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37: The One Where They Parody Saw [audio fixed again] (Game Theory)
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Hello listeners. You don't know me, but I know you. I want to play a game. In your ears are two earbuds. Connected to the earbuds are a podcast playing an episode about game theory. Hosting that podcast are two knuckleheads. And you're locked into this episode. The key is at the end of the episode. What is game theory? Why did we parody the Saw franchise? And what twisted lessons will you learn?
-See our New Youtube Show "Turing Rabbit Holes Podcast" at youtube.com/TuringRabbitHolesPodcast. Also available on all podcast players.
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Transcript
Introduction & Personal Update
00:00:00
Speaker
Hi everyone, Baka here. Before we begin, I'm going to tell you why I've been going by Baka for a few episodes. I was basically issue of privacy and identity. I'm transgender, and I didn't want to go public with it immediately, but I also didn't want to go by my old name.
00:00:17
Speaker
I'm now going by Sophia and she-her pronouns, and that's how I'll identify in future episodes of Breaking Math. You're probably wondering what I'm going to sound like. Well, voice is not affected by the hormones I take, so it's a skill you have to practice, and though I'm getting better at it, it's still rough. This message was recorded after the episode was, so I still use my old voice in the episode that you're going to be listening to so far.
00:00:44
Speaker
Sophia here, and this will be the voice that I'll use starting next episode. I'll be taking speech and elocution lessons starting in March, so it'll improve substantially. Nothing else will change about the show, and we'll still continue to put out the same type of content you've come to expect.
Season 3 Launch & Communication Channels
00:01:00
Speaker
Thank you for being a listener of Breaking Math, and here's kicking it off season 3 with an episode about game theory. Hello, listeners.
00:01:15
Speaker
I know you. I want to play a game. In your ears are two earbuds. Connect to the earbuds are a podcast playing an episode about game theory. Posting that podcast are two knuckleheads. And you're locked into this episode. The key is at the end of this episode. What is game theory? Why do we parody the Saw franchise? For what twisted lessons will you learn? Let the games begin.
00:01:44
Speaker
Episode 37, the one where the parodies saw.
00:02:16
Speaker
I'm Baka. And I'm Gabriel. And you're listening to Breaking Math. And just like many, many podcasts out there, we have a Facebook page. If you go to facebook.com slash Breaking Math podcast, you will see our wonderful Facebook page.
00:02:30
Speaker
And you can email us at breakingmouthpodcast.gmail.com. But what do we have on our Facebook page? So our Facebook page has some videos that we've made in the two years that we've come into existence. Some creative, some artistic, mathematical videos. We also have a video about Socrates and Mino that's been there for a while. I made a video that's up on there where it's just me drawing with a graphing calculator on the Mac.
00:02:58
Speaker
Oh, yeah, yeah, and you add a lot of color to it and basically it's sort of an abstract evolving mathematical shape. It's beautiful. You even added some music to it too, didn't you?
00:03:07
Speaker
Yeah. Put some gentle guitar music in the back. Yeah. Also, um, I don't play guitar. I found it. It was royalty free. Yeah. And also, uh, Baca has been, I consider you a professional at Photoshop. I don't know what you, what you consider yourself, but I've been amazed by your Photoshop skills. So in addition to doing this audio podcast, Baca has made, uh, so many beautiful mathematical posters, uh, things that teach mathematical concepts like the tensor poster, but also just, um, a lot of funny covers. Every one of our episodes has a very funny cover right now.
00:03:37
Speaker
There's a cover that is making fun of Saw. Baka actually took my face and photoshopped it to look like jigsaw.
00:03:45
Speaker
And you did mention posters. We are still selling that for $22 and 46 cents each. If you want, it turns out that Patreon can't do, what do you call it? It can only, you have to do repeated billing with Patreon. So get it through Patreon, but if you want to only get a one-time fee, just email us at breakingmathpodcast at gmail.com and we'll set you up. And that's just temporary until we get a store together.
