Political Science 30: Politics and Strategy, Lec 13, UCLA

I am returning to exactly the same form of the cops and robbers game that we worked on on Tuesday I've got too many pages here this looks like what I really need where we have cops they can be on the beat bring the donut shop we have robbers they can be at work they can be at home payoffs 2 and negative 5 from this cell 1 and 0 and this cell 5 and 0 here negative 5 more than 5 okay just so the main thing that we feel it on Tuesday was that this is a game that has no Nash equilibrium in pure strategies no matter what fell we end up in one of the players will have regrets you guys okay this is skiing myself some people creating their heads is it oh thank you good good yeah I like seeing all these furrowed brows and that's that's a problem isn't it this is what the payoff should be it doesn't make sense for the robbers payoff to be low in the case where they're out robbing people and the cops are in the donut shop this is the situation that the robbers like they get to get away with their crime and indeed if we can just think for a minute about my mistake the game I had on the board does have a pure strategy equilibrium right okay robbers have a dominant strategy in this case if the payoffs were really this if we have four robbers get a conscience and decide they really don't like crime whether they get away from that or not then they're always going to stay home and once the robbers play their dominant strategy the cops will have a dominant strategy to be in the donut shop so that game I had on the board there was one that didn't have a Nash equilibrium it had a dominant strategy equilibrium so okay deep breath this is the game right this is the game we had on Tuesday this is the game that makes sense for the cops and robbers story if the cops are on the bee and the robbers are at work the robbers are the ones that have regrets they wish they'd stayed home if the robbers stay home and the cops are on the beat the cops are the ones that have regrets they wish they'd gone to the donut shop no crime happening here the cops are in the donut shop and the robbers are home the robbers have regrets huh we could have gotten some action today and finally the point that you're raised if the robbers are out there rotting and the cops are in the donut shop the cops are gonna have regrets because whoops there's a lot of crime happening we're in the donut shop we don't like that so good now we're back to where we were on Tuesday we've got our game that has no pure strategy Nash equilibrium on Tuesday we did find the mixed strategy Nash equilibrium and what did we find when we look for the mixed strategy Nash equilibrium we found a probability for each player okay a probability distribution over each player's strategies such that given one player's probability distribution the other player didn't regret theirs and vice versa so given the probability that the robbers were at work the cops can't do better than choosing their equilibrium probability and given the cops equilibrium probability the robbers can't do better than getting them okay so the way we wrote the mixed strategy Nash equilibria was that we assign variables to each player's probability we had P be the probability the cops are on the beat and we found that to be one-half all right and we let Q be the probability the robbers are at work that was for xi okay so one part of our interpretation of the mixed strategy Nash equilibrium here is that this is a game where if you are predictable you'll be sorry okay you will be out of equilibrium you will have regrets it's not a self-reinforcing pattern for the players to be predictable okay but in looking for the mixed strategy Nash equilibria what we also found is not any poll probability will do okay that there is only one probability for the cops one probability of the cops being on the beat that would actually make the robbers willing to randomize choose randomly between their two strategies okay if the cops are on the beat with a probability greater than one half then the robbers are not going to be willing to choose at random whether they go to work or not if the cops probability of being on the beat is higher than one half the robbers would do better by staying home okay so if the cops are not on the beat with probability one half the random strategies by both players are not in equilibrium what's different what I'm emphasizing from Tuesday is the cops equilibrium probability having it be exactly the right number is not going to affect their payoffs if the cops pick too high a probability but we leave the robbers probability the same the cops are doing no better no worse okay remember how we found these probabilities we found them we've been on the cops probability because it was the one that made the robbers indifferent between their pure strategies given that the robbers are making the cops indifferent between their pure strategies the cops given the robbers probability are doing just as well by being on the B as being in the donut shop so they're indifferent between being on the beeping and the donut shop flipping a coin that is 50/50 flipping a coin that is weighted way or another okay so the name difference in interpreting mixed strategies is that neither player themselves does worse by picking the wrong probability rather the system won't be in equilibrium okay so there's less of that clear if I was a player who cared about this game I would do the mental work to get the