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Nash Equillibrium and the 'Prisoner's Dilemma'

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It seems as if the purpose of these theories is to state that a rational self-interested individual will inevitably conflict with another rational self-interested individual. I surmise that Rand would have had a problem with this. Here is the Prisioner's Dilemma entry from Wikipedia (I have bolded some particular parts):

Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one testifies ("defects") for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?

In this game, as in all game theory, the only concern of each individual player ("prisoner") is maximizing his/her own payoff, without any concern for the other player's payoff. The unique equilibrium for this game is a Pareto-suboptimal solution—that is, rational choice leads the two players to both play defectly even though each player's individual reward would be greater if they both played cooperately.

In the classic form of this game, cooperating is strictly dominated by defecting, so that the only possible equilibrium for the game is for all players to defect. In simpler terms, no matter what the other player does, one player will always gain a greater payoff by playing defect. Since in any situation playing defect is more beneficial than cooperating, all rational players will play defect, all things being equal.

The situation, along with Nash's solution, is aptly presented in the movie A Beautiful Mind in this 2 minute clip.

What are your thoughts on this theory? Am I misinterpreting it? It seems as if two rationally self-interested individuals would choose to cooperate by both remaining silent.

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What are your thoughts on this theory?
Well, a simple-minded equation "what I emotionally want = what I should focus on" is clearly wrong, and I don't think that Smith or certainly Rand would hold such a position. These so-called dilemmas exploit complexities of free will, non-omniscience and variable adherence to reason. The Prisoner's Dilemma itself requires accepting a contradiction (existence by force vs existence by reason), and when you start with a contradiction, you are gonna get a dilemma. I suggest trying to state the problem with respect to picking up chicks in a bar, where you avoid the basic contradiction.

The Wiki entry has numerous flaws, including major errors regarding what constitutes being rational.

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What are your thoughts on this theory? Am I misinterpreting it? It seems as if two rationally self-interested individuals would choose to cooperate by both remaining silent.

As David said, there is no omniscience. If two rational people were given all the facts, they both would remain silent. Since they don't know all there is to know, they would both, in terms of payoff alone, betray each other.

Of course one must wonder about the specifics. Are these people in fact guilty of anything? Are their actions morally a crime, or are they a victimless "crime" the police shouldn't be prosecuting in the first palce? If they are guilty of a real crime, how rational are they to begin with? How loyal are they to each other, and for what reason? Does one or the other know the police chief is banging his secretary and can that info be bargained for a slap on the wrist? I mean, once you start contriving unlikely scenarios, all bets are off.

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As David said, there is no omniscience. If two rational people were given all the facts, they both would remain silent. Since they don't know all there is to know, they would both, in terms of payoff alone, betray each other.

Perhaps I was under the supposition that both people knew all the facts, as I figured that was part of the 'dilemma.' But, assuming that they don't know what the other will do, I understand the situation better.

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As David said, there is no omniscience. If two rational people were given all the facts, they both would remain silent. Since they don't know all there is to know, they would both, in terms of payoff alone, betray each other.

Of course one must wonder about the specifics. Are these people in fact guilty of anything? Are their actions morally a crime, or are they a victimless "crime" the police shouldn't be prosecuting in the first palce? If they are guilty of a real crime, how rational are they to begin with? How loyal are they to each other, and for what reason? Does one or the other know the police chief is banging his secretary and can that info be bargained for a slap on the wrist? I mean, once you start contriving unlikely scenarios, all bets are off.

This is similiar to what has always bothered me about the dilemma. It disregards, or even eliminates trust between the two parties. Those business relationships I have that are best have strong levels of trust regarding intention and capacity. If crime were a business(rational endeavor) then things would go better for partners who had more trust and forethought.

Adding full context always evaporates supposed contradictions.

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Perhaps I was under the supposition that both people knew all the facts, as I figured that was part of the 'dilemma.'
Assuming the two are co-conspirators and actually guilty, they do not actually know what evidence the police have. It is reasonable to assume, based on the fact that the police are even talking about some plea bargain, that the evidence is insufficient, but the suspects do not automatically know that. Ergo one major source of knowledge difference. Second, because each has free will, even if A and B know the same things regarding the crime, arrest and evidence (to the point that they know that there is a risk of conviction), they cannot know in advance what the other will do, because they do not know the other person well enough.

