[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

[EventHorizon]

Woo yay! You even got past the red-herring 10th round! But then...

"so that means their chain of logic will go infinitely back (being Masters) and they will defect every time. But even if one starts the defect chain earlier, he doesn't make up for it and they both get sub-optimal."

I honestly don't know if that's nonsense or I just don't understand it. But I'm loving your answer anyway. Why wouldn't they just defect on all 50 rounds?

The same logic in my original post...

There is no punishment possible after the last round. If you defect on the last round, you could try to get a bonus of two at the end. The other person would defect too being the master of logic he is. This makes the last round amount to nothing. You could look at the second to last round as the last round and continue the pattern of mutual defection, but this lowers total reward since there will be a defection by the other player coming. So you can assume cooperation up to the last round.

Essentially you want to be the one to defect first if there is going to be defection, but defecting early lowers total reward...

I thought this paragraph included the 10th round by "look at the second to last round as the last round and continue the pattern of mutual defection," and didn't think it was worth mentioning specifically.

That being said, your actions don't affect theirs, so still defect on that last round. But the round before it will still be subject to tít-for-tat because defect then and both of you will defect on the final round and get no points.

So

they'll cooperate up until the final round where either both or one of them will defect, unless they are so masterful they realize that the other person is thinking the same and follows the same paths - in which case they would cooperate 100%

Essentially, this shows the circular logic at the end of my first post.

but if they both defect, they both get 49. 100 may be maximum, but 50 is optimal

Optimal with respect to what? If one person is going to cooperate all the time, isn't defecting on the last time (without time for retaliation) a better strategy (you end up with a higher score than otherwise). Perhaps only Super Masters of Logic +5 would do that (sorry for the sarcasm).

I don't see the reason for the "Woo yay!" No disrespect to unreality, I'm sure he came up with that on his own. But as I see it, we are basically in agreement (even with the somewhat arbitrary decision of having them cooperate on the last round).

The final two rounds become something like a paper rock scissors game (creating the circular logic I mentioned originally)

CC - Just cooperate... it's better than the two rounds of mutual defection.

CD - Well, if he's going to cooperate, I should defect on the last round to get the 51.

DD - I'll still get 50 If I defect a round earlier than him.

rinse and repeat.

There are a few pareto optimal solutions, included is the 50-50 (Nash bargaining solution / the fair outcome). I argue there is no best (without qualification) solution, but I am partial to the fair pareto optimal one.

[Guest]

I don't see the reason for the "Woo yay!" No disrespect to unreality, I'm sure he came up with that on his own. But as I see it, we are basically in agreement (even with the somewhat arbitrary decision of having them cooperate on the last round).

The Woo yay! referred to the earlier part of his answer,

"[at first i thought that they] cooperate until the tenth round when told the number of rounds, then defect every round after that because of the unexpecting-hanging (or "surprise tiger") paradox. No matter what they've been doing before, playing defect on the very last round (when known) will do them better (2/0 instead of 1/-1). However, then going back a round they would defect then as well since that has become the last round before defection. Except for every round that they both defect, they get 1 less point each than if they had both cooperated (assuming they both go back the same level of thought nesting)... anyway, I realized it wouldn't matter before round 10 cuz they do know that it has to end SOMETIME (right?) so that means their chain of logic will go infinitely back (being Masters) and they will defect every time."

This is, IMO, the correct answer (though I must admit my mind rebels against it somewhat, as unreality's seems to have done). It means that both players will defect all the way through and win nothing. Now comes the interesting part, where everybody says how that can't possibly be correct

[EventHorizon]

I guess I should have spent more time with the induction than just giving it a sentence or two...

Yes, always defect is the Nash equilibrium solution of the single-shot game (and therefore also for repeated-play versions). There is no way for the other player to decrease your score by deviating from their part of the Nash equilibrium. But there are many solution concepts beyond the single-shot Nash equilibrium.

I doubt two intelligent beer-loving Masters of Logic would voluntarily choose the outcome of getting nothing. Have you ever heard of the Folk Theorem of game theory? It states that any outcome with average payoff greater than the fall back position can be an equilibrium of the repeated-play game. This is essentially because no strategy will result in the highest payoff against all other strategies.

This is obvious by the following:

-Always defect will get the highest payoff against any static sequence of actions.

-Always cooperate (or t1t-for-tat) with a defection on the last move will get the highest payoff against "never forgive" (cooperate until the first defection, then defect)

Even if their only communication is knowing the outcome of the rounds, it can be used to signal willingness to cooperate.

Logic isn't the only factor at play here. It also depends on how risk-averse the participants are. This is because there is a communication channel, and therefore learning can take place so strategies can change/evolve.

Do you believe a Master of Logic, when confronted with t1t-for-tat with random forgiveness (however, not enough to make it worth sticking with always defect), would not explore possible strategies and try to end up with more tokens?

Or, similarly, would someone (who isn't completely risk-averse, and who doesn't believe the other is "unteachable") not try and "teach" to gain more tokens?

This reminds me of a quote from Confucius -- "Only the wisest and stupidest of men never change."

Sticking with always defect when something is better != wisest. Then again, logic isn't necessarily wisdom.

[Guest]

I guess I should have spent more time with the induction than just giving it a sentence or two...

Yes, always defect is the Nash equilibrium solution of the single-shot game (and therefore also for repeated-play versions). There is no way for the other player to decrease your score by deviating from their part of the Nash equilibrium. But there are many solution concepts beyond the single-shot Nash equilibrium.

I doubt two intelligent beer-loving Masters of Logic would voluntarily choose the outcome of getting nothing. Have you ever heard of the Folk Theorem of game theory? It states that any outcome with average payoff greater than the fall back position can be an equilibrium of the repeated-play game. This is essentially because no strategy will result in the highest payoff against all other strategies.

This is obvious by the following:

-Always defect will get the highest payoff against any static sequence of actions.

-Always cooperate (or t1t-for-tat) with a defection on the last move will get the highest payoff against "never forgive" (cooperate until the first defection, then defect)

Even if their only communication is knowing the outcome of the rounds, it can be used to signal willingness to cooperate.

Logic isn't the only factor at play here. It also depends on how risk-averse the participants are. This is because there is a communication channel, and therefore learning can take place so strategies can change/evolve.

