Game Theory

Uh, I'm Ben Poac. I'm Provost of Yale, but in real life, I teach economics. I teach Game Theory. Um, so I'm actually going to get rid of the jacket if you don't mind, um, and go back to my life as a professor for an hour. Um, I've actually never given a talk in anything like this before. Uh, kind of amazing church, a church with a small group of people in it. Makes me think of the Church of England, so that's probably appropriate for me. We're going to talk about Game Theory. Um, game theory is the analysis, or a way of analyzing, strategic situations. So, what are strategic situations? Strategic situations are any setting where the things that you care about, the outcomes you care about, depend not just on your own actions, but on the actions of someone else. All right? So, let's think of some examples, but we won't stay long with examples. Uh, in economics, an obvious example would be two firms who are competing in the same marketplace. So, think of Apple and Google, Apple and Samsung, right? They're both competing in the market to make, uh, telephones, to make these things. Whoops, just knocked off the mic. Make these things, and they have various strategies available to each other: pricing, quantity, innovation, suing each other, bribing judges, whatever else they do. All right? So that's a game involving players where they're competing, but it isn't only competition that we can model. We can also model settings of cooperation. So, think about teams working in the same firm, or, uh, looking around the room, groups of deans and vice presidents of Yale who are working together, uh, to try and get things done. So, so game theory is also the way you analyze how to herd cats, all right, roughly speaking. All right? So, when we teach game theory at Yale, uh, what we do, uh, is we try to play games in class and we analyze them as a way of seeing, uh, you know, teaching the technique, but also learning about the game. That's a technique of teaching that's now spread, uh, pretty much worldwide, and it's what we're going to do today. So, today I'm going to play a game. We're going to analyze that game. We're going to play the game. We're going to analyze that game, and we hope, I hope, we're going to learn something as we're going along. Okay? So that's what we're going to do. Um, uh, if some of you have seen this game being played online, if you've seen my online courses, uh, and, uh, if you haven't, a, a plug-in for looking at the Yale online courses, not just mine, but everyone else's as well. So, I need two volunteers for this, and, uh, rather than embarrass people, I'm just going to pick people out. So, I'm going to pick two deans who happened to be here who didn't know this was going to happen. Uh, I'm going to pick Robert Post, who's here, and I'm going to pick James Bundy, who's there, and I'm going to drag them out the front. Uh, so, so, uh, and the reason, one reason for picking on these two deans is this is a day to celebrate Yale, and Robert Post is the dean of the Law School, which is quite simply the best law school in the world, and he is quite simply the best dean of the best law school in the world. And James Bundy is the dean of the Drama School, which is the best drama school in the world, and he is quite simply the best dean of the best drama school in the world. So, why don't we have a little plug for our, our wonderful schools while we're here? Okay, so the audience provides the volunteers. I'm going to provide the props, which in this, on this occasion, is, uh, two wet sponges. All right, we're going to have a sponge fight. All right, so these are the sponges, and I'm going to wet them. And this being Yale, I will use nothing but the best. I'm going to use Yale University spring water. Did you know that Yale University had a spring? I don't. And we're going to, we're going to wet these, and, uh, the people who, who are custodians of this building, look away for a second, 'cause I'm going to dribble water on the floor. And Bruce is back there, and he's thinking the facilities bill is going to us. So, don't look for a second, Bruce. Right. And I'm going to squeeze these out and try and make sure they're roughly equal weight. Right? So, Joan, are they roughly equal? Right? Yes. Okay. Okay, so we have a blue sponge and a green sponge. I'm going to give the blue sponge to James Bundy, and I'm going to give the green sponge to Robert Post. And in a minute, I'm going to position them at either, well, I'm going to do it now. I'm going to put Robert here, here, here, and I'm going to put James here. Okay. I'm going to move my coat out of the way. All right, here we go. So, we're going to have a sponge fight, or, if you like, a duel. And the rules of the game are this: we're going to have alternating moves. All right? And when it's your turn to move, you are going to have to make a choice. And the choice is going to be whether to throw your sponge at your opponent or whether to take a step forward. All right? If you throw the sponge and hit your opponent, you win. If your opponent throws the, uh, in case his sponge and hits you, you lose. Right? Simple, simple rules. We'll need a few specifications. One important specification, uh, is, uh, well, the most important spec is this: each player only has one sponge, and once they've thrown their sponge, if they miss, the game continues. All right? So, we could work out the strategy this in great detail, but let's not. Let's just realize what's going to happen. So, if you throw your sponge and you miss, since you have to keep on stepping forward, uh, eventually, the other player is going to be right on top of you, and it's going to basically plonk his sponge on your head. Right? So, if you throw a sponge and miss, you're going to lose. All right? But we will play that through and see what happens. Two other rules, minor rules: each step has to be about a yard, right, to keep things fair. And this is a rule that I don't know will work in America, but let's see: uh, gentlemen, never duck. All right? All right? All right? Okay. All right, so we'll see that. Okay, so everyone understand the rules? All right, so we'll make James player one and Robert player two. I'm exploiting you, rather, I apologize, but deans are meant to be exploited by provosts. Get fit, hit in the role. So, uh, James, you want to throw your sponge or, or a step? Take a step. Okay, so, so a step. And now it's Robert's turn, and he's also going to step. And now it's James's turn, he's also going to step. All right, now it's Robert's turn. Step. At this point, let's just pause a second and let's ask people what they think people should do. So, it's James's turn. You can all see the situation. One of the things we think about in game theory is how to put yourself in the shoes of other people. So, imagine you're in James's shoes right now. Would you throw or would you step? Step. Everyone, step. Okay, they're all saying they don't trust your arm. I don't know, I don't know how you are at baseball as a. All right, let's have the same question now for Robert. Who, who thinks he should throw? Who thinks he should step? Step. They don't really trust their, the arms of the deans. Okay. All right, all right. And now, yes, yes, are they allowed to dodge? No dodging, no dodging, not allowed. Not allowed. Again, gentlemen, never do things like that. This is a strictly gentlemanly game. All right, so, uh, James's turn. Who, who thinks, throw? It's James's turn. Who thinks, throw? Who, who thinks you should throw? You think throw? Shannon says throw. Who, raise your hand if you think throw. Raise your hand if you think step. Well, it's up to you. Don't know. Know. We don't know that. We don't know that. We don't know. And it's a pretty, it's a pretty light sponge. We, we'll, we'll get there. We'll get there. Hold that thought, but we will get exactly that idea. Right? So, so, throw or step. Now, what do people think? Throw or step? Who, who thinks throw now? Who thinks step now? Well, well, up to you. Get more even. James, throw or step? Who, now it's, have a poll again. Who thinks throw? Who thinks step? It's interesting. The people at the front are saying throw. Back, got to be a coincidence. But, but, but, uh, you know, David Swinson is saying anywhere on the body. Anywhere on the body. David is saying throw it. He is the best judge of uncertainty in the, in the room. So, is he on aerodynamic? Oh, not