00:04:12
Speaker
Yeah, yeah, yeah, that's a workaround to the current difficulty with Patreon. That, or perhaps Patreon will fix that issue. But yeah, just email us and we'll let you know how you can get set up with a beautiful poster.
Beginning of Game Theory Discussion
00:04:24
Speaker
And also, of course, you can find our podcast among many other places. You can find our podcast at anchor.fm slash Breaking Math Podcast.
00:04:36
Speaker
I like Anchor. Anchor has a feature where listeners can leave a voicemail to their host. Would you like some voicemails? Sure, we'll use them wisely. Yeah, I'd love some voicemails. Send us your voicemail. We'll let us know how you're enjoying the episodes and what else you'd like for us to include on our episodes. Just go to the phone app, anchor.fm, and go to the breaking math page, and there should be a giant button that says voicemail. So I think that that's
00:05:03
Speaker
Pretty cool. Lastly, of course, we've got a website that needs updating, breakingmathpodcast.com. And you should go there if you want to see some of the applets that we've created in the past years. Yeah. They're just simple JavaScript programs that illustrate mathematical concepts.
00:05:19
Speaker
Or the hypercube, like if you're going to go to a rave, just put the hypercube on your phone, you know? Yeah, it actually would be a cool laser projection on a rave. Yeah, it certainly would be. And you might have noticed that we're back. Finally. Game theory. Actually, before that, we have the solution finally to the hat problem from a few months ago, right?
00:05:35
Speaker
That's right. A few months ago, I had given a problem about four prisoners and some hats. There's a couple of different prisoner hat problems, but I gave one of them and I did not give the solution. So it is now time for the solution. First, a quick recap of the problem.
00:05:51
Speaker
So in this problem from our previous episode there of course are four prisoners and I think the setup is that there's just not enough there are not enough cells for the prisoners the prison is full so the warden decides to give the prisoners a puzzle and if they get it right then they get to live and if that sounds almost like a game theory thing
00:06:15
Speaker
Anyways, the warden gives the prisoners a problem to solve, and if they quickly solve it, then they're all set free. And if they do not solve it correctly, then they are all executed, because this is a messed up problem. It is.
Exploring Game Theory Concepts
00:06:30
Speaker
And they're wearing two different colors of hats, and the prisoners have to escape.
00:06:34
Speaker
by one of them answering with their hat color with 100% accuracy. To rephrase the question, which prisoner knows their hat color and how do they know?
00:06:47
Speaker
Yeah, yeah, exactly. And so a few details. There are four prisoners. Let's just call the four prisoners prisoner A, prisoner B, prisoner C, and prisoner D. And here's the setup. Prisoner B is set up facing a wall. He can't see anybody. He's not looking at a mirror. He's just looking at a blank wall. Prisoner C is set up right behind B. So prisoner C can see prisoner B, but nothing else. And he's not allowed to look behind him.
00:07:14
Speaker
Prisoner D can see both prisoner B and C so basically all three BC and D are in a line and they are facing a wall and D can see D can see C and B. Sorry D can see There's no way to say that and not make it. It's okay. It makes sense. Prisoner D can see prisoner B and prisoner Sorry
00:07:41
Speaker
Okay, I could do this. Thank you. So four prisoners, A, B, C, and D, are arrested for a crime. Prisoner B is facing the wall, and so he's in the very front facing the wall right there with his nose to the brick. C is right behind B, so C can actually see B in front of him. Then D can see both B and C.
00:08:05
Speaker
Good job. You did it. You did it better than I did. Yeah. And as you said earlier, there are two colors of hats. So black and white, red and green, whatever. We'll just say black and white. And where's A? Okay. Oh, and A is set in a different room altogether. A can't see anybody, but they can all hear each other. So A is just off in solitary confinement by himself.
00:08:26
Speaker
Alrighty, so the solution. So here's the thing. The solution involves the warden asking a question and it specifically involves the fact that people do not answer right away. Oh, and we forgot to say that there's two different colors of hats.
