The (strange) Mathematics of Game Theory | Are optimal decisions also the most logical?

this video is sponsored by Wix a platform that makes it possible for anyone to build their own unique website for free to start this video you and I are going to play a game well we'll try our best now playing this game is fairly simple we each have a card labeled green on one side and red on the other then at the same time you and I are going to hold up both of our cards and we can show whichever side we want then based on the color combination that shows up and how we decide to play our cards we are each going to get some amount of money now here's the breakdown of those payouts if we happen to be showing the same color regardless of what it is then you will be awarded five dollars and I will get nothing if I show green and you show red then I get five dollars and you get nothing but if instead I show red and you show green then I get three dollars and you get to the goals to end up with the most amount of money possible and we're gonna play several rounds now notice that overall this game is definitely in your favor the numbers are almost symmetric but there's that one instance where I should give five dollars to make things even but instead I get three well you still get two now my question to you is what would your strategy be to win this game over several rounds because my chance you should be able to but how would you essentially guarantee it well let's first look at how you should be seeing this game based on your payoffs it's definitely best for you to show green because then when I play green you win five and when I play red you'll still win two whereas I will get either zero or three respectively so overall you should come out ahead whereas when you play red it's 5050 odds for each of us to win five dollars so here's where strategy kicks in if you start just playing green every time I'm simply going to play red every time because that's the three to split payoff thus I would win three every round while you win - and I obviously come out ahead now if you respond to that by saying fine you know what I'll just randomly play green and red each about 50% of the time because overall I'm favored to win then as a response I will only show green every time thus half the time I win five and half the time you win five so a game that you should win instead comes out to 5050 odds for each of us now of course if I start playing the same thing every time you could counter that by switching up your bets and it becomes this back and forth mind game but let's not do that because you can win this guaranteed in the long run by following a specific strategy and I'll just tell you the strategy is to play green sixty two point five percent of the time and red thirty seven point five percent in the time and it should be random now there is a range of percentages that give you the edge but what you see here is the most optimal so now let's see why this works if I decide to show only green than the sixty two point five percent of the time you play green I will win nothing and the thirty seven point five percent of the time you play red I will win five we do the math and my expected outcome is one point eight seven $5 per round since five dollars is awarded every round that means you would win three point one two five dollars on average per round with the strategy so clearly playing green isn't in my interest and I better try my luck with red but let's just do the math again when I only play red than the sixty two point five percent of the time you play green I win three dollars from that mixed result then the other thirty seven point five percent of the time when you play red I win nothing we do the arithmetic again and get the exact same numbers as before so there's no color or strategy that's in my favor now whether I play green all the time red all the time or any mix the expected payouts are always bigger for you the real beauty of the strategy is you could tell me exactly what you're doing and there's nothing I can do to improve my odds since that criteria is met we have found what is called the equilibrium more specifically the Nash equilibrium okay technically we're only halfway there because remember everything I said about your strategy you could say about mine if I was playing each color half the time then you could just play only green and gain more of an edge than before I'm not gonna show the math but for me to play optimally and reduce my losses I would play green thirty seven point five percent of the time and red sixty two point five percent of the time the same numbers as you but for different colors if I'm doing that we'd see the same payoffs as before but now there's nothing you can do to win even more and thus we have found the real Nash equilibrium where even if we reveal our strategies to each other the other person has no reason to do something else decade to go it was discovered that in fact every finite game even chess has at least one Nash equilibrium this equilibrium is typically not easy to find and I don't think we know what it is for chess actually but either way the interesting thing is that equilibrium actually isn't always the best outcome even though it can still be the smartest move imagine a similar game but this time if we hold up different colors we get nothing if we both hold up green then you get two dollars and I get one if we both hold up bread I get two dollars and you get one we play this several times and keep whatever money we win now this game is kind of weird when we don't allow for cooperation for example if you find I'm kind of being selfish and just