about his aerodynamics. We have to find out. All right, ready? Oh, game continues. Uh, sorry, I assume going to step and step and step and step. All right, so, uh, why I get you guys to sit here for a minute, because I'm going to use you in a second. Um, all right, so I'm going to victimize our, our volunteers a little bit more later. Um, let's talk about this game a little bit and see if we can learn something from it. And we were already beginning to draw out some lessons there, and we'll draw out more as we go along. So, first, why is this game interesting? So, one reason this game is interesting is, uh, it corresponds to a duel, a duel with, with, uh, uh, with muskets, I guess, a bit more serious than sponges. And, uh, uh, for the people in, in the room who are Russian literature specialists, you'll know that there are plenty of duels in 19th-century Russian literature, a little bit in French literature, uh, probably the most famous, I'm guessing, are the one in War and Peace, all right, and the one in Eugene Onegin, the Pushkin one, um, or Tchaikovsky, if you prefer that version. Uh, there are many, many others. In the, uh, War and Peace version, there are other examples, better examples. Oh, yes, up. Yeah, better, better, more up. Okay, higher, higher, better. It's a big room. Is that better? Good. Okay, so in the, uh, Tchaikovsky, in the War and Peace version, I think we are led to believe, believe that the hero, Pierre, throws, not throws his musket too early, and he kind of gets lucky and kind of wings D'oh, off, and, and, uh, I know that's a spoiler alert, I just, but that actually happens about page, that's about page 300 in the novel, which means it's in the first half of the book, right? So there's plenty more after that. Uh, in the, uh, Pushkin version, Eugene Onegin, Eugene Onegin kills Lensky. It's really the critical event of the, of the poem. Um, and I think the main lesson we should get from this for dueling is don't duel, right? Rather than how a duel, just don't duel. But if you did find yourself in a time machine in 19th-century Russia, this might help. But there's more to this idea than just duels. There are games elsewhere where the critical decision is when to do something. There are games when what matters is what you would do, and there are games where what is matter, what matters is when you would do it. So, in this game, there's no question about what you're going to do. You're going to throw the sponge, but the critical question is when you would do it. Let me give an example. How many of you have watched the Tour de France on TV? A fair number of you, actually, surprisingly few. But in the Tour de France, it's a bike race, right? In the Tour de France, in each stage of the Tour de France, one of the critical decisions that each rider has to make is they have to decide when to try to break away from the peloton, from the pack, right? And if you break away too early, in these long stages, for sure you're going to be reeled in because over very long distances, the peloton will go faster than you can go. And if you break away too late, then either someone else will break away, or just one of the sprinters will win, right? So, the key decision then is, is, is when to go, not when to throw the sponge, but when should I break away? There's even a movie, American movie called Breaking Away, all right? So, this is one of the main, uh, games within a game in bike racing. It isn't the main game within the game in bike racing. The main game within the game in bike racing is trying to figure out where to hide your steroids. But this is, this is all right. But this is important. Uh, let me give an economic example, uh, in, in, or business example. In business, you can imagine two firms who are both developing a new technology, and that technology could be a new way to, uh, uh, uh, book, uh, uh, air tickets online, for example, a better way of doing that than the existing one. So, both companies are busy doing R&D and trying to perfect this new software or hardware, whatever happens to be. And they have to decide when to launch, uh, their software. And the problem here is, if you launch too early, when the thing hasn't been perfected yet, it might not work correctly, and you're not going to get a second chance, right? Basically, if it doesn't work, no one's going to trust you. And if you launch too late, in this particular market, the market, I mean, you can think of other examples, but this particular market, then the other guy will have launched already. The other firm will have launched already. His or her firm will have established a foothold and probably is going to end up becoming the standard and being the whole market. So, this is an example of a market where probably at the end of the day, there's only going to be one player. So, getting the critical foothold matters. But if you go too early, you launch too early, you're going to not be trusted by customers. If you launch too late, someone else will have established them, established the standard. So, that's about launching a product, not launching a sponge, but basically the math of it is the same. All right, so we're going to analyze this game. Um, we're going to use, um, these to analyze the game, and I'm going to do a little bit of work on the board, but I, I, I want everyone to realize that when I say "we," I mean "we." So, I'm going to help along, but you're going to solve this with me. Okay? So, this is not going to be me lecturing. That's what we do. It, yeah, we get people to participate. We just did. So, we're going to try and learn this together. So, just to get set things up before we do that, let me just establish a tiny bit of notation, and draw a picture. All right, so, just a very simple bit of notation. I'm writing kind of small. I'm hoping it's going to be visible. Uh, let P1 of D, this is the only notation I'm going to use. This is the only kind of mathy thing. Let P1 of D be the probability. So, I'll call that prob, uh, that one hits at distance D. All right? So, I'm going to use P1D to say, having to write it every time, P1D is the probability that player one, who I guess was, who was player one, was, was James, hits were he to shoot his sponge at distance D. And similarly, let P2 of D be the same thing for player two. All right? So, I'm going to draw what I think these probabilities look like. I'm going to use a picture. All right, we're going to use this picture for a while. All right, so the notation is not as important as the picture. How low can I go before I'm going to lose the back of the, uh, oh, higher, higher. Okay, is that okay, Charles? No, higher, still way high. Okay, in that case, I'm going to delete the top line and I'm going to, everyone know what P1D is now. All right, I'm also going to push this back, perhaps, so you have a little bit of an angle of view to hold it up on the stage. That better at all? Uh, all right, so let's try and do it right at the top here. That okay? Is that okay, Charles? Yeah, okay, good. All right, so on this axis, I'm going to put D, which is distance, and on this axis, I'm going to put probabilities. And I'm going to make some assumptions about the way in which these probabilities of hitting depend on distance. And at least two of these assumptions I think are inoffensive, and the third one is offensive. All right, so, uh, the first thing I'm going to assume is that if you were at distance zero and you shot at distance zero, what would, what would be the probability that you hit if you shoot at distance zero? One. Okay, so one. So that's an assumption, but it seems a reasonable assumption. So, I'm going to assume that these probabilities start at one. If we were at distance zero, remember, I'm drawing this as a, as distance, but of course, the game is really going in this, this direction. Right? So, we're starting far away and getting closer, but I'm going to draw it with distance in the standard way, going left to right. All right, so the second assumption I'm going to make is that as you get further away, the probability of your hitting goes down, right? So, the further away you are, the less likely you are to hit if you, if you shoot at that distance. So, I don't know, it could look like, it could look like this, say, right? It doesn't have to be exactly like this. I'm just going to assume it slopes downwards. All right, now, I'm not going to assume that both players