Talk:Prisoner's dilemma/Archive 2

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Error in cigarette advertising example?

In the cigarette advertising example the article says: As the best strategy is dependent of what the other firm chooses there is no dominant strategy and this is not a prisoner's dilemma. The outcome is though similar in that both firms would be better off were they to advertise less than in the equilibrium. I don't see how this differentiates it from the standard prisoner's dilemma. In the standard prisoner's dilemma the optimal situation to both players is that they both cooperate so clearly, if you know the other one is cooperating of course you should cooperate too and defect otherwise. Clearly there isn't a dominant strategy. Thus, the optimal strategy does depend on what the other player chooses just as the article says for the cigarette example. Note that I don't like this way of expressing it in either case because the purpose of the strategy is to provide a decision faced with the uncertainty about what the other person is doing. If you know what the other player does the game is completely uninteresting.

Sorry - I just noticed that I mis-remembered the prisoners dilemma. I thought both would go free if both stayed silent (due to lack of evidence or whatever). In that case there wouldn't be a dominant strategy and it would be equivalent to the cigarette example.

Any 'Usual ranges' of values on the payoff matrix values?

it might be intuitive that the ultimate choice will depend on P (Punishment for mutual defection) in the payoff matrix: for example if both cooperate they will get 6 months each, both get 8~9 years if both defect, and the cooperate one get 10 years if the other one defects (of course the one who defects in this case goes free), then the expectations is the insignificant marginal benefits of actually defecting (just in case the other person/algorithm does defect) that their decision may just as well be cooperating.

From the above, here is an suggestion for a modification to the problem: the rules will be the same except P will have a range, ranging from some attractive values (Close to R) to some severe case (Close to S) so the inequalities T>R>P>S and 2R>T+S are still satisfied, but it's the actual value of P we are hiding from the two prisoners (The range is given and distribution may or may not be available)

Now what is the optimal individual(selfish) decision? does it depend on the fact that the prisoners know the distribution or not?

Misc comments

Shouldn't it be "Prisoners' Dilemma"? There's more than one prisoner, remember. -- Cabalamat 18:15, 27 Aug 2003 (UTC)

But its an individual's decision

i made a change regarding the tv game show example. i realise that some people in the US may subconsciously feel that a television game show is "real life" and not at all an "artificial setting". But i would think that once they think about it they would accept that a TV game show, controlled by some dictatorial media organisation and mega-bucks, with every aspect of the physical and social environment highly controlled, is hardly a very natural setting... Boud 14:43, 17 Dec 2003 (UTC)

Gotta say, I don't think the Assurance Game or Chicken are in any way variants of the Prisoner's Dilemma... Evercat 23:51, 17 Dec 2003 (UTC)

I don't know where to start editing this, but an important paper appeared in Science_(journal) in by Axelrod co-authored with W.D. Hamilton in 1981, and subsequent contests which should be mentioned as this predates the 1984 book by Axelrod.

Thanks for the reference. Dropped it into the list and made the Refs formatting consistent while I was up. Haven't done anything to compare it against article content, though. Dandrake 20:47, Mar 12, 2004 (UTC)

I've always felt that the basic flaw in the Prisoner's Dilemma is that it is so obviously an artificial set up. Would any real police force set free a confessed criminal? In other words it is solely an intellectual excercise and deliberately unrealistic, so it's hard for me to take it seriously even if it does prduce useful insights. Lee M 02:05, 15 Mar 2004 (UTC)

I guess you wouldn't have enjoyed the debate between Bohr and Einstein when they duked it out over quantum indeterminacy using a silly thought experiment in which a photon is released from a box that's weighed before and after. But they found it useful. Seriously, this thought experiment is a bit unrealistic, but it's analogous to a lot of real-life situations. including the tragedy of the commons; so economists who take game theory seriously have to worry about it. Dandrake 02:23, Mar 15, 2004 (UTC)

Well, maybe I'm just too literal-minded (although to look at my list of contributions you might not think so)...Lee M 01:46, 16 Mar 2004 (UTC)

This is actually a common prosecutorial tactic. A person involved at the lowest levels of a crime is offered freedom from prosecution or, more commonly, a greatly reduced punishment, in exchange for testimony against the defendants whom the prosecution really wants to convict. I think that Martha Stewart's stockbroker's assistant was in this category; he participated in the crime, but because of his testimony he won't go to jail. Anyway, if you don't like the witness setup, the PD situation arises elsewhere. For example, several companies selling a competing product might enter a price-fixing agreement. As long as each of them observes the agreement, all of them will profit by charging prices higher than what the market would provide. One "cheater," though, that reduces its price slightly below the agreed-upon floor, will gain market share and thus make even more profits. Of course, other conspirators see this happening and lower their own prices. Eventually the cartel falls apart. Not all cartels fall apart, but when they do, it's a PD situation -- where each decisionmaker pursues his own immediate interests, everyone is worse off than they would be if each had acted based on a sense of the common good or some other approach that differs from the classical interest-maximizing view of rationality. (Of course, because others can see and react to a decision, the cartel example is closer to the iterated version of PD.) JamesMLane 13:17, 18 Mar 2004 (UTC)

I'm not sure that this picture is really appropriate, since the Prisoner's Dilemma really has very little to do with literal prisoners. Perhaps an image of a strategy table for the game would be more appropriate. Adam Conover 05:22, Mar 16, 2004 (UTC)

The Prisoner's dilemma is a totally abstract concept, so you're not going to get a point-on picture. A board game like you suggest is going to be even harder to "get". I decided to opt for a picture that illustrates the two prisoners themselves and I think it's a good fit, personally. →Raul654 05:25, Mar 16, 2004 (UTC)

Actually, the picture you posted was exactly what I was hoping for. I entirely approve. Adam Conover 09:33, Mar 17, 2004 (UTC)

The diagram to the right of the first explanation (right after the Brief Outline heading) does not match the explanation. Lkesteloot 07:15, 16 Mar 2004 (UTC)

Duly fixed. Only took 6 tries. Damn it's late ;) →Raul654 07:25, Mar 16, 2004 (UTC)

Wow, that was fast! Thanks! Lkesteloot 07:36, 16 Mar 2004 (UTC)

Blah!

re: the paragraph on communication (1)

I feel the following section is misleading at best:

This illustrates that if the two had been able to communicate and cooperate, they would have been able to achieve a result better than what each could achieve alone. There are many situations in the social sciences that have this same property, so understanding these situations and how to resolve them is important. In particular, it demonstrates that there are situations where naïve reasoning in one's own best interest does not always lead to the best result.

