A Qualitative and Intuitive Explanation of Expected Value

(Cross posted to my personal blog.)

Alice: Expected value is one of my favorite things in the world.

Bob: That involves math right?

Alice: Yeah.

Bob: And you use it in your daily life?

Alice: Yeah.

Bob: See I can’t live life like that. I hate doing math. I prefer to just go with the flow.

Alice: Well, I take it back. I mean, it does involve math, but a) the math isn’t that bad, and b) no one actually does the math for day-to-day stuff.

Bob: Huh?

Alice: I’ll show you what I mean. I’m going to try to explain expected value in such a way where it is a) super intuitive, and b) qualitative instead of quantitative.

Bob: Ok.

Alice: There are three steps. 1) Consider both outcomes, 2) think about the probabilities, and 3) think about the payoffs. Let’s start with step one.

Bob: Ok.

Alice: This is an extreme example, but consider a hypothetical base jumper named Carol. Base jumping is incredibly risky. People die. But Carol doesn’t like to think about that possibility. She doesn’t want to “let fear control her life”. So she just “focuses on the positive”. She thinks about what a life changing experience it will be, but not about the possibility that it will be a life ending experience.

Bob: I see. That’s pretty crazy. You can’t just focus on the positive. You have to consider the negative as well.

Alice: Yeah, for sure. And just to reiterate, this is exactly they type of response I want to evoke in you. That all of this is really just common sense.

Bob: Ok.

Alice: So that’s step number one: don’t blatently ignore one of the outcomes. Duh. You have to actually consider both of them. (Well, most of the time there are many potential outcomes, not just two. But we’ll assume there are just two here to keep it simple.)

Bob: Makes sense so far.

Alice: Cool. Step number two is to consider probability.

Bob: Uh oh!

Alice: No, don’t say that! The math isn’t bad, I promise. And it’s important that you go into this with an open mindset.

Bob: Ok, fine.

Alice: So imagine this scenario. You’re walking down the street and see a food stand selling lentil tacos for a dollar. You’ve never really been a legume guy, so you think it’s pretty unlikely that you’d like these lentil tacos. Let’s say there’s a 10% chance. And now as a second thing, imagine that it was a slice of pizza instead. Pizza is usually pretty good, so let’s say there’s a 90% chance you’d like the pizza. It’s probably worth buying the slice of pizza, but not the lentil tacos. Because of probability.

Bob: Yeah I guess so. That’s how I’d think about it in day-to-day life. But I thought that for expected value and probability, you’d have to… like… use real numbers and do real math.

Alice: You thought wrong! I think that’s a common misconception, and a really important one. You don’t have to be all fancy with your numbers and math. In real life, even a mathematician would reason about this taco and pizza decision the way you or I would.

Bob: I feel like this isn’t “doing probability” or “doing expected value” though. It’s just normal everyday reasoning.

Alice: I disagree again. We haven’t gotten to the third component yet, but I would say that this “normal everyday reasoning” is in fact expected value based reasoning. I guess “based” is a key word. Maybe you can say that it isn’t true expected value because it isn’t actually doing the math. But the underlying concepts are still there, and that is what is important.

Bob: Hmm.

Alice: Let’s continue to the third component and revisit this.

Bob: Ok.

Alice: So the third component to consider are the payoffs.

Bob: Playoffs?

Alice: Very funny. Payoffs.

Bob: What are payoffs?

Alice: Let’s think about that lentil taco example. Imagine that instead of a lentil taco, it’s a lottery ticket that pays $1M, and that everything else is the same: it still costs a dollar and has a 10% chance of “paying off”. Would you buy the lottery ticket?

Bob: Of course! That’d be amazing!

Alice: Exactly! But how come?

Bob: Ah, I think I see what you’re saying. A 10% chance at a taco is whatever, but a 10% chance at a million dollars is a big deal. Eg. the payoff is way, way, higher. And that is something that you have to consider.

Alice: Exactly.

Bob: And these payoffs are a separate thing on top of 1) considering both outcomes and 2) considering the probabilities.

Alice: Right.

