to dinner with friends, instead of splitting the bill evenly, you could randomly select one person to pay the whole bill. If you do this for enough dinners, then the randomness will wash out, and you’ll end up paying the same total amount for all these dinners that you would’ve paid if you had split the bill evenly. This is true even if each dinner is a different meal with a different price and with a different set of friends.

So, “value = expected value” is true “in the long run” - as long as you have enough independent events. But what about when you don’t?

### Expected utility maximization

There could be times when you don’t want to maximize the expected value of how much money you have (or get) in a situation. For example, you might want to buy insurance, in which you pay (say) \$5000 a year in premiums, for a well-under-1% chance of a \$100,000 payout in the case where you need some very expensive medical treatment.

This is fundamentally because money isn’t something we care about in and of itself - we care about it because of what we can buy with it. Money becomes more valuable when one develops an expensive medical condition.

“Expected utility maximization” can be thought of as “maximizing the expected value of whatever you must fundamentally care about” - candidates for things we fundamentally care about might be “years of healthy life lived” or “total deaths averted” (when making government policy or giving to charity) or “magical happiness points” (an abstraction). See The Bayesian Mindset for more on this idea.

## Risk-neutrality

People often seem to make decisions based on intuitions re: whether they want to "play it safe" or "swing for the fences." For example, a fair amount of ink has been spilled about whether philanthropists should be "hits-based."

The idea of expected value maximization can sort of dissolve debates like this. Expected value maximization sometimes tells you to play it safe, and sometimes tells you to swing for the fences. If you're choosing between a 100% chance of 10 people helped or a 1% chance of 900 people helped, you can calculate the expected value of each and pick the higher one. In the latter case, "playing it safe" is better (100% * 10 > 1% * 900), but if it were instead a 1% chance of helping 1100 people, you'd want to "swing for the fences."

If you're strictly maximizing the expected value of X, this is sometimes called being "risk-neutral" with respect to X. You aren't looking to minimize your risk or to increase it; you're just looking for the best expected value.

## Footnotes

1. This is a standard result in statistics that I’m not bothering to explain or link here. If you really want the technical details, it’s worth getting a stats or probability textbook, and ~any textbook should explain this.