Communication is not a prerequisite here. Unless the two prisoners trust each other utterly, communication in advance won't help them get out of the dilemma. This because there is still the maximum payoff -- convince the other to keep their mouth shut while you confess, thus gaining instant freedom.

It also is completely implicit here what 'better' means. It should be made explicit that the total jail time for both prisoners together is least if both choose to stay quiet, but this isn't best for an individual prisoner. This is actually an important part of the dilemma.

This is repeated later on where it claims that 'naïve reasoning in ones own best interest' is to blame for the non-optimal result. This doesn't make sense as there is no reasoning that is actually the wise choice for a selfish individual.

Furthermore, the prisoner's dilemma is applicable far outside the social sciences. It is also very important to evolutionary biology, for instance.

--Martijn faassen 23:03, 16 Mar 2004 (UTC)

I think perhaps you don't understand what a prerequisite is. As stated in the paragraph there are two: to communicate, and to cooperate. Markalexander100 02:04, 17 Mar 2004 (UTC)

(To finish a particular debate before it starts: I don't want to discuss my understanding of certain words. Back to the problem.). Communication is not required to make the choice to cooperate with the other prisoner. Communication also doesn't necessarily lead to the choice to cooperate. The description is therefore misleading at best. In addition as I already mentioned, it is left implicit what 'better' means in this context.

With some good will you can definitely interpret the paragraph so that it is strictly spoken correct, but it certainly could communicate its meaning better than it does not.

I think actually the problem starts in the second paragraph of the description of the dilemma, which describes one particular way of reasoning and one particular outcome. This seems to be a particular resolution of the situation, not the part of the dilemma itself. Is this part of the classical description of the dilemma?

I'd propose rewriting the last paragraph of the "dilemma" and the paragraph under it to something as follows:

Let's assume both prisoners are completely selfish and want to minimize their own jail term. As a prisoner you have two options: to cooperate with your accomplice and stay quiet, or to defect and betray your accomplice and confess. The outcome of each choice depends on the choice of your accomplice. If your accomplice chooses to cooperate and stay quiet, the optimal choice for you would be to defect, as this means you get to go free immediately, while your accomplice lingers in jail for 10 years. If your accomplice chooses to defect however your best choice is to defect as well, since then at least you can be spared the full 10 years serving time and have to sit out 5 years, while your accomplice does the same. If however you both decide to cooperate and stay quiet, you would both be able to get out in 6 months.

The naïvely selfish reasoning that it is better for you to defect and confess is flawed; both would end up in jail for 5 years. Even sophisticated selfish reasoning does not get you out of this dilemma however; if you count on your accomplice to cooperate it may be smartest to defect. However, if your accomplice knows this and thus would defect, it would be smartest to cooperate. And so on. This is the core of dilemma.

If reasoned from the perspective of the optimal interest of the group (of two prisoners), the correct outcome would be for both prisoners to cooperate, as this would reduce the total jail time served by the group to one year total. Any other decision would be worse. In a situation with a payoff structure like the prisoner's dilemma individual selfish decisions are not automatically best if viewed from the perspective of the group as a whole.

The particular example of the prisoners may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature which have the same payoff structure. The prisoner's dilemma is therefore of interest to the social sciences such as economics, politics and sociology, as well as to the biological sciences such as ethology and evolutionary biology.

What do people think? --Martijn faassen 22:41, 17 Mar 2004 (UTC)

Understanding the wording of the article would be a good first step. Communication is required because it is the basis for cooperation. At the moment, we have two concise, accurate paragraphs; replacing them with four wordy paragraphs would be counterproductive. Markalexander100 01:48, 18 Mar 2004 (UTC)

(I don't want to go into a debate about my understanding of the English language, thanks.). Communication is not required as the basis of cooperation, as I pointed out several times. Cooperation can be iniated just fine without communication in this case. This is true for many instances of the prisoner's dilemma. The iterated dilemma simulations are a good example - the agents just choose to cooperate or defect without advance communication.

I feel the two paragraphs, while arguably accurate if you already know how to interpret them (what does 'best' mean in this context, what is cooperation), do not explain the dilemma well enough. The text also implies that there is a way to 'resolve' the dilemma, while the idea behind the dilemma is that for a selfish individual involved there is no safe strategy. This is the dilemma. The other party can always choose to defect, in which case the right selfish choice is to defect as well.

[cutting in] But the right selfish strategy is always to defect no matter what the other guy does, unless there's some way in which his action is influenced by yours. Evercat

This is a good description of one form of reasoning. If both reason this way however, you'll both be in jail for 5 years. Hardly the right selfish strategy for both of you. Isn't the dilemma wonderful? --Martijn faassen 22:00, 18 Mar 2004 (UTC)

Sensible strategies to resolve the dilemma arise when there is more context, such as in the iterated variety.

I feel more than a brief outline of the dilemma is needed before launching into a long discussion on the iterated variety, as otherwise one risks a reader's misunderstanding of the basic dilemma. --Martijn faassen 09:39, 18 Mar 2004 (UTC)

Communication is completely irrelevant to the Prisoner's Dilemma. One of the fundamental points of the problem is that neither player knows what the other is going to do in advance: it is impossible for them to communicate. Martijn is quite correct, therefore, to say that communication is not a pre-requisite for co-operation; not in the PD problem, anyway.

The first paragraph of the re-written text makes the mistake of assuming that one prisoner has the opportunity to base their choice on a knowledge of what the other one has done. This is a misunderstanding of the PD problem. The two prisoners make their choices simultaneously, in separate cells, with no possibility of communication — so it is incorrect to say "The outcome of each choice depends on the choice of your accomplice." It would be correct, of course, to say that in an Axelrod-type 'Iterated' PD game, a player's choice will be influenced by the other player's previous choice(s). But, obviously, that doesn't apply in a one-off PD such as the 'classical' scenario.

Apart from that flaw, though, the re-written text looks generally like an improvement to me. R Lowry 19:25, 18 Mar 2004 (UTC)

Thanks. I can see where you are coming from about my first paragraph. I think this was not due to a flaw in my understanding but due to a flaw in not being explicit enough in my own writing. The outcome (in amount of jail time) of the choice does depend on the choice of your accomplice, though, which is what I was trying to point out. The choice of course doesn't as you don't know what your accomplice chose/will choose. Simultaniety of decision is not required. As I wrote earlier way up on this page: "The only thing needed to create this dilemma is the non-communication (by a reliable source) of the other person's actual decision before you make your own."