Do you believe a Master of Logic, when confronted with t1t-for-tat with random forgiveness (however, not enough to make it worth sticking with always defect), would not explore possible strategies and try to end up with more tokens?

Or, similarly, would someone (who isn't completely risk-averse, and who doesn't believe the other is "unteachable") not try and "teach" to gain more tokens?

This reminds me of a quote from Confucius -- "Only the wisest and stupidest of men never change."

Sticking with always defect when something is better != wisest. Then again, logic isn't necessarily wisdom.

At the other end of the scale, cooperation on all 50 rounds is also out of the question, and the inductive logic that follows from that is quite inescapable. Even if it seems rather abstruse by the time it's been extrapolated back several rounds, it might not seem so to a Master of Logic. Always Defect is a stable result in the sense that neither contestant can improve their lot by throwing in a last minute defection, but it's such a bad result that players might try anything to avoid it.

There has to be a way out of that. Always Defect is the logical conclusion but it depends on the firm belief that the other player will play with the same scrupulous logic as you do, as would be expected from a Master of Logic. If instead the other player starts by cooperating, this premise is shown to be incorrect, and the door is opened to other possibilities. Perhaps the logical approach would be to take the 2 points, ignore the unexpectedness of the behaviour that caused them, and continue to defect for the remaining 49 rounds. After all, any "understanding" between players is subject to change at any time, so at any given time you can assess the game ahead without reference to what has gone before. That's obviously a pretty stupid approach but it might be what a Master of Logic does.

Like you say, logic isn't necessarily wisdom. I haven't the confidence to declare my answer the definitively correct one. If it isn't, I think there is probably no correct strategy, since the logical thing to do is behave illogically, thus defeating the logic which brought us to the Always Defect result. In which case the answer is still "0 points", since at the beginning of the game the Masters of Logic would give it a moment's consideration and then both their heads would melt. The only real wisdom is that of the Grand Master, who doesn't give out beer tokens for nothing.

[Guest]

Normal people can potentially make a choice which will make them worse off, whether through a sense of moral obligation, trust, logical error, lack of confidence in your reasoning, or just saying "what the hell". Masters of Logic can not. For that reason any conspiratorial undertaking they enter into (on the basis that anything's better than continual mutual defection) will inevitably fall foul of the inductive problem arising from knowing when the last round is, with 100% certainty. So neither of them is capable of making the first move toward cooperation. It isn't in their nature. Masters of Logic are actually slaves to logic.

I feel pretty sure of that now, so I'll put it in a spoiler in the OP. Feel free to disagree!

[Guest]

It's a good plan, but I must stress that no conspiracies of any sort are taking place. That includes conspiratorial "understandings" between the players. Neither will care if the other is left scowling at the end. I predict much controversy from this topic. It's far from over...

And with that, I bid you good night.

What controversy? There's absolutely not incentive not to cooperate on every round, especially since we're talking about perfect logicians here - which implies rational logicians that want beer ftw and won't be petty. You said yourself in the description of the game that the object is NOT to get more beer than the other person, but rather to get the most total. You can't get more beer total with +2 -1 rather than +1 +1, so cooperation is still the best choice. No wait, that's wrong, but still: getting the most for yourself = getting the most total when you're dealing with two players with equal logic and equal abilities. Logicians would realizing that defecting leads to no gains, and that total cooperation is the best choice. Why would anyone, perfect logician or not, NOT cooperate?

Edited January 9, 2010 by DarthNoob

[Guest]

What controversy?

Well, nobody's agreed with me so far, except unreality who then changed his mind

There's absolutely not incentive not to cooperate on every round, especially since we're talking about perfect logicians here - which implies rational logicians that want beer ftw and won't be petty. You said yourself in the description of the game that the object is NOT to get more beer than the other person, but rather to get the most total.

Maybe I wasn't clear enough, but each player is concerned only with their own points. Getting more beer for the other player is of no value either way. They aren't petty, just completely selfish. I can see there is a bit of ambiguity in the OP - sorry if that led you astray (though the following line should have cleared up any doubt). Anyhow I've added a footnote just to make sure.

You can't get more beer total with +2 -1 rather than +1 +1, so cooperation is still the best choice. No wait, that's wrong, but still: getting the most for yourself = getting the most total when you're dealing with two players with equal logic and equal abilities. Logicians would realizing that defecting leads to no gains, and that total cooperation is the best choice. Why would anyone, perfect logician or not, NOT cooperate?

Simply because the other guy isn't going to.

[Guest]

Well, nobody's agreed with me so far, except unreality who then changed his mind

Maybe I wasn't clear enough, but each player is concerned only with their own points. Getting more beer for the other player is of no value either way. They aren't petty, just completely selfish. I can see there is a bit of ambiguity in the OP - sorry if that led you astray (though the following line should have cleared up any doubt). Anyhow I've added a footnote just to make sure.

Nah, your OP was fine. I just misunderstood, and realized I misunderstood halfway through writing the post, thus "No wait that's wrong"

Simply because the other guy isn't going to.

Both players are equal to each other, which means unless random chance comes to play, they will reach the same outcome. That means even if they played some sort of game involving cooperation and defection, they'd still end up no better than 50 beers (half wins and half losses means 25*2 - 25).

Because both players are completely equal to each other (and they know they're equal - they know they're master logicians), like I said, getting the most for yourself = getting the most for the other person and yourself = maximizing the beers for all involved. There is no reason why "the other guy isn't going to."

Say B is a perfect logician, but A is not.

If A thinks that B will cooperate the entire time, then they might be tempted to defect on the first round, thus gaining two points:

A gets 2, B loses 1

But then B realizes A is not going to be cooperative, so B will defect to, and they get stuck at zero (or they get stuck at zero from the first round).

Logicians would know that this would happen (or some other less than optimal variation), so they wouldn't choose this.

Now, the last round might be dictated by random chance, so maybe it'll be 51/49 or 49/49 rather than 50/50. I think the logicians would use the same logic that dictated the first round and apply it to the last round to get 50/50, but I don't know. (However, it can't go back to the second last round, or third last, and so on, because of the risk it creates of more than one double defection)

If they don't know the number of rounds, then they will cooperate 100% of the time for the same reasons.

It just makes no sense not to cooperate. It bloody weirds me out!

