Game Theory: Nash Equilibria

we're gonna return to our discussion of game theory last time we distinguished various kinds of games we talked about sequential games as opposed to simultaneous games a sequential game being one where one player plays and then the other player responds and so on a simultaneous game where both players have to play at once really without knowing what the other person is doing and we talked about zero-sum or constant some games where there's a fixed amount and then it's a question of how that ends up getting divided up as a result of the gameplay and other cases where there isn't a fixed amount where everybody can get more everybody can get less sometimes you can take the same game as we saw and view it both ways so let's say we're talking about a game in which somebody wins somebody loses if you're just thinking about the win-loss column it looks like there's a fixed sum right there's one victory and it's question of which team gets it if you're thinking about the score on the other hand it might not be like that there can be a high-scoring game or a low-scoring game so let's say we're thinking about hockey or what what's a sport where I mean baseball maybe ties are impossible let's say well in that kind of case there is one team that's gonna win one team that's gonna lose from that point of view it's a zero-sum contest on the other hand it might be a high-scoring game I remember well going to a ball game the night I got my PhD I passed my oral exam on my dissertation and then went to a baseball game and it was great went out of the stadium they were there were firemen they're passing out free tickets so I figure I won't ask any questions I'll take a free ticket well it was very far up in the stadium but it was also then a wild game it was the Pittsburgh Pirates versus the st. Louis Cardinals and I think the final score was thing like 14 to 8 it was just crazy that's very high for a baseball score on the other hand I've been to games that were 1 to nothing so anyway you know it that if you're looking a run scored it can be it's not at all a fixed sum it can increase it can decrease yeah estrin of the kinds of games it matters a lot what kind of game were playing we have to know for example whether it's a simultaneous game or a sequential game if it's simultaneous I have to guess what your strategy might be I don't know what you're gonna be playing whereas on the other hand if it's sequential it's really important that I know what move you just made and I'm going to have my structure depend on that so I need to know and the same thing with a zero-sum game if if somebody else moves ahead is that going to come potentially at my expense or can we both win out of that is there an opportunity for mutual cooperation so to form a strategy for playing this game I need to know what game I'm playing now in a way that's really obvious it's like hmm how are we gonna play is it baseball football hockey chess matters right but it also matters in an organizational context I think uh okay I'm a new employment employee I notice that a lot of the people who've been hired recently got fired after a short time I don't want to be fired a short time so I want to think okay how do I avoid this how do I play this game what kind of game on my playing and it won't be that obvious and I have to know not only what kind of game this is but also yeah in detail how much information am I gonna have about what other people are doing but also crucially how much do my interests here overlap with the other players are we in competition let's say it might be a situation where we're all junior in this law firm and only one of us is gonna make partner well then gosh that's like a zero-sum game or a fixed sum game where yes there is one partnership available and we're all competing for that one slot or is it the case that maybe it's not like that and we can actually cooperate and help one another so it going it's going to matter I have to know how much my interests overlap with the other people and so I need to think yeah to what extent can I cooperate with other people should I cooperate with them to what extent should i compromise with them it's like yeah we're opposed and we can't really cooperate to get our mutual goal because we don't have any mutual goals but maybe we can compromise maybe we're just in competition okay if you are let's say you're playing in the Super Bowl you don't really cooperate with the other team you know you don't say hey why don't you get a certain number first answer let's get sick number you don't compromise and say well we're not gonna really try very hard until the third quarter as long as you don't try you're in competition you have to know that also sometimes you're just an outright conflict with the other person and any gain of yours is gonna come up their expense and vice versa so you've got to identify the situation now one way to think about this is to think about the other individual players what are their preferences like suppose we're in a relatively simple game where there are only four possible outcomes and I rank those outcomes in terms of my preferences one two three and four okay my first choice my second choice my third choice my fourth choice now these are just ordinal numbers as we've mentioned in games we can think about Cardinal utilities and often that's appropriate but sometimes we can think about the form just thinking well what are my preferences think of it in ordinal number terms well then here are the 24 possibilities for another player maybe we agree exactly they share my preference ordering precisely I say this one's first this one's next best this one's life and this is the worst basic yeah totally with you on all that it might be like that it could be the