Here's a suggested rewrite of the first paragraph (split into two).

Let's assume both prisoners are completely selfish and want to minimize their own jail term. As a prisoner you have two options: to cooperate with your accomplice and stay quiet, or to defect and betray your accomplice and confess. The outcome of each choice depends on the choice of your accomplice. Unfortunately you don't know the choice of your accomplice; even if you were able to talk you couldn't be sure whether to trust him.

If you expect your accomplice will choose to cooperate and stay quiet, the optimal choice for you would be to defect, as this means you get to go free immediately, while your accomplice lingers in jail for 10 years. If you expect your accomplice will choose to defect, your best choice is to defect as well, since then at least you can be spared the full 10 years serving time and have to sit out 5 years, while your accomplice does the same. If however you both decide to cooperate and stay quiet, you would both be able to get out in 6 months.

Further tuning is likely necessary. It's clear our communication about this is easy to misunderstand, which shows we should describe things carefully. --Martijn faassen 22:00, 18 Mar 2004 (UTC)

Yes. You're quite right, of course, to say "The outcome (in amount of jail time) of the choice does depend on the choice of your accomplice. [...] The choice of course doesn't as you don't know what your accomplice chose/will choose." I think I just misread what you were saying earlier (although, in my defence, the text did seem to indicate prior knowledge of the other prisoner's behaviour). All in all, I like the new improvements. R Lowry 07:39, 19 Mar 2004 (UTC)

Iteration involves communication- you know what the other one did last time. That's why it's a way out of the dilemma. Markalexander100 01:40, 19 Mar 2004 (UTC)

re: the paragraph on communication (2)

A question... Where does this:

This illustrates that if the two had been able to communicate and cooperate, they would have been able to achieve a result better than what each could achieve alone.

come from? If you allowed the two prisoners to communicate, and they did so, and went back to their seperate rooms to report to the police officers, how would their situation have been changed from before? Neither has gained any information, as the other's statements made during the conversation have no binding importance. The same argument of "no matter what the other guy does, I'm better of defecting (confessing)" still applies, doesn't it? Either confessing is the right thing to do beforehand and afterhand, or it's neither, right? This has been sort of hinted at somewhere above, but I just wanted to phrase my concern in a consolidated sort of way -- kine

The point is not that communication solves the dilemma; it's that communication allows (but not requires) cooperation, which solves the dilemma. Markalexander100 04:36, 20 Mar 2004 (UTC)

Or, more precisely yet, communication facilitates cooperation, which solves the dilemma. Even with communication, kine is right that one prisoner has the ability to betray the other. Conversely, they can cooperate without communicating. For example, they might have agreed in advance what to do in this situation, or each of them might feel enough loyalty to the other (or to their joint criminal enterprise) to act cooperatively without communicating. JamesMLane 05:44, 20 Mar 2004 (UTC)

"they might have agreed in advance what to do in this situation" which would presumably require communication. Certainly if they're motivated by loyalty, then there is no dilemma, but that's why the statement of the dilemma supposes that they're selfish. Markalexander100 06:06, 20 Mar 2004 (UTC)

As I have argued elsewhere on this page, the issue of communication simply isn't relevant to the Prisoner's Dilemma. The way the problem is set up is that the two prisoners have to make their decision independently, without negotiation or any other sort of communication with each other. That absence of communication is one of the defining characteristics of the PD: neither prisoner knows what the other is going to do, so they each have to make their decision alone, second-guessing the other.

The paragraph that is causing this misunderstanding (the last para in the 'Brief Outline' section), was identified as problematic a few days ago and some possible alternatives were presented (please see above, re: the paragraph on communication). It would be helpful if people could have a look at those and comment on how they could be improved upon. R Lowry 10:25, 20 Mar 2004 (UTC)

"the issue of communication simply isn't relevant to the Prisoner's Dilemma"

"absence of communication is one of the defining characteristics of the PD"

Those two statements are directly contradictory. Markalexander100 02:25, 21 Mar 2004 (UTC)

Well, maybe I chose my words loosely, but hopefully the point that I was making is clear enough: neither prisoner can communicate with the other, so they each have to make their decision alone, second-guessing the other. It is, therefore, nonsensical to speculate about what might have happened if the two prisoners "had been able to communicate and cooperate" - the PD problem simply doesn't allow for that to happen. R Lowry 23:56, 21 Mar 2004 (UTC)

I'm of the opinion that even if the prisoners are able to communicate, the dilemma would still stand if minimizing jail time is the only thing that matters to the prisoners. In reality the situation is messier, and in communication one prisoner could threaten to have his friends kill the accomplice's family if he even thinks of confessing, or alternately promise golden mountains if only the accomplice stays quiet, so non-communication is better to make them into 'ideal selfish agents'. Even non-communication could lead to such speculation though: if you're some unimportant flunky and you know your accomplice is the kingpin of crime, for instance. Anyway, these are all different ways in which the real life payoff matrix is never identical to the pure jail term payoff matrix; these considerations change the payoff matrix.

The real life payoff matrix will be identical to the jail term only if the only thing that matters to the prisoners is minimizing jail time. Communication then cannot help in resolving the dilemma. I'm not sure whether we need to include such a stipulation; the prisoner's dilemma is only an illustration of (and name for) this particular payoff matrix. --Martijn faassen 23:38, 22 Mar 2004 (UTC)

re: the payoff matrix

The payoff matrix is backwards. According to the article, if you confess involvement in the crime, and your partner denies it, you go free, while he gets 10 years in jail. Clearly should be the other way around.

--Egomaniac 22:34, 17 Mar 2004 (UTC)

I don't think so. The procecutor is trying to get you to confess (about your and accomplice's involvement). The procecutor presumably wants to maximize the amount of jail time spent. Anything is better than both prisoners staying quiet and the crime going unsolved.