are the same here. All right, so it could be the case that the other person has a different shape curve that does this. All right, and actually, for what I'm going to say today, it wouldn't matter if these things crossed, right? But all I care about is that they're both downward sloping. So, in this case, perhaps this one is P1 of D, it's player one's probability of hitting, and this is player two of D. All right, the way I've drawn it, who, who is the better shot, player one or player two? Player two. Everyone, okay, that player two is the better shot because at any distance we happen to talk about, were they to shoot at that distance, player two has a higher probability of hitting. They were okay with that? So, player two is the better shot. Now, again, I don't need that assumption. Uh, I just need them to be downward sloping. It could be that player one is better at shooting at close distances, and player two is better at shooting at long distances. That's fine. But we'll use this one today. Doesn't it work? All right, so, so far, I haven't assumed anything particularly difficult. I think here's the difficult thing. I'm now going to make an assumption that's not real, uh, but I'm going to make this assumption so we can analyze it today in what is most of your first ever game theory lecture. Okay, so I'm, I'm going to make an assumption that's not true, but will help us analyze it. It's hard enough even with this assumption. All right, sometimes it's useful to make a simplifying assumption. I'm going to assume that each player not only knows their own probability of hitting, but they also know the other person's probability of hitting. So, I'm going to assume, actually, I'm assuming even more than that. I'm going to assume that these are commonly known. So, I'm going to assume that player one knows their own ability, they know the other person's ability, and they know that the other person knows their ability, and so on and so forth, right? So, I'm going to assume that's all known. All right, so that's a big assumption, but it's, it's enough for today to solve. All right, I'm going to assume that these are known. All right, now imagine it, in fact, is the case that one of these players is known to be a better shot than the other player. Let's ask ourselves a question: who should shoot first? The known better shot or the known less good shot? Who should shoot first? The known better shot or the known less good shot? Well, um, um, who are we playing the game? Who should, meaning, normal, you put them in their shoes. If you put yourself in their shoes, when you're in that person's shoes, from that person's point of view, who do you think should shoot first? Who thinks the better shot should shoot first? So, hang on, let's have a poll. Who thinks the better shot should shoot first? Raise your hand. Who thinks the less good shot should shoot first? Who is hedging their bets? All right, so, uh, let me try and talk it through. So, the people, I'm guessing people who are thinking the better shot should shoot first, they're thinking something like this: they say, well, the better shot has a better shot of hitting the other person at this distance. So, uh, you compare two people at equal distance. The person who's more likely to hit has the better shot. So, the better shot should shoot first. Don't know exactly where they should shoot, but they, the point at which they should shoot is earlier, since the better shot. All right? And then the people on the other side, I'm guessing, tell me, are thinking something like this: they're saying, well, the other guy is a better shot than me, so I better shoot first to preempt him from shooting me. Is that right? And we could take this further. Then the, the better shot could think, hang on a second, I know this is all known, right? So, I know the other guy knows that I'm the better shot, and therefore thinks I'm going to shoot first, and therefore he's going to shoot first to preempt me shooting him. So, maybe I should move even earlier to preempt him from trying to preempt me. All right? And, and we could go on with this discussion for a while, and I think once we've opened up this particular Pandora's Box, you can see it's not going to be that easy, right? It's not obvious who should shoot first. At first, it seems obvious, got to be the better shot, but it's not obvious at all. All right, so here's what we're going to do. We are going to figure, and I'm going to emphasize that word "we" again, we are going to figure out who should shoot, and we are going to solve this exactly. We're going to figure out exactly when they should shoot. All right, so we're going to figure that out. All right, for, for the physicists and mathematicians in the room, and I can see some, you know, the answer already, but pretend you don't. All right, all right, so, all right, on the way, I'm going to point out two ideas I hope I remember to do this that are more general than this game. All right, so we're going to work through this game, but since I want to use this as an illustration for ideas you might use elsewhere in life, or when you're playing games, or wherever, I'm going to try and draw out two larger ideas as kind of take-home things for you all. All right, so remind me to do that if I forget. Okay. Oh, I brought Pushkin with me. Somebody can have a copy of. I was going to read it out, but I guess we haven't got time. All right, so to do this, uh, we're, I said we're going to do this. So, one of the things we're going to learn, this is not one of my big general ideas, that isn't a bad one. One of the things we should learn is when you have a difficult problem like this, it's a good idea to try to break it down into some smaller problems, because the whole problem just seems unmanageable. So, what I'm going to do is I'm going to break it down into two pieces, see if we can figure out what to do in two kind of easy pieces, and then use those pieces as building, building blocks to try and get the general answer. Okay, that's going to be my strategy here. All right, so the first building block, we'll call it Building Block A, and Building Block A is going to be a question. All right, so, uh, so for both these building blocks, I'm going to assume, I don't write this bit, I'm going to assume that no one has shot yet. All right, so assuming no one has, I could write that top, assuming no one has shot yet, right? Then, if player I knows at distance D, say there they are at D, that the other person, let's call the other person J, that J will, will not shoot, or throw, if you prefer, uh, next turn. Can I use the word "tomorrow" for "next turn"? It'll save, save us having to write "next turn" every time. We, we, we will not shoot tomorrow when it's player J's turn. Then what? Your player I, let's say you're Robert, and you've got your sponge, and James still has his sponge, and it's your turn, and for some reason you know, you, Robert, know that James is not going to throw next turn. Assuming that no one's throwing, it should you throw now or not? No, you should not throw. Right? You should not throw. Why should you not throw? Now, in that second, you've got a better chance next time, right? Because if you knew that the other guy is not going to shoot next turn, tomorrow, then you're going to get a better shot the day after tomorrow. Doesn't mean you should necessarily shoot the day after tomorrow, but you're certainly shooting the day after tomorrow is better than shooting now. Is that right? Everyone okay with that? This, this, this is meant to be the easy fact. Let's make sure this one's, this, this is good. Yeah. So, if I know the other person's not going to shoot tomorrow, I should not shoot now. I should wait till the day after tomorrow. We're good, David? We're good on that? Okay, I'm checking because he's my teacher. All right, then do not shoot. All right, all right, so that was fact A. So, fact B, you can kind of guess what it's going to say. It's going to say, assuming no one has shot yet, suppose that I knows at distance D that J will shoot tomorrow. Same thing with will, with will, with will not, replacing will, right? So, again, suppose you're Robert, no one shot the sponge yet, no one's thrown the sponge, and you, for whatever reason, you know that if you don't shoot now, James is going to shoot tomorrow. What should you do? All right, it depends. It depends. It does indeed depend. What does