holding up red every time then you might as well hold up red every time too it's better for me because I'm winning - while you win one but hey it's better than you holding up Green and us winning nothing you have no reason to do anything else and same goes for me so we found a Nash equilibrium but you could say the same thing about us both playing green if your showing green every time then I might as well do the same which means that's also a Nash equilibrium but if we really played this game what would you actually do it might be tough to answer but there is an incentive to randomize your play especially when you're unsure of what the other player will do the equilibrium for you is to play the color more beneficial to you or green two-thirds of the time and then red one-third of the time because now there's nothing I can do to improve my expected outcome if I only play green I'll get $1 two-thirds of the time and if I only play red I will get $2 one-third of the time so no matter what I do I'm expected to make about sixty six point six cents per round and for me I should play red two-thirds of the time and green one-third of the time just the opposite of you this here is another Nash equilibrium the weird thing though is that it would actually be better for us to completely randomize your color choices as and make it 50/50 because that will give us an expected payout of 75 cents per round instead of sixty six point six cents the problem is if I knew you were picking colors with 5050 odds then I would just always play the color more beneficial to me aka red or I'd at least play it more often because I'll win two dollars more often than one so 50/50 odds is better for everyone but it's not stable as they can incentivize the other person to change strategies so hopefully you can see how this kind of math allows us to look at a game and its payouts a different kind of way to create an optimal outcome for ourselves or even both parties and although the math may say one thing we also have to consider the human element of this and what our opponent will sometimes do like here's an example where knowing how well other people understand the game is key to winning it let's say everyone watching this video has to pick a number between 0 and 100 inclusive we will then find the average of everyone's guesses and whoever picked the number closest to two-thirds of that average wins so like if you say 34 and these are everyone else's inputs then you would win because the average of all of these including yours would be 52 and two-thirds of that is just above 34 which you're closest to so the question is what would be the Nash equilibrium here well since everyone has to pick a number between 0 and 100 we know two-thirds of the average cannot exceed 66.6 so there's really no point to put anything above that but if everyone realizes that then no one's gonna put a number above 66.6 which means now two-thirds of the average cannot exceed 44.4 and again if everyone realizes that then no one's going to input a larger value and if we continue this analysis we eventually get down to zero which is in fact the Nash equilibrium everyone should just say zero and then no one has an incentive to change their answer but the real question is how deep are others going to see into this well when this was actually implemented through an internet-based competition people who pick 0 did not win instead for this example twenty one point six was the winning number which means the average input that was submitted was around 32 it's lower than 50 so there were players who seem to have the right thinking but most probably did not or at least didn't go far enough however after people play this game over and over they do keep lowering their guesses until they all eventually reach 0 so it takes some learning but equilibrium is eventually reached and now let's see a two-player game we're betraying the other person is actually key to making the most money sort of for this game same as before their payoffs for the color cards we show if we show different colors than the person who is showing green wins $100,000 while the other wins nothing so why would anyone show red because if you both do you both win $10,000 if you both show green you both win $1,000 you will only play this once and you cannot divide up the money afterwards so what would you guys do in this game and let's say you can cooperate with the other person like you guys can talk and figure out a strategy what you'll likely come up with is both of you should just show red that way you eat get $10,000 no one will go home empty-handed and hey 10,000 each is better than 1,000 each however here's where the betraying part comes in if you guys agree to show red then it is now really in your interest to show green and basically screw or the other person because then you'd win $100,000 and the other would get nothing and also remember your opponent would be thinking the exact same thing okay someone needs to get mr. beast to have strangers play this game with each other cuz I'm betting we'd get some pretty good responses and also possibly some fights but either way you guys let me know at least what you would do in this situation like if I could offer these payouts and I picked two of you at random and you'd never have to see each other again would you cooperate would you betray the other person but you expect them to betray you and so on anyway getting back to the analysis there's actually something hidden within the math sort of that's easy to see but also kind of weird at the same time and that's that plain green is in fact always in your best interest regardless of what