Something I find very interesting, quite telling really, is that the scenario only states that they are "suspects", not that they actually did the alleged deed. The equation of "suspect" and "criminal" is part of the problem, and why there is such a thing as a "Prisoner's Dilemma". The other part is that people don't understand the underlying contradiction of being a Prudent Predator -- these "paradoxes" are part and parcel of one and the same corrupt meta-ethics.

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Perhaps I was under the supposition that both people knew all the facts, as I figured that was part of the 'dilemma.' But, assuming that they don't know what the other will do, I understand the situation better.

It's self evident. I mean no offense, but it's right there in the descritption of the dilemma. If both prisoners know all the facts, why separate them?

Now, assunming both are guilty, then the optimum result for the police is if they both betray each other, therefore thye are both locked up for a substantial term. The worst result for the police is if they both stay silent, because then the prisoners both get a slap in the wrist. Any other result has one prisoner getting off scot free. If the prisoners are aware of all the options, they'd both stay silent. the only chance the polic have at their best result is to keep facts from the prisoners, and to keep each prisoner's actions from the other.

That's a staple of cop shows, BTW. Each time they have two suspects, they question them separately. That way they can pressure them both more effectively. Say by telling suspect one that suspect two has already ratted him out, or thinks he's a moron, or is willing to give up the loot, etc etc. How does suspect one know whther or not suspect two did or didn't say any of that?

In reality, this tactic is used often enough. It's elementary. If you question suspects jointly, they can draw support from each other, warn each other off, etc. You want the suspects weak and vulnerable, the better to get information from them.

Back to the scenario. The best way to play out a scenario is to dramatize it. Give it actions and dialogue, and see whare it gets you. That's also how the flaws are more easily exposed, and the nonsense just bubbles up. Try it.

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The term 'rational' within neo-classical economics tends to have a very restricted meaning ('acting to maximise a utility function'). Theres a risk of equivocation when you start saying things like "Game theory says that if people are acting rationally blah blah blah" because you arent using the term 'rationally' in a way which other people are likely to accept.

Anyway Game Theory is perfectly valid mathematically and the relevant results are all formally provable. Its possible to show that you can construct a pay-off matrix which contains a Nash Equilibrium which isnt pareto optimal ('pareto optimal' means the best result for both parties). However the problem comes when you try to take your purely formal payoff matrix and apply it to the real world - the story of the prisoners dilemma is one attempt to 'translate' this mathematics into English, but these translations sometimes rely on thought experiments which are highly implausible and omit a lot of real-world details. So yeah, the maths is obviously valid, its the application of it which is often flawed (as others have pointed out, the classic statement of the prisoner's dilemma ignores pretty much all the complexities which would exist if that situation were to actually occur).

However even if you take the statement of the prisoners dilemma as it stands, theres a lot of empirical evidence from iterated simulations which shows that the best strategy is to try and cooperate, assuming that the game is going to be played multiple times.

Edited by eriatarka
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It's self evident. I mean no offense, but it's right there in the descritption of the dilemma. If both prisoners know all the facts, why separate them?

It is that the whole situation is a bit absurd to me, and I guess I had a hard time getting past that. Why would a rational person not remain silent? The only reason to do anything else is if he believes that the other individual would do something which would have an adverse effect on his interests. But I am assuming two rational individuals, (and so is the scenario) who both have the potential to think the situation through and fully understand the best decision. In such a case, why wouldn't both the prisioners remain silent?

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Why would a rational person not remain silent?
A rational person would know that the other person might not remain silent. Since this is an exclusive-or kind of choice, the fact that it's possible that the other person will rat you out is reason to not remain silent. Silence would be "rational" (in the reality-evading sense that we're using it) only if you are certain of silence from the other person. The fact that they both have a rational faculty which allows them to reason the situation out does not mean that they will actually use that faculty in the desired way.

This is one reason why Baltimore gangs are strictly enforcing a no-snitch rule, so that force dominates reason. You don't have to think, you just know that talking to The Man is a death sentence.

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It is that the whole situation is a bit absurd to me, and I guess I had a hard time getting past that. Why would a rational person not remain silent? The only reason to do anything else is if he believes that the other individual would do something which would have an adverse effect on his interests. But I am assuming two rational individuals, (and so is the scenario) who both have the potential to think the situation through and fully understand the best decision. In such a case, why wouldn't both the prisioners remain silent?