If you get 5 or 10 years in jail if you confess, then there is no dilemma for a selfish prisoner. Just deny. Either you get off free (your partner gets 10 years) or get just 6 months both. --Martijn faassen

The point is that the important confession is your accusation of your partner, not of your own involvement. That's the only way the dilemma makes sense -- to "deny" is to deny that your partner was involved, and to "confess" is to accuse your partner of being involved. Otherwise, how can the payoff matrix possibly make sense? --Egomaniac 18:15, 18 Mar 2004 (UTC)

That is a good point. I think I've seen the dilemma described in this way before (googling; yeah, found a description along those lines). So, you and he accuse each other, 5 years in jail for both. Only you accuse the other, you get off, 10 years in jail for him. You both stay quiet, both stay in jail on a minor charge for 6 months. This means that the first paragraph of the 'brief description' could be improved as well. I've been too focused the payout matrix; this is correct. For the theory it's fairly irrelevant how this matrix is motivated in particular but obviously we need to give a good description of it nonetheless. --Martijn faassen 22:00, 18 Mar 2004 (UTC)

I'm pretty sure that the payoff matrix is correct as it stands, in the context of the 'classical' example (ie, the one with the prisoners). The reduced sentence for confession is the prisoner's reward for being a 'grass'. (One concern: I don't think the payoff matrix should have the prisoner get off with no sentence at all when the other denies - I think it should just be a reduced sentence - but that's a relatively small detail.)

I do think, though, that actually the 'classical' example is a little confusing; I made the same mistake as Egomaniac when reading this article, and that misreading is not unusual in my experience for someone trying to understand the classical PD. It's actually far clearer, in my view, to express the PD - as Robert Axelrod, for example, does - in the form of a simple two-player game. When described in that way, the PD payoff matrix typically reads something like this:

	Cooperate	Defect
Cooperate	3, 3	0, 5
Defect	5, 0	1, 1

I certainly don't think that we should get rid of the classical scenario, because it is very much a part of the PD literature and should stay in the article. But I do think that the other, more straightforward, matrix ought also to be worked into the article somehow — possibly in the 'Iterated PD' section, along with the description of Axelrod's tournament. R Lowry 19:40, 25 Mar 2004 (UTC)

Just a note that I agree with User: R Lowry on working this matrix into the article somehow. Martijn faassen 21:01, 29 Mar 2004 (UTC)

@Cyrius (rv - new version is less clear than old version)

reverted version about paradox in prisoner's dilemma

It can not be "less clear", it makes a different point. The prisoner's dilemma is not primarily about selfish behaviour vs. cooperative behaviour, but leads to a paradox. Even if I don't care about the fate of my fellow prisoner, there are two completely logical rationales with contradicting outcome to what I should do. --Moon light shadow 21:28, 19 Mar 2004 (UTC)

No there aren't - there's only one logical course of action if one is selfish. The paradox is that both persons acting selfishly come off worse than if they'd both helped the other. Evercat 21:54, 19 Mar 2004 (UTC)

Thank you for making my point, Evercat. Having two choices is not a paradox. To restate: the problem is that a prisoner, selfishly trying to reduce his sentence, will increase it. And if you think the prisoner's dilemma isn't about selfish vs cooperative behavior, then I suggest you read the rest of the article. -- Cyrius 22:40, Mar 19, 2004 (UTC)

"a prisoner, selfishly trying to reduce his sentence, will increase it" - not exactly - by acting selfishly, he's always reducing his sentence compared to what he would have got. It's the action of the 2 people together that causes the problem... Evercat 23:35, 19 Mar 2004 (UTC)

Yes, there are. The argument for denying is just as selfish as the one for confessing. The argument for denying is that the situation is symmetrical, and because of that both will take the same choice, if there is a rational choice for the problem at all. You don't need to argue with being unhappy with the outcome of the other argument for confessing here. You also don't need to talk in terms of "cooperation" or "communication" to come to that conclusion.

Having two choices is not the paradox. The paradox that there are completely logical arguments for both choices and both being completely selfish.

I don't say that the dilemma can not be seen as a conflict between selfish vs cooperative behavior, but this is completely irrelevant for the paradox. --Moon light shadow 10:07, 20 Mar 2004 (UTC)

There is no valid selfish argument for denying, as it's easily shown that, no matter what the other guy does, you're better off confessing. The only (selfish) reason for denying is if one believes one's own action will somehow influence the other guy's. Evercat 12:20, 20 Mar 2004 (UTC)

Your common sence leads you on the wrong track, when thinking about a paradox. You can indeed proof that you are better off confessing, no matter what the other guy does. Now forget that for a moment! Forget that you may what them to cooperate! Forget that you labeled confessing as selfish and denying as cooperative! Then, please, read my paragraph again: "An other rationale reasoning goes like this: I know that the other prisoner is in the same situation as I am. If we both decide rationally, we will come to the same conclusion. So we will either both confess and serve 5 years, or we will both deny and serve 6 months. Accourding to this argument it is better for me to deny." --Moon light shadow 13:04, 20 Mar 2004 (UTC)

But this argument is clearly fallacious, since we've already proven that it's always best to confess. I think the fallacy is to confuse what it would be rational for both people to do if they were acting in unison, to what it would be rational for a single individual to do. Although it's true that "if both reason rationally they will reach the same decision", that does not mean that they will achieve the optimal result if they both reason rationally. Indeed, in this case, they get a worse result than if they'd both reasoned irrationally. If there's a paradox here, this is it. But it's just not the case that there are two equally valid ways of reasoning here. Evercat 13:16, 20 Mar 2004 (UTC)

There simply *isn't* a rational path towards minimizing your jail time. Defect or cooperate, both are in a sense rational, it's just that rationality doesn't get you very far. Nor does irrationality. It all depends on the other prisoner. Who has the same problem. This is what makes the prisoner's dilemma such a vexing dilemma. Martijn faassen 20:22, 25 Mar 2004 (UTC)

Is there enough consensus to feed my rewritten paragraphs (in italics above, with 2 amended ones lower down) into the main article? --Martijn faassen 23:38, 22 Mar 2004 (UTC)

No, for the reasons stated above. Markalexander100 01:34, 23 Mar 2004 (UTC)

Your main concern has been my inability to understand the wording of the text. Since I think I'm at least of reasonable intelligence, I think that is a good argument for the rewrite in itself. Martijn faassen 00:17, 24 Mar 2004 (UTC)

I think there is a general consensus that there a problem with the 'communication' paragraph. That has been mentioned now in several separate sections of this Talk page, by different people. Markalexander100 is the only one who has raised any objections to your rewrite, and I notice that they haven't yet tried to rebut my last reply to them (above). I would say you have a majority in your favour. R Lowry 18:48, 23 Mar 2004 (UTC)

Okay, I've gone ahead and landed the rewrite. The next candidate for work is the first paragraph, looking at Egomaniac's concerns there. Martijn faassen 00:17, 24 Mar 2004 (UTC)

At the beginning of the article, I think the table should go _after_ the explanation. Its hard to understand the table without reading the text. Luke 16:33, 29 Mar 2004 (UTC)

Just a note to cheer for User:R Lowry's improvements. Great progress, thanks! Martijn faassen 22:00, 31 Mar 2004 (UTC)

Aw, shucks. :) I'm worrying now that maybe the intro is a bit brief and could do with filling out a little...

