Game Theory: Nash, Dominant Strategy, & Prisoner's Dilemma

now let's talk about game theory what it actually is most of the questions you'll be given will have some sort of a payoff matrix as this is called table like this and you'll mostly be asked to find what's called the Nash equilibrium now there's two easy steps you can do to solve for it what we'll do is we'll sort of talk through why they even make sense but that way having them you'll sort of become procedurally fluent at it so before that though let's talk about what this thing even means the games we assume for microeconomics here are what are called simultaneous game so you play them together what that means is unlike a game of chess where one person goes and then the other person goes and then you go again unlike that this is kind of more like Rock Paper Scissors where both players are going simultaneously and then sort of the outcome is based on what they both do so here the two players of the game let's say here we have an example of a game the do players here are Eminem and Drake now these words over here the two options next to Drake these are called strategies so Drake can either release a single or release an album those are his two options it's kind of like rock-paper-scissors what he can do is he can either do release a single release an album those are his two options Eminem his two options his two strategy choices are you could make a video or you can do a concert you know so those are his two options notice that they don't have to be the same options but they certainly can be most examples you'll see will probably have the same options but they don't have to be by any means now let's say Drake releases a single and Eminem does a concert right so if that happens well then go to that row where it's a single in that column where it's concert and this is where you end up this three and four that that sell now the red number the one before the flash is always the row players payoff kind of like how many points they get if you will the profits so here Drake gets three Eminem gets four if they ain't up there notice though that one thing is these for this game any game here really it's not really gonna be a winner and a loser you're not really gonna have a winner and a loser because these games are more of you know you're just trying to maximize your own profits you're trying to maximize your own points you actually don't even care about the other guys if you have to choose between you getting eighty and the other guy getting zero versus you getting 81 with the other guy getting a hundred in the real world you might say oh yeah I'd rather do this because now I am eighty more than the other guy but according to our assumptions of game theory no no you'd rather if you had the choice you'd rather do this and get eighty one for yourself because you'd rather have that one extra you don't care about the fact that you're now 19 less than them instead of a team or you don't care about the other guy basically and that's why you know as you'll see that's incorporated into our rules here so now that we know understand how this works what these mean you know there's four possible outcomes of the game and those are these four cells if you will these four boxes so the Nash equilibrium is the one where we'll sort of end up as you'll see that's sort of what's expected to happen if both people are trying to maximize their own profits assuming that the other guy is also trying to maximize their own profit so here's how we can do this let's say we're Drake let's say we're on Drake's staff right now we're we have to help them make decisions here's what we'd be thinking we're at work and we're trying to choose between single or album which one should we do well here's the here's how we're gonna analyze it let's say suppose so we have some Intel saying that Eminem's doing the video in that case in that instance alone what would we want to do well if we know that he's definitely doing the video you can almost block this off it's almost like you've narrowed the world down to just this first column if Eminem's doing the video for sure we know that we are either gonna get 5 if we do the single or 10 if we do the album so in that case what's better well the albums better 5 beats 10 so here's what we know so far if we know so if we know that I mean I'm doing the video then we'll want to do the album so we wanted to the album but that's you know conditional on that video that he's doing the video now the only other cases what if Eminem was doing the concert well in that case it's kind of like you're blocking this off the world's narrowed down to I mean I'm doing the concert and in that case the single gets you three and the album gets you seven so in that case again the album is still better for us to do then compared to the single because seven is better than three so in that case we'd want to do yeah so notice if you're thinking about the overall steps that we did that's step one over here one column at a time that's why we looked at on this case and then in this case this is what we're role-playing as the row player it's weird because we're role-playing as a role player but we're kind of starting with if Eminem if did a video but we're thinking as Drake so either way one column at a time we want to circle the highest number before the slash because the numbers before this last represent that real players Drake right the red numbers so we're only looking at the red numbers for now we're ignoring all the black numbers here so just looking at the red numbers in the first column ten beats five in that column seven beats three so you're never comparing these two or these two you know so either way that's how you do it now notice in this case so if you've been listening carefully you might have noticed that Drake basically always wants to do the album in in either case regardless of what I'm in I'm doing in both cases one at a time the album was better than the single so in that case we're gonna say that the album doing the album is a dominant strategy for Drake so if any one of your two options strategies again if any one of your two strategies is always better that's a dominant strategy yes that's why there's an asterisk here so if if you're circling when you're doing that circling if it's the same single row essentially what you're doing here is one column at a time you're basically picking a row right five or ten your