[Guest]

I should point out that I've already been into the answer to this in some detail in the "others" section so if you don't want it spoiled, stay away from there!

Two Masters of Logic sit down for a friendly or not so friendly game of Iterated Prisoners' Dilemma. The rules of the game are quite simple:

Each player has 2 cards, marked "COOPERATE" or "DEFECT".

In each round of the game, each player chooses one card and plays it face down, then both cards are revealed.

If both players played "COOPERATE", they are awarded a point each.

If they both played "DEFECT", they get nothing.

If one player played "DEFECT" and the other "COOPERATE", the defector gets 2 points, and the cooperator loses 1 point for being a sucker.

The objective of the game is to amass as many points as possible*, as these will be converted into beer tokens after the game, and paid to the players by the Grand Master who is hosting the game.

It doesn't matter whether the players score more or less than each other, their sole objective is to maximise their own score.

The players do not know each other and may not confer or agree on a combined strategy, but they are both Masters of Logic, so will both play the very best strategy possible for their own gain. Each player knows that the other is also a Master of Logic.

The length of the game is not decided at the beginning, but is announced after the tenth round. On this occasion it happens to be 50 rounds.

How many points will each player get in total?

The Grand Master has not brought any beer tokens to the game. He knows both players will get zero points. They will play "DEFECT" on every round.

Later edit: After debating this extensively with Neida I still can't decide if it's right or not. It seems to come down to a matter of opinion.

*For clarity, each player doesn't care what the other player gets, and is only concerned with their own points

[Guest]

Part of what seems to be complicating the equation is the fact that octopuppy is drawing the "Unexpected Tiger*" paradox into the mix, which has no agreed upon solution. By saying that the final round needs to be treated differently (or that it's the only one that matters), we get into a cyclic examination of every round where each round becomes the "last" round in turn since we've already made a determination about the next round, so there is no longer an uncertainty about it. With no agreed upon way to handle the situation, I think that agreeing to disagree may be the only final solution.

There's a lot of sense in your post, but I don't really see why you've brought the "unexpected tiger/hanging" paradox into it. There is a superficial similarity but I don't think it arrives at a paradox in the same way. Once we've decided that they will both defect on all rounds it doesn't then become logical for either to improve their score by cooperation on any round. The reasoning isn't cyclic, but an inductive process that terminates with defection on all rounds. It's straightforward induction, and if there is a flaw in my reasoning I'm pretty sure it's not there. Personally I find the matter of what happens on a single round much more open to question.

[Guest]

I'm not saying that you can't make contradictions, just that you can't make contradicting assumptions. However, I'm prepared to admit that I haven't been clear on this and it isn't too clear a point anyway. What I have been trying to say is that you should only make one assumption to make a proof bullet proof. In this example I think you should either say:

"We both know a situation exists where we both cooperate and win 1 token. Assume I defect..." (note, in this case I would then follow with "then whatever reasoning led me to defect would also be followed by the other person, so we would win 0)

Or:

"Assume that mutual cooperation is the best solution. Is there any other possible final outcome given what I know that could give a better solution?" (note, in this case I would say no, as the only other possible final outcome (as opposed to intermediate step) is that we win 0)

What you can't do is say "Assume that mutual cooperation is the best solution, now assume I make a different decision" and then say that you've found a contradiction so your first assumption is wrong. This is simply the "assume it's heads, now assume it's tails" analogy I referred to earlier.

We are getting bogged down so I'll keep this brief. A phrase like "If s/he will play cooperate, the move that will get me the most points is defect." makes no assumption (not even that they will cooperate). It's just stating what move gets the higher score in the event that the other player cooperates. I understand you thinking that we should not be considering things which will not happen, and I'd like to move on into exploring that area rather than dwelling on this.

I never say that you get to decide what the other person will do. However, using the knowledge that "if there exists such logical reasoning for me to make a particular choice, the same line of reasoning will be followed by the other person" means that you don't have to decide what they will do - you are both simply locked into the same conclusion. You can use the fact that your decision is as restricted as theirs to qualify that - i.e. "if they were making the logical choice, then if I did anything different I would not be!"

All this simply means that both will make the same decision. But the generalisation that both will make the same decision is deceptive because it is incomplete. The more complete version is that both will make the logical decision. That makes it clear that there are no options of "we both choose this" or "we both choose that" to consider. When (and only when) we have ascertained what the logical decision is, then we will know something about the decision the other player will make.

OK, so let's move on to my other line of reasoning. First of all let me say that I think your answer is dependant on saying that if a Master of Logic is placed in a paradoxical situation then they could not make a choice. (I know that this is based on me believing that, by your argument, defection is also a cyclic argument as I've explained before, but bear with me for a moment.) If they were asked to minimize their losses then this would not be a paradoxical situation, but as they have to maximize their gains it is.