[unreality]

darth said what I was trying to say. If they do indeed play the same game, 50 and 50 cooperation will get them the most points

[Guest]

Both players are equal to each other, which means unless random chance comes to play, they will reach the same outcome. That means even if they played some sort of game involving cooperation and defection, they'd still end up no better than 50 beers (half wins and half losses means 25*2 - 25).

If they could choose a strategy collectively and be sure both would hold to it, then cooperation would be it. Trouble is they can't (for reasons about to be clarified)

It just makes no sense not to cooperate. It bloody weirds me out!

That's why I wavered in It's a total affront to common sense.

In order to not waver, it's important to establish certain things:

1) The probability of mutual defection on the last round is 100%. 99% isn't enough.

Let's look clearly at the final round in isolation. Say you're a Master of Logic about to play the final round. You would think "On this round, I will score one point more by defecting than I will by cooperating. This is true regardless of what the other player does." There are no further rounds in which cooperation may be possible, what has gone before is irrelevant, there are no other considerations. Your goal is to score as many points as possible. Playing the "cooperate" card would not have the effect of making your opponent do the same. If your choice could influence the other player's choice, then you would cooperate. But it doesn't. You know the other player will think the same, and that both players thinking this will lead to a score of zero. But there is nothing you can do about that. The only thing under your control is the card you play. All factors outside your control being equal, the card that will score more is "defect". Therefore you will play this card, with 100% certainty.

2) Preceding rounds also have a 100% chance of mutual defection.

Now suppose you're on the 49th round. You know that there is a 100% chance of mutual defection on the following round. The same logic means that there is a 100% chance of defection on this round. This is where Masters of Logic differ from the rest of us. An imperfect logician might fail to see this line of reasoning and play "cooperate" on an earlier round. The other player, knowing this, may choose a strategy of mutual cooperation since there is some chance that this might work. The uncompromising perfection of the mind of a Master of Logic hammers out any such possibility. Their clear goal and perfect reasoning mean that their actions are completely predictable, like a computer following its program. Once we have established what the programmed behaviour is, there is no chance at all that they will deviate from it.

[Guest]

I have a feeling we may have to agree to disagree...

1) The probability of mutual defection on the last round is 100%. 99% isn't enough.

Let's look clearly at the final round in isolation. Say you're a Master of Logic about to play the final round. You would think "On this round, I will score one point more by defecting than I will by cooperating. This is true regardless of what the other player does." There are no further rounds in which cooperation may be possible, what has gone before is irrelevant, there are no other considerations. Your goal is to score as many points as possible. Playing the "cooperate" card would not have the effect of making your opponent do the same. If your choice could influence the other player's choice, then you would cooperate. But it doesn't. You know the other player will think the same, and that both players thinking this will lead to a score of zero. But there is nothing you can do about that. The only thing under your control is the card you play. All factors outside your control being equal, the card that will score more is "defect". Therefore you will play this card, with 100% certainty.

False.

"I will score one point more by defecting than I will by cooperating. This is true regardless of what the other player does."

How is this true regardless of what the other player does? Defecting won't score more if the other defects.

If you have two doors, one leading to paradise, one to hell, a master logician would go through the door that leads to paradise. Why? Because it's logical to choose the better choice. (of course, this precludes the assumption that what is "better" has already been decided. In this case, the more beers the better)

A master logician would not choose the road that would lead to double defection, because that's illogical.

They will cooperate, because they know the other person will do so too. Let me repeat AGAIN:

Both players are equal in every respect, so unless chance comes into play, they will play the same thing every time, and they know they will play the same thing (ignoring chance). They know double cooperation will occur.

AGAIN: What is best for one player = what is best for both players total

It is logical to help each other out, believe it or not.

2) Preceding rounds also have a 100% chance of mutual defection.

Now suppose you're on the 49th round. You know that there is a 100% chance of mutual defection on the following round. The same logic means that there is a 100% chance of defection on this round. This is where Masters of Logic differ from the rest of us. An imperfect logician might fail to see this line of reasoning and play "cooperate" on an earlier round. The other player, knowing this, may choose a strategy of mutual cooperation since there is some chance that this might work. The uncompromising perfection of the mind of a Master of Logic hammers out any such possibility. Their clear goal and perfect reasoning mean that their actions are completely predictable, like a computer following its program. Once we have established what the programmed behaviour is, there is no chance at all that they will deviate from it.

False on two counts (and this is something I already countered in my earlier post, but I'll repeat):

Though I'm willing to admit there might be chaos in the final round (though I'm starting to doubt that more now), you cannot extrapolate this to earlier rounds, because master logicians would not be willing to risk double defection occuring more than once.

The other reason this is false is the one I just outlined: it's logical to cooperate, to help each other out, because that will lead to a 100% chance of 50 (or perhaps 49).

[Guest]

"I will score one point more by defecting than I will by cooperating. This is true regardless of what the other player does."

How is this true regardless of what the other player does? Defecting won't score more if the other defects.

If the other player defects, you will score: 0 if you defect, -1 if you cooperate. Defecting still scores 1 more.

A master logician would not choose the road that would lead to double defection, because that's illogical.

It's illogical to think you could stop the other player from defecting by playing cooperate. How does that influence the other player?

Let's say a player thinks "I'll play cooperate, because I am a Master of Logic and if I do so it will be because it is the logical choice. It being the logical choice, the other player will do the same.", they are depending on the reasoning that being a Master of Logic implies that their every decision will be logical. Being a Master of Logic is a consequence of your every decision being a logical one, it doesn't imply you can do whatever you like and whatever you do will be logical just because you did it. In order for playing cooperate to be the logical choice, it must be supported by some reasoning. What reasoning could there be? Playing cooperate has no direct advantage to your score, since it always scores 1 less than defect. The advantage of playing cooperate is solely that it gives the other player an incentive to also play cooperate. On the last round, that advantage does not exist, so the logical basis for cooperation is gone.

Though I'm willing to admit there might be chaos in the final round (though I'm starting to doubt that more now), you cannot extrapolate this to earlier rounds, because master logicians would not be willing to risk double defection occuring more than once.