it depend on? Depends on distance. What else does it depend on? Okay, so we're getting better than that. Depends on distance, and the reason it depends on distance, it depends on the probabilities, and it indeed depends on who's the better shot. All right, so it's going to depend on all those things. Can we be more precise? So, that's right, it, it does depend on distance, it depends on distance because the probabilities depend on distance, and the probabilities themselves depend on who's a better shot. So, it's right, it depends on those things. Can we be more precise? Probability gets shorter, closer. Oh, I guess you're 100% if he misses. So, second again, he's an engineer. Doesn't, shouldn't allow the engineers here. That's right. So, okay, so here, here's the answer. That's right. So, so what do you have to compare? You have to, Joan, yeah, go ahead. True. That, that's a good point, actually. But, but notice I've assumed, I've assumed that I've done that here. So, the assumption that you know these curves means you really do know what these probabilities are. So, in some sense, even we didn't play it that way, we've assumed that you know the weight of the sponge. So, so here's our difference between reality and the assumption. So, in reality, James probably doesn't know his own ability or the weight of the sponge, but we're going to assume for today that he does, just to keep things simple. All right, so let's, let's come back to that. It depends again. So, what does it depend on? If I throw now, state the question again before I answer it. The question is, if no one has thrown yet, and if I know that you are going to throw tomorrow if I don't throw today, sorry, I know that you are going to throw tomorrow if I don't throw today. What should I do? That was the question. And the answer is, it depends. And then it's, what does it depend on? Well, what do we need to compare? We need to compare what happens if I throw. If I throw, then my probability of winning the game is the probability that I hit the other person. All right? If I wait, knowing that he's going to shoot me tomorrow, or shoot at me tomorrow, then my probability of winning the game is the probability that he misses tomorrow. Is that right? So, I have to compare the probability of my hitting if I throw today with the probability of him missing if I leave it and let him throw tomorrow. That make sense, right? Compare apples with apples, or oranges with oranges. So, the answer is then, shoot or throw? Shoot if, all right, so let's, let's use our notation. Pi D, that's the probability that I hit if I throw today, is, uh, bigger. We'll make it bigger than or equal. Let's not worry about the equals for now. Doesn't really matter. So, we make it bigger than or equal to, uh, P, is greater than or equal to, two, the probability that he misses tomorrow. And that's 1 minus P J D minus one, because we'll be closer together tomorrow. All right, so this is the probability that I hit today if I, if I shoot, and this is the probability that if I wait, if I don't shoot, and he then shoots at me, this is the probability that he misses tomorrow. Am I okay with that? One minus makes it the probability of him missing, and D minus one because we're closer together tomorrow. All right, everyone okay with that? Yeah, somebody can wave if they're not. It's okay. It's okay to. Should I do it again, Joan? Good, good. Again? Yes. On. Okay, so this is the probability of winning by my hitting, and this is the probability of me winning by his missing. Okay, now I'm actually going to do some real math, and I know that some of you are not, you know, didn't excel in math in college, and pretty math nervous. So, this is the only actual math I'm going to do. Uh, so those people who are nervous, can you please hold on to your seats underneath the cushions, just for a second? I'm going to add, I'm going to add P J D minus one to both sides of this thing. All right, okay. Is that's the only math I'm going to do here. Okay, okay, so, so this is the same as saying, shoot if and only if P D plus P J D minus one is greater than or equal to one. All right, everyone okay with that bit of math? I just added something to both sides of an inequality. Yeah, I know the physicists are okay with that. I mean, everyone else is going to be hard to see in the last. Is that too? Is that going to be too hard to see? So, drag it back. All right, let's, can people, Charles, can Charles, can you see that? Up. It needs to be higher. Okay, we're going to raise it up. How much higher? If it was here, it would be okay. Okay, so, uh, we'll just rewrite it. Uh, shoot if, and I'm going to rewrite this rearranged equation, which is P D plus P J D minus one is greater than or equal to one. There it is, Charles. Okay, yeah, okay, all right. So, what do we know now? We know what you should do if you know the other person's not going to shoot. That is to say, you should step. And we know what you should do if you know the other person is going to shoot, which is, you should shoot if and only if this sum is bigger than one. Let's go back to our picture. Let's put some steps in. So, these are very little steps because it's a very little picture. I'm, I'm not insulting the foot size of the deans, it's just, you know, I've scaled it down. Okay, and let's see where would this be roughly? So, me just probably here, guessing. All right, right, so, um, let's ask the question, when is this equality, inequality met, and when is it not met? Okay, so at the beginning of the game, when people are out here, if we ask the question, is P, or P1D at the beginning? P, P1D plus P2, 2D minus one, is it bigger than one? Is it bigger than one out here? No, it's small. These are two small things. So, it's, it's going to be smaller than one. On the other hand, once we're in here, once we're in really close, the sum of these two probabilities is bigger than one. All right, everyone okay with that? And there's going to be a critical point when, for the first time, the inequality switches from being incorrect to correct, from being false to being true. All right, and we're going to call that distance D star. All right, so I, I've already drawn it approximately here, but pretend it, pretend it's accurate, right? So, what are we saying here? We're saying this inequality, which I'm claiming is going to be important, but none of you know why yet, I'm claiming is important, this inequality is not met, not met, not met, not met, not met, not met, not met, not met, not met, not met, and then it's met, met. Okay, okay, good. So, now let's make a claim and then prove the claim. So, I claim that the following is true. I claim that nobody should throw their sponge until D star, but whoever's turn it is at D star should throw. And were it to be the case that they didn't throw at D star, then the next person should throw at D star minus one. Okay, so, so here's my claim: nobody throws before D star, but at D star you throw, whoever it is. Okay, that's what we're going to try and convince you, try and convince you that that's true. All right, to convince you that this is true, we're going to use these two facts and these two stooges. All right, so, so I'm going to bring my stooges up again and convince them. We're going to, if this was ESPN, we would be playing things in slow-mo. I don't have, actually, in the drama school, they probably have that, but I don't have slow-mo in here. So, we're going to play slow-mo in kind of, imagine slow-mo. Here we are, back, back at the beginning of the game, and we're going to do slow-mo. And here we are. Who first? I've forgotten. So, James was first, he's player one, and he is thinking, what should he do? Should he throw or step? Now, we know the answer, but let's just walk through it slowly. All right, so James can think the following way: suppose, suppose Robert was not going to throw next go. Suppose it's the case that Robert is not going to throw next go, then which fact should he use? Should he use fact A or fact B? A. And what would his conclusion be? Step. Okay, that's one thing that Robert could think. Uh, so, one thing that James could think. But the other thing James could think is he could think, suppose Robert is going to throw next turn. Right? So, if James thinks that Robert is going to throw next turn, then he should use fact B, and the conclusion should be, well, he, if he's using fact B, James should throw