Minimising losses and maximising gains are the same thing! And your reason for saying defection is also a cyclic argument, if I'm finding it correctly, seems to hinge on the idea that "we also both know we can do better than 0".

Each Master of Logic thinks "If I were to simply observe this game, what would be the best strategy for any two Masters of Logic to play in order to maximize their winnings. If I can determine that then, knowing that I am a Master of Logic and that the other player is also a Master of Logic, we know we will come to the same conclusion and play that strategy."

Now it's fairly straightforward to realise (as most have) that the strategy that yields the most reward across both players is always cooperate. So let's assume that this is the best strategy to maximise individual winnings. Is there any alternative strategy that would give a greater yield? Now we see that any such strategy must rely on at least one Master of Logic defecting. However, by the logical induction that Octopuppy has used before, you also know that it is logical that if there is to be any one defection, then that would result in an always defect strategy. Thus there is no other stable strategy that would yield more than 50 tokens, thus always cooperate is the best strategy. Hence that is what I will play.

This (and the remains of your post) is just the one round scenario writ large. You can change the number of rounds or ratios of rewards and it amounts to the same thing. Both players must make the decision to cooperate for the benefit of the other, because if the other does the same it will be to their benefit. Given a choice between mutual defection and mutual cooperation, whether it is for one round or fifty, mutual cooperation is the better option. My argument is that they are not given that choice, because you cannot choose what somebody else will do.

I really want to move on into a new area with this. I understand you feeling that there is something wrong with my argument, and I also suspect there may be something wrong with it. But before we explore that, I want you to fully appreciate what is right with it, otherwise what follows won't be in the proper context.

So I would ask you to put those niggling doubts aside (we'll validate them later) and have another look at it ().

Note that I have avoided the language of free will. Only one scenario will occur and all others are impossible, so I must evaluate which choice, "I cooperate" or "I defect", best fits the goal of maximising my points. While it is recognised that the other player will also play the most logical move, it is also recognised that my current decision will not affect what that move will be. It is, in effect, fixed, although unknown to me. Because it is unknown, I must consider both possibilities, although when the current thought process is complete and I know what the logical choice is, then I will know what the other player plays.

This model of logical thinking is admittedly basic and in cases where my best move was affected by the other player's choice of move, or future moves were taken into consideration, it might get more complex. But fortunately this case is remarkably straightforward and results in no such problems.

An important thing to bear in mind is that my thought process will determine only my actions. I would not conclude that cooperation is the best choice for me, since it is the other player's cooperation that would benefit me, and my thought process will not determine that. In the end I would be better off if cooperation proved to be the logical choice for both of us, but that is no means for arriving at the conclusion that it is. It's an appeal to consequences.

I'd like you to consider all that with a receptive mind and not think too much about arguing against it, since we've been going in circles too long over this. I acknowledge your misgivings. That's all I have time for now, so you'll have to wait to see where I'm going with this...

[roolstar]

What a long (and complicated) thread to read... So here's my contribution to this subject.

Going back to the original OP, the 2 players are masters of logic (and both love beer). They want to maximize the number of beers, they don't look to get more than each other. The word "sucker" will not affect their judgements either (emotional triggers do not affect masters of logic). In that case defecting would be stupid, illogical and an EMOTIONAL decision.

That being said, Masters of logic would always cooperate. Defecting even once is an emotional decision

Now I realize that, since this thread is taking so long, people are considering these masters of logic as regular people with feelings, who may seek EGO results and even vengeance.

Well, in that case, NOTHING BIG really changes either. The only chance of defecting would be at the last round (unless these masters of logic are plain stupid). If that round is defined before hand, they may both defect or both cooperate that round and the total number of beers they accumulated will change by a maximum of 1 beer.

Now if we don't take the "masters of logic" part literally, we will end up in what the humanity has been through since early history. And one of the 2 brothers will kill the second to maximize its winnings and wealth. We may even find a World War I or II scenario. In real life, this game will never end. People will end (being mortal), but the game won't, it will continue with our children and grand-children playing it.

That is why now we know that our decisions are much more emotional than rational, and some people (egoistic, short-term people), may at some point choose to defect.

I assume that this puzzle may have something to do with the situation the world is right now. And if you don't know what I'm talking about, just watch the news.

I don't intend to discuss politics, I hate this subject (everybody assumes they are on the "right" side of the conflict). But in my opinion, the intention behind the United Nations was to provide an arena for this game to be played rationally. (My opinion about the UN set aside).

Remember this: "We decide emotionally, then justify it rationally!"

In 2 words, No one wil be sober after that game ends

Edited January 16, 2010 by roolstar

[Guest]

...That being said, Masters of logic would always cooperate. Defecting even once is an emotional decision

wtf? Take the argument I linked to in my last post. What's emotional about it?