probability right probably not in this game okay probably not in a mixed strategy case okay so that's recap what I want to do now is think a little bit about what else we've learned about this game okay in particular I want to think about outcomes in this game so I'm going to put my equilibrium probabilities here okay so with probability 1/2 the cops are going to be on the beat and with probability 1/2 they'll be in the donut shop using this definition of P if I'm on the beat with probability 1/2 the only other thing I can do is be in the donut shop so that has to be probability 1 minus P with probability Q now I'm the robbers I go to work with probability 411 and so what's the probability that I stay home 7-eleven is very good all right so in equilibrium what we're gonna see is any one of these possible cells okay their strategies are chosen at random okay they don't know in advance what their strategy is going to be but something is indeed going to happen okay with probability for over twenty two we're going to end up in this cell okay where the cops reanima we decide oh yeah okay today we're gonna work the robber is randomly decide yep we've got a lot of energy today too we're going to be here so we're going to see that some fraction of the time what am i doing to get that number I am multiplying the probability associated with a cop strategy and the probability associated with the Roberts strategy now if you're thinking that other classes where you've used probability and you're thinking can she really multiply the probabilities I can and the reason why I can is because these random choices are being made independently okay if there was some other factor that was affecting both the cops choice and the robbers choice if the these two random variables were related then I wouldn't be able to multiply the probabilities but in the mixed strategy story the idea is that they are choosing these probabilities independent of each other okay the mixed strategy choices are independent and random okay independent random and governed by these particular mixing probabilities okay so this probability here is the probability that cops are on the beat I'm writing it out in words just to be very clear times the probability the robbers are at work okay so we're trying to figure out the probability that we end up in any particular outcome and a mixed strategy Nash equilibrium what we do is we just multiply the probabilities associated with the two strategies that create that outcome okay so we can do that for all the cell's the probability that the cops are on the beat and the robbers stay home then is one-half times 7 11 7 20 second seen here 1/2 times 11 and down here the probability that the cops are in the donut shop and the robbers are at work that happens probability for over 20 – take a deep breath ask myself do these probabilities add up to 1 always have to write if you're getting the probabilities right well four plus seven is 11 plus 11 is 22 okay we're fine okay so in the mixed strategy equilibrium we'll see all of these outcomes with some probability and they're not the same okay it's more likely that the robbers stay home then they go to work but it's also possible that they go to work sometimes when the robbers are out there robbing people sometimes they get away with it sometimes they don't okay so if you're asked as you will be in next week's homework to interpret a mixed strategy Nash equilibrium if you're asked okay what's going to happen here so one thing you can say is that you'll see all these possibilities you'll see robbers active sometimes not active other times you'll see the cops working hard sometimes not working hard other times and you can say something about what's relatively more likely it's more likely on if this is happening over and over again we'll see more days when there's no crime because the robbers stay home then we'll see days when there is crime because the robbers are at work okay so the kind of answer I'm looking for I'm going to get your question in just a second the kind of answer I'm looking for when I ask a question what do you expect will happen I'm actually looking for the probabilities associated with all these possible cells yeah there's there's no PS any okay this is um this is let me go over this we did this at the end of class on on Tuesday we can have there's four possibilities right we can have I'll draw a little diagram I drew man this is not a game diagram this is I'm just saying is there a PS any or no is there mixed strategy nash equilibrium or no the game we got up here cops and robbers or possibilities is a game that has no pure strategy nash equilibrium and it does have a mixed strategy Nash equilibrium okay the way we saw that there is no pure strategy Nash equilibrium is we just looked through there's only four possibilities right and so we looked at each cell and we asked ourselves are either of these cells equilibria in the sense that is there any cell we can be in where one player wouldn't wish they did the opposite okay so in this cell given that the cops are on the beat the robbers wish they'd stay home the robbers have regrets given if the robbers stay home in this cell the cops have regrets this payoff is higher this cell given that the cops are in the donut shop the robbers which they've gone out and done some robbing and given that the robbers are out there robbing the cops have regrets if they're in the donut shop okay so it's just finding that there's no pure strategy Nash