00:08:44
Speaker
No, no, we said that. I mean, there's two of each color of hat. Oh, yeah. We didn't say the distribution. So there's two, let's say two black and two white. Thank you. I hope our listeners are keeping up with all this. I'm sure they're doing fine. Okay. So essentially when the warden asks if anybody has a solution, nobody answers right away. However, this in itself gives some information.
00:09:12
Speaker
Well, first of all, there is a solution. There's two solutions. The first one is that if B and C, so the people closest to the wall have the same color of hat, then Dean knows that they have a different color hat or Dean knows that he has a different color hat. Yeah. Yeah. Yeah. That's right. That's right. Then you know what? That that's a really, really good point because D can clearly see B and C.
00:09:35
Speaker
Yeah, so that's one of the answers. But let's say B has a white hat and C has a black hat or vice versa, so they have different colors. Then D does not know for sure what color hat he has on at all. It could be either one, it's a 50-50. So then the solution is that C knows that D hasn't spoken up and therefore must not know what the answer is. And the only occasion in which D doesn't know is when B and C have a different color hat on.
00:10:02
Speaker
However, C can clearly see whatever color B is wearing, whether it's black or white, and therefore C can deduce that his hat is the opposite of B's hat. Yeah. And so you might be thinking, well, that is a communication, you know, not hearing because they can't do this simultaneously and always have the answer. And that is true. So it is a lateral thinking puzzle as well. Yeah. And why is A even part of this problem at all?
00:10:31
Speaker
I think it's so they have two of each had. Okay. You know what? Good point. So it makes the problem seem more complicated than it is. Yes, absolutely. But a can turn out to be extraneous information, which sometimes is the case in problems. So that's, that's the solution. The, the, the answer, uh, C is the only prisoner who has a chance at answering correctly. Yeah, that's right. That is in the occasion that B and C have a different color hat on.
00:10:56
Speaker
So that's the solution. And a shout out to the YouTube show, SciShow. They're pretty big YouTubers. If you haven't gone to them, go to them. They have a fantastic YouTube show that we watched on Game Theory.
00:11:06
Speaker
And without further ado, what is game theory? Game theory. This is a huge, huge question. And answering this question is part of why it's taken us months and months to release this episode. It's a very large encompassing field that deals with describing human behavior in terms of costs and benefits.
00:11:30
Speaker
Well, not just human behavior, any decision-making behavior. It's basically the data that you're given is you're given a list of players and preferences. So if you're playing chess, each move might have a different preference to the other ones. And it's figuring out those preferences that is a huge part of some types of game theory.
00:11:54
Speaker
Yeah, and you had mentioned when you are given information, let's say you are the observer, that's different than you are the player. And that also has implications for what game theory means is who is given what information. Information is a huge component of game theory. It's its own field altogether.
00:12:12
Speaker
But we're in this episode, we're going to set up the basics of game theory, just go through some basic concepts.
Advanced Game Theory Concepts
00:12:18
Speaker
And of course you heard the intro. We're going to be doing that with the help of this character, whose name is not Jigsaw, but broken Lego, broken Lego. Yeah. Wow. It's kind of like jigsaw is a jigsaw broke. I guess a jigsaw is a broken picture. Well, it's a component, but it belongs somewhere. But, but a broken Lego is just, it would hurt to step on.
00:12:40
Speaker
Yeah, and that's a form of torture, so you'll see as we go. Here's the episode. Lego, you listening to this are in a dungeon with one other person I have kidnapped named Skyler. It is your job to withstand my wicked games together through an understanding of game theory. But you'll have our help. That's right. We're going to tell you some things relevant to game theory so that you can solve the puzzles.
00:13:10
Speaker
and will tell you after Brogan Lego describes each perilous challenge. You now find yourself in a different room from Skylar, separated, alone, afraid. You are both strapped to dentist chairs surrounded by an array of menacing dentistry tools. On a speaker by your ear, copyright-free music plays.