Now I realize that, since this thread is taking so long, people are considering these masters of logic as regular people with feelings, who may seek EGO results and even vengeance.

Quite the reverse, I think it is their lack of feelings that causes them to act in a mutually destructive fashion. On the face of it that sounds odd, particularly coming from me as I generally advocate rationality. But these Masters of Logic are not real people and do not behave like them. When dealing with real people, the most rational choice is quite different (see my answer to by way of contrast). Knowing that there is even a small chance that the other person will act out of a sense of obligation, kindness, or decency changes things drastically. But with these players there is no chance. They are totally selfish and don't even care about reciprocal consequences outside of the game. Please note, this is not because rationality is bad. It's not their fault, I just defined them that way by stating "their sole objective is to maximise their own score". That leaves no scope for other motivations. I don't intend for this puzzle to be a model for politics, or at least if it is, it's a model for politics at its worst. More usefully, you could see it as a contrast with human behaviour, and knowing how and why it differs may be instructive.

In that respect I'm reminded of the recent film "I, Robot". I quite liked the plot, which pointed out a fundamental flaw in Asimov's Laws of Robotics. If we ever succeed in making artificial intelligence, we should be very careful about what objectives we give it, as good-sounding objectives don't necessarily produce the results you'd expect.

[Guest]

the ultimate problem with this dilemma is that both the always defect and the always cooperate strategies are both equilibrium points, and therefore an argument can be made either way.

octopuppy, while it is true that on any individual round I cannot force my opponent to pick cooperate, if I pick defect, my opponent will no longer have a reason to cooperate. if my opponent picks defect the same rule applies.

we can further assume the each player will want the most beers, and by defecting on any round before the last one will reduce that total.

again the last round is questionable, but in no way does the last round force either player to want to defect and therefore reduce the beers they get prior to that. to put it simply, you seem to be arguing that defection is the best for a single round. okay I agree. but your implication that it means that defection is good for every round is a bit convoluted. for example, if my opponent defects the 49th round and I cooperate, I can punish him by defecting the 50th round and making sure he doesn't get any more. In particular any defection prior to the 48th round doesn't really make sense, as that would reduce the maximum yield from cooperating.

[Guest]

octopuppy, if your heart is so set on this, how about we have a code off similar to the rock paper scissors algorithm? the person with the most beers at the end of the contest wins.

[unreality]

v like this? v

if anyone is interested we could host a quick algorithm contest similar to the Rock Paper Scissors one (although it probably wouldn't be as interesting, but maybe I'm wrong) - I've already copied some of the code from the RPS program and can pretty easily change it to the Prisoner's Dilemma if anyone is interested

phillip makes an interesting argument that says that defect is best for a single round but NOT for a bunch of rounds. I think we're not arguing over that... like I said a few pages back, via induction or just common sense (though that can be misleading here), what applies to one seems to apply to all (for perfect Masters of Logic only - NOT for normal people) so the whole question can be resolved by looking just at a single round

[Guest]

octopuppy, if your heart is so set on this, how about we have a code off similar to the rock paper scissors algorithm? the person with the most beers at the end of the contest wins.

Not on your nelly, I know who'd win! But this is strictly a Master of Logic V Master of Logic scenario, when playing other types of player different logic applies.

Right, on to what I was promising earlier: Some new perspectives. There's a few ideas in here so I'm colour coding them to avoid confusion about where one ends and the next begins.

First, here'a a variation to show where the limit lies in my opinion. Suppose the scoring system for one round were exactly the same except that we now penalise mutual defection.

So:

If both players played "COOPERATE", they are awarded a point each.

If they both played "DEFECT", they both lose 2 points.

If one player played "DEFECT" and the other "COOPERATE", the defector gets 2 points, and the cooperator loses 1 point.

Following the logic in my we get nowhere. If the other player cooperates, my best move is to defect (gets 2 points instead of 1). If the other player defects, my best move is to cooperate (gets -1 point instead of -2).

If you thought the original scenario probably favoured cooperation, this should really settle it. Mutual Defection has lost its status as a Nash Equilibrium. Now Mutual Cooperation and Mutual Defection are just two equilibria, both unstable, one offering a point to each player, the other offering the loss of 2 points.

My position on this is that it is actually a dilemma for Masters of Logic, whereas the main scenario isn't. Defection was the clear choice before, but now there is no reason for them to favour one move over the other.

At this point Neida and others may be shouting at the monitor "BUT THEY WILL BOTH CHOOSE THE SAME!!! If you play cooperate, you'll win a point, if you play defect, you'll lose 2 points! It's a fact!"

