Liz: Okay, ready?
Ben: Maybe No, no, I wasn't ready. Okay.
Both: Rock, paper, scissors.
Ben: Oh, okay, paper beats scissors. Hi, I'm Ben Klemens. I'm Liz Landau and this is Pod, Paper, Scissors.
Ben: Sorry. Are you regretting playing scissors?
Liz: Oh my God. Okay, I know this is only a game, but I really regret playing scissors. I should have played rock.
Ben: Oh, Liz, I don't think it's something after regret. And in fact, let's spend the next episode talking about why you should not have regretted playing scissors.
Liz: Alright, well, last week, we were talking about my horrors of my dating life in LA, and how that might relate to Game Theory.
Ben: Yeah, so the one example that I think we might want to review right now. You and an X were both invited to a poker game.
Liz: Yeah, I actually decided not to go. Because in a very small setting like that, and everything was too fresh. In my mind, I decided that I would be happier not going knowing that he was going.
Ben: Yeah, so we describe this in terms of Nash equilibria. So this is the theory part of the show. So one option would have been that you went, and your ex didn't go. And that would be an equilibrium, in the sense that you would have been happy with the outcome because you got you got to go. And your ex would have been like would not have regretted the outcome because your ex wouldn't have changed in his mind and said, "Oh, well, I know Liz is at the game, but I'm going to go", like that wouldn't happen. And you'd have another equilibrium that we talked about last time where he would go, and you wouldn't, and that's also an equilibrium. Nobody's going to change their decision based upon what the other side is doing. But what I didn't talk about, was that there's _another_ equilibrium to the game.
Liz: Another Nash equilibrium?
Ben 2:13 Yeah, there is. So it's the _mixed strategy_.
Liz: Yeah, you know, dating and relationships is so complicated. We might as well turn to game theory to see how we could do it better. I mean, it's not like there's a map to love.
Liz [singing]: It's the love map
It's the map that you follow when you're lookin' for love
It's the love map
It's the map that you follow when you're lookin' for looove
Ben: I think one of the ways to better understand your relationship to your ex is going back to Rock paper, scissors. So if you Yeah, cuz Yeah, we're talking about regret. Let's say that, you decide that your strategy for rock paper scissors is just to play rock all the time? Well, I'm going to play paper and then you're going to regret it right? You're going to say, No, that was not the right strategy. Let me try again.
Liz: When you do something repeatedly, and the other person has a strategy that they use over and over again, you can then detect that strategy and reexamine and change your own behavior. So if you're playing rock three times in a row, then the fourth time, I'm gonna play paper, knowing that you're probably gonna do rock again.
Ben: yeah, exactly. So don't...do that. Right. And you could really picture this, this sort of, like cycling of regret, where you say, I'm gonna play rock all the time. So I'm like, Okay, well, I'm gonna play paper all the time. And then you say, Well, okay, then I regret playing rock all the time. I'm gonna play scissors all the time. And then I'm gonna say, Well, I regret paying paper all the time, I'm gonna play rock. And then we could just go in little circles the whole time.
Liz: I don't think this is really about regret. I think this is about seeing what another person is doing, and then changing your behavior based on what they did.
Ben: Fair enough. But we're looking for some way where you don't have to change your behavior where you can say, ahead of time, knowing what's going on, knowing what I expect, then I'm comfortable with what I'm doing. And just playing rock all the time is not gonna. Not gonna work for that front. But yeah, I mean, okay, we'll cut to the ending. What do people actually do? They try to play rock a third of the time paper a third of the time and scissors a third of time, right? So that's a mixed strategy. So the pure strategy would be to just play rock all the time. The mixed strategy is to say, No, no, I'm going to mix the three possible strategies to the fullest extent possible.
Liz: But you can't do it in such an obvious way that you play rock than paper than scissors than rock than paper than scissors because the other person's got to pick up on that also.
Ben: Oh, absolutely. If that would be a deterministic strategy to right. So, ideally, you could actually just totally randomize every step along the way.
Liz: Oh, so I guess if you had a 12 sided die?