Knowing all options, and knowing what the other prisoner does, the rational choice is for both to stay silent. Work it out thus:

If A betrays B, then B will betray A in payback and to lessen his sentence to 5 years. therefore it's not in A's best interst to betray B, as he'd be buying a 5 year sentnece. Likewise with positions reversed.

If A stays silent, then B stays silent. Becuase if B betrays a, then A will hcange his mind and betray B. thereofre it's in A's best ineterst to stay silent. Likewise with positions reversed.

If options and facts are kept from both A and B, then they will both turn on each other, assuming payoff is the only criterion, because they both think that will get them off. In truth they would each get 5 years, which means the police lied to them (the police lie to suspects all the time, that much is not contrived), and fooled them both.

If the dilema assumes all options are known but not all facts, then it assumes that staying silent is the selfless thing to do. That is, you sacrifice yourself for the toher in hopes the other will likewise sacrifice himself for you. It is a asacrifice because staying silent may help or hurt you, depending on what the other does, while betrayal helps you by setting you free at best and giving you half the maximum sentence at worst.

Now let's dramatize:

Dagny and John are apprehended on suspicion of antisocial activities by Wesley Mouch. Kept apart, they are informed if both stay silent they each get six months, if one betrays the other one goes free and one gets ten years, if both betray each other they both get five years. If there is any question in your mind that they'd both stay silent, then re-read AS :)

BTW the scenario does not contemplate a confession. Why not? Because a confession is an entirely individual action. Confess and you get some jail time, say 7.5 years. But your confession does not affect your buddy, right? Or it screws up the matrix of options. What if A confesses and B stays silent. Does A get 7.5 years and six months while B gets six months? What if A confesses and B betrays A? Does A get 7.5 years and five years while B gets five years? It no longer makes sense, right? You'd have to tweak it more.

As wit all other of this type of scenario: do not bother to examine a folly, only ask yourself what it accomplishes.

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This is similiar to what has always bothered me about the dilemma. It disregards, or even eliminates trust between the two parties. Those business relationships I have that are best have strong levels of trust regarding intention and capacity. If crime were a business(rational endeavor) then things would go better for partners who had more trust and forethought.

Adding full context always evaporates supposed contradictions.

That is a part of the contrivance. You are not dealing with rational businessmen, you are dealing with thieves. You are not dealing with honest decisions in the first place but in guilty persons attempting to avoid a punishment for crimes committed.

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Just a point of clarification on this puzzle: both 'prisoners' are supposed to know all the facts about the payoffs of the various options, but the reason they are separate is so they cannot coordinate their actions. Each makes his choice independently and without knowledge of what the other is doing. The reason that the only "rational" choice for the individual is to rat is because if you rat, the two possible outcomes for you are no sentence or five years, whereas if you do not rat the two possible outcomes for you are six months or ten years. Even though it is more likely you will be serving the five years if you rat (unless you get lucky and the other guy is a sucker), if you stay silent you are almost certain to get burned.

If anyone is interested though, that solution only holds for a single instance of this scenario. They have put this model into a computer with various other values for the outcomes (in my bio class we did it as "points" where by cooperating each gets three, by defecting each gets one, and when one defects on the other the defector gets 10 and the other gets 0) and the best strategy over the long haul, according to the computer simulation, is called the tit-for-tat strategy. You open by cooperating, and then just do whatever the other person did to you last time. That strategy and its variants win. Game theory and its related mathematical models are very important in the study of how cooperative behavior can arise through evolution.

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If anyone is interested though, that solution only holds for a single instance of this scenario. They have put this model into a computer with various other values for the outcomes (in my bio class we did it as "points" where by cooperating each gets three, by defecting each gets one, and when one defects on the other the defector gets 10 and the other gets 0) and the best strategy over the long haul, according to the computer simulation, is called the tit-for-tat strategy. You open by cooperating, and then just do whatever the other person did to you last time. That strategy and its variants win. Game theory and its related mathematical models are very important in the study of how cooperative behavior can arise through evolution.
Sort of. The optimal strategy for an individual depends on the rest of the population - theres no intrinsically 'best' strategy independent of environment. In general, the tit-for-tat strategy performs very well, but you can have evolutionary stable populations which contain the three strategies 'tit-for-tat', 'always cooperate', and 'always defect', and depending on the population ratio, the 'always defect' entities might do best