complete opposite they say no I think the fourth is the best third second then what you call the first I think that's the worst option but then it can be any of those things in between right and so in a certain sense before you even get into the details of thinking about this you've got to think what is my relation to this other player how do my preferences line up with theirs so let's think about an example you've got interests that I'll put you in relation to for example my interests in teaching this course to what extent to those interests overlap are they the same now in a certain sense they're the same right I mean if we look at it from one point of view we could say well I want you all to succeed so we both agree that the best option is you get an A Okay second best option you get a B etc but now you might think but still we can all achieve that if we just say get on A's well get what we want let's go home so it's more complicated than that now that brings up a couple of things one is we both care about other things than just the outcome in terms of a grade I care how much you learn maybe you care how much cooler I hope you do you might care about other things like how much work is involved and so on so that's complicated first of all but secondly we can think well then yeah to what extent to our preferences about all of this really agree suppose we think workload and the amount of work you do for this course what would your first choice be a little Wow you're no wonder that explains a lot but no but though okay suppose you say look I'm a busy person but less work for me the better is that my preference well no now is my top choice that you just worked like dogs on my course alone well not really actually I'd like you to work hard at it but on the other hand I'm not really thinking that best choice is a bunch of people who do nothing but study what I'm trying to teach and so we're gonna partially overlap here but not entirely overlap and that seems to be really vital to recognize to take a very simple example if I have that preference ordering one two three four and then Alice has the same ordering look we can cooperate perfectly right we have the same goals we have the same preferences we have the same likes and dislikes so Alice and I are naturally on the same team no problem Bob ah we're in perfect conflict he ranks them in exactly the opposite order now maybe we can compromise we can say well let's settle on two or let's settle on three but any gain for me in terms of getting something I'd like better comes at his expense right so we're directly in conflict but what about Carol she ranks these options one for two three now notice we agree about some things we agreed that one is the best option and so that's an important thing to recognize if I'm thinking okay here's another player in this game we don't agree about everything but we do agree about that also we agree that one is better than two and two is better than three the only difference here is she thinks four is a lot better than I think it is so that tells us okay we've got a lot of ground for agreement as long as four isn't the issue we're in good shape we can cooperate a lot if on the other hand four is the question ah then we may be in conflict and then Dave say his are three for two and one well we agree about a little bit right we agree the threes but of poor but otherwise we disagree on pretty much everything and so if I'm thinking who can I work together with and Who am I in cosmic twist but the answer is gonna be Alice no problem we can work together and we can cooperate to achieve the same goals Bob at best I can compromise with Bob we're in a situation of conflict Carol we can mostly agree there's just this one issue we have problems about Dave we mostly disagree there's one thing we really do agree about and so the point is think about this sort of thing in an organization and think about what you're trying to get what you prefer think about what other people want now in our private lives when does this even come up I mean it's natural to think about this in terms of politics Here I am in the legislature I have my certain views I'm thinking who can I work together with on legislation who am i arguing against who can i form a coalition with on this issue or that issue this is vital the same thing can be the case in an organization you're trying to accomplish something who are your allies in that organization who are your enemies and for the whole organization who are their allies who are your enemies and so on but like where does anything like this come up in your own ordinary day-to-day life yeah disagree on thing okay good yeah we've talked about friendship right and these quality matching relationships well what are my chances being good friends with Bob they don't seem very good we would agree about everything hey Bob says maybe we should get to know each other what I have lunch yeah and I says well what kind of food do you like Oh Chinese hate Chinese what of it no this is a bad example for me cuz I pretty much like the food so it's hard for me to think of something I wouldn't like but basically yeah Bob and I aren't gonna see eye to eye about anything Alice on the other hand if anything the danger might be we just sit around nodding at each other right you know I said wow this is great yeah that's great oh and and so on and everything is like but still that's pretty good for a friendship office um yeah we agree about things Carol yeah vlog is for doesn't come up yeah and so and so you're right in forming friendships this is pretty because a friendship what is it from this point of view at least more than this but it's at least a kind of Alliance a kind of coalition you're forming with somebody else hey you and I agree about