Prisoners' Dilemma and Nash Equilibrium Why two not-so-loyal criminals would want to snitch each other out
Prisoners' Dilemma and Nash Equilibrium
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- On the same day, police have made two at first unrelated arrests.
- They arrest a gentleman named Alan.
- 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 dealing 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 are 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 the chance
- to chat with each of this 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 2 years if nothing else happens.
- But then he starts to realize that
- these 2 characters look like.
- He starts to have a suspicion for whatever reason
- that these were the 2 characters that actually committed
- a much more serious offence, that they had committed
- a major armed robbery a few weeks ago.
- And all the District Attorney has to go on
- is his hunch, his suspicion. 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 gonna get two years for drug dealing,
- that's kind of guaranteed". But he says
- "look, if you confess, and the other doesn't,
- then you will get 1 year
- and the other guy will get 10 years".
- So he's telling Al, "look, we caught Bill too just randomly today,
- if you confess that it was you and Bill who performed that armed robbery,
- your term is actually going down from 2 years to 1 year.
- But Bill is obviously going to have to spend a lot more time in jail,
- especially because he's not cooperating with us,
- he's not confessing".
- But then the other statement is also true:
- If you deny 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 1 year. So this is like telling Al
- "look, if you deny that you were the armed robber
- and Bill snitches you out,
- then you're gonna get 10 years in prison
- and Bill is only going to get 1 year in prison".
- And if both of you essentially confess, both confess,
- you will both get 3 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, and they both get 2 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 pack.
- 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 the "pay-off matrix", a pay-off matrix.
- So let me do it right here for Bill.
- So Bill has two options, he can confess to the armed robbery
- or he can deny that he had anything,
- that he knows anything about the armed robbery.
- And Al has the same two options.
- Al can confess and Al can deny.
- And since it's called the pay-off matrix,
- let me draw some grids here.
- And let's think about all of the different scenarios
- and what the pay-offs would be.
- If Al confesses and Bill confesses then they're in scenario 4,
- they both get 3 years in jail, they both would get
- 3 for Al, and 3 for Bill.
- Now, if Al confesses and Bill denies
- then we are in scenario 2, from Al's point of view,
- Al is only going to get 1 year,
- but Bill is going to get 10 years.
- Now if the opposite thing happens,
- that Bill confesses and Al denies
- then it goes the other way around.
- Al is going to get 10 years for not cooperating and
- Bill is going to have a reduced sentence of 1 year for cooperating.
- And if they both deny, they're in scenario 1, where
- they're both just going to get their time for the drug dealing.
- So Al would get 2 years and Bill would get 2 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 2 years.
- But what we'll see is that 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, it's actually rational for both of them to confess.
- And a 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 picked the optimal choice
- given the choices of, or given whatever choice the other party picks.
- And so 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 he confesses, what's better for me to do?
- If he confesses and I confess, then I get 3 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 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 1 year, or deny and get 2 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 Al to pick,
- taking into account Bill's 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's confessing or denying.
- If Al confesses, I can confess and get 3 years,
- or I can deny and get 10 years.
- Well, 3 years in prison is better than 10,
- so I would go for the 3 years.
- If I know Al is confessing.
- But I don't know that Al's definitely confessing, he might deny.
- If Al's denying, I could confess and get 1 year,
- or I could deny and get 2 years.
- Well, once again, I would want to confess and get the 1 year.
- So Bill, taking into account each of the scenarios that Al might take,
- it's always better for him to confess.
- And so this is interesting.
- They're rationally deducing that they should get to this scenario,
- this Nash equilibrium state,
- as opposed to this globally optimal state.
- They're both getting 3 years by both confessing
- as opposed to both of them getting 2 years by both denying.
- The problem with this one is 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 my scenario
- by changing my, by changing what I wanna 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 1 year.
- Likewise, if Bill had thought that maybe Al is likely to deny
- he realizes that he can optimize by moving in this direction
- instead of denying and getting 2 and 2
- he could move in that direction right over there.
- So this is an ustable optimal scenario,
- but this Nash equilibrium, this state right over here
- is actually very, very, very stable.
- If they assume... this is, it's better for each of them to confess
- regardless of what the other one 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 the strategy
- 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 wanna do that.
- Or you could move in this direction,
- where it would be Al changing his decision.
- But once again that gets 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 would pick something
- that it's not optimal globally.
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