if his James's probability today plus Robert's probability tomorrow, there it is, I've just drawn those lines, is bigger than one. Is it bigger than one? No. So, he should step. So, in this case, whether James thinks that Robert is going to throw tomorrow or thinks that Robert is not going to throw tomorrow, you arrive at the same conclusion, namely, you should step. And therefore, you should step. Okay, so, all right, all right, so let's do this once more in Robert's shoes, but you'll see the same idea. So, I'll do it faster. I can speak fast. Uh, here's Robert. Robert is thinking, what James is going to do. If Robert thinks that James is not going to throw tomorrow, then by fact A, Robert should step. And if Robert thinks that James is going to throw tomorrow, then should use fact B, and look at these lines on here and say, throw if this line plus this line is bigger than one. But they're not bigger than one, they're smaller than one. And therefore, either way, he should step. So, he should step. Okay, everyone okay with that argument? Okay, okay, so, just, I, I promise I'd point out some things I'm going along that are more general lessons. Right? So, the thing we just saw was an example of what's called a dominance argument. A dominance argument says, if I think A, if I think I don't use A, if I think P, then I should do X, and if I think not P, then I should do X, therefore I should do X. All right, dominance arguments, pretty straightforward, but people get them wrong. Okay, so, so if, if you do the same thing whether P or not P is true, then you should do that thing. Okay, if I had time, I'd give you examples of movies where they don't do it, but never mind. Okay, so, all right, so fine, so, so Robert should step, and then James will go through the same argument. It'll still be a dominance argument, so he should step, and then Robert should step. We're going to go in fast motion for a second, and this will go on being true until we get to about here. Come, come, come, come, come, come, come, stop. Okay, okay, so here they are. So, we're going to get, so we're going to get, not shoot, not shoot, not shoot, not shoot, not shoot, not shoot, not shoot, not shoot, not shoot, not shoot. So, uh, let's see, two, one, two, one, two. So, uh, uh, here we are, finally at D star. All the way through this argument, including the last step, which was, which was, uh, James's argument at D star plus one, the dominance argument said, no matter what I think, I should step. So, I step. Now, however, we arrive at D star, and here's Robert at D star, and Robert's thinking through the same reasoning. So, here we are, we're in Robert's head. This is a good head to be in. I, I read some of his stuff. So, managing yourself in Robert's head, and Robert is, is putting himself in James's head, trying to think this through. And Robert is saying, if I think that James is not going to shoot tomorrow, then I should use fact A, and that tells me not to shoot today. But if I think that James is going to shoot tomorrow, then I should use fact B, and this time, for the first time, when I work my way through fact B, it tells me I should shoot today. Right? Because I'm at D star. So, now these two arguments are pulling in opposite directions. If I think the other guy isn't going to shoot, I should step. If I think the guy is going to shoot, uh, then I should shoot. So, I'm stuck. I'm stuck. All right, so it was easy up to now. We had, don't shoot. But now we're at D star. Now, if game theory could say nothing other than these dominance arguments, you probably shouldn't take my course. All right, so it better be the case I've got some way out of this dilemma. This is a dilemma, but I have a way out of it. So, here's, here's how we're going to get out of this dilemma. We're going to figure out, remember, we're in Robert's head. All right, echo in there. Right, we're in, right here we are. It's like a church, right? So, in Robert's head, we're going to try and remember this dilemma would be solved if Robert knew what James was going to do tomorrow. So, we're going to work out what James is going to do tomorrow. But we're not going to do it, as was suggested just now, uh, uh, by, by, by Mr. Mor, by just taking one step forward. We're going to take lots of steps forward. In fact, we're going to go to the end of the game. So, what's the end of the game? The last possible step in the game. So, come forward, come forward. No one shoots. No one. Come, come, come, come. Personal space, but further forward. All right, okay. So, this is the end of the game. All right, no one shoots. No one shoots. No shoots. Eventually, they're on top of each other. Their noses are touching. It's uncomfortable. All right, all right, okay. And here we are at the end of the game, and this turns out to be Robert's turn again. And so, if Robert finds himself at distance zero from James, if we get here, what should Robert do? Sorry, yeah, what should Robert do? He should shoot. Why should he shoot? Because he's going to hit with probability one. So, he should shoot. Is that right? All right, so that was easy. So, if we get to this stage, Robert's going to shoot. Let's go one stage back in time. No, no, no, no, it was must have been his turn. It's your turn now. So, the previous turn must have been James's. So, James is going to go back in time. Okay, so here we are, one. Now, we're the distance one apart. By the way, we had a shoot here, right? Can now they're distance one apart, and it's James's turn. And now, what does James know? That Robert's going to do tomorrow. If James doesn't shoot, he knows, we just put it on the picture, he knows that Robert is going to shoot. And therefore, James should shoot. He should make that decision based on fact B. Is that right? James knows Robert's going to shoot tomorrow, so James should shoot if his probability of hitting now plus Robert's probability of tomorrow is bigger than one. And since Robert's probability on his own is one, for sure, one plus something is bigger than one. That was another second bit of math. I lied. Okay, there are two bits of math in the course, right? So, one plus something is bigger than one, so he should shoot. Is that right? Right? So, so James would shoot at this point, right? All right, so, so now let's go back one more stage. And now we're at stage, two, two away. All right, and now we're in Robert's head, and what does Robert know? What does Robert know? He'll shoot. He'll shoot. And therefore, what should you do? Shoot. Right? Because, why, why should you shoot? Because you go through fact B, and you find the sum of the probabilities is bigger than one, so he should shoot. So, let's go back to another step. All right, and here we are at stage, uh, one, two, three. Here we are at stage, uh, uh, three, which is actually our D star minus one. Here's James. This is what we were worrying about, right? Right? So, what does James know at distance three? What have we just worked out? He knows that, James knows that Robert's going to shoot. And so, James should use fact B, and using fact B, he should, he should shoot. All right, so now let's go back one more stage, and now we're back where we were, where our dilemma was. Now we're back at D. I've worked my way back to D star, and but what's different now? Now, when I arrive at D star from the end, remember last time I was at D star, I had a dilemma in Robert's head because I didn't know whether James was going to shoot or not. But now I do know what he's going to do. What's he going to do? And therefore, what should you do? There we go. So, we've just, we've just shown. We've shown that at D star, Robert shoots. In fact, what have we shown? We've shown until D star, nobody should shoot. But at D star, and if you get there any stage afterwards, you should shoot. All right, so thank you to my, my two stooges again. Sit them down again. Thank you. You can go. Just sit. Thank you. All right, thank you. You expect them to be great actors and great interpreters of the law. So, that's, that's good. This is the, the law of game theory. All right, so what did we just show? Let's just make sure we understand what we just showed. We showed that what I claimed, we showed that no shot should occur until D star, and that at D star, which of course depends on people's abilities, but at D star, whoever's turn that happens to be should shoot. And we used two big ideas. I didn't.