Another thing: I've rearranged the order of the classical payoff matrix so that the cooperate / defect sections match the second, 'canonical' one (without actually changing the values - they seem ok to me.) As far as I can see, it was simply a case of changing deny and confess around, but please someone shout if I've got it muddled. R Lowry 20:56, 1 Apr 2004 (UTC)

Some comments about the recent additions to the intro:

Some of the first paragraph in essence repeats some of the preconditions set out in the description of the classic dilemma, though most loosely. I'd also like to see something added about ..but if you choose to cooperate, then the temptation to act selfishly and not cooperate is high..

The key aim, then, of the prisoner's dilemma problem, is to see if cooperation can evolve 'spontaneously' among egoists: that is, to ask if self-interested individuals can come to learn, over time, that their interests are better-served by cooperating with those around them, rather than by solely pursuing their own advantage without concern for others.

I have some difficulties with this text. The prisoner's dilemma itself has no aim. The prisoner's dilemma illustrates a particular situation that by itself is not resolvable in a rational way to either selfish party's satisfaction. I guess the aim of the dilemma could be to illustrate a rather surprising problem: logic doesn't get one out of a seemingly simple situation.

The followup question is that we happen to have all kinds of intuitions nonetheless on how to tackle this. We do feel we can resolve this. It only becomes possible to resolve such situations when either:

outside factors come into play which actually adjust the payoff matrix. This is not really the dilemma anymore, though.

patterns come into play. Memory, or at the least history (agents may not have an individual memory but their culture or genes may have). The dilemma is played repeatedly (perhaps in different circumstances). Strategies, either rational or evolved, could arise in such a case. Stable conditions can arise where populations of tit-for-tat style cooperators drive out invading defectors. The evolution of morality, etc.

I also don't think the word 'spontaneously' contributes much. A key question is whether strategies can exist or evolve, but they need not be spontaneous at all... Anyway, a bit of a rant. I don't really know what to suggest as an improvement (or I'd done it already), but some further tuning would be a good idea. Martijn faassen 22:49, 7 Apr 2004 (UTC)

My understanding of the introductory text is that is should briefly give a flavour of what the article is about, so repeating in general ('loose') terms some of what follows later on seems to me like a fair way to start the article.

Regarding your comments about the second paragraph, I've tried a small rewrite...

The key problem, then, arising from the prisoner's dilemma, is whether or not cooperation can evolve among egoists: that is, can self-interested individuals come to learn, over time, that their interests are better-served by cooperating with those around them, rather than by solely pursuing their own advantage without concern for others.

R Lowry 09:56, 9 Apr 2004 (UTC)

Examples

I've added an example of the Prisoner's dilemma I've been pondering on today. It came to me while watching the boring race in the Tour de France today. I'm interested in your comments on this, in my opinion, interesting real-life example of this dilemma. -- Solitude 19:03, 15 Jul 2004 (UTC)

I think you would be interested in the article [Social Science at 190 MPH on NASCAR's Biggest Superspeedways that talks about a similar analysis of 'drafting' in NASCAR racing. Also, and I'm not sure if the previous article uses the term, but these are both examples of Co-opetition where you have to co-operate with competitors to stay ahead of the pack.

I also added that to apply PD in actual police and prosecution procedure is likely to cause miscarage of justice. In u.k. and many other countries, it is banned procedure. God save U.S of A. FWBOarticle 17:35, 23 Jul 2004 (UTC)

Water shortage: Bad example

I think the payoff matrix for the water hoarding example does not satisfy the requirement that T > R > P > S. One of these is replaced by an equality. If you cooperate you survive. If you are the hoarder you survive. Also, it is a multi-player game whereas PD is (strictly speaking) a two player game. Paul Beardsell 02:30, 24 Jul 2004 (UTC)

Actually that requirement for the payoff matrix looks wrong to me.

I thought there was an additional constraint:

T > (R+P)/2 < S

i.e. alternately cooperating and defecting doesn't result in you being better off- quite the contrary.

This is not a requirement of the standard Prisoner's Dilemma since it is a one-shot game. It may be applicable to the iterated version, but I've deleted the reference in the article since it was in a section talking specifically about the standard, one-shot version. DavidScotson 14:54, 12 Aug 2004 (UTC)

I agree that water hoarding is a bad example of the PD, as it involves many people taking many conecutive moves, rather than a one-shot decision. It's really an example of the Tragedy of the Commons where everyone tries to take too much because they suffer no ill effects from their own overindulgence, yet because everyone else is in the same situation you end up with squabbling lines of wannabe hoarders facing empty shelves and on average everyone loses despite their actually being more than enough water to go round.

Tragedy of the commons is a related area, but unless their are only two players, it isn't the same thing, quite.

Interestingly, tragedy of the commons didn't occur very much in England for centuries; the commons were under joint ownership, and hence they had more effective governance.

First sentence incorrect

The first sentence of the current version of the article is:

The prisoner's dilemma is a non-zero-sum game that illustrates a conflict between what seems a rational individual behavior and the benefits of cooperation, in certain situations where short-term gains produce later wrongs.

This is incorrect. It refers to the iterated PD. In the PD there is only one "game" and only one outcome. This mistake is made several times in the article.

Paul Beardsell 02:36, 24 Jul 2004 (UTC)

Re-written. Paul Beardsell 07:01, 26 Jul 2004 (UTC)

The first sentence is still incorrect, it now says "The prisoner's dilemma is an example of a non-zero-sum game where the best individual strategy cannot be decided but where the best solution for both players considered as a group is obvious."

I think the jargon-y reference to 'non-zero-sum game' is inappropriate in the opening sentence, but my main problem is with the second half. The best individual strategy can be decided. Defecting, regardless of your partner-in-crime's decision, will always be the best move for an actor who is rationally self-interested (read "rationally self-interested" in this one-shot game situation as "totally selfish and unconcerned about future repercussions, whether external e.g. revenge or internal e.g. guilt"). In game theory terms defecting is a dominant strategy and so both players will always defect.

