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Current time:0:00Total duration:9:21

Prisoners' dilemma and Nash equilibrium

PRD‑3 (EU)
PRD‑3.C (LO)
PRD‑3.C.3 (EK)
PRD‑3.C.4 (EK)
PRD‑3.C.5 (EK)
PRD‑3.C.6 (EK)

Video transcript

on the same day police have made - at first unrelated of arrest they arrests a gentleman named Al and they caught him red-handed selling drugs so it's an open-and-shut case and in the same day they catch a gentleman named Bill and he is also caught red-handed stealing drugs and they bring them separately to the police station and they tell them look this is an open-and-shut case you're going to get convicted for drug dealing and you're going to get two years and they tell this to each of them individually they were selling the same type of drugs just happened to be that but they were doing it completely independently two years for drugs is what's going to happen assuming nothing else but then the district attorney has a chance to chat with each of these gentlemen separately and while he's chatting with them he reinforces the idea that this is an open-and-shut case for the drug dealing they're each going to get two years if nothing else happens but then he starts to realize that this these two characters look like he starts to have a suspicion for whatever reason that these were the two characters that actually committed a much more serious offense that they had committed the a major armed robbery a few weeks ago and all the all the attorney Gentile the district attorney has to go on is is his his hunch his suspicions he has no hard evidence so what he wants to do is try to get a deal with each of these guys so that they have an incentive to essentially snitch on each other so what he tells each of them is look you're going to get two years for drug dealing that's kind of that's kind of guaranteed but he says look if if you if you if you confess if you confess and the other doesn't other doesn't then you will get one year then you will get one year and the other guy the other guy will get will get ten years so he's telling al look we caught bill - just randomly today if you confess that it was you and Bill who performed that armed robbery your term is actually gonna go down from two years to one year but Bill is obviously going to have to offense a lot more time in jail especially because he is not he is not cooperate with us he is not confessing but then the other the other statement is also true if if you if you deny and the other confesses and the other confesses now it switches around you will get 10 years because you're not cooperating and the other your co-conspirator will get a reduced sentence will get the one year so this is like telling Outlook if you deny that you were the armed robber and build snitches you out then you're going to get 10 years in prison and Bill's only going to get one year in prison and if both of you if both of you essentially confess both confess both confess you will both get three years you will both get three years so this scenario is called the prisoner's dilemma because we'll see in a second there is a globally optimal scenario for them where they both deny where they both deny and they'll both get two years but we'll see based on their incentives assuming they don't have any unusual loyalty to each other and these are you know these are hardened criminals here they're not brothers or related to each other in any way they don't have any kind of loyalty pact we'll see that they will rationally pick a non or they might rationally pick a non-optimal scenario and to understand that I'm going to draw something called a payoff matrix a payoff matrix so let me do it right here for Bill so bill bill has two options he can confess he can confess to the armed robbery or he can deny that he had anything that that anyone that he knows anything about the armed robbery and Al has the same two options al can confess al can confess and Al can deny and since it's called a payoff matrix let me draw some grids here let me draw some grids and let's think about all of the different scenarios and what the payoffs would be if al confesses and Bill confesses then we're in scenario 4 they both get they both get three years in jail so they both they both will get three for Al and three for Bill now if al confesses and bill denies al confesses in bill denies then we are in scenario two from Al's point of view Al's only going to get one year al is only going to get one year but bill is going to get ten years bill is going to get ten years now if the opposite thing happens if bill confesses and al denies then it goes the other way around Al's going to get ten years for not cooperating and Bill's going to have a reduced sentence of one year for cooperating and then if they both deny if they both deny there in scenario one there in scenario one where they're both just going to get their time for the drug dealing so I will get two years and bill will get two years bill will get two years now I alluded to this earlier in the video what is the globally optimal scenario for them well it's this scenario where they both deny having anything to do with the armed robbery then they both get two years but what we'll see is it is actually somewhat rational assuming that they don't have any strong loyalties to each other a strong level of trust with the other party to not go there and it's actually rational for both of them to confess and the confession is actually a Nash equilibrium and we'll talk more about this but a Nash equilibrium is where each party has picked a choice given the choices of the other party so when we think of or each party's to pick the optimal choice given the choices of the other or given whatever choice the other party picks and so from this from Al's point of view he says well look I don't know whether bill or bill is confessing or denying so let me let's say let's say he confesses what's better for me to do if he confesses and I confess then I get three years if he confesses and I deny I get 10 years so if he confesses it's better for me to confess as well so this is this is a preferable scenario to this one down here now I don't know that bill confessed he might deny if I assume bill denied is it better for me to confess and get one year or deny and get two years well once again it's better for me to confess and so regardless of whether bill confesses or denies so this once again the optimal choice for alta pick taking into account bills choices is to confess if bill confesses Al's better off confessing if bill denies Al's better off confessing now we look at it from Bill's point of view and it's completely symmetric if bill bill says well I don't know if al is confessing or denying if al confesses I can confess and get three years or I can deny and get 10 years well 3 years in prison is better than 10 so I will go I would go for the three years if I know al is confessing but I don't know that Al's Nestle definitely confessing he might deny if al is denying I could I could confess and get one year or I could deny and get two years but once again I would want to I would want to confess and get the one year so bill taking into account each of the scenarios that al might take it's always better for him it's always better for him to confess and so this is interesting there rationally deducing that they should get to this scenario this this Nash equilibrium state as opposed to this globally optimal state they're both getting three years by both confessing as opposed to both of them getting two years by both denying the problem with this one is this as an unstable state this is an unstable state if one of them assumes that the other one has if one of them assumes that they're somehow in that state temporarily they say well I can always improve I can always improve my scenario by changing my by changing what I want to do if al thought that bill was definitely denying al can improve his circumstance by moving out of that state and confessing and only getting one year likewise if bill was thought that maybe al is likely to deny he realizes that he can optimize by moving in this direction instead of denying getting too into he could move in that direction right over there so this is an unstable optimal scenario but this Nash equilibrium this date right over here is actually very very very stable if they assume if if this is it's better for each of them to confess regardless of what the other ones does and assuming all of the other actors have chosen their strategy there's no incentive for bill so if assuming everyone else has changed their strategy you can only move you can only move in that direction if you're bill you can either you can go from the Nash equilibrium of confessing to denying but you're worse off so you won't want to do that or you could move in this direction where which would be al changing his decision but once again that guess gives a worse outcome for Al you're going from 3 years to 10 years so this is the equilibrium state the stable state that both people will pick something that is not optimal globally