Love and Data is the continued theme in this mini-episode as we discuss the game theory example of The Battle of the Sexes. In this textbook example, a couple must strategize about how to spend their Friday night. One partner prefers football games while the other partner prefers to attend the opera. Yet, each person would rather be at their non-preferred location so long as they are still with their spouse. So where should they decide to go?
Data Skeptic
[MINI] The Battle of the Sexes
(upbeat music) - The Data Skeptic Podcast is a weekly show featuring conversations about skepticism, critical thinking, and data science. - Do you know our topic for today? - You said it was called Battle of the Sexist. - That's right. Do you look it up at all? - I did. Some kind of game theory thing. - That's right. Any thoughts beyond that or should I dive right in? - Didn't understand a thing. - So you're gonna have to be really specific. - Did you understand when they talked about the concept of equilibrium? - I know what equilibrium is. I don't know what context you want me to understand it in. - Well, tell me your perspective on it and then we'll see how well that fits with the way a game theorist uses the word equilibrium. - Equilibrium means things have changed enough and that they are at a stable point. - That's very good. Yeah, stability is precisely the key to what an equilibrium is. Game theory is the study of how two people interacting are gonna decide how to interact. - Well, I think more specifically, you mean come do a decision together. - Well, yes and no. So in game theory, we talk about people as agents. Each agent makes up their own decision on how they're gonna act. But if you're just an agent on your own and the other agents don't matter, then you just do whatever you want. It's like playing solitaire at a bus station. Nobody else matters. But in general, our interactions do matter. So it's not precisely the case that we come together and make a decision together. But a lot of times it benefits us both if our decisions, our choices are sort of synchronized in some particular way. - I mean, even if they're not making decisions together, they're thinking about what the other person's thinking about. - Right, let's say for example, you and I were gonna both open restaurants right next door to each other. We both rented out two restaurant spaces. If I open a pizza restaurant, it would probably be a dumb decision for you to also open a pizza restaurant. 'Cause now we'll just have all the people who want pizza will be split between our two restaurants and all the people who want some other food will go somewhere else. - Sure, but keep in mind Burger King and McDonald's are always next to each other. - True, but a whopper is not a quarter pounder. - They're burgers. Who said my pizza was gonna be the same as yours anyway? - Fair enough, good point. The key about an equilibrium is the way that two strategies fit together. If I adopt a strategy and you adopt a strategy, is there a reason that one of us would shift? So if I decide I'm gonna open the pizza restaurant and you say I'm also gonna open the pizza restaurant, I might wanna change the cuisine in my restaurant because I'm worried about too much overlap. So then if I change, the question is will you also change? If you say, nope, I'm fine, I'll make the pizza, you make the gyros, then we're good. But if you also change to match me, then I might wanna change again. So it's kind of unstable. So a lot of the places people first get exposed to this sort of game theory is when they talk about prisoners dilemma. Did you ever do prisoners dilemma in college? - Yes, I was a political science major. - Oh, you guys studied a bit differently though, I think, right? What's your recollection of it? - This is an international political economy. I don't remember other than they just seemed to act on their own while also thinking about what the other was gonna decide. - Yeah, exactly. So I can set it up for you. In prisoners dilemma, there are two people captured who are probably guilty of a crime together. They put them in separate rooms, so you can't communicate anymore. They're gonna try and push each of the people to confess to the crime. Each agent has the choice of either confessing or not confessing, right? If you both confess, you're both gonna go to jail for, let's say, six years. Each, so 12 years total in jail. If you both refuse to confess at all, maybe they can't get you on the serious charge, but they have enough to convict you of something, a lesser crime, so you're both gonna get one year in jail. But if one of you confesses and the other does not, like somebody rats out the other person says, "I'll make a plea deal," then the person that made the deal goes to jail for zero years, and the person that didn't make the deal goes to jail for, like, let's say, seven years. So what would you do in that situation? I mean, obviously, it's better if they agree that one of them will confess. Well, actually, not obvious. You said six and six, and then where's one confess, the other serves seven. They're probably both not gonna confess. - So this is now, you're getting to one of the key points that always actually messed me up when I started learning about game theory. I was always looking at it from a more global perspective, like, well, let's say you and I finally lived out that fantasy of robbing a bank that we talk about a lot, right? - We don't talk about that. - And, okay, you know, plausible deniability, no, we don't talk about that at all. Not even a little bit. But, hypothetically speaking, let's just say two people named Schmeil and Schminder, Rob a bank one day. When I look at the potential jail time, I look at it and say, well, I don't want Linda to have any jail time. I would rather have it all on me. But then it would mean, in my situation, I'm gonna serve seven years of jail time, you'll serve zero. Assuming we both know that that's the strategy we're both gonna execute on, which might not necessarily be the case. But even if the lesser strategy of, we each do one year in jail, that's only two years total in jail, that's best for the couple, except that no one's looking after Yoshi. So those numbers I was throwing out about how many years in jail, if we look at those as precisely the amount of penalty, it's very mathematical, but it doesn't always