But is it? If what I say is correct and there is a genuine dilemma, if there is no logical path to the decision, then it's like choosing heads or tails. Without a clear logical choice there is a 50-50 chance of either. If the other player has chosen a move at random, cooperation will win either 1 or -1 points, and defection either 2 or -2 points. Both average at 0. Defection is the more high-risk option but also high reward so is not particularly favoured. If the other player is slightly more likely to cooperate, you would defect, and vice versa. So the "pick one entirely at random, with no bias" strategy is stable (for a single round game, I hasten to add).

But instead of choosing something at random, what's to stop both players simply realising that Mutual Cooperation is the way to go to get maximum points? If this were logical then it would be so for both, and we can go back to being able to predict what the other player does.

One thing I've insisted on is that in decision-making players cannot think "if I choose X, the other player will too". It's true, but they cannot think it. Why? Because it's illogical. More on that later, but first here's a couple of scenarios to illustrate the limit of that.

Neida asked earlier if it would make any difference to me if we said at the start that a rule of the game is that both players must make the same choice. I said it would though it's a bit bizarre, since each is deciding on behalf of the other. But no more bizarre than some stuff I've come up with, so let's consider it more. We'll hook the players up to a W. Heath Robinson-style contraption involving wires, pulleys, and preferably an unnecessary steam engine or two, so as to ensure that it is physically impossible for them to move differently. In essence there are now three moves that a player could play: Cooperate, Defect, and Unable To Play Since The Other Player Is Trying To Do Something Different. We know that the latter will not happen, but it's there for clarity. Now players can safely say "I will be unable to cooperate unless the other player also cooperates" so although they are not causing to other player to cooperate, they are at least preventing them from doing anything else, so it's just as good. In this case I am sure that both would cooperate.

An even more borderline case is that of the cloned Masters of Logic which I mentioned earlier but will cut-and-paste to save everyone bother.

Let's suppose the clones are virtual reality clones in a deterministic environment (quantum mechanics simulated by pseudo-random deterministic functions), and let's say the environment has 180º rotational symmetry so their physical state is exactly the same, bar rotation (I can't claim credit for any of these scenarios by the way, I pinched them all off this guy). This is functionally equivalent to playing against yourself in the mirror, although the causality is different.

I've already justified cooperation in that case by virtue of the fact that both will act the same because they are identical, not just because they are logical. So from the cooperation position they can reason "I could play defect, but if I do, the other player would too" without depending on the other player's hypothetical defection being logical.

That example's really on the borderline, I was starting to think that this would be a defection scenario, but today I'm thinking they would cooperate.

So what is the difference between these and the original scenario, where we can be just as certain that both players will do the same?

As I mentioned earlier, thinking "if I choose X, the other player will too" is true, but in my opinion Masters of Logic cannot think it while making decisions. This comes down to more than there simply being no direct causality, as the clones illustrate. It's because the generalisation depends on the fact that both players will pick the most logical choice. Saying "I should cooperate because if I do the other player will too, and that's better for both of us" would make cooperation the most logical choice if it were true, but is also only true if cooperation is the most logical choice, so it relies on the assumption it seeks to prove.

Alternatively you could think "If cooperation is the logical choice, it will make us both better off". To rely on that is an appeal to consequences, but in that respect it's rather interesting.

While appeals to consequences are generally considered to be fallacies, when a Master of Logic makes a choice, they must make something that looks suspiciously like one of those, as in "If decision X is the decision I will make, then it will benefit me more than if decision Y is the decision I will make. Therefore decision X is the decision I will make." Normally we don't consider that a fallacy because we have the illusion of free will, but a Master of Logic does not. However, the previous statement follows from the fact that a Master of Logic will make the decision which benefits them most, that being their objective.

But wait! We know perfectly well that mutual cooperation is what benefits them most. However mutual cooperation is not a decision either can make. It relies on the other player cooperating.

Why can't we rely on the fact that both will do the same thing? Because while we are considering what action we will take, we have to give all possible actions the benefit of the doubt. We can't know which is the logical choice until the thought process is over, and then and only then can we determine that the other player will do the same. The clones can think "If I cooperate, the other player will do the same", because they are clones and this inference is independent of whether cooperation if logical. But until the thought process is complete and cooperation has been identified as the most logical choice, a Master of Logic cannot conclude the same. They cannot think "If I do something illogical, the other player will do the same". It may seem absurd for a Master of Logic to consider doing something illogical, but when choosing between two actions, one of which is logical, they have to consider doing the other since they do not know which is logical until they have considered both.

[Guest]

You're (original poster) wrong.

They won't both get 0.

If they use the molehill-for-tat strategy, which has been shown through open experiments as the best strategy for a multiple-play game of Prisoner's Dilemma, then they will both obtain 50 points.

That is how you maximize your points without any knowledge of the intentions of others.