equilibrium is usually just a process of elimination yeah okay the other things we looked at were games that have no mixed strategy Nash equilibrium but do have a pure strategy Nash equilibrium and we looked at the prisoner's dilemma as an example of that in a little while I'll do an example of a game that has both there are games that will have neither but we're not going to cover them in this any game that we can write down this in this kind of forum where we can list all the strategies for each player will have one of these types of equilibrium okay okay I want to see another thing about actually not part of the outline but it's a an important point to make about what everything I mean by saying that the mixed strategy choices have to be random okay they can't be predictable in any way okay it cannot be the case that the cops are the easy ones to think about since the cop the cops are supposed to be going on the beat with probability 1/2 in order for this pattern of random behavior to be in equilibrium they can't do something though like be on the beat every other day or be on the beat in the morning of the doughnut hunt in the afternoon if there's any predictable pattern the robbers are going to get that okay so the cops have to be truly doing something that cannot be anticipated okay if you think about other contacts that might fit this kind of game income tax auditing is one that's a pretty obvious example of that that the IRS would prefer not to have to audit people's income tax returns it's a big headache everybody hates it it consumes a lot of resources people not everybody not me not you guys I'm sure but there are people out there that prefer to cheat on their taxes then do the right thing okay and that sort of scenario would lead to this kind of mixed strategy Nash equilibrium and the IRS goes to great pains to be truly random in the way they audit they choose which returns to audit okay that randomness can be frustrating sometimes if you're a all business owner of a type that is very unlikely to cheat on your taxes say for all sorts of reasons your record of doing it the kind of business you're in is one where your record-keeping is pretty transparent it's really unlikely if they get you they still my audits you anyway because it's very very important that they are unpredictable okay if they are predictable then the people who want to cheat on their taxes can figure out a way around their strategy okay the same story could be told from the robbers point of view okay the robbers can't have a pattern for when they're choosing which of their four out of every eleven days they're going to be active okay it has to be something like a coin flip like a spinner being spun the way I talked about on Tuesday okay so it's not in the outline but I'm going to put it over here how do I say it mixed strategy this isn't actually the word that people would normally use that doesn't seem like there is a word people would normally use for this mixed strategy implementation must be truly random no predictable patterns okay another thing we can say about and under the subject of interpreting the mixed strategy equilibrium we could talk about the outcomes okay the outcomes now we really have to talk about a probability distribution here if someone asks you what's going to happen in a game where a mixed strategy Nash equilibrium is being played you can't give them a short answer you have to qualify it well this could happen with that probability of cetera okay it's a little bit easier not entirely to talk about payoffs in the mixed strategy Nash equilibria what you talk about now are you got an expected payoffs right okay we don't know what's gonna happen but we can in advance put an expected value on the equilibrium payoff for both the cops and the robbers in this example okay so let's um let's do that on this board here as I often do I use a capital letter u to you for utility to symbolize a payoff I'm gonna say the expected utility of the cops here now we're looking at the expected utility from the mixed strategy Nash equilibrium remember to find Q the mixing probability we were looking at the expected utility the cops got from choosing one strategy versus the other okay where we only had two possibilities to consider if we were looking at the cops expected utility of being on the beat you'll remember that was the argument of the function on Tuesday then we only had two possibilities to consider now we're looking for the expected utility that is associated with annex travel Ibrahim when we have to consider all four possibilities okay that's the only difference it's still the payoff from each of the four cases multiplied by the probability we end up in that case okay so from the cops point of view with probability for over twenty two I could be on the beat and the robbers are at work and that gives me a payoff of – I kind of like catching those guys with probability 720 seconds I can be on the beat and there's no robbers to catch that's kind of a bummer pretty boring wish I had a donut with probability 422 here oh god I'm in the donut shop all hell is breaking loose in my precinct that is a payoff of negative five for me and with probability 720 seconds I'm enjoying my donut the robbers are sitting at home that's my favorite payoff okay so this is the expected payoff of the cops and I won't work through all the mechanics here when I worked through to my office and you guys can verify it I got an expected utility of fifteen eleventh just by