Point out the second one, so I forgot. So I'm going to do, do so now.

The first big idea was this dominance idea. And the dominance idea was, if you, if, if in circumstance P, you should do X, and if in circumstance not P, you should do X, then you should do X, right? I'm saying that adamantly because it seems obvious, right? And that was one big idea.

And the other big idea was, when you get stuck in this thinking, this way, a really good way to solve out a game is to go to the end of the game and work backwards, right? It's good to anticipate what people are going to do, and the best way to do that is to go to the very end of the game and then work yourself back forward again. All right.

Now, that idea is called backward induction. Backward induction. And backward induction is the most important thing you're going to learn if you take a game theory class. It's hard to do, actually. It's hard to do because it's hard to make yourself do it. It's hard to have the instinct that I really should work backwards and not forwards. Time works forward, right? Our logical way of thinking works forward, but when you're trying to play a game with somebody, trying to put yourself in other people's shoes, whether putting yourself in your shoes, trying to anticipate what they're going to do while they're anticipating what you're going to do, it's easier to work backwards.

So backward induction combined with dominance tells us when we should shoot here. All right. So let's just go back to the beginning and just make sure we understand the game before we draw out any more general lessons. People, okay, at this point, have I, you don't particularly look like deer in the headlamps, which is a good sign. Say, I haven't lost too many of you, is that right? Good. Okay.

So, yeah, it always happens that when you get stuck, that if you reason from the other end, yes, yes, it's because if you look at those inequalities, it's monotone. So I'll show you after, but basically, as you get closer, both people's probabilities are getting higher, always. So that pair of probabilities, you're always, you're always going to cross one out and replace it. One of them will survive, and the other one get, will, will be crossed out with something bigger. So once you cross D star, once you cross this inequality, you can't go back. It's, it's, it's, it's a strictly monotone sequence. That was a physicist asking the question, so I assume I answer is in the appropriate language. Okay. All right. Good.

So, okay, so what do we show? We showed that it isn't as simple as asking, should the better shot shoot first, or should the worse shot shoot first? And it isn't even a question of preemption and preempting preemption. It's a question of understanding both probabilities, working the thing backwards, and finding out that there's a critical moment to shoot. Now, be careful here. I'm not saying that that critical moment to shoot doesn't depend on abilities. It does depend on abilities. So it's very tempting to say, look, a critical point to shoot here, and this corresponds to when you see the white in the other person's eyes. Not quite true, right? Because you know, when you see the whites in other person's eyes depends on your eyesight, and when you should shoot depends on both your abilities, right? So, so I'm not saying there's a unique time to shoot, but I am saying it isn't a question of who the best shot is. It's a question of both abilities put together.

Now, let's throw in some dust. Let's make this harder. All right. So we now all know that if you're playing this game, you should shoot at D, if you knew everything you knew and if you'd taken the course. But you might say, that's fine, but in, in, it's fine if I'm reasoning, I'm playing this game against another person who's in the room, who's taken the course. But what if I'm taking the, what am I'm playing this game, you know, it's life and death. It's, it's Nien and Lasky. Neither of them took the game theory class. As far as I know, they didn't take the game theory class. I'm looking around for a Russian scholar. I see a German scholar, but not a Russian scholar. So, but as far as I know, they hadn't taken a game theory class before the beginning of all.

All right. So, and you might, might think, what if I'm playing against somebody who's a little crazy, right? So what if I'm playing against a crazy person, right? I could be, I may have gone to Yale, but I could be playing against somebody who went to Harvard, right? Who, who never had a chance to take a decent game theory class. And so, so, um, how would it, how would it change our argument if I thought the other person was a little crazy? Let's still assume that we know each other's abilities, but now I suspect the other person just isn't capable of reasoning through in the way we just did. How does that change the argument? Would that make you shoot earlier? Would that make you shoot later? Would that make you think? Good, good.

So even if you're playing against a crazy person, even if you're not confident in the other person's rationality, or even not confident in the other person's confidence in your rationality, or any other statement of that form, which is often true in the real world, even in that circumstance where you don't know you're playing against a sophisticated person, it doesn't change the argument. It doesn't change the argument in here. It doesn't change this argument, and it doesn't change that argument because that argument was a dominance argument. That argument said, I don't care who I'm playing against, whether I'm playing against a Yale student or a Harvard student, I don't really care if the person's human, right? The person could be a robot for all I care, right? All I care about is whatever is going to happen, I should, I should not shoot. Therefore, I should not shoot, right?

So, in, in the first part of the argument, the dominant part of the argument, there was no interesting thought of the form, if the other guy is rational, and if the other guy knows I'm rational, like that. It was just a dominance argument. It's an incredibly robust argument, right? But in here, while we were doing the backward induction argument, there I was thinking the other guy is rational, and I was kind of thinking that he can figure out that I'm rational, or that he can figure out that I can figure out that he's rational, or she, right? So it's used he, and it's easier. So, uh, in, in this piece of the argument, Robert at D star had to figure out what James was going to do at D star minus one. And that argument required him to have confidence that James would figure out what Robert would do at D star minus two, etc., etc., etc. So, not did it require thinking the other person, putting myself in the other person's shoes and thinking they're rational, it required putting myself in the other person's shoes while they're putting themselves in my shoes, and they know that I'm rational, and so on and so forth, right? So the backward induction argument is much more, um, fragile to irrationality, right?

So Lori Sanis gave a talk yesterday about irrationality, which is an incredibly important topic these days. You know, it's possible that on Monday, they might give the Nobel Prize in Economics to Robert Shiller, which would be very nice, who's worked on such topics in finance. Irrationality here doesn't affect this part of the argument. It does affect this part of the argument. All right.

Now, nevertheless, nevertheless, the fact you should not shoot before here is a pretty strong argument, right? Because frankly, by the time you figured out the other guy's irrational, it's too late. You don't really care if he shoots early here. You're perfectly happy if he shoots early here. All right. So this argument, this particular example, even if I think the other guy isn't rational, I shouldn't shoot before D.