basically picking between the first row or the second row so if it's the same Road the bottom of the second row is what we picked every time that means that that's a dominant strategy that row that album is a dominant strategy for Drake we're halfway done the other half is basically the same thing but now roleplaying is the other guy so now here we have to be careful because now instead of one column at a time we're going one row at a time and now we're only looking at the black numbers we're ignoring all the red numbers looking at the numbers after the slash and we want the best one one row at a time and so essentially in each row we're picking which column is best because the columns what Eminem would be choosing between concert or video so if we're Eminem and we somehow know that Drake's doing the single what's better the video gets us to but the concert would get us four so just in that one row four it's better so we'd want to do the concert the only other case is what if Drake did the album well then it's nine or seven and that guy is actually the nine for the videos better so notice in this case Eminem in one instance he wants you to the concert and then the other he wanted to do the video so Eminem has no dominant strategy and so right off the bat there is no dominant strategy equilibrium a DSE a dominant strategy equilibrium is only there if both players have a dominant strategy then it's called the DSC and so if even if one of them doesn't have a dominant strategy there's no DSC but the dominant strategy equilibrium aside what's the Nash equilibrium so even if we don't have a dominant strategy equilibrium we could still have a Nash equilibrium and here we do the Nash equilibrium long story short is once you're done doing all the circling any cell out of the floor that has both number circles which this guy is the Nash equilibrium cell in this game if we were starting from the beginning now that you hopefully understand a little bit of the reasoning behind why these steps even make sense now you can sort of fluently apply them in each problem and without being thrown off conceptually if you're given this problem all you have to do is do the circling one one column at a time then one row at a time and then and you saw that has both number circled that's a Nash equilibrium and in the process you'll know if there's any dominant strategies or not and whether there's a DSC or not now let's look at another example suppose you and your friend are deciding whether to go to Starbucks or Big B and so you both like them equally and let's say these are the numbers usually these numbers will be given to you for any problem and let's find the Nash equilibrium well we could just apply the procedure at this point so one column at a time so in this first column so again we're all playing as you so if they're choosing Starbucks what do you want to do Starbucks cuts you ten Big B gets you zero so you'd want to do Starbucks in that case cuz you know you don't want to drink coffee alone on the other hand the only other case if they're doing Big B again you're choosing between 0 and 10 so in this case though you'd want to do Big B so you don't have a dominant strategy because you didn't want to do the same row every single time we sort of circled this row here this right here halfway done now let's analyze what your friend would be thinking your friend is thinking hey if you do Starbucks then now we're only looking at the red numbers by the way now we're choosing between the 0 and the 10 now we're picking a column and so the best column is the 10 over here right they want to do Starbucks in that case or if you are doing Big B again Starbucks gets them 0 big me get some tents and they'll want to do that now if we look back and try to answer the question what's the Nash equilibrium there's actually two different boxes with both numbers circled so as we just witnessed it's actually possible to have more than one Nash equilibria that's the plural of it so here there's actually two different Nash equilibria so we're not restricted to just having one in fact there might even be games with zero Nash equilibria by the way a game of this type is called a coordination game usually it's kind of like problems like should we all drive on the left side of the road or the right side of the road philosophically it doesn't matter as long as everyone's doing the same thing so here really doesn't matter in this case whether you go to Starbucks Starbucks or Big B as long as you're both going to the same place you both are happy so that's why there could be multiple Nash equilibria finally let's look at how game theory actually applies to economics usually the game that we look at in economics that's related is called the prisoner's dilemma it's really just a type of gaming the type being the numbers are kind of arranged in a way where something really funky will happen in a bit so let's say we're given this game basically it's a game between Coke and Pepsi and blue is Pepsi Reds coke let's say they're deciding between should I saw my you know product at $1 a can or $10 a can let's say those are the only two options that they have notice though that even though these over here are numbers 1 and the 10 we're not really going to do anything with them numerically if you look at where they're positioned really strategy option so treats us no differently than a word that represents the strategy option so that's just one thing those don't go in get into the circling mix so either way looks like what happened looks like what the Nash equilibrium is well if Coke is for sure gonna set their price at $1 a can what's Pepsi gonna want to do well the dollar I can get some ten and ten dollars a can they'll only get five million right instead of 10 million or something so they're better off selling better price that one dollar can getting ten million bucks on the other hand what if cokes you know really expensive well then Pepsi if they're also expensive look at Dell get 20 million but if they're only $1 can they'll get 25 so again we're just looking at this common circling the higher of the two numbers before the slash so 25 wins either way hey it's the same row that we circled every time so setting their prices at $1 to Ken to actually a dominant strategy it's always better for Pepsi to do that now we're halfway done now let's look at the other half if your coca-cola now it's one row at a time so if Pepsi's for