00:13:35
Speaker
Also, there's a clown picture. It's creepy. On your left, you'll notice a lever. When you hear this sound, you have 30 seconds to decide what pull that lever. If you and Skyler both pull the lever, you will both be tortured for five hours. If neither of you pull the lever, then you will only be tortured for one hour each.
00:14:02
Speaker
If Skylar pulls a lever and you don't though, then you will be tortured for ten hours and Skylar will go free. If you pull it and they don't, then you will go free. The torture is cruel as it is unusual. The music coming out of the speaker by your ear will be replaced with the sound of people chewing and slurping. It's gonna be awful. Like I didn't even like recording it.
00:14:33
Speaker
Oh, that's awful. I can't even imagine, you know, funny story. My stepdad eats breakfast cereal and he slurps all the time when I'm sitting by him. It drives me nuts. And I'm not really allowed to say anything, but I usually just pick up and go or put some earphones on. But then I get in trouble for putting in earphones. And it's not that I don't want to be social, it's that I don't want to hear the guttural sounds of slurping. That is real torture.
00:15:02
Speaker
Yeah, it's called mesophonia when you don't like the sounds of annoying things. What kind of twisted person is broken Lego to subject someone to this? Well, the kind of twisted person who's gonna help us learn some math. Yeah, hopefully, though, I in fact, I'm confident that with the knowledge that we're about to drop our listeners will will get out will get out of this with the best possible outcome.
00:15:28
Speaker
Now, one thing about the lever is that the lever, of course, is just a mechanical thing. And with or without the lever being labeled, the answer to this problem will be the same one if the question posed is, what is the best strategy for players that don't know one another yet? And basically, we're simplifying this universe so that this is the only challenge. It's not completely realistic, but you try coming up with a format like this.
00:15:54
Speaker
Exactly, exactly. In fact, this particular scenario is very, very, very similar to a well-known problem called the Prisoner's Dilemma. Yeah, so let's say that the torture is just jail time and that you and Skyler had committed, you know, some crime together. A theft or something, yeah. Yeah, and so the police say that if you confess
00:16:19
Speaker
And Skyler doesn't I mean if neither of you confess then you'll both only be tortured for one hour or you'll both only go to jail for one hour if you both defect then you'll both go to jail for five hours and That's not a very long time in jail, but let's say years and if Skyler pulls a lever and you don't or confesses rather Yeah, and so you can see they're the same thing if Skyler confesses and you don't then you'll go to jail for ten years and Skyler goes free and
00:16:48
Speaker
Yeah, so there's a few things that are worth considering with this particular setup. The relationship that you have with Skyler, or in the case of the prisoner's dilemma, the relationship between the prisoners. Do they trust each other?
00:17:04
Speaker
Yeah, because if they do have some kind of cooperation, then you could see that if both of them decide to stay silent, then the total amount of jail time or torture time, or if you don't pull the lever, is going to be at its minimum for both of them together.
00:17:24
Speaker
But the thing is, let's say Skyler pulls the lever. Is your best strategy if Skyler pulls the lever to pull the lever or not pull the lever? Well, if Skyler pulls the lever, your best strategy is clearly to also pull the lever. Yeah, and if Skyler doesn't pull the lever, what is your best strategy? Still to pull the lever. So no matter what, pulling the lever is both of yours best strategy if you're being completely selfish.
00:17:49
Speaker
Yeah, exactly. And it lands you, and if you both pull the lever and remember, then you're both tortured for five hours. So even though you could cooperate and make it better for both of you, the best strategy to do based on what you know is to pull the lever. And this is actually why climate change is such a difficult problem. We'll be talking a little bit more about
00:18:13
Speaker
that kind of thing in cooperative game theory. But basically you have all these economies that don't have a common interest in their ecology. And so you have all these different actors basically making selfish decisions and landing everyone together in worse places than they could be individually than if they all worked together.
Game Theory Puzzles & Real-life Applications
00:18:36
Speaker
Yeah, yeah, exactly. Now, this is also a Nash equilibrium. Is that right?