Now, let's push that idea further. I've played this game in class with undergraduates. I've played it in, uh, my business school class with business school students. I've played it with parents. I've played it with alumni. I've played it with Deans. Uh, and, uh, uh, so I have some idea about how people do on average in this game. Now, assuming that people are following the reasoning I just said, and assuming that ability is distributed roughly randomly, and also making this somewhat unreal assumption that people know each other's opponents, how often should I, how often do I expect to people, how often do I expect to see people hit on average? Well, D could be the weaker person shooting, it could be the stronger person shooting. If it's happening at D star, one is, the sum is one. All right. So what's on average? On average, how many hits should I see? 50%, right? So if I get a large enough sample, there's a bit of hand-waving in that argument. Someone was going to point out, allowing me a little bit of hand-waving, right? For discrete versus continuous, roughly speaking, 50% should hit. I see far fewer than 50% massively fewer than 50%. I would say I see 10% hits. Why? Why do I see 10% hits when the, when we just agreed that even if everyone's crazy and irrational, no one should shoot between before D star, and D star is telling us that half, half the shots should be hits? Why do I see such a low hit? People aren't experienced. People aren't experienced. Although that might make them shoot later, I mean, you know, in some sense, right?

So I think there's two things going on. Let me take them in turns. This one first, overestimate. So I think, I think people do two things. They, they overestimate their opponent. They also overestimate their own abilities. All right. So there's a well-known psychological bias that people overestimate their own ability to do something. They probably also overestimate the other person, but they certainly overestimate their own ability. Things. So there's an overconfidence bias, and there's stacks of literature on the overconfidence bias, right? So if you, if you Google overconfidence and start reading papers, you can be there a long time, right? So it's pretty well established that people are overconfident in their abilities. Amateur golfers will try that shot which they've seen Tiger Woods do, where he slices it between the trees, right? And I don't play golf, but I'm told amateur golfers are never going to hit that shot. All right. Uh, I don't actually, I don't know the answer to that, Meg. It's possible. Um, uh, uh, there probably is a literature on that because almost all these things have been tried, uh, uh, uh, with, uh, gender in there. Uh, if I had to guess, I would say men are more overconfident, but I don't know the data. So I, without seeing the data, I have the same suspicion you have.

So one thing is overconfidence. That's a well-known bias. I think there's another thing going on here, and I think it's going on specifically in America. And I think is, is what's going on is another thing going on here. I think that at least in America, there is a thing I want to call the proactive bias. Right? People like to be proactive. When your kids in America, you are taught that being proactive is a good thing. I know this because I have kids. I have a 10-year-old and an eight-year-old. I have a one-year-old too, but he hasn't gone to school yet. My 10-year-old and my eight-year-old, both of whom are girls, are being told, uh, uh, being proactive is good. Seizing the bull by the horns is good. I also have more evidence about this. So just to make it clear, being proactive would be throwing the sponge rather than letting things happen to you, right? So people think being proactive is good. And you see this on on SportsCenter for those people who watch SportsCenter, you'll see these guys in the sweaty, um, usually it's guys, it could be actually, I don't want to be sexist about this, think it's males or females, they're in this sweaty locker room afterwards, and they make this statement, and they say, uh, it's great because we control our own destiny. And coming from England, controlling my own destiny just sounds kind of scary to me, I have to say. If, if I wanted to control my own destiny, I wouldn't have gotten married. I didn't say that. I didn't say that.

So I, I worry that this proactive bias causes people to throw the sponge too early, right? That, that taking the bull by the horns sounds like a good idea, but when you think about it, running away is an awfully better idea, right? Right?

So, uh, if I have one last lesson for you, other than, you know, the takeaway lessons from today's, today's talk, go through them. One is dominance arguments. I won't rehash what they are. The second is backward induction. Sometimes it's worth going to the end of the game and working backwards. And the third is this, and it's really just for the Americans. I think the Europeans in the room don't make this mistake. And that is this: the point is not to go down swinging. The point is not to go down. I'll leave it at that. I'm done. I can take a few minutes of questions, but I want to make sure you get me to the next talk. So I won games. I have solved completely. If the opponent is Gandhi, then I know he won't shoot. That's right.

So Gandhi, you just wait and plunk it on his head. The question. Yeah. So the question was, what if you're playing against Gandhi, right? So Gandhi is not going to shoot you because he's a saint. So if you were going to win the game, and you are so inclined, and Shan is a cynical kind of guy, he'd just wait and plunk the sponge on Gandhi's head. An image I'm now going to have when I go to bed tonight. Do I like the casino? I never go to a casino. Um, um, um, I, yeah, I don't really like. Out of interest, I'm not a big casino player. I kind of like playing cards. Casino, I mean, roulette wheels seem a little bit too random to me. But I, I also, I worry a little bit about irrationality in the casino. So I had a student once who was studying, uh, behavior in, uh, not not roulette, but in, uh, 21, Blackjack, thank you. And, and it turns out that people do exactly the wrong thing there. Uh, they, they, um, what do they do? Uh, they, um, they quit when they're behind, rather than quitting when they're ahead. And it turns out since you, since you use five packs and so random sample, if you've been doing well, you know, any sensible updating will say you're going to do worse in the future. And if you're doing badly, you're actually going to do better. There's actually a regression to the mean. So people do exactly the wrong thing. And I'm scared I would do the wrong thing too. And, and my wife would not like it.

Are examples of nature? Yes. It's a good question. It's a very good question. So the question is, can we see games like this in nature? So let me, as like a politician, let me answer a slightly different question. We'll come back. So, so, so, uh, game theory is enormously used now in biology, in, in behavioral biology. It's, it's a major tool. And it turns out that there are wonderful connections between the game theory we developed in economics, uh, and notions of equilibrium in economics, and notions of evolutionary stability in, in, in, uh, in biology. And they're not exactly the same idea, but one is kind of a sub of the other. And, what you can do, uh, in evolutionary biology, which is kind of cool, is if you, if you know the basic form of a game that a particular, uh, species is playing, be careful because the actors now aren't rational, they're, it's gene selection, but still, if you know the form of the game, but you don't know the exact payoffs of the game, you can actually, uh, look at the behavior, look at the behaviors you see in the population, and back out what the exact payoffs must be, which is a lovely example of what's called identification in economics, right? So, so biologists do incredibly cool things with game theory. Actually, at this point, they do cooler things than frankly economists do with game theory. It's really cool.