Ben: Oh, yeah, yeah. Finally, the 12 sided die comes in handy IRL. Sure. And it. And you know, there's a lot of debate about whether people really do randomize in life, whether there is such a thing. And I believe it, I think that people make kind of arbitrary and haphazard decisions based on you know, like, really inconsequential things that just pass by, you know? Yeah, if you just play rock, then paper, then scissors, rock then paper then scissors. That's also a deterministic strategy. So really, we're looking for a situation where to the fullest extent possible. You try to play randomly. And I think I think that's really what people do to the fullest extent possible, which, you know, lots of lab studies have shown people are really bad at randomizing. To digress for just a second, one of the ways in which people are bad at randomizing is that if you're playing each a third of the time, that means a third of the time, you're playing the same thing twice. And when people try to randomize, they lower that probability. Like if you play rock, it feels on random to play rock again, the second time, but yeah, it's got a one in three chance.
Liz: You know, I've been thinking a lot about regret recently. And when it comes to using rock, paper, scissors, for example, as a way to talk about regret, you know, even if we can show mathematically that the best way to play it is using each option, exactly with equal probability, and randomize, you still might lose in any given round, and it still feels like that was your fault. Even if it wasn't, you can know that you used the best strategy, and yet still feel regret about what you actually played.
Ben: Yeah, it can be hard to say that you played—like, you don't regret the strategy. Like there's no point saying, Well, no, I shouldn't have played it a third/a third/a third, I should have done something else. No, no, the best thing you could have done was a third/a third/a third, and then not regret the strategy and then regret the outcome and say, oh, it didn't go out the way I wanted. I, I, I've been kind of thinking about this, too, that it feels like a different kind of regret, from a lot of other sorts of regrets where maybe you didn't think it through fully beforehand. This is game theory as self help, I hope you're enjoying this. Maybe you didn't think it through beforehand, maybe you didn't have the optimal strategy. But sometimes, even the best, most optimal strategy still only works like a third of the time, and that's the best you're going to get.
Liz: Yeah, I was thinking about the stock market. So back when I was in graduate school, one of my professors told us that the best thing that we could do as grad students saving for our future was to buy mutual funds, and mutual funds that are tracked with the indices like the S&P 500, NASDAQ, etc. Because buying the indexes, time and time again, has been shown over the long haul to be the most optimal strategy in investing. And yet we have thousands of permutations of funds, of managers of funds making different kinds of funds, like everybody is trying to sell you the best way to invest your money. And when you have events like the crash of 2008, and the brief crash that we went through this spring with the Coronavirus and markets tanking, it's easy to tell yourself, oh, no, like, I was wrong. I went with the wrong fund. I went with the wrong strategy. But it could be that in the long haul, it is the right strategy since you have to weather the storm.
Ben: Yeah, I would agree. I think a lot of people they sort, confound the strategy and picking the right fund for themselves, and then how that fund turns out and it's hard to think through, you know, "Oh, if only I was on the other track." You know, we left one other thread loose. We'd promised we talked about the movie _A Beautiful Mind_ about Johnny Nash.
Liz: Oh, yeah. My best friends who I rode the bus with once at Princeton.
Ben: Yes, yes. So, listeners, I want you all to know, Liz went to Princeton. [laughs] So having established that we can go to the movie, which is famous among game theoreticians, because it's supposed to be the moment when Johnny Nash, like, there's a scene where he has the epiphany, you know, this is what game theory is, and—
Liz: Oh, yeah, that's when he's in a bar. And Johnny and his male friends are looking at some females across the room, especially a certain blonde woman who everybody wants to go and talk to. And his epiphany is that if everyone, quote goes after the blonde, end quote, then she will just walk away. But if each of the young men go after a single brunette, who I suppose they are less actually interested in, then every man gets a woman, and everyone is happy. I see a lot of problems with this setup, don't you?
Ben: It is deeply problematic. And we'll start with—it never made sense to me, like, you know, they're all these, like, stand up comics from the 70s and all and they're like, yeah, went up to a brunette and blah, blah, blah. And it's like, really, we're, this is the distinguishing feature of a woman.
Liz: Yes, as a redhead, I can confirm that there's all kinds of misconceptions of people based on their hair color.
Ben: Yeah, so okay, so we'll start with that. And so let's let's change the game. So instead,
Liz [singing]: Only a ginger, can call another ginger, ginger.