crunching through okay same for the robbers okay expected utility for the robbers of the mixed strategy Nash equilibria it's the same probabilities right same four cells we can end up in but now I have to put the robbers payoffs in here because I'm looking at it from the point of view of the robbers okay so for twenty seconds times I'm out doing my thing and the cops get me plus seven twenty seconds times cops are out there but I'm at home my payoff in this so plus another one of those probabilities 720 seconds I'm at home the cops are in the donut shop I don't really care about the cops in this case when I'm at home and probability for 20 seconds I get my high payoff of five here when I'm at work and the cops are in the donut shop the that actually ends up being zero you can actually kind of see how it should be zero here but these two payoffs are the same distance from zero and they occur with the same probability so it just balances there so the expected payoffs here are 15 11 okay a number a little over 1 and 0 1 place where you might use the expected payoffs of a mixed strategy Nash equilibrium is in thinking about whether the equilibrium payoffs are Pareto optimal or not ok so what do you do if you want to ask that question ok or Pareto efficient is this Pareto efficient ok well let's look at that is there some certain outcome that can happen in this game that will make one player better off without making the other one worse off let's just go through systematically since I can't see the 15:11 Sauveur there and I'll forget it in two seconds I'm just gonna copy them over here so I can see them ok so does this outcome Pareto dominate the expected payoffs for the mixed strategy one is one player better off in this outcome and neither player worse off no right you're shaking your head who's worse off the robbers are worse off exactly yeah it's the hard thing about this is seeing that that's not that hard okay so this does not Pareto dominate it okay what about this cell this is pretty dominate it no right cops horse off this one we're not before what I'm doing right now for the comparison no okay what I'm doing and now that's a very good question what your lane says am I using the probabilities I mean blabbering a little bit on your question at all when I'm comparing the outcomes here with the expected payoffs I'm not okay what I'm asking myself is are the expected payoffs what we get from these random strategies and then mixed strategy equilibrium is that outcome Pareto dominated by any of these outcomes is there something that could happen that would be better off that would be better for one player and not worse for the other player in this game okay so what I'm doing here is I am comparing the cops expected payoff to their certain true payoff here I said okay well that's higher that looks good I'm comparing it to the robbers certain payoff I said oh that's lower so this doesn't Pareto dominate it same story here okay the cops are worse off here what do we got here this is a Pareto improvement right okay so what can we conclude here it's an important enough conclusion I'm switching colors here this is Pareto dominated by I'll use words here the arrows are kind of ambiguous what that would mean okay so the mixed strategy Nash equilibrium is Pareto dominated by cops and donut shop robbers at home okay how am I seeing that here we are in this kind of this happy world the robbers are staying home the cops can relax the robbers are no worse off here okay their payoff is no worse than in this risky world okay the cops are much better off okay the cops are getting a very high payoff here and that's I wanna say that's always a feature of mixed strategy Nash equilibria but you might imagine that this cops and robbers game is like indicated with the tax uhm example and you'll get a different example in your homework next week is a metaphor for a lot of monitoring situations one player is trying to get away with something that the other player is trying to monitor them for okay those games often have mixed strategy Nash equilibria and they are often Pareto dominated by one of the outcomes so this game like the prisoner's dilemma has that kind of bothersome quality which is that the only Nash equilibrium the only Nash equilibrium we have is one that's in mixed strategies and it is Pareto dominated by something that is not in equilibrium this cell is not in equilibrium because if the cops are in the donut shop the robbers are going to wish the robbers are gonna have regrets that they weren't out there committing crime okay so that's a worthwhile substantive point but the other point that I'm making here is just process point that when you're asked whether the outcome in a game with a mixed strategy equilibrium is Pareto optimal what you do is you compare the expected payoffs from the mixed strategy equilibrium okay which is going to depend on it's usually going to associate a positive probability with all four outcomes compare that to the certain outcomes in each cell and that's what we find here okay yes no no Rose says can you determine which of the boxes it's going to be in and that is the key and sometimes they're kind of frustrating feature of mixed strategy equilibria you can't tell okay somebody asks you smart UCLA grad this is the situation that's going on in my city I know these are the payoffs what's going to happen and you're gonna say anything can happen okay that's a feature of these situations where the equilibrium strategy