00:18:40
Speaker
Yeah, and Nash equilibrium is where everybody can only do worse by changing their own strategy while the other side does nothing. So basically, it's where you could only go downhill. Interesting. Yeah, and I find this fascinating because obviously, in the context of what's happening here, there's you, the listener, and then there is Skylar.
00:19:03
Speaker
Or there's the two prisoners who don't know each other and they obviously have no way of communicating or strategizing. So they don't know what the other person is going to do. So it's a matter of probabilities. And if we assume, if in this case we assume that both you and Skyler or the two prisoners are selfish, then clearly the only thing to do is to pull the lever.
00:19:25
Speaker
Yeah, and of course, you could introduce probability theory into this analysis too. For example, there's so many things you could do. So what we're basically saying is that game theory is just the simple thing of what states of the game, so what set of choices by the players are such that everybody can only do worse. So that's a question in game theory. So I hope you're getting more of a sense of game theory.
00:19:55
Speaker
Yeah, absolutely. So with this in mind, it sounds like the best move forward is to definitely pull the lever.
00:20:06
Speaker
I see that you and Skylar have found one another, and you are separated by a pane of plexiglass. In front of you, too, are vials containing different poisons from the other, as well as sinks with faucets. The sinks drain into a cup, on a chain which you both have access to. Your poisons are the antidotes for each other.
00:20:30
Speaker
So this is an interesting setup here. So in this game there is you, the listener, on one side of the plexiglass and then there is Skylar on the opposite side. And you both have a poison in a vial and you also have this setup where you can both pour your poison into a drain that goes into a
00:20:53
Speaker
A cup. So basically, to make it more simple and less mechanical, you do have different poisons and you have to either decide to mix them together or not mix them together or just drink one of the poisons. And remember, the poisons by themselves are deadly, but together they're fine. And if you just put water in the cup, then you could just drink water just fine.
00:21:13
Speaker
So as listeners here, or as listeners, do we know exactly what proportion of the poison to pour in order to create the anecdote? Oh yeah, like it's just 100% or 0% of the poison. So basically you either fill it with poison or water. This is a very interesting one, or rather it's a little on the nose, should we say.
00:21:36
Speaker
Well the reason why it's interesting is because it's not because of that because obviously the solution is to like if you see if you see Skylar pouring in their their poison then obviously you're gonna point pour in your poison too and if Skylar doesn't then you're gonna pour in nothing too.
00:21:53
Speaker
Okay. If you both want to survive though, that's the thing is that if Skylar is homicidal and suicidal or the other one is not, then it changes the Nash equilibria of this problem. And this problem is really to discuss Nash equilibria. So again, what are Nash equilibria? It's a decision with the maximum, should we say, statistical benefit out of all the choices. What is statistically the best choice?
00:22:21
Speaker
Well, it's not unnecessarily statistically the best choice. Um, it's the absolute best choice. Like there cannot be, uh, something is not a Nash equilibrium. If somebody couldn't change their answering and, and still be at the same level, it has to be, if they change it, they have to have a worse outcome. So that's why these are stable the way they are. There's a few other considerations with this particular problem. Uh, for instance, um, are either you or Skyler forced to go first?
00:22:51
Speaker
Um, I mean, I, I let's just say, well, let's explore both things with Skyler being not, not suicidal, homicidal. And with Skyler, yes, being homicidal, suicidal.
00:23:04
Speaker
Okay, so assuming that nobody is suicidal, homicidal, and both you and Skyler want to get out of this scot-free, then really it shouldn't really matter who goes first. No, because you guys have the same goal, so you guys are going to do the same thing. Yes, yes. You'll both pour your poison into the cup and you'll drink it and not die. Or you'll both just pour water into the cup and drink it and not die. Okay, exactly. So those are two Nash equilibria associated with that state of the game, or those set of rules in the game.
00:23:33
Speaker
Now, in the case that, let's say that Skylar is both homicidal and suicidal and wants both of you to die. Let's say that Skylar goes first. That's the best for you in that situation. Yes, because if he pours the poison in and if you are not homicidal or suicidal, then you just pour yours in as well and it neutralizes the poison.