reflect the way agents are thinking, right? Because I have a preference that you never be in jail. In fact, I would go over you. But if it was just some stranger I robbed a bank with, I don't necessarily care about the stranger if they do any jail time or not. Obviously a year in jail for me is a year in jail for me. What would it be like if you did a year in jail? That would be really bad for me also. Would it be as bad as my year? Maybe it would be only six months is bad, or something like that. - Odd, but they try to quantify it. - Well, that's the key to a lot of economist work in game theory. So I always struggled with this one because I was unhappy with the textbook, how simple sort of the utility functions were. But you can always restate the utility functions in whatever way suits you, and then do your math that way. So I've learned that that was not a valid complaint I had. So the equilibrium solution to the classic prisoner's dilemma is defect defect, meaning both of the accusations refused to cooperate with the police and they both do a long-term in prison. And that seems silly, right? Like why didn't they both try and cooperate? Well, it's because they each are motivated to switch their strategy. So if it's like, oh, the other person's gonna cooperate, then I can defect and I'll get off better. But then the person on the other side of the table is thinking the same thing, like, oh, if you cooperate, I can defect. So there's really no reason to think the other person would ever cooperate because the incentives aren't set up for them to do so. So it's unstable because it only works if they both go out and eliminate this risk and most likely they won't. So the stable position is for both to defect. There are also equilibriums that use some random element. Did you know Game Theoryists talk about the classic game of rock paper scissors? - I do now. - You wanna do one with me? - No. (laughing) - Well, if I could talk you into playing rock paper scissors with me, what would your strategy be? - I would make whatever symbol I felt like when we counted to three. - You seem like more of a scissors person to me, am I right? - That may have been the first one. (laughing) - So would you do scissors every time? - No, it just rotate through. - What would be your strategy of rotation? - I don't know how I felt. Like, if you did the kind of rock at one time, I just might do the rock the next time. I'm like, huh! - Well, if I noticed you doing that, I'll do the paper the next time. - No, that would just be one time, then the next time, I would think it's something else. - So you'd be pretty much random. - Eh, I don't know, per the magician, he's suggesting that our thoughts aren't random. - Can you give some background in the magician you're referring to? - So Kyle and I went and saw a magician. And he just talked about from the high level how magicians try to fool people in the audience. - So yeah, that was the most recent lecture at the Skeptic Society series that goes on a Caltech that we were at. And the equilibrium solution getting back to rock paper scissors is pretty much what you were saying, it's to randomize. And the reason is, if you do anything that's not random, it would have a pattern. And if there's a pattern, I might detect the pattern and then exploit the pattern to beat you in the game. So the equilibrium solution is to just be random in rock paper scissors. And we call that a mixed equilibria. Why, I don't know, I've always been frustrated with that. We should just call it probabilistic equilibria or something. I don't know why it's called mixed. I've never gotten a straight answer, but it is. So mixed just means has a random element, okay? Let's move on from there. Battle of the sexes is the idea that there's a couple. Now, it's gonna get a little convoluted. The couple doesn't have cell phones. They can't text or communicate or whatever. And there's a mix-up. And they can either go to a football game or to the opera. In this example, classically, it's assumed that there's a heterosexual couple in all these examples. It's assumed that the woman prefers the opera and the man prefers the football game. - That's not the definition of heterosexual couple. That's the definition of sexist. - What do you mean sexist? - We assume the guy wants to go to a football game and the woman to the opera, that is sexist. - Right, in fact, I think it would be the opposite in our relationship, right? (laughs) How many times have you been to the opera? - I've never been in my whole life. - I got to beat two to zero. How many times have you been to a football game? - I have been many times. So you have me beat many to zero. So I think our incentives would be opposite the typical telling of this game. But let's just go with it or we could switch it out for comedy versus horror movie or something. But there's a case where there's an asymmetric preference in the relationship that one of the partners doesn't appreciate the primary event of the other. Yet, the couple, much like us, would like to be together the most, right? Being together is what matters. Not so much where you are. I'd go to the football game with you. - I don't know, I wanna see an opera. - Okay, I should take you. If we both go to the opera together, I get three points, you get two, for being with me. If we go to the football game, you prefer the football game, you get three points, I get two. If we go our separate ways and I end up the opera alone, I get one point, which means you're at the football game, you get one point. If by chance we go the opposite way and I go to the football game hoping you're there, you go to the opera hoping I'm there, we both get zero, that's the worst case scenario. Convoluted, we can't plan in advance, we can't call each other. What would your strategy be if you were confronted with this decision? - You go to the one you want. - So that is one of the equilibrium strategies because if you go to the one you want and I may or may not be there, right? If I synchronize with you, you get a high reward. If I don't, you get a low reward, but not a zero reward. And the vice versus true for me. So me switching my choice between football and opera does affect your reward, your points, but it doesn't incentivize you to switch your decision. See how that works? - No, what do you mean? - You want to get the most points, right? - Well, we'll