Ben: So instead, let's say there's, there's a preppy or a woman who offers to, you know, buy her date a $30 dinner. And then there's a less preppy woman who's offering to buy her date a $15 dinner. They're trying to work out what they're going to do. And Nash in the movie, he says, Yeah, so nobody goes for the $30 dinner. Everybody goes after the lower value dinner.
Liz: Oh, so in your setup, the women are offering meals with certain monetary value.
Ben: Yeah. And they're grad students. So they're desperate. So you know that they'll take it. Tinder food stamps. The equilibrium that he proposes is that all the guys don't go after the higher price dinner. And they all agree to go after the lower price. And _it's not an equilibrium_. It's not a Nash equilibrium. Because if you know that all of your colleagues are going to go for a $15 dinner, then why don't you go for a $30 dinner, right?
Ben: so it's not an equilibrium. And as soon as the movie came out, it was a running gag among like every game theorist in the world, like, so instead, the correct way to do it.
Liz: Wait wait, let me ask you. Okay, bad. So what would make it an actual scenario in which it were a Nash equilibrium?
Ben: So yeah, so every game has an equilibrium. That was Nash's big epiphany, as per last last time, it wasn't about, you know, we need to overturn Adam Smith, as they talked about in the movie. You think it's like how
Liz [exaggerated epiphany voice]: Adam Smith was wrooong!
Ben: Yeah, that was funny. Now, I don't know if Nash even would have thought about that in those terms. Yeah. It he's portrayed as a bookish dude, and the point is, his big epiphany was we can apply Brouwer's fixed point theorem to two player games. So it's okay,
Liz: my eyebrows are fine.
Ben: So anyway, how we would apply it here is that there would be a mixed strategy. If you go after the $30 dinner. And let's say there are just two of you, if
Liz: And when you say go after, do you mean, you get the dinner if you flirt with a particular person.
Ben: Right, so if if I flirt with the, the woman with a higher value gift certificate, and you don't, then I get the meal, if we both flirt with the woman with a hair value gift certificate, then she rejects both of us, in the premise in the movie. So it's a it's a coördination problem, where we have to, we have to find a way in which one of us goes in one direction and the other goes in another direction. Coördination problems, as you can imagine, are super common throughout the world. If I go for the higher value gift certificate, two thirds of the time, and go after the lower value gift certificate a third of the time, that actually works out and you do the same.
Liz: Okay, Ben, so in this A Beautiful Mind 2.0 scene, we're supposed to imagine is only if one man approaches her that she will be like, Sure. Let's go to Outback Steakhouse with this $30 gift certificate.
Ben: Yeah, and in the movie that this was like weird-ass computer graphics where the guys walk up to her and they like go poof. [Liz laughs] But anyway, it's zero pay off. If we both go out, go to the same woman. So we're sitting at the table and I say to you, alright, I'm gonna go for the higher value gift certificate which is $30 with two thirds likelihood, which means if you go for it, you're going to get you have a 1/3, one in three chance of getting $30. So that's worth 10 bucks in expected value. I'm going to go for the other one with 1/3 chance, which means if you go for it, you have a two thirds chance of getting $15, which is the same, it's, it's 15 times two is 30 over three is also $10. So you're indifferent, you're like, Okay, that's fine. And since I'm indifferent, I'm going to do the same thing, I'm going to go after the higher value with two thirds likelihood, and the lower 1/3 likelihood. And the math works out, both of us don't want to change our minds, given what the other person is doing. And that's the equilibrium. That's where knowing what what each of us are, knowing our strategies, there is no regret before the before the fact. We're not going to regret our strategies, even though sometimes we're going to kind of collide and both come out with nothing, or we're going to come out with the lower value gift certificate. It's not because we're being coöperative. It's not because we're being chummy, and I'm saying, "Oh, go ahead, get the higher value gift certificate". Ahead of time, we had to mix our strategies in order to come up with with something that's mutually not regrettable.
Liz: But wait a minute, then I'm with you on this math in this very suspect, very well structured, new scene in a beautiful mind. But what good is this in the real world?
Ben: Well, we do randomize all the time. And that does have this effect that sometimes we're going to get the lower value gift certificate, sometimes we're going to get nothing. And that doesn't mean that we had the wrong strategy ahead of time. You know, the weatherman says there's a 15% chance of rain. Do you do you actually still have the picnic? Or do you call your friends and call it off? We can still have picnics, right, here in in COVID times?