for both players is to be unpredictable they can't predict each other's choice and we can't predict their choices with game theory we can predict what pattern of choices what probabilities governing the random choices will be a self-reinforcing pattern but that's all we can do okay that actually reminds me of an another point I wanted to make this point about ex post mistakes we're going to see the same thing that we saw in our analysis of sequential games with nature notes extra board here all right and a mixed strategy Nash equilibrium no player will have ex-ante regrets given the other players choice hey I'm putting this in parentheses because that's just what a Nash equilibrium means the idea of Nash equilibrium every time in any context has this idea of looking at one player's choice holding the other player's choice constant and then flipping the roles okay so given that the robbers are going to work with probability for xi I the cops cannot do any better than being on the beat with probability 1/2 okay it turns out I am NOT doing any better or worse by being on the beat with probability 1/2 and I would be by being on the beat with probability 1/4 or always being on the beat if the robbers are going to work with probability for xi I'm indifferent between my pure strategies but I can't do better ok so no player will have ex-ante regrets ok the flip side of it is though no matter how the random choices okay cops random choice to be on the beat robbers random choice to go to work no matter how those random choices work out one player will and with ex-post regrets okay so given that you're choosing randomly with your mixing strategy I don't regret the fact given that you you robbers out there are going to work with probability for 11s I wasn't wrong I'm the cops to flip that fair coin probability 1/2 to either be in the donut shop or not ok but we could certainly end up in cells like this one where I have some regrets or even this one here I'm regretting being the donut shop and you you robbers you you're out there robbing people and here I regret being on the beat when you're not doing anything there's nothing for me to catch ok so in this game there's always going to be ex-post regrets the way things work out but because they're playing the Nash equilibrium by definition they're not making an ex ante mistake okay that was the way it was in the sequential games where the mistakes had to do with a choice choice in scare quotes there by nature here the the choice is still by nature the choice is true still truly random okay the robbers are picking their strategy randomly so it's a choice by nature being implemented by the robbers but there's still this fact about it that if we end up here okay so this is a cell where the cops have do it in a slightly different language made an ex post mistake here okay crying happened we were in the donut shop ah that's awful but if the choice to be in the donut shop was made with the 50/50 probability the cops weren't wrong they weren't doing something irrational giving their incentives they just were unlucky okay and over here look at it from the robbers point of view okay here's a cell where the robbers have ex-post regrets just using very slightly different language to make the same point if it works out that my spinner I'm the robbers now told me to stay home that day and the cops spinner randomly told them to go to the donut shop I'm gonna think it was a mistake okay but only a next post mistake it's only a mistake after I know the fact of what the cops have done this is only a mistake after the cops know the fact of what the robbers have done before they knew how that was gonna work out they were making their best choice or they weren't weren't making a worse choice so there let me add one qualification over here this is usually I switch colors for qualification this is true for any game with a mixed strategy Nash equilibrium okay this part that there will always be exposed regrets this is only true when there is no pure strategy Nash equilibrium okay if we're in a game with both types of equilibria even we're playing the mixed strategy Nash equilibrium it's possible that we can end up with ex-post regrets or not and that actually seems like a good place to start talking about or start looking at an example of a game that has both types of equilibrium yeah okay so Stephanie is asking is in game theory in general or ex ante regrets ever possible not by strategic players okay that's that's I would say fair answer in life people do make ex ante mistakes and even if you think people making ex ante mistakes is a big part of life game theory can be helpful to identify what those ex ante mistakes are okay but in our games with people playing their nash equilibrium strategies there will never be an ex ante mistake and alright the point i've been emphasizing is that even with these super rational people never making that kind of ex ante mistake here's a situation where somebody's always going to have made an ex post mistake somebody's always going to regret what they did even if it was the best choice they made with information they had at the time they made the decision other questions on cops and robbers before I go to the next game okay do my usual thing of erasing from the outside in in case questions develop well the game I'm gonna look at next is sort of a coordination game it's I guess the disk coordination game it's the chicken game and I can't actually remember whether I know I haven't