just say points is happiness. - Yeah, points is the measurement of utility of happiness. So you'd like to get the max happiness, which occurs when you are at the football game and I'm also there with you in our scenario. But you can't control what I'm gonna do. In this case, I'm not saying in our marriage, I'm just saying in this case, you don't-- - I roll. - You don't know which decision I'm gonna make. So you said, I'm gonna go to my preferred location, which is the football game. You don't know if I'm gonna be there or not. If I show up, you get three points. Hooray, we're together and we're at your preferred place. If I don't, you still get the one point. That's based on you deciding I'm going to the football game. Let's analyze your other choice. You could have said, I'm going to the opera. What happens there? If I show up, you get two points. If I don't show up, you get zero points of happiness. If it's 50/50 where I'll go, that expected payout then is one point for you. - Well, it's either gonna be zero or two. - Right, so if it's 50/50, 50% chance of getting two, 50% chance of getting zero, the expectation is eh, on average, I would get one point. - Yeah, I mean, if we did this a hundred times. - Right, yeah, that's another excellent point. Game theory is all about repeated games. We're gonna do this every weekend. And for some reason, we're never able to plan in advance and coordinate where we're gonna be. And for some reason, bolder cell phones are always dead on this day and we don't have email or yada yada. So we have to kind of do this without communicating. Pardon the convolution, but that's what gets us to a simple example. For you, if you say I'm going to the football game, you're guaranteed one point. If you say I'm going to the opera, you should on average get one point. So there's no motivation for you to change. - Change from what to what? - From your strategy of, I'm straight away going to football all the time versus I'm gonna go to opera all the time. I have the same equilibrium strategy available to me. I can just always go to my event, in which case, we'll both be synchronized. We're at a stable point where we each get one point and neither of us is incentivized to switch our strategy because if we both switch, it'll screw everything up. If I'm like, oh, Linda always goes to the football game. I'll go to, and if you're simultaneously thinking, Kyle always goes to the opera, I'll go there too, then we miss each other and we both get zero. - But the reward's greater. - Uh-huh, that's right. The overall reward could be greater if we could coordinate. But again, remember, I've taken away communication for this case. - They're not coordinate. Another strategy could be Kyle, at least in my head, I think Kyle wants to go to the opera. I'll go to the opera and make some happy. And if you assume he's there, I could be like, well, he's probably there. - Yeah, that's the key observation here because that's why I was doing all that stuff about rock, paper, scissors. There are also mixed equilibria solutions here. If you go to your preferred place, 3/5 of the time and go to the other place 2/5 of the time, that's also an equilibria strategy because we will both wind up with a pretty high payout, actually a better payout, on average. And there's no reason for either of us to change our strategies in that if we both adopt that random flighting of how we'll go someplace. But if you say, oh, I'm gonna go to, I'm gonna mix it up, maybe that's why it's called mixed. I'm gonna mix it up, go sometimes opera, sometimes football. If I always go opera, then you have a bad strategy. You kind of figure that one out though, right? You didn't come up with 3/5, but you knew the randomization part and you saw that we could both share a greater utility. And isn't that what marriage is all about? - Yeah, but you could also think of it like stock options. - How so? - Well, you want some stable, less risky ones, but the less risky one is you just go to your thing in which you're happy, and it doesn't matter if the other person's there. And then sometimes the payoff is worth it, and then you could put a percentage of your risk in the one where you're like, oh, well, the other one's so much more fun if they're there, even though the event isn't what you wanted. - Your point. - And you could do that. So that's a high risk situation, but a bigger payoff. So it's just like stock options when you're choosing your 401k. - Yeah, that's a great mixed equilibria suggestion. - You're a bit of a game theorist, Linda, I didn't know. We should talk more about this. - I think he knows what my game theorist. - All right, let's wrap this up. One last thing just to add some color. In the classic battle of the sexes game, we talk about the options of going to either the opera or the football game. What would you say is the best date I ever took you on? - My favorite is a weekend in New Orleans. - Yes, indeed. And I'd also say I really liked the time you took me to the stars concert. - I thought you took me. - Well, that was before you knew you were my girlfriend. (laughing) I think you were my girlfriend then, but you just didn't know it yet. But later you took me again as a-- - You liked the later one. - That's right, I liked them both. - Okay. - Anyway, thanks again for joining me, Linda. - Goodbye. - And thanks to the listeners for participating in our "Love and Data" series. I have a quick request. If anyone has any stories about how data, or even maybe technology has affected their own love lives, we would love to share that on the show. I might put together a little compilation, something kind of like the moth, and share some listener feedback. So we have a widget called Speakeasy that's on the Facebook page, where you can leave a message right through the browser. Or we have a number that you can call and leave a voicemail. It's 310-906-0752. I'll give you one more time. 310-906-0752. And that's in the US, so I don't know, I don't know, I should just figure it out somehow. 'Cause it's like plus one or something. Anyway, leave us a voicemail at either of those locations if you've got a great story. I would love to get at least five or six of these, and then put a little bonus episode out with some of the content. So if you have something to share, please do. And thanks for listening. (upbeat music) (upbeat music)