Why would there be chaos? The mind of a Master of Logic is well-defined, isn't it? Their objectives are well-defined, and their reasoning is perfect. If they will defect in the last round, they can be 100% sure of this in advance. Extrapolating it to earlier rounds is not a matter of choice. If they will defect in the last round, the next-to-last is now a round played with no future consequences. You may think "better not defect in the last round then, if it leads to earlier defection". Trouble is, when the players get to the last round, that won't matter any more. By then, the rest of the game will be in the past. They will then, in effect, be playing a single round game with no incentive to do anything but defect. So they will defect. They can't escape the knowledge that this will happen.

[Guest]

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.

Let me address this part of the question specifically. We assume that there will always exist some 'very best strategy' given any set of circumstances, otherwise the question itself is flawed. Since both players' circumstances are identical at the start of the game, the 'very best strategy' for each player will be identical and since both players are Masters of Logic, their moves will also be identical. Given that each player knows that the other player is also a Master of Logic, how can they fail to see this symmetry? And, having knowledge of this symmetry, how can 'the very best strategy possible for their own gain' not be the one that cooperates every round?

We could endlessly follow a circular and pointless logical train that resembles "if A does this then B should do this, but since A knows this, A should instead do this, therefore B should do this, but since A knows this, A should instead do this..." But why bother? A Master of Logic would begin with the assumption that both players' choices will be identical throughout the entire game and find the pattern that gives him the most points. Which happens to be 'both players cooperate every round'. This is the 'very best possible strategy' and both players will utilize it.

The only flaw I can see with this is that the strategy cannot be arrived at by following a logical train that begins with "If, on turn x, I do this then..." The strategy must burst forth, fully formed, from the foreheads of the Masters, as it were. But my argument is that this is the only way in which a 'very best possible strategy' can even exist at all. Any logical train that doesn't eliminate asymmetrical results outright will never arrive at a best strategy but will circle endlessly.

I think what you are trying to say is that once a strategy is chosen, it ceases to be the best possible strategy since the other player knows what you will do and can therefore exploit it. So the best strategy must be the one that is the least exploitable, even if known by your opponent beforehand, which is to defect every round.

I personally disagree. I believe that the Masters would be able to simultaneously arrive at the strategy that awards each of them 50 instead of 0. So I suppose we'll have to agree to disagree. Irregardless I've enjoyed thinking about this puzzle very much. Thank you for posting it!

[Guest]

If the other player defects, you will score: 0 if you defect, -1 if you cooperate. Defecting still scores 1 more.

And you gain 1 if the other cooperates...

It's illogical to think you could stop the other player from defecting by playing cooperate. How does that influence the other player?

Again, you're assuming the other player would defect. It's illogical to defect, so it's also illogical to think the other player will defect

What reasoning could there be?

I've already outline this: what is best for one player = what is best for both players, so a master logician would play that which would optimize beers for both players (which is double cooperation, which they know will occur)

Playing cooperate has no direct advantage to your score, since it always scores 1 less than defect.

Again, you're assuming the other player doesn't cooperate.

The advantage of playing cooperate is solely that it gives the other player an incentive to also play cooperate. On the last round, that advantage does not exist, so the logical basis for cooperation is gone.

again, and here I do copy and paste: "what is best for one player = what is best for both players"

(because both players are trying to reach the same goal, so cooperation - the usual meaning of the word, that is - is logical]

Why would there be chaos? The mind of a Master of Logic is well-defined, isn't it?

If a master logician observes a coin about to be flipped and he is told he has to bet on one outcome, he has no logic, so master logician or not, he has to guess. Chaos is possible... though maybe not in this situation. I was just saying that I'm not completely sure about the last round (but by now I'm pretty sure that the last round will proceed like the previous ones, that is, double cooperation)

Again, i think we might have to agree to disagree, unless you refute my "what is best for one player = what is best for both players"

EDIT: Tuckleton also posted a variation of my opinion, basically the same thing from a different perspective, though I want to delve into...

The only flaw I can see with this is that the strategy cannot be arrived at by following a logical train that begins with "If, on turn x, I do this then..." The strategy must burst forth, fully formed, from the foreheads of the Masters, as it were. But my argument is that this is the only way in which a 'very best possible strategy' can even exist at all. Any logical train that doesn't eliminate asymmetrical results outright will never arrive at a best strategy but will circle endlessly.

Let's say the masters saw more than one possible optimal solution, e.g., A defects and B cooperates for 25 rounds (does that add up to 50? Or is that 25? Well, whatever, just read!), and then vice versa. Given multiple solutions, they wouldn't know which to pick, and because chance/chaos has been introduced, they wouldn't know what the other would pick. Here is Octopuppy's point about logic =/= wisdom: they have no reason to realize that most people would sensibly choose the option of all cooperation. Chances are they'd pick different paths, and end up with an arbitrary amount of less-than-optimal points.

But here's the thing: they take this on a round by round basis, so in every individual round, double cooperation makes the most sense, so no other solutions will ever be considered, and Octopuppy's conventional wisdom is unnecessary, as logic will do

Edited January 11, 2010 by DarthNoob

[Guest]

...how can they fail to see this symmetry?...We could endlessly follow a circular and pointless logical train that resembles "if A does this then B should do this, but since A knows this, A should instead do this, therefore B should do this, but since A knows this, A should instead do this..." But why bother? A Master of Logic would begin with the assumption that both players' choices will be identical throughout the entire game and find the pattern that gives him the most points. Which happens to be 'both players cooperate every round'. This is the 'very best possible strategy' and both players will utilize it.

Masters of Logic would see the symmetry but I maintain that there are limits to how they can reason with it. Let's say they are both on the final round and considering what to do. Both know that mutual cooperation is better than mutual defection. And let's say you're right and both have chosen cooperation.

Now, let's consider what is to stop one of them, call him player A, from deciding to defect and thus get 2 points instead of 1. If he assumes symmetry, then he might think "if I choose to defect, player B will as well". But is that logical reasoning?

Masters of Logic behave alike, not because of some magical link between them, but purely as a consequence of their reasoning being perfect, and based on the same criteria. So both will make "the logical choice".

So when both players A and B defect, player B did not defect because player A did. Player B defected because defection is the logical choice.

If cooperation is the logical choice, player A can now be certain that player B will cooperate. So he can choose to defect without fear that B will do the same, thus making defection the logical choice for player A. But the logical choice for one will be the logical choice for both and so a contradiction arises. This contradiction only comes about if you assume cooperation to be the logical choice.