Bob: But I don’t understand something: with the lottery ticket, the payoff is clear. It’s a million dollars. But how can you say what the payoff is for eating a good taco?

Alice: You just wave your hands and estimate.

Bob: Really?

Alice: Yeah! I mean, if you want to be more precise you can, but is that worth it? This is relevant to your previous question about not actually doing the math.

Bob: How so?

Alice: Well, by getting deeper into the math, let’s say it’ll move you slightly closer towards the “right answer”. But it also takes time and effort to do the math. So there’s a trade-off: is it worth “paying” that time and effort in exchange for a being slightly closer to the right answer?

Bob: I see. I guess it’s a matter of how much it costs vs how much it actually moves you.

Alice: Yup, exactly! An analogy would be looking for a restaurant to go to. Say you’re in NYC and there are hundreds and hundreds of restaurants. And say you’ve been through 50 of them already. You could continue to look. Spending more time looking will move you closer towards finding the optimal restaurant. But looking also takes time and effort. And you’re hungry. So is that time and effort worth it?

Bob: Gotcha. And so coming back to expected value, in everyday life, it usually costs too much to sit down and deal with numbers, so that’s why even a mathematician will usually just ballpark everything?

Alice: Right.

Bob: But for big decisions you’d want to start messing with the numbers?

Alice: Right again.

Bob: That makes sense. Although I do have a question. It seems like the payoffs are often times in different units. Eg. for the lottery ticket it’s in dollars, but the lentil taco it’s in… enjoyment? How can you compare the two?

Alice: Just wave your hands and don’t worry about it.

Bob: I know that’s not your actual answer Alice!

Alice: Haha, true. Well it is my answer in 99% of cases (and I’m waving my hands with this 99% number right now). But sometimes yes, the different units can be a little weird, and there is a more “official” way of handling it.

Bob: Which is?

Alice: Well, think about dollars. Why would it be nice to win $1M?

Bob: Uh, because it’d be awesome?!

Alice: Why would it be awesome?

Bob: I mean, I would be able to retire.

Alice: And why would that be awesome?

Bob: Why would being able to retire be awesome?

Alice: Yes.

Bob: Isn’t that self-evident?

Alice: Maybe. But I want to hear your answer.

Bob: Well, I’d get to, I don’t know, relax at the coffee shop and finally start pursuing my novel. And exercise more. And maybe volunteer at that poetry club at the local high school, mentoring the students.

Alice: And why would that be awesome?

Bob: You keep asking that same question! What do you mean?

Alice: Why would it be awesome?

Bob: I guess because that stuff makes me happy.

Alice: Exactly.

Bob: I still don’t see where you’re going with this.

Alice: What about eating pizza?

Bob: Ohhhh, I think I see. Pizza is awesome because it tastes good, and experiencing tasty things makes me happy. Ultimately, in some sense, everything boils down to happiness. You can do your annoying thing where you take some payoff, and whatever “unit” it’s in, when you ask why it matters, the answer ends up being because it makes you happy.

Alice: Yeah. You can quibble about the word “happiness”, but essentially that’s it.

Bob: Ok so when payoffs are in different units, the “official” way of handling it is to convert it into happiness, and in everyday life, well, I guess we do the same thing intuitively.

Alice: Right. And the “official” term is utility, not happiness.

Bob: Gotcha.

Alice: So, to summarize where we’re at, expected value has three components: 1) consider both outcomes, not just one, 2) think about the probabilities, 3) think about the payoffs.

Bob: Yeah, that does make sense. And then if you want to do it the official way, you have to do math right?

Alice: Yeah. The math is actually pretty simple. Take the lottery ticket example. There are two possibilities. You win, or you lose. Say there’s a 10% chance at winning and the payoff is $1M. And when you lose, you get back nothing. So we’d do 10% * $1M = $100k for the first outcome, and 90% * $0 = $0 for the second, and then add them together to get $100k. More generally, you go through each potential outcome, ask what the probability is, what the payoff is, multiply the two, and add it all up. In this example, when you add it up you get $100k, so it would be worth paying up to $100k for this lottery ticket.

Bob: Oh wow, that’s actually not that hard!