enough things and like enough of the same things we can enjoy being together and if you disagree about too many things it's gonna be hard other places going out to eat with family okay good good yes right because even if you think okay it's not like I don't like that just think you do have preferences and so yes some families have different ways of dealing with this than others in mind we tend to say what do you want to do well I actually I've talked about these problems before I had to cut a little segment where I mentioned my wife strategies so so yeah I bet it can be a problem right if everybody agrees then it's great you know exactly what you want to do if they disagree oh maybe you have to reach some compromises when my kids were little they disagreed all the time and so there was always this need for a compromise in some families somebody just says I lay down the law we're just doing this so you've got to have some way of resorting to well if you don't agree resolving the conflicts reaching compromises taking turns whatever it is other examples yeah good allocation of resources exactly you're on some committee or you're part of an organization and let's say it's question of budget or it's a question of manpower and what you're gonna devote your attention to let's say you go in thinking the most important thing for our committee to talk about is this and that it's that and so on and somebody else may have a very different idea so I've been on this committee recently we were asked so what's what are your top priorities what do you think we ought to be doing I said well I think the most important thing is this and immediately someone else those famous person the committee said no I think that's totally wrong it should be this and so several others started yeah you know saying something similar and others said but wait if we you can't do that until you've done this and and in short it was an attempt to say they're not so much money at stake just where do we devote our attention what issue do we focus on and that's the kind of thing that arises in all sorts of contexts in organizations in clubs but just within a family right somebody says hey you know we're going on vacation what do you want to do and it's a similar kind of issue people have very different ideas about the ideal vacation for some people it's lying on the beach for some people it's climbing thousands of vertical feet in the Rocky Mountains but for some people with some other things so anyway yeah it's it's a very different conception and you have to sort all that out okay well let's think about some other questions here we've talked about the difference between these kinds of games and in order to understand the structure of a game in general I'm gonna use this kind of diagram and now you can see why I've moved two slides instead of drawing the blackboard because it's hard to keep the colors of things straight with chalk even if I bring in my own colored chalk but here I can put things in colors and I hope you can see that the idea is there's a very simple game there are two players in this game a and B which will does mate over there a is the sort of row player and a has two options in this example just well going right or going left but it doesn't matter option one option two and then here B is the other player B's represented in terms of the columns that again has two options so this is a simple 2x2 game some games are like this most interesting games are much more complex but then we look inside and say well what's the outcome suppose a goes to the right and B goes to the right well then we get something like here that indicates an outcome the first one here I've designated as blue because that's the outcome for a the second one of the pair is the outcome under that scenario for B and we can think that through for each of the options ah you play right he plays left huh you get this and here I've represented these as ordinals first second and so on but they could be cardinal numbers too maybe here you get five points here you get zero points and so on okay so we're gonna see that kind of structure a lot and that's the meaning of that sort of diagram now let's think about a super simple game this is called two finger Mora and you might have played this as a kid or you know you're trying to decide how to resolve a conflict between the two of you and so you say okay odds or evens and you say you know on three one two three and then you see whether the total of fingers you put out our otter even and so in this case let's assume instead of any number of fingers it's really just one or two hi I do remember playing this hello elementary school but we had a more complex version where you could do anything so sometimes it'll be none and sometimes it was five and we got to practice our arithmetic doing this but it wasn't in the efficient form of the game because all that matters is odds or evens anyway we might represent it this way okay player a called evens so what happens well you have a choice here of putting out one finger or two fingers in playing the game and a wins if it's an even number total so suppose a puts out one finger and B puts out one finger one plus one is two it's an even number and so a wins B losses okay and we'll represent that with a 1 here and a minus 1 there but what if one puts out 1 and the other player puts out 2 then a loses and B wins and the same thing if B puts out one finger and a 2 well again a loses B wins and then if they both put out two fingers it's even again so a wins B losses and so that's a super simple game but you can see already that we can represent the outcome in this sort of structure notice it's a zero-sum game take any of these cells and add it up it's just 0 win loss minus 1 plus 1 so that's a good