Now, this particular game, I don't know. There's some biologist in the room. I'm looking at Tom. I, I don't know exactly what game this would correspond to in nature. But there are games that, uh, a better example for nature is the game rock, paper, scissors. And the rock, rock, paper, scissors. So rock, paper, scissors, everyone know that game? Rock, paper, so rock beats, uh, scissors, and scissors beats paper, and paper beats rock, that game, right? So rock, paper, scissors is a nice little zero-sum game that you can find examples of that look a lot like that in nature. And you can twiddle them a little bit. So not exactly rock, paper, scissors, and they actually have no evolutionary stable, uh, point. And you can show in examples of that, uh, cycles occurring in, in populations of, actually the most famous example is, uh, uh, tree lizards, and mating behavior of tree lizards. And you can show, you can predict out from those cycles what the exact payoffs of the rock, paper, scissors game they were playing was. So you can do really cool things in biology with this. But Tom knows more about that than I do. So I'm going to, I'm going to, I'm on dangerous territory. Tom, do you want to comment? Ah.

How do I use game theory in my day-to-day work as a Provost? I, I, um, yeah, so it's a good question. So I, I, I do a little bit. I mean, I think I think I've been doing this so long that it's kind of hardwired for me to think this way. So I do tend to lay out things and kind of work them backwards. And I do tend to sort of try to go down one branch of the tree at a time and see if they're all the same. If there's a dominance argument there, I can see a hold of. But also a large piece of game theory as applied to economics or to organizational behavior is the theory of incentives. And as you well know, Tom is the Dean of the Graduate School. He's a wonderful Dean of the Graduate School, and he steers the graduates in a wonderful way. Some of that is about incentives, that's the game theory part. And some of it is about your relationship with the grad students, which is more the psychology part. So I'd say Peter and I have the combined have the skills, but Peter has the more important skill from me. And well, in this example, you use, you know, the shape of the skill curve that each player. Yeah.

But let's say I go in the game and I know my skill curve, but I don't know my opponent's. Now, my opponent may have an average skill curve, may have a highly skilled curve, may have a skill curve, right? How do I develop a strategy? Good, good. So, so it's, it's harder, but not impossible. So it's harder because now we'd have to think about the population of skill curves out there. And I, I want to be able to think of my opponent as being drawn, perhaps randomly drawn, from some population of skill curves. And, so I'm going to have some, I'm going to index each skill curve, and I'm going to take, I'm basically going to take expectations at every point in the game over those skill curves. So essentially, the reason I didn't do it here is I need to carry a whole bunch of integrals, and I'm taking, I'm taking distributions over curves, which is a messy thing to try and do. But basically, the basic idea is still the same. I'm, there's going to be, I'm going to be carrying some expectations around. Because now I'm not, now it isn't that I know what the probability of this guy hitting tomorrow is, it's that I, I, I know only the expectation of that. I know with expectation what that, what their probability of hitting is if they were to shoot tomorrow. Uh, so that means that the math is a little harder, but the basic idea of looking forward and working it out is still runs through. I, I, I'm underselling the math a little bit, because not only do I have to worry about the distribution of the distribution of skills out there and worry about taking expectation over those skills, but that distribution is going to change as the game progresses. I'm going to be able to infer from the fact that the guy hasn't shot yet that he's not as skilled as I thought he might be, right? So that distribution, you think about some bell-shaped distribution of skills where these are the high skills and these are the low skills, some of those high skills are going to, I'm going to be able to conclude that they're not in the game as the game proceeds when I see the guy not shoot. So I have to, but so I have to keep track of all that stuff. I have to do Bayesian updating on this distribution. I have to get some expectation and so on. But up to a lot of kind of math, nice math, but out of math, the basic reasoning still goes through. That's, so, so the reason I didn't do it here is you can kind of hear there's a lot more math involved, but it's going to be okay.

Well, doesn't it change if, if it's a life or death game? My objective is don't lose. That's let, and it's great if you don't win it. Yeah. So, so doesn't that change my, my? It actually, so that's an interesting point. So, so in this example, if there's just two monetary prizes, one for winning and one for losing, and the winning one is strictly better than the losing one, then it actually doesn't change the strategy. In, in this simple game, but, but that's too quick an answer. Imagine there's three outcomes. Imagine there's an outcome, so imagine the game is not, is being played not exactly as we played it with sequential moves, but with simultaneous moves, right? So in each period, people simultaneously have to decide whether to throw or not, right? So now a third outcome is in the game. Now there's the outcome win, there's the outcome lose, and there's also the outcome both miss, tie, or draw, right? Now, now it's more complicated. Why is it more complicated? Because when there are three outcomes, we have to take into account things like risk aversion, just as we would the many money outcomes. If there's only two monetary outcomes or two outcomes, this analysis is fine. But as soon as we go from two to three, we have to start worrying about risk aversion and things like it. Again, kind of in David's territory, we have to do a little bit of, of what David does for a living to solve those things. But it's, it's a great question. Yeah.

Probably make this the last question because you have other talks to go to. But she distinguishes among three levels of relation. The tech is very illustrative. It tends to be fragile. So the amount of on a lot exists. Yes, I agree with that. I, of course, this is apology. So we're talking to a sociologist now, as you can hear. So, uh, I agree with that. So, so by the way, today was high church because, well, look around you. So, so, so what are we thinking? So, so, so what we're doing in, in mathematical economics, or the more mathy end of the social sciences, is trying to tease out our intuitions and trying to tease out how different assumptions lead to different conclusions. We are not, and you're absolutely right, we are not trying to say this is a tight prediction of what's going to happen. Some things will be pretty robust. This, this argument about dominance is pretty robust, right? But most arguments are not that robust. And one of the things we have to do in, in, in most areas of theoretical economics, but the same is true in theoretical areas of sociology, is we need to tease out exactly what are the assumptions we're making, and would these results be robust to changing them? So the great thing that, and this is not the most inspiring thing to end on, but I'll end on it despite that, the great thing about doing mathematical social sciences, which I, I guess is what I've done in my career, is it allows you to think through a kind of thought experiment rather than experiment. It says I can't really, um, so I'll be having some discussion, let's say with my wife, who's an English professor, and she'll say, yes, but you've assumed this, this, and this. And it allows me to say, okay, you're right, all those, I did assume this, this, and this, A, B, and C. But look, the argument still works if I drop assumption A. The argument switches exactly at this point. If I drop assumption B, and if I drop assumption C, yeah, you're right, the whole thing's completely nonsense, right? That's a great conversation for, uh, scientists and humanists, or in this case, social scientists and humanists, to be having. It shouldn't be a hostile discussion. It's a discussion about focusing in on the details of an argument. It's a great Yale scholarly thing to be doing. And Martin Shubik did it much better than I ever will. He's a Yale professor, one of the founders of game theory. There are many people around Yale doing things better, better at that than I will. But you have a chance to see some of more, some, some more of them later on this afternoon. So thank you for coming, uh, and we'll see you later on this afternoon. Thank you.