talked about it yet it's a pretty close relative of battle of the sexes when we get it up you'll see how it's the same in the small way and which is different okay so the most innocuous version of chicken is there's two kids and in there's two kids and they're riding their bikes toward each other and whoever swerves first is the chicken if you were first haha when they just were and I feel great and you're like haha I chickened out it's a bummer okay so the choices and the chicken game are swerve don't swerve just as cops and robbers can be a metaphor for all sorts of monitoring situations battle of the sexes could be a metaphor for lots of coordination decisions for example like nominating a primary using primaries to nominate a presidential candidate chicken could be a metaphor for crisis bargaining situations countries rattling their sabers and trying to look as threatening as possible to get some concessions for their neighbors both of them trying to do the same thing okay so swerve and don't swerve this is the chicken all right so let's get some payoffs in there all right so a player B I don't swerve and you do haha you're a chicken you don't serve and I do yeah I'm a chicken it's true a lot of the time what'll happen is we'll both swerve okay yeah everybody smirks uh-huh but swerve there let's do it again sometimes though what has happened is we're both really tough we don't swerve and that's real bad okay crash we cry as daddy Lane yeah thank you thank you a lane points out I've got my negative sign I'm getting so into my role-playing here of acting out payoffs for you guys I'm not being serious about my my payoffs here okay so in this cell the one who swerves is the row player they get the negative pay off the one who does not work not for evading is what makes you feel good in this game okay both swerving is bad okay so in the way this would be applied in in I our scenario a and B could be Indian Pakistan have in the last decade been playing this game with each other repeatedly and every once in a while everybody gets worried that they get perilously close to this you know outcome are they really going to start a nuclear war in South Asia they are this doing the somewhat more benign example where probably the worst outcome would be a concussion alright so this game does have pure strategy Nash equilibria it's got one right down here okay your player be given that I didn't swerve okay you're glad that you did okay it's embarrassing that I'm calling you a chicken right now you don't like that but it would be way way worse to fall and wreck your bike I'm player a okay given that you swerved I'm really glad that I didn't okay otherwise it would've just been one of those boring do-over is like a tough nananananana that's you so neither player has regrets here and same story up here you can just completely switch the roles given that I swerved you're glad that you did because you'd rather be able to call me a chicken than not given that you didn't swerve I'm glad I did because I'd rather back down and be embarrassed than have us crash okay so this game has to pure strategy Nash equilibria when I said a minute ago that it's a close relative of battle of the sexes what makes it similar to battle of the sexes is that there are two Nash equilibria and one is better for one player the other is better for the other player what's different from battle of the sexes is battle of the sexes the Nash equilibrium happens when they do the same things okay so you're in equilibrium when you truly coordinate chicken if you want to make a fine distinction you could say it's on this coordination game okay we're in equilibrium when we do different things we're not in equilibrium where we do the same thing okay we're either here what were there these two cells are not equilibrium okay in this cell given that you swerved you guys be the role player I wish that I didn't it would have been so fun and given that I swerved you wish that you didn't okay here given that you didn't swerve I wish I had given that I didn't swear you wish you had alright so I'm kind of giving away the punchline here but let's verify whether there is a mixed strategy equilibrium here and I'm going to just go through the same process that I did on Tuesday okay so what am I looking for when I'm looking for a mixed strategy equilibrium I'm looking for the probability that a swerves and the probability that B swerves okay so let Q equal the probability that a this works so now we have to go back to the little recipe that I had in my outline on Tuesday for how to find the equilibrium value of this probability okay it's an equilibrium value if it's the probability that makes be indifferent between swerving and not swerving one players equilibrium probability is the probability that makes the other player indifferent between her pure strategies okay so let's figure that out the expected utility to be a swerve is Q the probability that a swerves times zero okay Q is the probability that a chooses that we're in this row if I choose swerve in that situation I'll get a payoff of 0 plus 1 minus Q the probability that a doesn't swerve my payoff will be negative 1 okay so that reduces to Q minus 1 the expected utility to be of not swerving is now I'm comparing my payoffs in this column using the probabilities that a plays these two strategies okay so not swerving gives me a payoff of 1 with probability Q Billy Q a swerves I get my payoff of 1 probability 1 minus Q of negative 10 making that