the other person does like let's say they show green that means that you showing green wins you $1,000 and you showing red went to $0 all the other person gets 100,000 so green is definitely the better choice here if they show red and you show green you win $100,000 but if you show red you win $10,000 you are always worse off by showing red and the same goes for the other person you guys would both much rather show green and that in fact is the Nash equilibrium the weird part is that both of you showing red is still better for everyone compared to both showing green the Nash equilibrium isn't the best payout but by playing red there's a real risk of betrayal and us you don't have a stable system if you're thinking wow this game seems like quite the dilemma then you're right this is actually the prisoner's dilemma probably the most famous game in game theory except instead of years you'll spend in jail and confessing or not to a crime I turned it into money and just showing a red and green card the analysis is just about the same without cooperation I would always show green but when you can talk to your person it gets interesting and it turns out they actually did a simulation of this years ago but it was an iterated version where several teams wrote programs that would essentially play this game over and over not just once because it wasn't a one-time play solely playing the Nash equilibrium isn't what won but rather it was programs that played the tit-for-tat strategy a program playing this would start by cooperating just like playing red and hoping to get the $10,000 payout for both people they would then mimic the previous move of the other player so if the other player first cooperated then it would cooperate again if at any point the other player betrayed and essentially showed green trying to get that $100,000 outcome then for the next move the tit-for-tat strategy would say to betray and if the other person kept betraying and so would the other program and since cooperating is more beneficial than both playing the equilibrium the strategy proved to be very strong not only has it been seen in computer code but it was also seen in World War one where any death by an enemy sniper would be retaliated but periods of no death would be met with no retaliation and thus there was an implied truce which helped create peace between trenches see this is why game 3 has become so big it can help us analyze and understand strategies and optimal outcomes in real-life scenarios game 3 can be used to analyze international conflict and find best strategic plays for both sides it's used a lot in economics to analyze auctions business mergers or the fair division of assets and had seen in pursuit of Asian games where each player wants to choose best strategies to either catch the other or avoid being caught so although game 3 mean I help you do any better at traditional games it can really open your eyes to the mathematics of decision making and how to create optimal outcomes for yourself and/or others and before we end this I just want to thank Wix for sponsoring this video the thing I've talked about in another video was held during my first job I actually tried building a business on the side and how during that time I taught myself how to write an HTML CSS PHP and JavaScript so I could build a website and let me tell you that took a lot of time that could have easily been saved with Wix anyone can build their own website for free and no programming experience is required you can start a blog online store a personalized portfolio for business purposes and really anything else you can think of in fact without spending any money let me show you how easy it is to get your website up let's say I want to make a blog to go along with this channel that's of course educational and meant for students I can pick the type of blog I want and after putting in some additional information you get to pick from tons of layouts that offer just the right feel for your site and once you're set up everything is very customizable so it's very easy to edit titles and pictures that give your site the personal touch that you want they actually put this as the default but honestly I think I'll keep it then if I want to maybe make some money on the side by offering some online tutoring it's literally two clicks away I can then design a pricing plan that fits my needs and make the page look exactly how I want a lot of what you'll need is even built into their default settings then buying a domain linking any payment methods creating customized email addresses and everything else you could need is all available on Wix if you're trying to go really professional you can even upgrade to a premium plan which is used by professional developers to save time so they can focus on more important business matters to get started right now you can click the link in the description and join over 100 million people who have used Wix to create their own amazing website and with that I'm gonna end that video there if you guys enjoyed be sure to LIKE and subscribe don't forget to follow me on Twitter and join the Midd Facebook group updates on everything hit that bell if you're not being notified I'll see you all in the next video

Channel: Zach Star

Views: 211,442

Rating: 4.9362497 out of 5

Keywords: majorprep, major prep, game theory, mathematics of game theory, decision making, rational strategy, strategy, mathematics of strategy, how to strategize, prisoners dilemma, nash equilibrium, what is game theory, applications of game theory, applied math

Id: dHi8BVZFHdA

Length: 15min 8sec (908 seconds)

Published: Thu May 09 2019