sure setting their price at a dollar a can you're choosing between ten or five which is higher well 10 is bigger so you're gonna want to do that which really just means that you want to do $1 it can and if Pepsi is doing $10 you can done coke once again they're choosing between 20 and 25 25 is better so they're again want to want to do $1 again so in fact that column was always picked right that column so that is a dominant strategy now for a coke and by the way that's totally symmetric in this case so that's why we need that if it was a dominant strategy for them in this case it'll be a dominant strategy for them either way both players have a dominant strategy so we have a dominant strategy equilibrium where they both set it at $1 I can that's also a Nash equilibrium and in fact every dominant strategy equilibrium is also definitely a Nash equilibrium but not the other way around either way here the Nash equilibrium is that both parties will set their product at $1 a can and make 10 million in profits doesn't seem like an issue notice in all the past games that we've might have done or you know usually the Nash equilibrium is a good thing that's usually where both parties are kind of you know as as long as they're gonna be able to be and here if we were to compare this with this this cell with this cell we can't really say whether one's definitely better than the other because here yeah 25 is bigger than the 10 so Pepsi would prefer to be here but coke you know coke this isn't better for Coke right so you know it's not really a dilemma yet if we're sort of saying oh yeah why why did this end up being the Nash equilibrium instead of this it's like you know who cares it depends on who you are Coke or Pepsi but if you compare to this cell over here huh they both there's an option out there where they both could make 20 million dollars apiece instead of this 10 million apiece so they both would prefer if you were to switch and go to this equilibrium but the Nash equilibrium says even empirical evidence kind of shows that this is what will happen and this is called the prisoner's dilemma the name comes from the fact that originally you can you know it was kind of used by police's to get prisoners to confess you'd want to confess even though you could technically get more by not confessing but same thing in economics now so Pepsi and Coke technically here's the thing if they both colluded with each other that's the word colluding and setting $10 a can it's as if they're a you know an oligopoly acting like a monopoly in that case you know they'd be able to set you know $10 a can really high price they're probably selling a very low quantity in that case because the price is so high but then they can get more money but what ends up happening is unlike if they were you know colluding with each other each person is kind of thinking whom but if I lower my price I'll just get so many more customers so I'll make more I'll make 25 instead of 20 so each person wants to kind of cheat on the other person and that's why they end up here where they both end up getting $10 instead of $20 right so that's the prisoner's dilemma now so you know two companies that are in sort of prisoner's dilemma competition with each other they could kind of try to tacitly collude that just means sort of silently trying to collude with each other here's the thing though the government's government's kind of made it illegal to actually do that you can't have you know a contract saying I Pepsi will you know sell coke you know my product at $10 a can and I coke will do the same thing you know those kind trucks are illegal it's that trust between them is illegal and that set of laws that refers to that is called antitrust law so antitrust basically makes colluding or cartels as they're sometimes called it makes that illegal now the only que the only time where it's possible even for these these firms to kind of sustain collusion over a long period of time is if the game is repeated so the game needs to be repeated many many times and then you'll kind of think okay you know what you know if I'm sort of nice to them and they're nice to me then we could maybe get more rather than trying to short-term screw each other over but then screw yourself up in the process too so either way that's actually called tit for tat and that's just a strategy you can use if the game is repeated most introductory economics classes don't go into that so we won't worry about that but that's basically what it is now here's the thing though the fact that you know they end up here instead of here might seem like a sad thing to us right like oh they're and they're only gonna make ten instead of twenty but from the government's perspective it's actually a good thing and that's why in fact they've made this illegal and you know I'm sorry yeah exactly this is legal and so that's why if they sort of have a contract and they're trying to do this so not only does the Nash equilibrium say that they're gonna end up here it's actually illegal for them to even do this that's why they're further more likely because they don't want to break the loss to end up here and here's why if you look at the big picture if this is your supply and demand curve perfect competition again will produce here a monopoly you make the Mr produces here at this price and notice that monopoly has all this deadweight loss right so if you're the government and if you somehow notice that you're you know one industry you know the soda industry went from a monopoly to a duopoly now they're competing well notice they're gonna want to have some price competition right and that actually lowers the price and that's why oligopolies usually always end up somewhere in between where a monopoly would end up monopolies are here perfect competition is here and oligopolies are somewhere in the middle so that's why they have some deadweight loss but it's not as much deadweight loss as a monopoly that's why by making this illegal the government's actually trying to lower the deadweight loss because hey if they're here that's like a monopoly on it's a lot of deadweight loss so that's why the government has antitrust laws it's to lower deadweight loss and to make markets more competitive

Channel: DiagKNOWstics Learning

Views: 3,710

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Keywords: Game Theory

Id: wZXgrvrwnpw

Length: 17min 53sec (1073 seconds)

Published: Fri Feb 02 2018