http://en.wikipedia.org/wiki/Evolutionaril...dilemma_and_ESS

Edited by eriatarka
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Sort of. The optimal strategy for an individual depends on the rest of the population - theres no intrinsically 'best' strategy independent of environment. In general, the tit-for-tat strategy performs very well, but you can have evolutionary stable populations which contain the three strategies 'tit-for-tat', 'always cooperate', and 'always defect', and depending on the population ratio, the 'always defect' entities might do best

http://en.wikipedia.org/wiki/Evolutionaril...dilemma_and_ESS

But if that were the case, you would no longer have an evolutionarily stable strategy, and cooperation will disappear from that population. Unless what you're saying is, the environment may change later to make cooperation favorable again, or it is a frequency-dependent strategy and will become less successful if the proportion of cheaters gets too high.

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But if that were the case, you would no longer have an evolutionarily stable strategy, and cooperation will disappear from that population. Unless what you're saying is, the environment may change later to make cooperation favorable again, or it is a frequency-dependent strategy and will become less successful if the proportion of cheaters gets too high.

No I'm saying that theres probably an equilibrium state which contains all 3 types of strategies (I think anyway, too tired to try and prove it). There's a negative-feedback mechanism at play because if the percentage of Always Defects increases too much then the Always Defect strategy starts to do badly because they always get the worst result against each other. The Always Defect approach only works if theres a significant number of Always Cooperate's in the population because thats when they do best. And Always Defect performs better against Always Cooperate than tit-for-tat does.

In a population that contains a lot of Always Cooperate's then the best strategy for an individual is Always Defect rather than tit-for-tat. And this doesnt mean that the Always Defect will take over entirely, since this situation is basically analogous to Hawks and Doves which has an ESS containing both types of entities: http://en.wikipedia.org/wiki/Hawk-Dove_game

Tit-for-tat is a good strategy because it's robust - it performs well in most types of environments. But given prior knowledge about the population make-up of an envioronment, you will sometimes be able to find a strategy which outforms it (in that particular enviornment).

Edited by eriatarka
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Just for my edification, what does a "evolutionarily stable equilibrium" state represent, exactly?

An evolutionarily stable strategy is a strategy that can persist in a population despite the intrusion of competing strategies. It doesn't have to be the "best", or "win", it just has to persist given the current conditions. Obviously it's a very context-dependent thing - no strategy is going to be stable and persist in all times and all conditions. It can also be a frequency-dependent phenomenon, as erikatrka was explaining (got what you were saying now, thanks). I don't know that there is such a thing as an evolutionarily stable equilibrium, unless you mean a population where the frequency of alleles is not changing, which is always going to be a temporary thing. That would be something along the lines of a Hardy-Weinberg equilibrium, which is really just a useful hypothetical that doesn't actually occur in the real world.

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An evolutionarily stable strategy is a strategy that can persist in a population despite the intrusion of competing strategies. It doesn't have to be the "best", or "win", it just has to persist given the current conditions. Obviously it's a very context-dependent thing - no strategy is going to be stable and persist in all times and all conditions. It can also be a frequency-dependent phenomenon, as erikatrka was explaining (got what you were saying now, thanks). I don't know that there is such a thing as an evolutionarily stable equilibrium, unless you mean a population where the frequency of alleles is not changing, which is always going to be a temporary thing. That would be something along the lines of a Hardy-Weinberg equilibrium, which is really just a useful hypothetical that doesn't actually occur in the real world.

Thanks Kat. This raises my concern with the concept. I too am starting to think that it is a hypothetical that doens't occur in the real world, but that woudl make it "not so useful". If the options under discussion in the Nash equilibrium are inherently ones where rational man can change the "current conditions" then any discussion about what works for now, will always be overshadowed in the long tem by the discussion of what actually is the best, surviving strategy - or if there is one -given the reality of ever fluctuating conditions.

Not addressed to you Kat, but in general: to me the Nash equilibria don't overturn the idea of rational self-interest, but rather just show under what conditions do things like cooperation reflect a self-interested strategy.

Edited by KendallJ
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Thanks Kat. This raises my concern with the concept. I too am starting to think that it is a hypothetical that doens't occur in the real world, but that woudl make it "not so useful". If the options under discussion in the Nash equilibrium are inherently ones where rational man can change the "current conditions" then any discussion about what works for now, will always be overshadowed in the long tem by the discussion of what actually is the best, surviving strategy - or if there is one -given the reality of ever fluctuating conditions.