example of a zero-sum game well now let's look at this from a point of view a can't control what B is gonna do but a has preferences about the outcome so a is trying to figure out a strategy and a says well okay if that player plays 1 hmm let's see I would rather actually I've why have i drawn the arrows in this direction i don't know well that was stupid did I keep making that mistake yeah okay this is a disadvantage of to doing slides like this it makes sense to you when you do it and then you look at it later and think crap now I can't change it if it were on the blackboard I could just say oh yeah I met it'd be this way but anyway I'll go with what's here for the moment basically think of the arrow here as representing better than okay so from a point of view this is a lot better outcome than that right I'd rather win than lose so this one's better than that one on the other hand if the other player plays two fingers well I like winning better than losing so this option is better for me so notice that if B plays one then this is the better option for me if B plays too this is the better option for me now in other words what I should do depends on what the other player is doing but of course this is a simultaneous game I don't know what the other player is going to do it would be very different if I could say okay we're gonna play two finger moral I call even you go first then it's like no problem you do this I do that and I win but in this case I don't know so later we're gonna learn a term for this a has no dominant strategy a dominant strategy would mean it's the best thing to do no matter what the other player is doing at least it makes you no worse off no matter what the other player is doing and here there's nothing like that totally depends on what the other player does well if we turn around and look at this from B's point of view we see something similar because then B can say well wait a minute I you know look I'm now representing odds so if you call one I want to call two and if you call - I want to call one and so it looks as if in this situation I also have no dominant strategy I want to adapt what I'm doing - what you're doing but of course I don't know that so that's why it becomes something that feels like an even game if we put those arrows together we see they don't settle anywhere they just go around in a circle suppose I knew what you had done and I said okay you keep what you're doing constant I might well change somebody would have an incentive to change and could change it so this game if you look at that cycle this game has no equilibrium well this idea of an equilibrium in a game was developed by John Forbes Nash jr. who was one of the key founding figures of game theory there's a movie A Beautiful Mind about his life and he accomplished a variety of things in mathematics and economics this was only one of them but it was a major contribution and we still call a certain kind of equilibrium in game theory a Nash equilibrium the idea is at an equilibrium point no player has anything to gain by changing strategy given what the other players are doing so every player is making the best decision they could make given the decision of the other well you can see that in this game there is no pure strategy Nash equilibrium that is to say it's not like there's any play you can make to say AHA that's always a winner for me if you go into this game of two finger more off thinking kind of a Homer Simpson but I always play one it's always the best thing to do that's not true it depends on what the other person is doing and so there is no strategy that is like this there's no point for that matter where neither player could do better by changing what they did well a pure strategy is something I just mentioned it provides a complete definition of how to play the game it tells you what to do deterministically in each of those scenarios and so it determines the move you make in any situation you could say the set of those available to you is your strategy set and that would be one way of looking at this game say well alright what's my pure strategy play one play two that's about it neither one is gonna lead to victory no matter what however there are also things called mixed strategy equilibria where I think ah now in fact if you're playing this game what should you do suppose you're playing it repeatedly against the same person what's your best bet good trying to catch a pattern in what the other person's doing right because if you can figure that out and by the way there are I will get in a moment to rock-paper-scissors there are actually rock-paper-scissors or competitions there were like big contests thousands of people enter and go for prize money and you think well but I don't get it I mean how do you win at rock-paper-scissors the answer is you try to analyze the other person I look at you and I think are your rock person you assume right and I try to guess because actually the rational thing to do in a game like this or in rock-paper-scissors is to not let yourself be predictable so it is just to be completely random in what you do and to do each of them roughly a third of the time but are no identifiable pattern but just as a random number generator is a very hard thing to actually construct they we approximate those but they're not really random the same thing is true in human behavior it's actually really hard to be random kind of amazing when you think about because a lot of people seem kind of irrational but but it's actually very hard to be random if you're trying to be random and so you try to guess the other player but what are you trying to do in addition to guess what they're gonna do when you try to be random yourself so that if you have a person can't guess what you're going to do