Liz: Oh, yeah. As long as you wear a mask and sit six feet from other people, you can still have a nice little picnic with your friends.
Ben: So even still, yeah, you don't want to get drenched. Right. So ex ante, you have to make this decision and say, alright—
Liz: hey, just because I haven't seen my cousin's kids in a while doesn't make me an ex auntie.
Ben: [Laughs] That's good. That's good. But the weather says there's a 15% chance of rain. Am I going to go to the picnic or not, right? And then the economist would call this an expected value calculation. So there's some expense, there's some value to having a picnic, when it's not raining. There's some like probably negative value to having a picnic when it is raining. And how do you balance those two? And different people have different preferences on this right? Some people if it's a 15% chance of rain, they're super conservative and they'll say, no, no, I'm worried. And some people they'll wait for like 80% chance before saying, okay, okay, screw it, we're gonna get rained on our sandwiches are gonna get wet. In either case, regardless of sort of your risk tolerance, you got to make a decision ahead of time, and then see what happens. And some percentage of the time, yeah, you're going to be wrong. And is that something that we need to regret? I don't know.
Liz: I think we all have in our minds a sort of sliding scale for what we're willing to risk, especially when it comes to the weather. Like, when I hear 15% chance of rain, I think well, that's quite low. So I'm gonna go outside without a umbrella. But it's like, 60% chance, like more than half, then I do carry an umbrella. But different people might have different ideas about what these probabilities mean. And sometimes it'll say 60% chance of rain on the Internet, I walk outside and it's already raining. And I'm like, it's 100% chance of rain.
Ben: If you walk outside and it's not raining, do you say it's a 0% chance of rain?
Liz: Somehow, I don't actually think that but
Liz: I know that that makes no logical sense. But it just it emphasizes that people have a poor understanding of what risk and probabilities actually mean. Also, I think that if it says 15% chance of rain, and a couple is going to have their wedding outside, they decide to go forward knowing the 15% chance of rain and in the middle of the ceremony, it starts storming and the chuppah blows away and the glass is broken because a wind gust comes and throws it against a tree I think they will still regret it, even though they had all the information available.
Ben: I don't know. I think think that sounds like...fun. I don't know. But it
Liz: Or it could be the best day of their life.
Ben: Yeah, yeah, I think regardless of what happens, you know, five years later you laugh at it. You know, comedy is tragedy plus a year.
Liz: Speaking of regrets, and risk and probability, I just read this really interesting short story by one of my favorite speculative fiction authors, Ted Chiang. And in this story, it is a world in which there are "prisms", you can buy a prism and look at yourself in a different universe. In fact, you can look at multiple prisms and communicate with a version of yourself in slightly different circumstances, and see what would have happened if you had made different choices in different universes. And I think a lot of us have sort of touchstone moments in our lives where we had to make a decision about taking one job or another, going to one school or another, moving to one place or another, being with a person or not, and always wondered, like, Who would I be if I had done something different? Do I regret the choices that I've made? And this story really gets at the heart of this question of, are we as people, the sum of decisions that we've made? Or do we occasionally make decisions that are not reflective of our true nature?
Ben: Yeah, I read this too, and I give it a hearty endorsement. It was really great to, Chiang is a really great author, he gets the fame that he deserves.
Liz: Yeah, this is from the book _Exhalation_.
Ben: In the context of what we're talking about here, it's an another nice way to think about how people randomize. Like, what if, what if, every time you have to have to every time you play rock, paper, scissors, the world splits into three universes.
Ben: And in each in one universe, you're playing rock, and one paper and one scissors. But yeah, yeah, it's it's hard to really determine what, what it means to randomize when there's only one outcome, right? They say, yeah, it's a 15% chance of rain, but we don't have basically, if it's a 16% chance of rain, six universes, one of which it rains and the other five it doesn't.
Liz: I might have a PhD in astrophysics. If I had played rock instead of scissors, years ago. Oh, regret, regret.
Ben: Anyway, yeah. So tune in next time, we're going to talk about, it's going to be the travel episode.
Liz: Oh my gosh, it's more important than ever right now to consider when you're traveling, how many other people are going to be in the train with you in the plane with you, at the rest stops if you're driving.
Ben: but we'll talk about that next time on
Both: Pod, Paper, Scissors.