Causality matters here. Player A can not cause player B to play differently on the current round, so reasoning based on such causality will not enter into the decision making.

See superrationality for more on this subject. It's a controversial topic but I stand by the reasoning above.

More problematic would be the case of a Master of Logic making two exact clones of himself (copying the exact structure of his brain including any thoughts and opinions knocking about inside), which play each other immediately after creation. Provided their experiences have not differed to a significant amount, each clone can be fairly confident that the other clone will do what they do. In this case I suggest cooperation would be the logical choice. If clone A thinks "if I choose to defect, clone B will as well", it is now justifiable.

Interestingly, there is still no causality between the players!

However, if they recognise free will as an illusion and consider their actions to be determined by prior physical conditions, their identical starting conditions imply that their choices will be the same. This is a bit paradoxical because they are now in effect considering a choice that, by their own admission, they don't really have. This is less like making a choice, than trying to perceive the choice that they are destined to make.

But because there is a correlation between their choices which is independent of the logical value of those choices, it's valid reasoning.

[Guest]

I've read this topic with interest and it's prompted me to post again (for the first time in a long time). I'm sorry to say Octopuppy that it's to counter your argument... ;-)

Now, let's consider what is to stop one of them, call him player A, from deciding to defect and thus get 2 points instead of 1. If he assumes symmetry, then he might think "if I choose to defect, player B will as well". But is that logical reasoning?

Masters of Logic behave alike, not because of some magical link between them, but purely as a consequence of their reasoning being perfect, and based on the same criteria. So both will make "the logical choice".

So when both players A and B defect, player B did not defect because player A did. Player B defected because defection is the logical choice.

I'll also refer back to another comment you made where you said that deciding to defect would always leave you 1 point better off regardless of how the other person acted, hence defection is the logical choice. The flaw in this is that you are considering the other person's choice as an independent event, whereas I think everyone (including yourself) agrees that it isn't.

This is because of the argument of "As we are both Masters of Logic who will apply the same reasoning to any situation and therefore reach the same conclusion, there is a 100% probability that we will each do the same thing." Therefore the argument "If I defect and the other person cooperates then I will be one point better off than if I cooperate and they cooperate" just doesn't come into it. By the symmetry logic, the case of one defecting and the other cooperating simply doesn't exist (remember we are talking about Masters of Logic here, not people who deal in grey areas, as you have pointed out a few times).

Once they've realised this, then the choice is simply "Either we will both defect in which case I get nothing, or we will both cooperate in which case I will get one token." It is clearly more logical to cooperate, even on this last round. Defection is totally illogical.

If cooperation is the logical choice, player A can now be certain that player B will cooperate. So he can choose to defect without fear that B will do the same, thus making defection the logical choice for player A.

This is where common sense takes over and says that cooperation must be wrong, but this is probably the biggest flaw in your reasoning. You have continually stated that Masters of Logic are not driven by choice or other reason, but purely by logic. Yet here you are saying that because A knows something he can choose to defect - thus stepping away from his logical roots. As soon as A decides to make a choice based on understanding what B is likely to do (knowing that B will be in an identical decision), he is moving away from logic and introducing uncertainty and risk. As you have said, a Master of Logic would not be able to do this as it goes against their definition.

As for your comment on the cloned Master of Logic, I think this just emphasises this point. Masters of Logic (by your definition) make decisions based purely on logic, not on life experiences, etc. so two Masters of Logic should act identically to one cloned Master of Logic.

Of course, in the real world Masters of Logic (and clones) do not exist! This means that in the last round it will all be down to how risk averse the individuals are, how they assess each other, how much they trust each other, etc. and how much of a poker face they put on. In fact, it may even mean that one defects in the penultimate round to attempt to get 50 (as has been commented earlier with an appropriate comparison to a game of rock/paper/scissors). However, two logical and intelligent people (who each knew the other person was equally logical and intelligent) would know that defecting any earlier than this would only pretty much guarantee themselves of getting less than 50 in total, so would avoid it.

I actually took part in a game similar to this in real life as part of a training course. It was interesting to watch how those that understood the game properly played the cooperate role at the start (understanding this was the best approach), but those that didn't understand it properly (the majority of people) were greedy and defected at every step. We never got to see what happened on the final round, as by that time the cooperative people had changed to defection because it was clear that was all the other person was going to do throughout and an "if you won't let me win then I won't let you win" attitude set in. In every one of our games, the "grand master" equivalent ended up getting the payout! Speaking to the trainers afterwards though, they said that they have had occasions (albeit rare) where both sides have understood and won the maximum prize - i.e. cooperated all the way. I might post the actual game we played as a separate topic to see how people approach it, as it was different to this but the underlying principles were the same.

[Guest]

Did you see my post at the bottom of the page?

The bottom-of-the-page phenomena is an accursed one! -.-

[Guest]

Sorry DarthNoob, didn't mean to ignore you. I've been a bit busy on account of posting this on RichardDawkins.net as well. Actually I decided to be even more counterintuitive there, and say the number of rounds was a random number between 10 and 100 and the players weren't told until the game ends. Thus provoking a storm of dissent from people who'd read somewhere that the inductive reasoning from the last round back only works when the players know the number of rounds.

All things considered the challenges have been more challenging at this end.

Again, you're assuming the other player doesn't cooperate.

Fair enough. It probably makes more sense (considering last round) to start with mutual cooperation and then ask "why not defect"?

I was just saying that I'm not completely sure about the last round (but by now I'm pretty sure that the last round will proceed like the previous ones, that is, double cooperation)

The last round is the key. If defection occurs there, it occurs for certain and everything else follows. So we may as well confine the discussion to that for now.

Again, i think we might have to agree to disagree, unless you refute my "what is best for one player = what is best for both players"

See riposte to Neida (when I think of one)

Let's say the masters saw more than one possible optimal solution, e.g., A defects and B cooperates for 25 rounds (does that add up to 50? Or is that 25? Well, whatever, just read!), and then vice versa. Given multiple solutions, they wouldn't know which to pick, and because chance/chaos has been introduced, they wouldn't know what the other would pick. Here is Octopuppy's point about logic =/= wisdom: they have no reason to realize that most people would sensibly choose the option of all cooperation. Chances are they'd pick different paths, and end up with an arbitrary amount of less-than-optimal points.