and consequently yeah it's a best mixed strategy here will be just be played them with 50/50 probability in any case in this case if you follow the arrows it becomes clear this game has no pure strategy Nash equilibrium it will have an equilibrium where both players just play 50% this 50% that and in general then are just mixing it up in a completely random fashion well Nash does have this existence theorem this is one of his fundamental results which says every game with a finite number of players in which each player can choose from finitely many pure strategies so it's important it be a finite game have at least one Nash equilibrium okay every game that's finite in both senses has a Nash equilibrium but as we've just seen it's not always a pure strategy Nash equilibrium sometimes it means you're mixing things up probabilistically you're doing things at random so sometimes it's rational to be irrational or at least to behave randomly more formally we could say a mixed strategy it's just a probability assignment over the pure strategies play this 50% of the time this 50% of the time and that would allow you to select one of these at random and so we've always got one of these but it might be mixed as it is in this case and lots of times there are these mixed strategies especially in competition you don't know what the up is you don't want the opposition know what you're doing so go back to that football example we were talking about last time you're the offensive coordinator it's first and ten just got the ball on your own 20-yard line what do you do he wants to throw it screen pass yes that's that's the standard Texas move circa night 2010 sorry halfback dot okay do you want to have any of these as your pure strategy don't say first and ten yeah we always do this can't lose do with that no you want to mix it up right you know what the defense to know what you're gonna do and so all of those are plausible things that you might do on first down you want some probabilistic mix over that and you want it to be random now notice in a football game in certain places it does tend to be random or at least you can approximate that at certain other points it's very hard to be random in this way you know you're in the last 45 seconds of the first half you've got the ball you're trying to go 80 yards to the touchdown do you do the halfback dive almost certainly not right probability of that is way now maybe it doesn't go to zero because you want that option in case you want to surprise the other team they go into a prevent defense and you think nobody's up here near the line maybe we can do that but in general you're going to say yeah certain things are going to become much more probable along paths the paths toward the sidelines etc other things are gonna be much less properly so in short your strategy in that kind of game will be a mixed strategy all the time really but the mix is going to change depending on the game situation now sometimes they are very useful as in football or two-finger Morra and basically that's when you don't want the other player to know what you're gonna do now outside of sports what are examples like that where you don't want the other player to know what you're gonna do it's to your advantage that they not know what you're up to yeah good negotiations right so so you get a job offer you graduate with a degree in philosophy or human dimensions of organizations from UT Austin you're a hot commodity in the job market right all these people are clamoring to hire you and you get this offer and now they say I mean they might just tell you here's the salary we're offering but however highly recommend that you take it however suppose it's more nuanced than that there's a negotiation they say well tell me what kind of salary would you accept or what kind of salary are you making now now if you're a new graduate that may not be applying but if this is like your second job they might ask that question and then you think oh that's like a sequential game where they're asking me to move first now sometimes it's a huge advantage to move first sometimes it's a huge disadvantage and in this context we'll talk a lot more about those later but here you might think that's a real disadvantage I mean what if they're thinking yeah we're gonna offer this person $60,000 a year and I say well yeah III was hoping for 40 or more right so that's a situation where it's up on the other hand they think maybe 60 and you say well I was hoping you know for that beach house in Venice California - get away - so I won't accept anything now less than a seven-figure salary okay goodbye actually I do know of a case in academia where someone was offered a job and came back saying well first of all that's not high enough salary secondly I want more blah blah blah I want this I want a guarantee time off Bubba and they wrote back and just said oh so you got to be very careful about these negotiation moves and you've got to guess what the other player is willing to offer what they will view is unreasonable what might be lower than they're actually expecting etc so it's tricky you don't want the other player typically to know what you'll really do you don't know want them to know the minimum salary you'd accept and they don't want you to know how much money they've actually allocated in the budget for this job you engage through this dance where you try to figure those things out are there other cases where you don't want the other player to entirely know what you're up to yeah good exactly so any time you're in competition any time you're in a situation of conflict you tend not to want the other party to know what you're doing might be true in wartime you don't want the enemy to know where you're going to attack and in fact Eisenhower