a little bit neater I get Q I get negative 10 here I get positive 10 Q sounds like 11 Q 11 Q here right 11 Q minus 10 yeah the bells are ringing watch me on this stuff okay so now what I'm gonna do is I'm gonna take these two expressions what I expect to get from this strategy what I expect to get from that strategy as a function of the A's mixing probability and I'm going to find the one the one probability that a can mix with that will actually make me indifferent between my pure strategies and again the reason why I have to be indifferent between my pure strategies is that if this is a higher number if the expected utility from swerving is higher than the expected utility from about swerving well I'm gonna swerve okay and if this is higher I'm gonna not the only way I'm going to be willing to make the choice truly randomly is if what I expect to get from each outcome each certain outcome is exactly the same all right so let's find that Q it's the value for which Q minus 1 equals 11 Q minus 10 okay we'll bring that around here we'll get 9 equals 10 Q Q equals 9 tenths okay so there's half of my mixed strategy Nash equilibrium now I want to draw your attention back to what happened when we went through this process with the prisoner's dilemma okay remember in Tuesday we went through this process with the prisoner's dilemma and we got a probability that was I think negative okay if you get a probability here for a mixed strategy Nash equilibrium that is less than or equal to 0 or greater than or equal to one what that's telling you is that there is no mixed strategy Nash equilibrium it's telling you there is no reasonable probability that will make your opponent indifferent between their pure strategies and of course that makes sense in the prisoner's dilemma when you've got a dominant strategy there's nothing anyone can do to make that strategy not dominant it's always going to be better than the other one okay so when you get an unreasonable answer here yes it's always good to check your algebra I bet you guys are better with algebra than I frequently am check it anyway but if you get a negative number here or a number greater than one it's not necessarily a mistake it's just it's a message it's saying yeah this is a game that doesn't have a mixed strategy equilibrium it's fine okay so that in equilibrium is 9/10 sometimes what people do I do this I think it's a a good practice is Q by itself I'm defining that as the probability that a swerves and if it this little star up here to denote to this equilibrium value okay we started off we said Q is any old Q any old probability but in fact there's only one Q for which this is true okay so we set it up this thing is true for any old Q this is true for any old Q but once we set them equal to each other now we're saying now I'm looking for Q star Q star is the one that will make my expected utility from swerving exactly equal to not so I when I'm going through this kind of analysis I find it helpful to keep straight a variable that is truly a variable from the solution that fits a particular question okay so this is the solution that fits the question of what is the next strategy Nash equilibrium okay so in cops-and-robbers what we did we found q and then we turned around and we found p we said well p in this context would be the role-players probability so I switched q and p here from cops and robbers remember I said that that's okay you can use whatever variable you want for either player just make sure to keep straight which one it is in this game though because the strategies and the payoffs are perfectly symmetric if 9/10 is the probability of a swerving that makes be indifferent guess what 9/10 is going to be the probability of B swerving that will make a indifferent okay you can work it through and it might not be a bad idea to UM work it through when you're recopying your notes or something like that but you will indeed find that the probability the swerves yes well follow me on the rule here let that be the probability and mixed strategy equilibrium is indeed 9/10 okay the reason why I don't need to go through the whole process again for B is if I started to do it okay let's just say I started to do it here and I'm actually I'm gonna leave that up there and I'm gonna do it now for a okay so the expected utility of a of swerving now is going to be a function of Bees probability P okay so now I'm a I'm asking what is my expected payoff if I swerve okay well it's P the probability B swerves times 0 plus 1 minus P the probability of the B doesn't swerve x minus 1 okay see it's going to be exactly the same and for not swerving down here it's going to be P times one plus one minus P times negative ten and I wish I hadn't erased the expected utility of B also wish I'd put the expected utility of a here cuz that's what it should be what I've got here in a in red are A's expected utilities as a function of P and they look exactly like bees utilities as a function of Q I will go ahead and put these expected utility back up here of not swerving it is Q times one plus one minus Q times negative ten so if you look at these two pairs the value of Q that satisfies the blue ones is going to be the same as this it's never wrong to work it out both ways and it honestly doesn't take that one either but it's also fine to look at the game and say that when the strategies and payoffs are completely symmetric the reasoning is going to be completely symmetric to okay since we never get any choices