Not addressed to you Kat, but in general: to me the Nash equilibria don't overturn the idea of rational self-interest, but rather just show under what conditions do things like cooperation reflect a self-interested strategy.

I didn't think anyone was using it to argue against rational self-interest. Also, an evolutionarily stable strategy does exist in nature - any behavior that persists in a population long-term and increases the fitness of its practicioners could be an ESS. The key is, these concepts are not meant to describe human interaction necessarily. The original application of game theory to ecology was to answer the question, "How could cooperative behavior evolve and persist (in non-human animals) if those who cooperate risk a fitness penalty against those that would take advantage of a cooperation?" This is geared towards describing patterns in vampire bats, or ground squirrels, not people.

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I didn't think anyone was using it to argue against rational self-interest.

I woud look at the opening post in this thread and question that assumption. It seems to be part of the discussion.

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I didn't think anyone was using it to argue against rational self-interest.

Kendall is right. My original post was about this particular aspect of the theory. The theory is used to argue against rational self interest. The essence of the theory seems to be: "Two rational self interested individuals will inevitably have a conflict of interest." If you browse through any introductory economics text, the dilemma and the Nash equilibrium will be discussed in some detail. The full implications of it are extended to prove that free-markets do not always "work".

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The theory is used to argue against rational self interest. The essence of the theory seems to be: "Two rational self interested individuals will inevitably have a conflict of interest." If you browse through any introductory economics text, the dilemma and the Nash equilibrium will be discussed in some detail. The full implications of it are extended to prove that free-markets do not always "work".

Nash equilibria and the Prisoner's Dilemma can also refer to specific mathematical concepts, which can arguably be useful for various decision models. In other words, these concepts need not be packaged together with social commentary that leads to terrible policy conclusions such as the presumed necessity of antitrust regulation. Unfortunately, too many neoclassical economists like to assume that all individuals are short-sighted utility function maximizers.

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The problem with prisoner's dilemma as an analysis of economic behavior is it tricks one into thinking that the trading is between the two prisoners.

In fact, it is between the prisoner and the cop. The "rational" outcome, that is, that one prisoner snitches out the other, is a free exchange between prisoner and cop, and results in the prisoner optimizing his outcome (minimizing his term), and the cop optimizing his outcome (maximizing the amount of jail time dispensed).

Economically, the prisoner starts out with a ten year term, assuming he's guilty. The cop starts out with four years dispensed jail time (the least he will give, if both prisoners keep mum).

By offering a deal to the prisoner, the cop allows him to "gain" five or ten years of free time, and in return the cop will "gain" ten years. Both prisoners, according to the rules, deal with the cop, each of them gains five years, and the cop gains ten years. This results in the maximum aggregate "gain" to all participants (20 years), and also optimizes the gain of the participant with the lowest outcome (in this case, both prisoners tie for worst outcome, a gain of five years).

If the prisoners were allowed to pool their economic power, that is, deal with the cop as a collective rather than individuals, they might agree to keep silent and minimize their aggregate jail time. The result is a suboptimal result (16 years), which goes to show that individuals acting in rational self interest gives better aggregate results than collusive behavior.

What do you know? Free markets do "work!"

Edited by agrippa1
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Two suspects are arrested by the police. The police have insufficient evidence for a conviction, and, having separated both prisoners, visit each of them to offer the same deal: if one testifies ("defects") for the prosecution against the other and the other remains silent, the betrayer goes free and the silent accomplice receives the full 10-year sentence. If both remain silent, both prisoners are sentenced to only six months in jail for a minor charge. If each betrays the other, each receives a five-year sentence. Each prisoner must make the choice of whether to betray the other or to remain silent. Each one is assured that the other would not know about the betrayal before the end of the investigation. How should the prisoners act?

Here are the options:

Defect, Silent :: Free, 10-years

Defect, Defect :: 5-years, 5-years

Silent, Silent :: 6 months, 6 months

A rational "player" will remain silent. Not for the rules or consequences of the "game", but because he is rational. "A rational man sees his interests in terms of a lifetime and selects his goals accordingly." "Integrity does not consist of loyalty to one's subjective whims, but of loyalty to rational principles." I think this means that even if a player knew the other would betray him, as a rational man with integrity, he could not betray the other player.

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