for example before d-day went to a lot of trouble to make the Germans think that he was going to attack in a different place and that's a common thing in wartime has we learn when we get to sunsoo war is based on deception and it's all about that but you might say it's not it doesn't have to be a fancy wartime thing anytime there's a situation of conflict you don't want the other parties to know what strategy you're playing say you're just kids playing hide and seek if if you always hide in the bedroom closet you know the other player knows where to find you you don't want to play the same play all the time yeah you have an example in the war oh sorry I stole your thunder but yeah war is excellent any other situation of conflict of competition of a situation where you want to keep the other party viewed as your opponent guessing now notice it's not toast so great if you want people to cooperate like you know you got a friend you're going to meet for lunch you say hey let's get together for lunch yeah sounds good I'll be done with class sure okay guess where I'll be I'll go to some place near campus you try to figure out where I am that's I think that's weird right that's not gonna work out very well for your friendship so this is something that's typically going to be good in a situation of conflict or competition not very good in a situation of comp well of cooperation okay and indeed sometimes if we're trying to cooperate then doing this would be terrible we're gonna see an example later about driving should we drive on the left or drive on the right it's very important we agree and cooperate if it's kind of like I don't know I'll do it 50/50 probability you guess where I'm gonna be that's very very bad I know sometimes it does feel like that it's interesting that the more people in driving I mean I'm interested in situations in driving that aren't really specified by the rules like suppose you're in a setting where two different lanes have to merge into one lane how do you do that it's it's not obvious what strategy you want to follow and what I've noticed is that in different places there are very different patterns that emerge from drivers depending on how frequently you encountered this normally driving in New York City is not a fun thing I don't really recommend it but this is one place where New Yorkers are super good they have to do this everybody just immediately takes turns everybody knows that's the that's the most efficient way to do it in Pittsburgh if you have to do that it's going you yell at the other person you shout at them Pittsburgh is a very friendly place but that means that a situation of conflict or competition it's also really unfriendly place and so you basically maximize potential for conflict in Texas when I first moved here it's like oh you go no you go first it was kind of like a hey we're both so nice it's less so now Californians are moving and they're ruining it but in any case it's it's a situation where yeah you've got to evolve some way of doing this making it predictable for the other party let's talk about another game it's a little more complicated than this rock-paper-scissors okay now you could be like actually I think it's part simpson who challenged to play this game says I'll go with rock nothing beats rock but of course that's not how it works if we want to represent this we could represent it in this kind of way we could say well alright a has now three choices right rock paper scissors so does be and here's how it comes out it's just a tie to draw if people played the same thing but then it's goes well paper covers rock rock smashes scissors scissors cut paper and so we can represent who wins and who loses in a simple diagram like this again where one means you win minus one means you lose so I won't go through all the nine squares but trust me that's sort of how it looks and then if we say well what are aids preferences here you can say well gosh okay paper here let's say B plays Rock I'm not in control of that decision if I'm a I can control this okay they play rock well I'd rather win than have a draw and I'd rather win the lose so it looks like then papers might play but if they play paper I should play scissors and if they play scissors I should play Rock so again I don't have a dominant strategy totally depends on what they're gonna do right and similarly they think it through and they think it through the same way so once again there's no Nash equilibrium there's no place where those settle and we say aha all be arrows point us here everybody prefers this to certain other square no matter what square we're in somebody has some incentive to move to some other strategy well we can complicate this further as on the Big Bang Theory rock-paper-scissors- lizard-spock where the rules go like this it's very simple scissors cuts paper paper covers rock rock crushes lizard lizard poisons Spock Spock smashes scissors scissors decapitates lizard lizard eats paper paper disproves Spock Spock vaporizes rock and as it always has Rock crushes scissors okay and so we can put that game into the same matrix it doesn't really matter how many options there is easiest to just see it like this and if you look at the arrows here about what beats what you can see there's a connection between the fact that there's no place where all the arrows point right they keep leading you around in circles and the same thing is going to be true when we set it up as a game table there won't be any Nash equilibrium that is to say any pure strategy Nash equilibrium you'll have to mix strategies you'll have to keep the other person guessing now once you've got that complication you can say well now I can make it arbitrarily complicated here's rock-paper-scissors lizard-spock spider-man Batman