Of the three statements made in the sentence, I think two are incidental or tangential to the topic (at least to the summary) and the third, as detailed above is factually incorrect. User:DaveScotson 23.02, 28 Jul 2004 (UTC)

We disagree on the best strategy "in game theory" terms and I suspect we disagree strongly. Furthermore I think your understanding of the problem is wrong or you understand the wrong problem. But it is 20+ years since I studied this formally so I am going to do a little research before I tell you you are wrong again! Paul Beardsell 23:03, 28 Jul 2004 (UTC)

I not sure we disagree strongly, we're perhaps just not communicating clearly. When I say best strategy I mean that the individual cannot improve his outcome by changing his decision (see Nash Equilibrium). It is important to note that the best outcome (as opposed to strategy) for an individual player is to defect when the other player is co-operating, which is also the worst outcome for the other player. This outcome never occurs because co-operating is an irrational strategy and so you never get the chance to betray the other player.

The mutual co-operation outcome is only optimum if you are concerned about the players' combined gain (or overall efficiency), though it is also, from each selfish individual's point of view, a better result than both defecting. User:DaveScotson

The PD is a one shot game and neither prisoner knows the other's decision until afterwards so there is no opportunity to change your decision afterwards. The best strategy is always trivially determinable if you know what the other player will do but trivial things are not interesting from a mathematical perspective and in this game and all other interesting games you do not know what the other player/prisoner will do. If cooperating was provably irrational or provable rational then the game would not be as interesting: There would be no dilemma. I think you have another game in mind - one of Hofstadter's or Martin Gardner's variants for the iterated PD, perhaps. The individual's best strategy in the PD does not admit to mathematical Game Theory analysis. If it did then GT would tell us the optimum strategy! But it does not. You think, however, that cooperating is irrational. Not according to GT which helps us not at all, one way or the other, in the standard, one play PD. Paul Beardsell 14:46, 30 Jul 2004 (UTC)

When I mentioned changing your strategy I meant in the Nash Equilibrium sense, i.e. if you were to initially choose co-operate as a strategy and then, before committing to it, look at your alternative you would see that you are *always* better off defecting, regardless of the actions of the other individual. If, in the same situation, you have already chosen to defect, then switching to co-operating would *always* be the worse decision, again regardless of what the other person does. As I mentioned above, this makes defecting the dominant strategy for both players.

If we were placed in a prisoner's dilemma situation right now (say with a 2 Million dollar 'prize' if we betray a co-operating opponent, 1 Million each if we co-operate, 1/2 million if we both defect and nothing if we are betrayed) what would you do? I would defect, knowing that it didn't matter if you were rational or irrational, whether you truly understood the dilemma or not. It wouldn't even matter if you were going to toss a coin to make the decision as I would *always* be better off defecting than if I co-operated. Now, to my mind that means the PD has been successfully analysed by Game Theory to the point were I have a clear and effective strategy. If you would reach a different conclusion, or even reach the same conclusion for different reasons I'd be interested to hear why.

Not *always*. If you "cooperate" and I happen to "cooperate" (even if I choose to do so by the toss of a coin) we both go home richer than if we had both used your analysis. Paul Beardsell 04:22, 31 Jul 2004 (UTC)

(If it helps matters, I too have studied Game Theory, though only six years ago in my case, as part of my degree in economics. I am therefore 99% certain that there are factual problems with article as it now stands e.g. I believe the Hofstadter example is a valid restatement of the PD. I'm familiarising myself with how the Wikipedia works and checking some other sources to convince myself of that final 1% before editing the article. If you can provide any links to support your case, that Game Theory does not indicate that defection is always the best strategy for an individual in a PD situation then I'd be happy to look at them.) DavidScotson 00:51, 31 Jul 2004 (UTC)

The two prisoners each are subject to the same rules and payoff matrix. Each, using your reasoning and Game Theory, decides to confess. But this is not the optimum decision: Each would be better off if they both deny rather than both confess so Game Theory has failed them. But each realises the other prisoner, being rational, is likely also to have reached the same conclusion so they decide to deny. But then the temptation is to confess to get out today, free. So each decides to confess. Each realises the other will have reached the same conclusion. So each decides to allow the other their own degree of super-rationality and each thereby decides to deny. One of them is likely to then betray and each realises this so each now reverts to confess! This is the non-optimum decision. Paul Beardsell 03:33, 31 Jul 2004 (UTC)

Let's for the time being leave aside the question of whether your new winner's game is equivalent to the PD. Let's return to the standard PD and the payoff matrix in the article. Paul Beardsell 03:52, 31 Jul 2004 (UTC)

That there is a Nash Equilibrium only says that there is an equilibrium. A Nash Equilibrium is not necessarily an optimum solution. And the PD is an example of that. Paul Beardsell 03:59, 31 Jul 2004 (UTC)

Hofstadter reference

The Hofstadter discussion re the swapping of two closed bags is from an analysis of his of the iterated PD. It is not "Another explanation". I intend to remove or drastically re-write this. Paul Beardsell 07:00, 26 Jul 2004 (UTC)

Defies Analysis?

There is a thread running through the article about whether the Prisoner's Dilemma is easy to analyse. While the discussion here and many of the prominent Google results for the term shows there is confusion about what the Dilemma is and means, I don't think that confusion is a defining characteristic.

Some quotes from the article:

"What distinguishes the prisoner's dilemma from other games is that the scoring (the "payoff matrix") does not allow for easy analysis."

"Even more difficult is the analysis of the iterated prisoner's dilemma"

"The prisoner's dilemma is the simplest game in Game Theory which defies analysis."

I don't really understand what is meant by 'analysis' here. Its usage suggests either an esoteric technical term which I have never encountered, or a non-native speaker trying to express that the outcome of the PD is counter-intuitive or paradoxical. (I don't really agree that it is either of these things purely because of the narrow way in which the game is defined. However, it might appear to be if you approach it from a common sense point-of-view i.e. try and place yourself in the position of the prisoners and think what you would do.) DavidScotson

By "analysis" I meant mathematical analysis. Maths cannot tell you what the best option is: There is no mathematical solution to the best selfish solution. "Mathematical analysis" is, I suppose, a technical term. Game Theory is a branch of mathematics. In the standard rules (and non-iterated) PD neither prisoner can decide on his own best selfish strategy using "common sense" either unless by "common sense" you mean you make an assumption about the action of the other player. And if you do not understand that then probably the article is at fault. This is why the word "dilemma" is used! Paul Beardsell 23:11, 28 Jul 2004 (UTC)