It's an intriguing question. The "solutions" in this case are (if you accept my thinking on the last round) Nash Equilibria.

Aside: This bit of the aforelinked article seems to support my position quite neatly, since Always Defect is a Nash Equilibrium and an inductive argument from the last round backwards shows that all other strategies aren't. I'm not offering that by way of Proof By Wikipedia, mind you, since it all comes down to whether you accept the bit about the last round. But I was quite pleased when I stumbled on it.

Back to what you were saying, it would be interesting, or at least, challenging, to try and think of a similar system with no Nash equilibrium, or several. I think Masters of Logic would not know what to do in the former case and be seriously taxed in the latter. Who's clever/learned enough to give us an example?

Edit: d'oh! Masters of Logic playing Prisoners' Dilemma with indefinite rounds would have more than one Nash Equilibrium. Mutual Always Defect and Mutual Tіt For Tat for instance. Choosing which to play is a bit of a no-brainer.

[Guest]

See riposte to Neida (when I think of one)

:-) Glad to see it's got you thinking at least!

Some additional points while you are thinking (and maybe to pre-empt your response)!

First off, it hasn't been said yet, but it's important to note that your "always defect" solution results in cyclic reasoning just as much as you say the "always cooperate" solution does, it's just not quite as obvious. For "always cooperate" the cyclic bit is "I know he will cooperate, so I will defect, but that means he will defect, so I should cooperate...". For "always defect" the cyclic bit is "I will win 0, but my goal is to maximize my winnings, so is theirs and we both know that, we also both know we can do better than 0, so always defect isn't the best solution, so I should cooperate, but then I should defect on the last round, but then I should always defect..."

When something is difficult to prove or disprove using convential means, it is often useful to take the approach of making an assumption and proving/disproving the assumption. That may well be what a Master of Logic would do here. The logic would go something like this:

1. We are both equally skilled Masters of Logic and so will therefore always reach the same conclusion given the same circumstances and data.

2. (Based on reasoning I gave in my previous post) I should always cooperate.

3. There is a question over whether I should change and defect in the final round.

4. Make an assumption that if I change and defect in the final round then my winnings will increase (or at least be unchanged).

5. From (1) I know that if I change and defect, so will the other MoL.

6. This would result in me winning 0 beer tokens instead of 1. Therefore my winnings will decrease.

7. The assumption is proven to be incorrect.

8. If changing and defecting will not increase my winnings, then I should stay with cooperation.

I think this is the only approach that doesn't result in cyclic reasoning of any kind. Note that you could follow on with "assume cooperating will not decrease my winnings, from (1) other MoL will also cooperate, winnings are not decreased, assumption proven."

I know where you are coming from on the Nash Equilibria, but I believe that would only apply to a slight variation of this problem. I.e. if the two MoLs were told they had to win more than their opponent then this would kick in - neither would win, but they would ensure that they do not lose. However, in the problem as it is currently posed (maximising their own gain) there is an equilibria situation that results in them both "winning" and so they would aim for that. Namely they would aim for the solution that results in a draw and maximises their combined winnings. As the result is a draw, maximising combined winnings directly equates to maximising personal winnings, which is the point DarthNoob made that prompted your comment I quoted above! If you have trouble accepting that they would aim for a draw then use the assumption approach again: assume it is not a draw, that means that at some point they acted differently given the same circumstances and data, which contradicts point 1 above and disproves the assumption, therefore it must be a draw. (Note this same reasoning also proves why t i t for tat won't work.)

I do think that this situation is similar to many traditional paradoxes. For example, the Achilles and the tortoise paradox you could argue a Master of Logic would get caught up in because they will continually reason that every time Achilles reaches the point the tortoise was at, the tortoise would have moved forward again. This is cyclic reasoning and the Master of Logic would more than likely realise that this approach is flawed, come at the problem a different way and realise that Achilles does indeed pass the tortoise. Similarly, with your problem, a Master of Logic would realise the flaw in a cyclic reasoning approach and look for an alternative, which is likely to be along the lines of what I've put above.

Edited January 12, 2010 by neida

[Guest]

I'll also refer back to another comment you made where you said that deciding to defect would always leave you 1 point better off regardless of how the other person acted, hence defection is the logical choice. The flaw in this is that you are considering the other person's choice as an independent event, whereas I think everyone (including yourself) agrees that it isn't.

When considering whether the choices are "independent", to be clear, they are both dependent on the same thing, but not on each other. The fact that they are dependent on the same thing means that they will evaluate the same (unless 2 equally "logical" choices are available*, in which case that's not guaranteed, though not a relevant consideration here)

Therefore the argument "If I defect and the other person cooperates then I will be one point better off than if I cooperate and they cooperate" just doesn't come into it.

Certainly this asymmetric outcome is purely hypothetical. It can not and will not happen. But in making a logical decision we need to consider hypothetical scenarios. Indeed, of the scenarios "we both defect" and "we both cooperate", one of these must be entirely hypothetical since the decision of a Master of Logic will only have one possible outcome (the logical one, whichever that is). So considering the other scenario is no less hypothetical than considering the asymmetric case. It also cannot happen, and is therefore unworthy of consideration. So if, say, mutual defection happens to be the logical choice, then a Master of Logic needn't even think about mutual cooperation.

Clearly, there's a problem with that. You cannot reason without considering everything that "might happen". But what does "might happen" mean when only one thing will happen? It means things that would happen if your decision differed from what it will actually be.

On that basis a Master of Logic can still verify the "most logical" status of mutual cooperation by considering what would happen if his decision differed. And it fails the test. If the player then follows a line of reasoning which suggests that the other player will also defect, that cannot be used to imply that defection is not the logical choice, since it depends on the confirmation that defection is the logical choice (otherwise the other player could not play defect). It's paradoxical. An assumption that cooperation is the logical choice only leads to contradiction.

It's tempting to just restrict ourselves to a simple view of two choices, mutual cooperation or mutual defection, and say which is better. But this means that the player's decision will determine whether defection or cooperation is also the "logical choice" for the other player. This is implicit within the notion of having a choice between the scenarios. It means that we can change the nature of logic by our decisions. I doubt that a Master of Logic could see it that way.