For those who don't know what the molehill-for-tat strategy is, it is this:

First hand play co-operatively.

Then you simply play what your opponent played the previous hand.

EDIT:

Apparently t-i-t is replaced by molehill

EDIT 2:

The major problem with this game is that it is a repeating scenerio where the previous choices are known and can have a direct impact upon the next choice.

By allowing it to repeat means that co-operation is the best choice. If it was a one time thing, defect is the best choice.

Edited January 18, 2010 by TheChad

[Guest]

OK, I am strapped for time at the moment (hence the lack of responses), so I will try to be quick...

I'd like you to consider all that with a receptive mind and not think too much about arguing against it, since we've been going in circles too long over this. I acknowledge your misgivings. That's all I have time for now, so you'll have to wait to see where I'm going with this...

I have tried to do this throughout and always try to do this. At the very least it makes sure you are clear what you are arguing against! :-)

I'm fairly sure I do understand your argument, it's just that I disagree with it. You say that you can not determine what the other player will do, so you must make the best choice on the basis that you have no idea what they will do, and it is only at that point that you will know what they will do. I disagree - I think a Master of Logic would not deny himself this incredibly valuable information and completely ignore it for the purposes of figuring out the best solution, only to then realise that they've done themselves out of 50 (or 1) tokens. (At the very least I would say that if they came to this conclusion they wouldn't stop there - they would realise they've done themselves out of beer tokens and so look for a better solution, which is what would make it cyclic.)

Before getting on to your other points, can I ask you to consider this? You say you can not know what the other person will do until you've figured out the most logical answer, as you only know they will also come up with the most logical answer. You also say that Masters of Logic are deprived of free will. Therefore, I'd suggest that your arguments of "If I choose this, then..." are flawed by this reasoning. This relies on free will. I.e. in your argument the Master of Logic is thinking "If I could choose what I liked and I picked this then..." But you are equally as constrained as your opponent - you can only choose the most logical answer. This is why I am saying the reasoning should be more as follows: "If I pick cooperate, then I could only have picked cooperate if it was the most logical solution. If it was then my opponent would also pick it. That would result in 1 token each."

Now admittedly, you wouldn't normally reason like this, as most things are more simple - it's either right or wrong. However, in a situation where the outcome is dependant on the decision you make, you have to take into account the decision itself. Again, most people would struggle with this, but for a Master of Logic it should be straightforward.

Basically I think if you want to apply constraints, you have to apply them equally. Knowing that each person has the same constraints is incredibly valuable information to the Master of Logic and it would not be logical to ignore it.

[Guest]

If they use the molehill-for-tat strategy, which has been shown through open experiments as the best strategy for a multiple-play game of Prisoner's Dilemma, then they will both obtain 50 points.

This isn't correct. If you alternate defect/cooperate then you alternate winning 2/losing 1. Over 50 rounds this would give you 25 tokens. However, as you say that your first step should be to cooperate and then do what you opponent does after that, if both players did this then they would simply always cooperate, which would give them 50 tokens.

[Guest]

First, here'a a variation to show where the limit lies in my opinion. Suppose the scoring system for one round were exactly the same except that we now penalise mutual defection.

So:

If both players played "COOPERATE", they are awarded a point each.

If they both played "DEFECT", they both lose 2 points.

If one player played "DEFECT" and the other "COOPERATE", the defector gets 2 points, and the cooperator loses 1 point.

Following the logic in my we get nowhere. If the other player cooperates, my best move is to defect (gets 2 points instead of 1). If the other player defects, my best move is to cooperate (gets -1 point instead of -2).

If you thought the original scenario probably favoured cooperation, this should really settle it. Mutual Defection has lost its status as a Nash Equilibrium. Now Mutual Cooperation and Mutual Defection are just two equilibria, both unstable, one offering a point to each player, the other offering the loss of 2 points.

I think this is the same solution, but intuitively easier to see the answer. However, as the arguments are basically the same (or at least very similar) we'll just get stuck in the same place unless we can agree on the previous question. I'm sure neither of us would be particularly fascinated by talking about a problem that we believe is based on a false premise!

[Guest]

Got a bit more time, so thought I'd respond on these points...

Minimising losses and maximising gains are the same thing!

No they are not. If you were asked to minimise your losses then you know that by defecting you can never lose anything, so that's a safe option no matter what. Maximising your gains is different - choosing an option that doesn't lose you anything here is not the objective of the game. If there is another strategy that would give you more then that would be better. Not that it necessarily applies here, but it would even be better if there was some risk associated with it. For example, consider the situation where you can roll a dice 10 times - if you roll a 1 you lose 1 token, if you roll anything else you win one token. You can also choose not to roll and you don't win or lose anything. A strategy to minimise your losses here is to not roll. A strategy to maximise your winnings would be to roll every time (even though there is a small risk that you may end up with a loss, this is the most logical strategy to achieve the stated aim.)