Alice: Haha yup! I think the hard part is when there are more possible outcomes. Here there are only two, but it gets slightly harder when you have to do this for like seven or eight outcomes.

Bob: Even that doesn’t sound hard, just tedious.

Alice: True dat. So I guess I was wrong. I think the hard part is moreso when you don’t know the probabilities, or when you don’t know the payoffs. Which is almost always the case in practice. When that happens, you have to work on estimating the probabilities and payoffs, and that can be difficult. Yeah, that is a better way of putting it.

Bob: Right. Like for pizza, I feel like it’d be really hard to figure out what the payoff is.

Alice: Sort of. I like to think about that sort of thing in terms of money. Like, how much is a good slice of pizza worth to you? $10? I don’t think that’s too hard to estimate. And these sorts of monetary estimates work just fine. You don’t have to convert them into utility.

Bob: Huh, I guess so. I’d have to think about it more. As I come across examples in real life, I’ll give it a shot.

Alice: Awesome!

Bob: So why are you so passionate about expected value? I feel like it’s just a normal thing that everyone already does. Eg. those three things you mentioned, 1) consider both outcomes, 2) consider the probabilities, and 3) consider the payoffs. You gave great examples. It is rare that someone actually ignores the risk of eg. base jumping. It is rare that someone ignores the probability eg. that they’d like whatever food they’re buying. And it is rare that someone actually ignores the payoffs, and doesn’t eg. jump on the possibility of getting such an awesome lottery ticket.

Alice: I think you’re right. And that is something that I really wanted to get across in this post. That expected value is a totally common sensical, intuitive, normal thing. The reason why I’m passionate about it, I guess, is because often times, this sort of thinking just goes out the window.

Bob: What do you mean?

Alice: Well, to really elaborate I think we’ll have to take a break and continue in a separate conversation, but I’ll tell you a quick story.

Bob: Ok.

Alice: So I was playing poker once with this guy. He is a big time online player. But we were in Vegas, and WSOP is the only legal place to play poker online, and he said he plays on some different sites.

Bob: Huh.

Alice: Yeah. So I thought that that was crazy. If you’re a professional, how can you risk playing on an illegal site? I’ve heard stories of people losing all of their money on these sketchy illegal sites. Eg. you have money deposited and then they don’t let you withdraw it.

Bob: That makes sense.

Alice: That’s what I thought too. So I said this to the guy, and his answer is something I’ll never forget.

Bob: What did he say?

Alice: Expected value.

Bob: What?

Alice: I mean those weren’t the words that actually came out of his mouth, but that’s the essense of what he said, and I’m embarassed that I didn’t see it at first.

Bob: I’m not seeing it either. You’re gonna have to spell this one out for me.

Alice: Sure. This guy was a very good player. Poker was very profitable for him. Yes, there is a chance that the sketchy site doesn’t pay him out. Let’s call it a 15% chance to be extra conservative. So playing poker on that site for him, it’s an 85% chance at a very profitable thing.

Bob: Ohhh I see where you’re going!

Alice: Yeah. So an 85% chance of making a lot of money, that’s not bad at all! Especially because he’s got money saved in the bank traditionally. And then there’s the point that there are many of these sites available, and he plays on many of them, not putting all of his eggs in one basket, so to speak.

Bob: True. But what about Black Friday where all of those sites shut down?

Alice: Again, expected value.

Bob: Ok ok, let me try this time. So what you’re gonna say is that, yes, there is some chance of an event like Black Friday happening, where even if you don’t put all your eggs in one basket, something bad can happen. Similar to the stock market crashing. But the question you have to ask yourself is whether this is an acceptable risk. Think about that as one of the possible outcomes, think about the probability, the payoff, the other outcomes, and what your alternatives are.

Alice: Exactly!!!

Bob: Gotcha. I am seeing how sometimes people just dismiss some of the underlying principles of expected value.

Alice: Yeah. I think it happens all of the time. It is really, really harmful, and that is why I am passionate about it.

Bob: That makes sense! I’m feeling similarly! I want to hear more about these types of situations where people dismiss the principles behind expected value so that I can avoid them!

Alice: Awesome. We’ll talk about that next time.