00:23:57
Speaker
Or they or vice versa or yeah, I guess if they're homicidal or suicidal they wouldn't pour water at all if they went first Yeah, they would only pour poison in hopes or they pour water in hopes that you pour poison But I guess they would in that situation that wouldn't make sense because they should know that you are gonna That you would know how to cancel that out. You wouldn't put poison into water. I
00:24:20
Speaker
This episode is all about game theory, and how it has been used to try to understand the mathematics behind decision making, and one of the key components of game theory is logic. To that end, our partner Brilliant.org has a course about logic.
00:24:37
Speaker
I'd love how the course starts by taking you through logic puzzles in the chapter, the Logical Detective, and all the way through Spot the Fallacies, Whaler and Venn Diagram Basics, and Demorgan's Law in the Silla Jims and Sets chapter to support your free education in mathematics and physics.
00:24:55
Speaker
Go to www.brilliant.org slash breakingmath and sign up for free. The first 200 breaking math listeners can save 20% off the annual subscription fee, which we have been using. And now back to the episode.
00:25:11
Speaker
Yeah, exactly. Well, and then one other consideration, if we're just kidding, keep track of rules here. If Skylar pours water, then there is no outcome on your... Well, I guess you could also pour water as well. I didn't think about that. But notice that this is not a Nash equilibrium. So again, it sort of depends on who goes first. But assuming that you both have to make a choice and it's either to pour water or to pour the solution in a shared cup,
00:25:37
Speaker
Then, whether Skylar is homicidal, suicidal, and wants you to die or not, as long as Skylar goes first, you should be fine. Now let's suppose both of you are homicidal and suicidal. Then, then, if you can actually cooperate in both for the worst possible outcome.
00:25:54
Speaker
Now, we say cooperation, but that's different than a cooperative game. A cooperative game is defined in a very specific way. We'll go into that in the next section. But cooperation happens at Nash Equilibria, basically.
00:26:10
Speaker
at places where, at solutions where the other person, okay, actually, so it's not a Nash equilibrium. If the goals are different, because let's say the goals are different, no matter what, somebody can make their outcome better because one person doesn't want to kill and one person does want to kill. So there's no Nash equilibrium there at all. But when they have the same goals, the Nash equilibrium show up. Uh, there's two each.
00:26:33
Speaker
Yes, I want to mention one other thing as well. Let's just say that they don't have the same goal. But as we said earlier, assuming that Skyler goes first, you should still, you can still get a preferred outcome for yourself, but just not for Skyler because you know, Skyler's preferred outcome won't happen.
00:26:51
Speaker
Yeah, if you and Skyler are perfect players, then whoever goes first loses. Correct, correct. Exactly, exactly. Or yeah, or if, as you said, both of you are perfect players and you have opposite goals.
Game Theory in Complex Systems
00:27:03
Speaker
Now again, it is kind of hard to imagine, you know, a scenario when you're playing with a homicidal suicidal person, but just for the sake of this game and for the occasions when that does happen, let's just consider it.
00:27:16
Speaker
Gabriel, are you telling me we have to sometimes consider weird things in math? Weird nonsensical problems? Yes, exactly, exactly.
00:27:26
Speaker
Okay. And then, uh, look, let's just say that, uh, you go first and let's say we're still on the occasion where you are not homicidal or suicidal. And Skyler is, in other words, Skyler wants both Skyler and you to die. If you go first, then again, no matter what you're going to lose. Cause if you pour your poison in, then Skyler can just pour water in and you're both forced to drink it and you both die.
00:27:53
Speaker
Yeah, and you might be thinking, what is like this in real life? And there's actually an example where homicidal-suicidal changes the game rules in almost the exact same way. And that's traffic lights. Because traffic lights, it's kind of the inverse of this problem. The Nash equilibria are at opposites because either you stop at a stoplight and the other person goes, or the other person stops at a stoplight and you go.