And your reason for saying defection is also a cyclic argument, if I'm finding it correctly, seems to hinge on the idea that "we also both know we can do better than 0".

Yes, that's right. Not sure what your problem is with that? Are you saying that a Master of Logic wouldn't recognise that?

Let's be clear on this - it's not simply a recognition that there's an outside chance I could get more than 0, it's a recognition that there exists another equilibria where both players would do better than that and you know that both players should strive for the maximum they can get.

This (and the remains of your post) is just the one round scenario writ large. You can change the number of rounds or ratios of rewards and it amounts to the same thing. Both players must make the decision to cooperate for the benefit of the other, because if the other does the same it will be to their benefit. Given a choice between mutual defection and mutual cooperation, whether it is for one round or fifty, mutual cooperation is the better option.

I agree that the reasoning can be applied to the single round game. I don't think that means it can be written off. I also think it is stronger for the 50 round game than for the single round game, so maybe would argue that you have to apply it to the 50 round game and then use induction back to be sure it still applies for the single round game. This is because, in the 50 round game, they each know that any deviation from the equilibrium from either player will result in less money to both. E.g. if the players were to exercise any free will (i.e. if they were not masters of logic) then if one player deviates in the 10th round, they will both defect from then on and both players will end up worse off. Similarly, if either deviates in round 1, they will both defect from then on and both players will end up worse off. The fact that Masters of Logic would both deviate at the same time if it was logical to do so does not effect the outcome here, as the difference between them both deviating in the same round as opposed to deviating one round apart is negligible in comparison to the overall prize available for the game.

My argument is that they are not given that choice, because you cannot choose what somebody else will do.

You do not need to choose what someone else will do. You are simply given knowledge that you have an identical constraint and that constraint is you must make the most logical choice. Therefore what is most logical for you is also most logical for them. This is really quite straightforward logic in itself so I really don't see what the issue is with this.

My arguments at the end of that last post that you referred to are not based at all on trying to figure out what your opponent will do, or on trying to decide something for them. They are simply based on the observation that a particular set of choices would result in the best outcome for both players, therefore being Masters of Logic, they would both choose this same set of choices.

[Guest]

I'll quickly answer a few little points...

No they are not. If you were asked to minimise your losses...

In the context of this game, I would consider losses as negative gains, and gains as negative losses. Otherwise the objective is not to maximise gains or minimise losses, it is for players to maximise their score. That makes no distinction between the importance of avoiding negative points or trying to get positive ones.

[And your reason for saying defection is also a cyclic argument, if I'm finding it correctly, seems to hinge on the idea that "we also both know we can do better than 0".]

Yes, that's right. Not sure what your problem is with that? Are you saying that a Master of Logic wouldn't recognise that?

Yes, because it depends on something which remains to be proven. If one player will play defection, the other can only get less than 0 by varying their strategy from this.

I agree that the reasoning can be applied to the single round game. I don't think that means it can be written off.

No, nor I. All I'm saying here is that the large scale losses on a larger game all hinge on the single round game result.

You do not need to choose what someone else will do. You are simply given knowledge that you have an identical constraint and that constraint is you must make the most logical choice. Therefore what is most logical for you is also most logical for them. This is really quite straightforward logic in itself so I really don't see what the issue is with this.

The issue I have is that until it can be established that a choice is the most logical, you cannot infer that the other player will do that.

...then get to the big one:

Before getting on to your other points, can I ask you to consider this? You say you can not know what the other person will do until you've figured out the most logical answer, as you only know they will also come up with the most logical answer. You also say that Masters of Logic are deprived of free will. Therefore, I'd suggest that your arguments of "If I choose this, then..." are flawed by this reasoning. This relies on free will. I.e. in your argument the Master of Logic is thinking "If I could choose what I liked and I picked this then..." But you are equally as constrained as your opponent - you can only choose the most logical answer.

I'm very glad you brought that up. I was hoping you would and you have not disappointed me. This is what has troubled me most about my own position, and it continues to do so (incidentally, that's really what was about). So I decided to make the logic bulletproof and remove the illusion of free will from After further consideration, I'd say I failed to do so. That reasoning doesn't look like there's any free will in it, but there is really. Consider the statement: "If s/he will play cooperate, the move that will get me the most points is defect." The other player's move is considered unknown rather than free to choose, so that's fine. But when considering my own move, how do I know the best move is to defect? Only by choosing it from all possible moves. But that depends on there being more than one possible move, which takes us right back to free will.

At this point I pretty much gave up on the idea of avoiding the language of free will in decision making, so in my most recent post I've reverted to it a bit. We needn't actually buy into the notion of free will, rather we can get around it by considering our future actions as unknown while they are dependent on the current thought process, which amounts to much the same thing.