So I have explained what I meant but that helps nothing if that is not what is understood by the reader. Perhaps we should replace "defies (mathematical) analysis" with "the optimum strategy is not discernible." Paul Beardsell 04:02, 31 Jul 2004 (UTC)

Perhaps I am missing something, but the one-shot PD is perhaps one of the easiest games to analyze. The central point is that 'confess' is a dominant strategy. No matter what the other player chooses to do, I am always better off confessing. Hence, the Nash equilibrium is trivial to derive: both confess. Apologies if I am just being obtuse about your meaning about "defies analysis".67.180.24.204 04:14, 31 Jul 2004 (UTC)

As I have said perhaps my terminology could be bettered: Mathematical analysis where this is the search for a Nash equilibrium is quickly successful. But (according to my understanding) a Nash equilibrium is not always an optimum solution and the PD is an example of that. Am I correct? Paul Beardsell 04:27, 31 Jul 2004 (UTC)

Well, yes the Nash equilibrium payoffs in the PD are lower than possible. But, I'm not sure what you mean by an "optimum strategy" as opposed to the Nash "confess always" strategy. Since confession is dominant, it is always the optimal strategy. The larger point is that difficulty is not what distinguishes the PD from other games. There are plenty of much more difficult and unintuitive games around, such as those with no pure-strategy equilibrium. (Discussion refers to one-shot games.) 67.180.24.204 05:10, 31 Jul 2004 (UTC)

But, as I know you understand, each would be better off if both "deny". Game theory and Nash do not help us get there. I'm being picky but I think you mean "confess" not "confess always" because, as you say, you're discussing the one-shot game. Paul Beardsell 05:41, 31 Jul 2004 (UTC)

By confess "always", I meant that it is the optimal strategy regardless of the other players choice.67.180.24.204 06:43, 31 Jul 2004 (UTC)

If I ever find myself in the PD situation I think I will assume that the other prisoner is very smart, that he is not a sociopath and that he is not obsessed by Nash equilibria: I will "deny" and trust that he will too. I think this is the truly rational solution and that it is also truly acting with the most selfish intent. It is the optimum selfish solution on which both parties could agree. The asymetric options are not agreeable to both parties and so each should selfishly realise that these are therefore not admissible as solutions: Each should recognise that the temptation to betray is what drives each to the sub-optimal solution. (Which happens to be the Nash equilibrium.) This should allow each rational selfish prisoner to reject the possibility of either party "confessing". And I have seen this operate between rogues, even enemies, in my school days. Game theory seems not to allow this conclusion to be reached. Paul Beardsell 05:41, 31 Jul 2004 (UTC)

If you trust that the other person denies, then you should confess. This is essentially tautological. A decision to do something else essentially means that the payoffs are mis-specified. For example, you might achieve some intangible psychic reward from denying when the other guy does. Well, then that ought to be built into the payoff matrix. Or, I might care about the other guy's sentence. Again, the payoff matrix should be adjusted. The specified PD payoff matrix essentially _requires_ that you confess if you know the other guy denies. One can think of the PD Nash solution almost as a theorem or a matter of pure logic. The cooperation you observe in your school days may either be because of opportunities for retaliation (eg a repeated game), or because of the payoff issues I just mentioned. In either case, it is not a true one-shot PD. 67.180.24.204 06:43, 31 Jul 2004 (UTC)

Expanding on the prior point. The payoff matrix in the PD is typically specified by years in jail. This is fine, so long as the players utility is a function only of sentence length -- the typical assumption. If utility contains other elements like altruism, or a taste for cooperation then the payoff matrix really ought to be adjusted from years, to some ordinal utility metric. Otherwise, we may rationally observe 'cooperative' outcomes in what appears to be a PD. But, this would simply be because the elements of the game are not fully specified. 67.180.24.204 06:49, 31 Jul 2004 (UTC)

But what if all I care about is my sentence and the other prisoner is similarly motivated. I know that the other player cannot be expected to deny if he thinks I might confess, that he should realise I am in the same position, that he should realise I realise he realises I realise ... and therefore we will logically be compelled to do the same as the other. Thus I exclude the asymetric outcomes. I choose the option with the lowest sentence and deny as will my fellow prisoner. Paul Beardsell 10:12, 31 Jul 2004 (UTC)

Suppose you know for sure that the other prisoner will deny. Now, look at payoff matrix. Are you better off confessing or denying? You are better off confessing. Yes, you absolutely would like to enter into a binding agreement with the other guy to both deny. But, in this case, you can't enter an enforceable agreement. There is no way to convince the other guy you will deny. And, even if you could, he would still want to confess to lower his sentence. That, of course, is why it is a dilemma.Wolfman 17:18, 31 Jul 2004 (UTC)

As I had hoped to have demonstrated, both prisoners realise that Game Theory does not help them get the most positive outcome. All that GT can offer is the suboptimal both confess outcome. Realising this, each should see that whatever one of them decides to do rationally must also be the decision of the other presumably rational prisoner. Should one choose confess because the other is thought to be going to choose deny means that the other has the same temptation and both will end up confessing. This is an unwanted result so both deny. Paul Beardsell 18:35, 31 Jul 2004 (UTC)

And that is why Game Theory is of no help. Paul Beardsell 18:11, 31 Jul 2004 (UTC)

I think I understand your position now. Essentially you are rejecting game theory as a useful method of analysis for the PD. Please correct me if I am wrong, here. But given this is an encyclopedia, an article on the PD would seem to call for a standard analysis. In my view, this article is not really the place for a critique of game theory. Indeed, the PD is commonly understand as a common and classic application of game theory. If you can find a link to someone who has published your critique, perhaps that would be the way to go. But as for an _original_ critique, this is not the place. If you feel strongly about your analysis, you might consider sending it to the Journal of Games and Economic Behavior for publication.Wolfman 19:41, 31 Jul 2004 (UTC)

You are right - I need to quote a reference. But I assure you my argument is anything but "original research" and is essentially a regurgitation of the Applied Mathematics 101 treatment of the prisoner's dilemma as taught to me in 1980. This is old hat. And it is why this game has "dilemma" in its title and other standard games in Game Theory do not. Paul Beardsell 22:13, 31 Jul 2004 (UTC)

Paul, I would be very interested to read a mathematical account of your analysis which predicts cooperation in a one-period game. This is a completely different solution concept than the Nash-equilibrium economists instinctively rely on. I teach the PD every year for a day in my freshman economics class. And if mathematicians rely on a different solution concept for this game than economists, I really need to get on top of that.Wolfman 22:52, 31 Jul 2004 (UTC)