00:28:19
Speaker
So you guys have opposite things where versus here, it's either both poor in the poison or neither poor in the poison that's in front of them. But let's say somebody is homicidal and suicidal when it comes to traffic lights, then they can run reds basically. Yes, yeah. And then it would be a perfectly analogous example to this.
00:28:44
Speaker
Well, almost analogous, because remember, the case in which both of them just stop at the stoplight is just a weird case to consider. It is a weird case. But again, it's a time where you have to imagine a weird thought experiment that doesn't quite make sense, but you need it to be mathematically consistent. Yeah. And what's interesting about that is that in game theory, you have these matrices where
00:29:06
Speaker
It's like a matrix where each grid cell is divided into two by a diagonal and player 1 is on the top and player 2 is on the bottom or vice versa, whatever. And the rows are labeled by options for player 1 and columns are labeled by options for player 2.
00:29:26
Speaker
Now in each triangle, each one triangle that's half of a square, you put the rank of the decision. So this is just any number. It could be A, B, C, D. It could be 1, 2, 3, 4. It just has to be ordinal. And that is the raw data of game theory. That's the data structure, if you will.
00:29:47
Speaker
Yeah, so in this example, again, I guess we didn't really talk about who's going to go first, but we've already laid out what happens for whoever goes first. Oh, yeah, for sure. I guess we'll just have to wait and see what happens.
00:30:11
Speaker
You have both now learned that something that seems fair can be very unfair if you are not aware of the other person's motivations and your increasing sense of paranoia about Skylar. You must work together now.
00:30:27
Speaker
Since plenty is the root of all evil, you have been tasked with distributing it fairly amongst yourselves. This task is split into two parts. The first part is this. In a pen in front of you are a bunch of angry black goats. I happen to know that Skyler can share 10 angry goats in an hour by themselves and that's
00:30:52
Speaker
can only share five. The first part of your task, who operate together to share goats.
00:31:06
Speaker
All right. So real quick, this is where we're going to talk about cooperation. Yeah. So, uh, this is another aspect of game theory. There, there's the competitive aspect that we talked about earlier, and then there's the, the, um, cooperative aspect where the goal is a little different. Now a cooperative game is where there's some communal pot, basically. Um, and, uh, this communal pot. So for example, it might be, uh, the in a corporation, it might be their bottom line or something like that.
00:31:52
Speaker
So basically, it's where you can have some number that represents the total product of the coalition. And a coalition is just a bunch of people doing something together. So back to the scenario.
00:31:56
Speaker
Yeah, or gosh, what's another example?
00:32:09
Speaker
The first part of your task is complete.
Cooperative vs Non-Cooperative Games
00:32:13
Speaker
Notice that, because you work together, dodging and shearing goats, you have shorn 20 goats. Whereas, had you two worked alone, you only would have shorn 15 total. 5 for... and 10 for Skyler. This is exactly as I expected.
00:32:34
Speaker
Now, part two. One gold coin is being distributed for every goat you have shorn. You must distribute the gold barely between yourselves. If either of you cheat the other one, then the remaining ghosts will blow up. Good luck. Oh, wow. This is interesting. So this is about compensation for the job done then. So there's still an award. It's just about who gets awarded what, right?
00:33:03
Speaker
Yeah. It's so it's like, I mean, it's like if you and your friends are going, Barry picking who gets how many of the barriers, like if, like if somebody is lazy, how many did they get? Okay. So I kind of think of this as sort of analogous to, I wouldn't say, you know, at the end of a football game, we've got the most valuable player. They are recognized, but this is about like, if the team gets a fixed amount of money, you know, which, how do you distribute? How do you figure out which player gets the money that this is exactly that? In fact,
00:33:33
Speaker
Yeah, it's very close, yeah. And so let's define fair, because we're just saying words, we should define things mathematically. So what is a fair share of the collective payoff? And to talk about the collective payoff, we're going to talk about the characteristic function of a coalition game. So the number that represents the total success of the group, that's the characteristic function. So the characteristic function for a coalition
00:33:58
Speaker
So in this case, the characteristic function for you and Skylar was 20 goats. The coalition function for you alone would be five goats, and the coalition function for Skylar alone would be 10 goats. I mean, the coalition, the characteristic function. Yeah. And of course, it doesn't completely like, the fact that you work together, you produced more than just 15. Otherwise, it'd be a much simpler problem.