So we have two unknowns: the other player's action, and mine. They will evaluate the same, but only because they will both be the most logical. Do we use this knowledge when determining which is the most logical, or do we give them a separate status, based on the fact that one is dependent on the current thought process and the other is not? The answer to that largely depends on what you think logical decision-making is. I'll not say any more, except that this is the new direction I wanted to take. I dare say more thoughts will come along eventually.

[Guest]

OK putting all the other thoughts to one side for a minute...

I'm very glad you brought that up. I was hoping you would and you have not disappointed me. This is what has troubled me most about my own position, and it continues to do so (incidentally, that's really what was about). So I decided to make the logic bulletproof and remove the illusion of free will from After further consideration, I'd say I failed to do so. That reasoning doesn't look like there's any free will in it, but there is really. Consider the statement: "If s/he will play cooperate, the move that will get me the most points is defect." The other player's move is considered unknown rather than free to choose, so that's fine. But when considering my own move, how do I know the best move is to defect? Only by choosing it from all possible moves. But that depends on there being more than one possible move, which takes us right back to free will.

At this point I pretty much gave up on the idea of avoiding the language of free will in decision making, so in my most recent post I've reverted to it a bit. We needn't actually buy into the notion of free will, rather we can get around it by considering our future actions as unknown while they are dependent on the current thought process, which amounts to much the same thing.

So we have two unknowns: the other player's action, and mine. They will evaluate the same, but only because they will both be the most logical. Do we use this knowledge when determining which is the most logical, or do we give them a separate status, based on the fact that one is dependent on the current thought process and the other is not? The answer to that largely depends on what you think logical decision-making is. I'll not say any more, except that this is the new direction I wanted to take. I dare say more thoughts will come along eventually.

This is all getting very complicated. I think I can argue against my own previous argument much more easily by saying that if you could reason along the lines of "If I make this choice it must be because it's the most logical..." then you could use that to come to almost any conclusion. Now, arguing back against that argument I would say that I don't think that's the case here, as we're simply using the reasoning to determine what else will happen before we can assess the result of making that decision. However, above all else, I don't think it really matters!

When discussing the approaches that involve discrete assessment of every single step of the game and every "if this, then this..." scenario, I think I may be arguing the right point, but I haven't been completely convinced and I would not argue that they are bullet proof in any way. Instead I have have been making those arguments to counter yours and show that I don't think your arguments are correct either.

Which is where I was going with the "bigger picture" solution. If the discrete assessment approaches don't work, then stop trying and look at the problem from a different angle. In particular, if you think of the bigger picture then it becomes clear that always defect is not the best outcome.

I think the critical element in accepting the bigger picture argument is that the Master of Logic would recognise that a decision which may appear logical on a micro level would be illogical when considered at the macro level. To demonstrate that this would be the case, consider this:

In a one player game of 10 rounds, a Master of Logic should say "win" or "lose" in each round. If they say "win" they win 1 token. If they say "lose" they get no tokens. At the end of the 10 rounds, if the Master of Logic still has no tokens, they win 20 tokens.

In this game, if the Master of Logic works at a micro level, i.e. in each round in isolation he considers the best outcome, then he will win 10 tokens by the end of the game. If he works at a macro level then he will win 20 tokens by the end of the game. Personally I think that a Master of Logic would not miss something like this and so would end up with 20 tokens at the end of the game.

I think what I'm getting at for both the single game and the 50 round game, is that the Master of Logic may well consider that at the micro level they would win more by defecting, but when stepping back and looking at the macro level they would see that cooperation is the best choice, because it is only then that they would take everything into account.

Coming back to something you said earlier - you mentioned superrationalityand linked to the Wikipedia page on it (as I have). I actually came across this independently as I must admit I didn't read it properly when you first linked to it. However, I think that maybe the key point is how does a Master of Logic think? If they think in a superrational way then they will always cooperate. If they think in a game-theoretical rational way then they will always defect. Although there is some debate over whether superrationality can exist in a human, I would think that a Master of Logic would be superrational - even the name pretty much suggests that and you've said yourself that a Master of Logic is not subject to human misgivings. The Wikipedia article then does a better job of explaining the answer than me and even touches on your cloning example (maybe that's where you got it from?) (For anyone who hasn't understood this paragraph, read the Wikipedia article.)

So, I think at the end of all this, either one of two things can be true:

1. You intend a Master of Logic to be game theoretical rational instead of superrational. In this case we're just going to have to disagree and as we're talking purely about a definition I don't think it makes much difference either way.

2. You disagree with the whole concept of superrationality and this topic (and our debate) has actually been about tackling this subject. Now that I'm familiar with it, I'm happy I argued my case so strongly and will happily go along with the consensus on this (that always cooperate is right) without any further debate.