I actually took part in a game similar to this in real life as part of a training course. It was interesting to watch how those that understood the game properly played the cooperate role at the start (understanding this was the best approach), but those that didn't understand it properly (the majority of people) were greedy and defected at every step. We never got to see what happened on the final round, as by that time the cooperative people had changed to defection because it was clear that was all the other person was going to do throughout and an "if you won't let me win then I won't let you win" attitude set in. In every one of our games, the "grand master" equivalent ended up getting the payout! Speaking to the trainers afterwards though, they said that they have had occasions (albeit rare) where both sides have understood and won the maximum prize - i.e. cooperated all the way.

All the way? I'd be really surprised not to see a defection at the end.

I might post the actual game we played as a separate topic to see how people approach it, as it was different to this but the underlying principles were the same.

Could be good. I suggest the rewards change continuously so that defection is more heavily rewarded at the beginning and amounts to very little difference at the end, for maximum unpredictability!

* Another way they could vary is if there is an incentive to vary.

Say we tell 2 Masters of Logic to write down an integer from 1 to 100. If they write different numbers, they each get that many beer tokens. If they write the same number, they lose that many tokens. If they believe they must arrive at the same decision, they would cut their losses and write the number 1. But it is easy to find a way to almost certainly get a different number. If the other player appears older or younger, write down your age, otherwise use some other equally arbitrary method. The numbers are not all logically equivalent (risk and reward vary depending on the number) but it pays to ignore that and focus on getting an arbitrary result.

Not that this has much to do with the question at hand but it occurred to me in passing.

[Guest]

When considering whether the choices are "independent", to be clear, they are both dependent on the same thing, but not on each other. The fact that they are dependent on the same thing means that they will evaluate the same (unless 2 equally "logical" choices are available*, in which case that's not guaranteed, though not a relevant consideration here).

I agree, they are not dependant on each other, but that doesn't negate the fact that each person knows that they will decide on the same thing, as they will both use the same circumstances and data to come to their decision. This knowledge (which is factual, not presumptive if we are talking about Masters of Logic) allows them to use this as part of their analysis of determining the best strategy.

Certainly this asymmetric outcome is purely hypothetical. It can not and will not happen. But in making a logical decision we need to consider hypothetical scenarios. Indeed, of the scenarios "we both defect" and "we both cooperate", one of these must be entirely hypothetical since the decision of a Master of Logic will only have one possible outcome (the logical one, whichever that is). So considering the other scenario is no less hypothetical than considering the asymmetric case. It also cannot happen, and is therefore unworthy of consideration. So if, say, mutual defection happens to be the logical choice, then a Master of Logic needn't even think about mutual cooperation.

I don't think this is quite correct. A Master of Logic will also use probability in his/her reasoning. For example, if you asked a Master of Logic to predict the outcome of rolling two die, they would say "7", as that is logically the answer that is most likely to occur.

In this case the Master of Logic would assess all four likely outcomes:

1. I defect, they cooperate: I win 2

2. I defect, they defect: I win nothing

3. I cooperate, they cooperate: I win 1

4. I cooperate, they defect: I lose 1

They then assess the probabilities of each event occurring. In this case the probability of points 1 and 4 are zero as it would break the rule that each will make the same decision. The Masters of Logic are then faced with two possible scenarios. Only one of them leaves them better off so they make that choice.

Now I know that my argument above isn't foolproof as it does make a bit of a leap to say the other two scenarios are still possible. As you have pointed out, only one outcome is possible - the one the Master of Logic will actually choose! However, I think the two step approach of ruling out impossible outcomes before choosing the outcome that gives the greater reward out of all those remaining is the right principle.

On that basis a Master of Logic can still verify the "most logical" status of mutual cooperation by considering what would happen if his decision differed. And it fails the test. If the player then follows a line of reasoning which suggests that the other player will do the same, that cannot be used to imply that defection is not the logical choice, since it depends on the confirmation that defection is the logical choice (otherwise the other player could not play defect). It's paradoxical. An assumption that cooperation is the logical choice only leads to contradiction.

See my latest post which I think addresses this paradox (the cyclic reasoning) and also (I think) confirms the answer from a different approach. EDIT: One additional point on this though is that it does not "depend on the confirmation that defection is the logical choice" as you have suggested - it simply depends on the fact that the other person will make the same choice as you because of the same reasoning - therefore if there existed some line of reasoning that showed defection was the logical choice then both would do it, which would make it not the logical choice. This is not a paradox - it simply shows that no such line of reasoning exists, as otherwise it would lead to a contradiction (again this is the disproving an assumption approach I used in my previous post.)

It would be interesting to see people's thoughts on that one!

By the way, thanks for this! I'm enjoying the stimulating debate and challenging my own thoughts - reminds me why I joined the forum in the first place!

Edited January 13, 2010 by neida

[plasmid]

I tend to agree with octopuppy's analysis of this puzzle, but must admit that symmetry argument is not to be taken lightly. I think that it can be addressed as follows, though.

The symmetry argument goes along the lines of: either both masters will cooperate or both masters will defect (for after all, they are both masters of logic), and of those two options the mutual cooperation option would produce the best result, so they must choose that one.

I counter with the following argument. Suppose you are a master of logic. You're such a perfect master, in fact, that you know with certainty what your opponent will play, because you're both equally perfect masters of logic after all. (If this were not the case, then the whole symmetry argument would break down.) Now there are two possibilities: either you know for a fact that your opponent will choose to cooperate, or you know for a fact that your opponent will choose to defect.

Consider the possibility that you know that your opponent will cooperate. Then what would your best move be? It would be to defect and gain two points rather than cooperate and gain one. But then the outcome of the game would be that your opponent chose to cooperate and you chose to defect, which is impossible for two masters of logic. From this, you can conclude that your initial guess (that your opponent would choose to cooperate) would lead to an impossible state and must be incorrect. The only remaining possibility, then, is that you know that your opponent will choose to defect. Then what would your best move be? It would be to defect and lose nothing rather than cooperate and lose beer tokens. You reach the conclusion that the only possible outcome is that you both choose to defect.

So basically, my argument is that it is impossible for a master of logic to choose to cooperate (if there is only one round in the game), and therefore it is impossible for both masters to cooperate. The shortcoming of the original symmetry argument is that it fails to realize that mutual cooperation is an impossible outcome before choosing the optimal strategy of the remaining outcomes that appeared to be possible.