00:34:24
Speaker
Yeah. And we still have to define the, which fair means in this case. And the way that Shapley defined it is that it's a share that is the average of all the ways, um, the coalitions can be formed with that player would be do better than coalitions without them. Who is Shapley again? Uh, Shapley was a game theorist. I don't know much about him. What was Shapley's first name or was that his first name?
00:34:46
Speaker
I'm not sure. I think it's his last name though. Okay. I'm just curious just because I want to look up him. Lloyd Shapley who introduced it in 1953. How about that? Okay. I just wanted to, you know, just for food for that. Oh yeah. So in this case, the fair share, we calculate that directly. So, um, you without Skyler, um, that is five goats worth of shearing, right? That's correct.
00:35:13
Speaker
And Skyler Without You, that's 10 goats worth of sharing. That's correct. And both together is 20 and neither is zero. Oh, that's another thing about the characteristic function. The characteristic function of no players has to be zero. There can't be an intrinsic payoff. You have to account for that.
00:35:28
Speaker
Okay. Otherwise, how would it throw things off if you had an intrinsic payoff? I mean, I'm sure you would just like, you know, normalize it. It wouldn't work with the definition very well because we're trying to measure the output of people because if we're measuring an output without people, we're getting background noise. So obviously with all the information we have right now with our task of distributing a coin evenly, this is where, just like with many other instances, it's mathematics to the rescue.
00:35:58
Speaker
Yeah, so remember, it's a share that's the average of all the ways that groups formed could do better with you. So listening of all the groups that could be without you, it's either Skylar or nobody, right? So Skylar by themselves would do 15 goats, right? But you two together would do 20, right? Yes.
00:36:17
Speaker
So that is 10 goats better than they would do alone, right, in that case? Yes. So 10 is the first number for the average. With the nobody, your entire contribution is five goats worth, right? Correct. I'm sorry, with nobody? With just you, with just our listener.
00:36:36
Speaker
Yeah, versus nobody. Yes. It's five goats worth. Yeah, that's correct. So the average of 10 goats and five goats is seven and a half goats. Yep. And that is how many goats. So basically the seven and a half gold coins basically is your share. And the rest of it is the other person share. How many gold coins go to Skyler? Skyler.
00:37:06
Speaker
Since there's 20 gold coins total, they get 12 and a half. Okay. And then what is the equation that you used to come up with that?
00:37:15
Speaker
So you take every group that you're not a part of and you calculate how much better the group would do with you in them and take the average of all those values. And that is your fair share of the collective payoff. You know, I think we just gave the listener the answer. So, well, I don't think broken Lego said we couldn't do that, right? Well, I mean, broken Lego is not trying to teach a concept.
00:37:41
Speaker
Okay, so we are then so okay my cat get out of the bathroom Maybe it wasn't shut tight or the cat jumped in and like turn turned the knob I hope not. I don't that meowing you here is math cat. Oh, yeah math cat is named Moose which means moose is short for moose of age Mouse eater So yeah
00:38:08
Speaker
There you go, so you have the answer. Now it's up to you to implement it. Yeah, go do that. Congratulations. You have solved three puzzles having to do with the differences between cooperative and non-cooperative games. Next time, we will analyze the consequences of this in more detail as well as the concept of common knowledge.
00:38:38
Speaker
I'm Baka. And I'm Gabriel. And this has been Breaking Math. Okay, so where can people go again to find us on all the internet? So you can certainly send us an email. We certainly enjoy those at breakingmathpodcast.gmail.com. You can also send us a voicemail directly. Go to the anchor.fm app and go to Breaking Math podcast. Let me double check that that is it's Breaking Math podcast. I think it is. I double check.
00:39:03
Speaker
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