wizard clock okay where we get a much more complicated thing but again notice no position here has all the arrows going into it or all the arrows going out and so yes I don't know if I want to go through all this but you see Spock stops wizard wizard stuns Batman Batman scares spider-man spider-man disarms clock block breaks rock-rock interrupts wizard wizard burns paper paper disproves Spock Spock the phuddle spider-man spider pen defeats lizard lizard confuses Batman because he looks like Killer Croc Batman dismantled scissors scissors cut wizard etc etc so anyway you can once you get the hang of this you realize oh yeah I can just keep adding nodes as I like in any case we've been talking about games that are competitive we're mixed strategies end up being the best play however there are lots of games that aren't like that in a coordination game you're rewarded for cooperating with the other player okay and you suffer when you don't cooperate that is to say certain matches of strategies are going to lead to a mutual benefit other strategy combinations are gonna lead to mutual harm and so you're rewarded when your strategies match the strategy appropriately of the other player now that doesn't mean exactly the same the nature of the match is going to depend on the particular game nevertheless the idea is you gain by coordinating with the other players so what are some examples of this kind of game yeah Marit s you gain by cooperating with the other player you don't what you know will I be home for dinner tonight I'll keep her guessing not a wise strategy or you know ah date night yes I'll be somewhere fun it's up to you to figure out where I am that's not a very likely successful strategy either you want to cooperate other examples yeah advertising or oil pricing ok advertising explain oh all right good good good yeah if we think about market chairs then it looks like marketing is something like a zero-sum game but actually that's not the best way to look at it right I don't want to market in a way that's going to reduce the market for sneakers even if that means they're gonna be here at worse than I am hurt by that it's like well we'll lose money but they'll lose a lot more ordinarily that's not what I want I would rather advertise in a way that isn't that yes makes people buy my product but if it lends greater credibility to a whole genre and leads to sales increases for all of us that's a good thing right so usually in a marketing campaign I'm not interested in a sort of zero-sum approach so much as I am something that helps everybody in this category and me especially I want to be helped more but on the other hand I would rather have a larger share of a larger pie rather than a larger share of a smaller pie and so I've got some incentive to cooperate and the same thing can be true in oil pricing as you mentioned our company is gonna price that it's gasoline at a certain level well we would rather make it more than less but on the other hand if it's too much more nobody will buy our desk they'll go to our competitor so there's an important sense in which we have to coordinate appropriately with the other people all right here's another simple example now this is I think an apocryphal story it did not really happen but the story goes like this in 1895 there were only two cars in Ohio guess what happened they ran into each other because there were no rules of the road this is not true but in any event that's a common Internet story so think about that kind of game the driving game where we think okay we could drive on the right or drive on the left now if I'm driving on the right and you're driving on the right also that's good right we're for promoting safety if I'm driving on the right and you're driving on the left we're gonna be in trouble and the same thing we both get our first choice if we either both drive on the right or both drive on the left so if we think about our preferences hey clearly prefers if he's driving on the right that you do the same but also prefers that if you're driving on the left you drive the same and so what a should do depends on what Bea's gonna do however in this case we do get Nash equilibria in fact we get two of them B is gonna have the preferences in favor of us doing the same thing as well and so in the end if we draw our arrows it looks like this in the end these are the best scenarios right if we're both driving on the Left I don't have any incentive to change that was my best strategy given what you did and similarly if you're driving on the right my best thing is to drive on the right I have no incentive to change so this is a situation where we get to pure strategy Nash equilibria now there is a mixed strategy Nash equilibrium where we both just totally randomized but that's obviously a terrible outcome to this game we want to be predictable and so we can maximize our chances of this if we both drive on the right or both drive on the left here in the isafe we drive on the right ah but we could be in Australia or Great Britain where we drive on the left by the way I've been in Scotland driving on the left shifting with your left hand is a very odd thing but anyway what happens when another car approaches you on a one-lane road all your instincts as an American are you've got this way of course all their instincts aren't you coming right luckily we did not collide they just mocked me for being an American but anyway that illustrates the importance of sticking with the Nash equilibrium all right next time we'll look at prisoners dilemmas and more interesting games

Channel: Daniel Bonevac

Views: 2,230

Rating: 4.8688526 out of 5

Keywords: John Forbes Nash, Nash Equilibrium, Game Theory

Id: FCGburaWXyc

Length: 44min 48sec (2688 seconds)

Published: Wed Mar 27 2019