I have to say, now I actually know what the "proper thinking" on this is, I'm glad that this didn't turn out to be a trivial matter with a really simple solution!

[EventHorizon]

...

First, here'a a variation to show where the limit lies in my opinion. Suppose the scoring system for one round were exactly the same except that we now penalise mutual defection.

So:

If both players played "COOPERATE", they are awarded a point each.

If they both played "DEFECT", they both lose 2 points.

If one player played "DEFECT" and the other "COOPERATE", the defector gets 2 points, and the cooperator loses 1 point.

Following the logic in my we get nowhere. If the other player cooperates, my best move is to defect (gets 2 points instead of 1). If the other player defects, my best move is to cooperate (gets -1 point instead of -2).

If you thought the original scenario probably favoured cooperation, this should really settle it. Mutual Defection has lost its status as a Nash Equilibrium. Now Mutual Cooperation and Mutual Defection are just two equilibria, both unstable, one offering a point to each player, the other offering the loss of 2 points.

My position on this is that it is actually a dilemma for Masters of Logic, whereas the main scenario isn't. Defection was the clear choice before, but now there is no reason for them to favour one move over the other.

At this point Neida and others may be shouting at the monitor "BUT THEY WILL BOTH CHOOSE THE SAME!!! If you play cooperate, you'll win a point, if you play defect, you'll lose 2 points! It's a fact!"

But is it? If what I say is correct and there is a genuine dilemma, if there is no logical path to the decision, then it's like choosing heads or tails. Without a clear logical choice there is a 50-50 chance of either. If the other player has chosen a move at random, cooperation will win either 1 or -1 points, and defection either 2 or -2 points. Both average at 0. Defection is the more high-risk option but also high reward so is not particularly favoured. If the other player is slightly more likely to cooperate, you would defect, and vice versa. So the "pick one entirely at random, with no bias" strategy is stable (for a single round game, I hasten to add).

But instead of choosing something at random, what's to stop both players simply realising that Mutual Cooperation is the way to go to get maximum points? If this were logical then it would be so for both, and we can go back to being able to predict what the other player does.

...

Ahh... Chicken. If you could, rip the cooperate card so the opposing player can choose to either lose 1 or lose 2. If your opponent thinks you are crazy (so you will only play defect)... the rational choice is to play cooperate to avoid the worst possible outcome. But if the opposing player is thinking the same, you'll both owe the Grand Master 100 beers. So, who will try and stop the bleeding first? It depends on how risk tolerant the players are and what they believe about the other player.

Neither mutual cooperation nor mutual defection is an equilibrium of the single-shot game (of the repeated game, the folk theorem comes into play... but mutual defection cannot be an equilibrium (without "knowing" incorrect information about the opponent)). CD and DC are the only pure strategy Nash equilibria of the single-shot game (which both happen to be stable). There's a mixed strategy NE in there as well (though it is unstable... like all mixed strategy equilibria) as Octopuppy pointed out (50-50). As long as both players believe the other will play according to it (and both players know it), there is no incentive to deviate from the mixed strategy equilibrium (e.g., it can last more than the first round given the right information).

There is no best strategy without knowing the opponent's strategy (just like I have proven for the prisoner's dilemma in a previous post). So I still believe there is no "best"/logical strategy without extra information (knowledge about the other player, or some history of play, etc). There are definitely some bad (dominated) strategies, but there are lots of good ones too. And as Neida pointed out, Masters of Logic are definitely not myopic... so they'll look at the big picture.

Of course, some extra information won't necessarily result in a best strategy (like knowing the other player is also a Master of Logic).

One thing I've insisted on is that in decision-making players cannot think "if I choose X, the other player will too". It's true, but they cannot think it. Why? Because it's illogical. More on that later, but first here's a couple of scenarios to illustrate the limit of that.

Coming back to something you said earlier - you mentioned superrationalityand linked to the Wikipedia page on it (as I have). I actually came across this independently as I must admit I didn't read it properly when you first linked to it. However, I think that maybe the key point is how does a Master of Logic think?

Logically? (sorry... I had to )

Does it not just depend on the information available to them? If both players know that they both will act according to a specific logic, and that logic includes knowledge that the other player has the exact same available information and will act according to the same logic, why would it be illogical to use that information to know that mutual cooperation would be best (due to the obvious symmetry)? It seems to me that superrationality is simply acting rationally when in the presence of certain extra information (though perhaps with inhibited free will mentioned previously to keep the assumption "everyone will play the same" correct).

What if both players are told that the other is superrational and that both players have exactly the same information, but one retains his free will? Wouldn't the best strategy be to cooperate until the last round (letting the other continue to believe incorrectly that he will always cooperate), and then defect on the last round?