in game theory that come from anything other than the payoffs the equilibrium mixing probabilities have to be the same when the payoffs are the same for similar walls okay okay so what do we think is gonna happen in checking all right so I'm gonna swerve with probability 9/10 you're gonna swerve with probability 9/10 9 times 9 is 81 81 one hundredths I'll write it as a percent 81 percent of the time it's gonna be a do-over okay you have to really like chicken to be getting that many kind of dud results okay but the thrill of it is in that one tenth of the time when I don't swerve and you do and that is nine percent nine times 110 times ten I'm just multiplying the probabilities again nine percent of the time haha you're a chicken and nine percent of the time also I'm gonna feel humiliated ah not a chicken and then there's that horrible one percent of the time when when we crash I think I'm not going to work it out here but not a bad problem some of you guys are coming to me and wishing for more problems as I did before the midterm I'll give you more problems to do out of the book but here's something for you guys to do either over the weekend while you're processing this or later when we review this for the midterm yeah is to evaluate is the mixed strategy Nash equilibrium in chicken Pareto efficient I won't tell you now but I'll try to remember to include that in the answer keys that I put up for the final study sheets and remind me I don't other thoughts on chicken yes Lilya is asking I think what you're asking is do what do we think is focal in this game say what would it be whatever we've got three equilibrium here we've got two pure strategy Nash equilibria and one mixed strategy Nash equilibria and I'm gonna give an answer somewhat similar to the answer I gave when this question arose with assurance can remember assurance at the game that's sort of like the prisoner's dilemma except that when one player cooperates the other one wants to – and I said in some context you might think that the fact the one equilibrium is Preto efficient by itself would make that equilibrium focal sometimes it does but not always and I used the hockey helmets example of one as one where the Pareto inferior example was focal and I think really the reason why people didn't wear those hockey helmets for so long was just because that's how they'd always done it and one way that things become focal is through the force of history because it's based on our shared expectations that same kind of reasoning you can apply to these games with both types of equilibria sometimes it seems like the Nick strategy equilibria are so weird and so random that they shouldn't really be focal okay that we should end up at some one of the part of the pure strategy Nash equilibria sometimes that is true okay not always chicken is not my idea of a particularly good sport but lots of sports if you really think I mean like a sport that somebody plays for fun soccer football where it's one team against another those games if you use game theory on them will only have mixed strategy Nash equilibria and if you think about it it sort of makes sense what's the fun in watching something where you know it's always going to be this way okay so that wouldn't be an example of games where the mixed strategy Nash equilibrium its focal and I actually I think that in chicken it truly when chicken literally the little boys riding their bikes toward each other the mixed strategy Nash equilibrium is what they're playing but otherwise it's no fun yes one of these is focal because I'm well-known to be a maniac and you're a reasonable person and we both know that we're going to end up here what's the fun of playing okay or similarly if the roles are reversed in the crisis bargaining situation if you think about that kind of situation being played between all sorts of neighboring pairs must neighboring countries don't get into these crisis ex escalation and I'm stationing troops on the border you're performing a nuclear test most countries don't do this okay so for most countries you know either one is dominant or the other one is we're not having these contests okay but occasionally you'll have one where it's not clear and the mixed strategy equilibrium is the focal one good question the questions on this yes yes Stephanie is asking when I'm when I'm asking whether the mixed strategy Nash equilibrium is Pareto efficient what am I asking for so let me sort of remind you of that what I'm asking you to do is to calculate both players expected payoffs from the mixed strategy Nash equilibrium where you're going to use the probabilities here to calculate the expected payoffs but then to take those expected payoffs and compare them to the pairs of payoffs okay so you use the probabilities here to sort of revisit a question Elaine asked earlier you do use those probabilities and calculating the expected payoffs then you're done with them okay once they're in the expected payoffs you're comparing the expected payoffs to the certain payoffs that could happen in any of the four cells okay um there is one other set of things I want to say about mixed strategy Nash equilibria and another set of things I really want to say about cops and robbers but I think what I want to do is I'm going to save that forum for Tuesday okay so do your homework if you didn't get the homework on Tuesday it's up on the website and I'll see you next week oh yeah

Maurice Vega

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