I have been looking for a reference and I had thought several would be readily available. I remain confident that I haven't made this up! The best example I have found of the single-shot cooperation I have found so far is here where the MAD of the cold war did not happen. But their explanation is not the one I advance - they speculate that each side projected their own leader as mad enough to complete MAD despite the payoff matrix saying otherwise. Still looking. Paul Beardsell 23:58, 31 Jul 2004 (UTC)

The mathematician who invented GT provides the "original critique" that Wolfman requires. "To von Neumann, the Prisoner's Dilemma was a paradox that all but destroyed what he hoped Game Theory would accomplish" [1]. Von Neumann did not consider the PD a classic example of GT but an anomaly. Wolfman says: "Essentially you are rejecting game theory as a useful method of analysis for the PD." Yes, I am. And so did von Neumann. Paul Beardsell 02:14, 11 Aug 2004 (UTC)

I'd rather see a link with a bit more weight behind it to support the claim that the PD "defies analysis" or is not a "useful method of analysis" as that article edges on an anti-'left' (his terminology) rant at times. However, putting that issue of reliability aside, the link you give appears to claim something else anyway. As I read the article it is saying that (JVN says) the PD is a paradox, where the rational choice for the individual leads to a sub-optimal outcome for all, which I'm not sure anyone has questioned. And if you read the article's footnote on Nash it seems to echo most of the changes to the article that I still intend to make at some point. DavidScotson 12:23, 12 Aug 2004 (UTC)

By "defies analysis" I meant that GT does not lead to the optimum solution. If that is not what others mean by that phrase then I will change the phrase to something else but my point will stand. The Nash equilibrium is only that: an equilibrium. That the Nash equilibrium coincides with the optimum solution in many games does *not* mean that technique can be relied upon to determine the optimum solution. Also the NE is being used by you (and others) as a synonym for "rational choice". Unless by "rational" you and others believe (a) there is no dilemma and (b) "rational" is that which agrees with the NE. No, there is a dilemma the analysis of which GT does not help us to the optimum solution, and the NE is not necessarily optimal. Paul Beardsell 12:56, 12 Aug 2004 (UTC)

Are we not just going in circles now? Do you believe that I am wrong in my understanding of the Prisoner's Dilemma (as taught in Game Theory classes around the world) or are you saying that the Wikipedia should have an article about the Prisoner's Dilemma with only disparaging references to Game Theory? DavidScotson 14:42, 12 Aug 2004 (UTC)

I cannot say if your understanding is wrong. Indeed I have not really been keeping track of who has said what. But some of what has been said is wrong. "Disparaging"? Where you got that from I do not know. What is certain is that neither is this article the place to only make favourable references about GT. I am repeating my points above not because the argument is circular but because my points are not being addressed. Paul Beardsell 20:56, 12 Aug 2004 (UTC)

Overuse of "classic"

This creates a false impression. It is a special example of a non-zero game. Most non-zero games do not have the dilemma at the core of the PD. And, if they do, they are called the PD. Paul Beardsell 22:31, 31 Jul 2004 (UTC)

Also, it's funny you should mention the phrase "classic example". This appears to my eyes to be a linguistic tic that afflicts the Wikipedia and I've been spotting it everywhere recently. It appears nothing is just an example of anything, it is always a classic example of whatever even when, as in this case, it is actually an outlier or anomaly and most definately not a classic example in the sense that it epitomises the entire class of things. More mundanely, there's probably many places where that phrase should be replaced with "famous example" or "commonly used example" or other less superlative and more correct language. DavidScotson 12:32, 12 Aug 2004 (UTC)

Yes good point and I have another: Political bias has nothing to do with maths. Paul Beardsell 12:56, 12 Aug 2004 (UTC)

Is this Correct?

If an iterated PD is going to be iterated exactly N times, for some known constant N, then there is another interesting fact. The Nash equilibrium is to defect every time. That is easily proved by induction. You might as well defect on the last turn, since your opponent will not have a chance to punish you. Therefore, you will both defect on the last turn. Then, you might as well defect on the second-to-last turn, since your opponent will defect on the last no matter what you do. And so on. For cooperation to remain appealing, then, the future must be indeterminate for both players. One solution is to make the total number of turns N random.

This seems a bit fishy - http://en.wikipedia.org/wiki/Unexpected_hanging_paradox

Yes, it is correct. Since the only Nash equilibrium in the last period (N) is to defect, you take that as a given in period (N-1). So, in period (N-1) the only NE is to defect. This feeds back to (N-2) etc.

Now, if the ending point is uncertain, this argument breaks down. In fact, if the game end with a fixed probability r in each period, then this is just like an infinitely repeated game with a somewhat higher discount rate. In that case, the Folk Theorem for repeated games holds: any payoff in the convex hull of payoffs (above the minmax) can be sustained as a NE (as the discount rate, including game ending prob, goes to zero). Essentially, that means that 'both cooperate' converges to a repeated game NE. But, it is crucial that the precise ending date be unknown. Wolfman 04:26, 20 Sep 2004 (UTC)

Ethics of PD....

Whilst reading an above discussion it occurred to me that it might be useful to have a short section about the pratical aplication of the PD according to the sociatal context: I mean that the PD reffered to would be legal in USA, whilst in the UK would be legal - though it's not necessarily so black and white - and their specific requirements as to what payoffs may be legally used. It may be that a discussion of this sort would not be appropriate but I did wonder whether the discussion should at least refer to the ethics of PD (after all, although PD may seem logical it may, to some extent, be undermined by other factors - such is the case in UK). One minor point - 'friend or foe' the US game show had a corrresponding UK show called 'Trust' (aired on Channel four, but now off air) - it corresponded in the logical structure of PB variation employed within the US format (as opposed to any stylistic correspondance). Though i'm not too sure whether this site is specifically aimed towards a US audience, or more broadly towards an English speaking audience ( - oops, have only just signed up to wikipedia, where has it been all my life?!)

Mutual high beam

I disagree with the high beam example. "It's better for you to leave your high beam on at all times, but no good if everyone does it." Leaving your high beam on disrupts the ability of on-coming traffic to navigate precisely. The safe navigation of other drivers in your immediate vicinity is advantageous to you as any accident is likely to impact you in a negative manner.

Usually true, however the risk is relatively low on dual carriageway roads with crash barriers in between. And the risk isn't usually *that* bad- all drivers have probably had to suffer this at some time or other.