There is much more to be said about the case of having multiple rounds. But I'll have to wait for another day before getting to that.

As an unabashed side advertisement, anyone interested in this thread might also be interested in a thread going on in the Games section: "Rock Paper Scissors Algorithm Contest", especially if you have a knack for programming. In fact, the idea of having a similar sort of algorithm contest in the future based on the prisoner's dilemma or some other sort of game theory topic had crossed our minds...

[Guest]

A few posts above I think Octopuppy said that if a Master of Logic cloned himself and then played against himself immediately that it would be logical for him to assume that his clone would make the same choices as him and choose to cooperate as a result.

Why would it be logical in this instance? Not because his decision to cooperate has a direct impact on what his clone chooses, but because he knows that whatever logical paths he follows to arrive at this conclusion will be exactly mimicked in his opponent's mind. So he is coming to a decision not based on the knowledge that his choice has an impact on his clone's choice, but because he knows that the reasoning his clone uses to arrive at his own choice will mimic his own in every respect, including that very thought.

It seems to me that we aren't arguing what we think we are arguing.

It all comes down to if one Master of Logic can be considered to be identical in every respect to another Master of Logic.

In my opinion that answer is yes. I believe that the whole point of using Master's of Logic in puzzles is to simplify the problem so that all emotional and environmental factors can be ignored and only a single answer will then result.

I've approached this problem based entirely on the idea that the players are Master's of Logic (and therefore identical) and that they know that their opponent is also a Master of Logic. Based on these criteria I believe your problem to be equivalent to one in which a Master of Logic cloned himself and had the copies play against one another.

I think the discussion here is entirely dependent on what the question means and not what the question actually is. In which case discussion is pointless. If Octopuppy says that within the confines of the question, Master's of Logic aren't equal and therefore can't assume that their opponents will think like they do... well then that should be the end of it right?

Edited January 13, 2010 by Tuckleton

[Guest]

Consider the possibility that you know that your opponent will cooperate. Then what would your best move be? It would be to defect and gain two points rather than cooperate and gain one. But then the outcome of the game would be that your opponent chose to cooperate and you chose to defect, which is impossible for two masters of logic. From this, you can conclude that your initial guess (that your opponent would choose to cooperate) would lead to an impossible state and must be incorrect. The only remaining possibility, then, is that you know that your opponent will choose to defect. Then what would your best move be? It would be to defect and lose nothing rather than cooperate and lose beer tokens. You reach the conclusion that the only possible outcome is that you both choose to defect.

This reasoning is based on the "disproof of an assumption" approach that I've used above, but there is a flaw in it. When you make an assumption, by definition there are various conditions set by that assumption. If you then go on to contravene those conditions, you are invalidating your own assumption, so you can't use that to show a contradiction. For example, if I were to say "assume I tossed a coin and it came out heads" then I couldn't follow that up with "but what if it was a tail" (or something similar) - the assumption sets the condition that it was heads.

The flaw in your approach is that you assume a final decision has been reached and then say "what would happen if I changed that decision" - contravening the condition that it was a final decision. In case that's not immediately obvious, I'll go into it in a bit of detail.

If we analyse your reasoning to determine what assumption you are actually making then we see the following:

1. Assume that my opponent will cooperate.

2. In order for my opponent to have come to the decision to cooperate, this is equivalent to saying, "assume there exists some logical reasoning with a finite number of steps that, once completed, will lead my opponent to cooperate."

3. If such a line of reasoning exists, then Masters of Logic would all follow the same line of reasoning in order to reach that conclusion.

4. This means that I will also have to have completed this finite logical reasoning process.

5. If, at this point, I now consider what my best move would be, as you've suggested, then I am breaking one of the conditions of the assumption - i.e. that I have already completed all of my reasoning.

Note that this invalidates the assumption - it doesn't in itself contradict it for the same reason I gave in the heads and tails example.

So the steps you take after this point are thus invalidated. The only reason you get to an impossible situation is because you have invalidated the conditions of your own assumption.

Another way of putting this is by saying that if you assume the other person has made a decision then this is the same as assuming both of you have made a decision. What you are then basically saying is "Once I've made a decision on what to do, then if I change that decision I would be better off." But if you can change your decision then you haven't made it in the first place! By extension, neither has your opponent, which contravenes (NOT contradicts) the conditions of your original assumption!

Take a look at the disproving an assumption approach I took a few posts up and see if you can find any holes in that. I'm not saying there aren't any, but I can't see any problem with it at the moment, so it would be good to see if someone else can!

If Octopuppy says that within the confines of the question, Master's of Logic aren't equal and therefore can't assume that their opponents will think like they do... well then that should be the end of it right?

I agree, but I believe that Octopuppy's reasoning relies on the fact that Masters of Logic are equal and will think like they do. If they aren't then we are talking about a real world situation where anything can happen depending on how they both consider risk, trust each other, etc, which I've commented on before.

[Guest]

^ what plasmid said (he said it better than I did)

See my latest post which I think addresses this paradox (the cyclic reasoning) and also (I think) confirms the answer from a different approach. EDIT: One additional point on this though is that it does not "depend on the confirmation that defection is the logical choice" as you have suggested - it simply depends on the fact that the other person will make the same choice as you because of the same reasoning

What I wanted to bring home there is that this correspondence only occurs as a consequence of their decision being logical, so you can't apply it in a hypothetical situation where their decision proves to be illogical.

- therefore if there existed some line of reasoning that showed defection was the logical choice then both would do it, which would make it not the logical choice. This is not a paradox - it simply shows that no such line of reasoning exists, as otherwise it would lead to a contradiction (again this is the disproving an assumption approach I used in my previous post.)

The fact that both will defect doesn't necessarily make it illogical. In order to improve on that, you have to be able to make the jump to mutual cooperation by choosing to cooperate. As I mentioned, the concept of being able to choose between the two is intrinsically flawed since you can only choose your own actions. The other player will play the logical move regardless of what you choose to do, so you can justifiably consider their position as fixed but unknown (unknown until you have ascertained what the logical move is).

I know this makes Masters of Logic seem a bit blinkered, so in for a penny in for a pound, let's see if they are completely stupid.

When writing my last post I couldn't escape a vague feeling that I'd shot myself in the foot by bringing up the clone idea. Clearly cooperation is the best strategy between clones. Just to make it absolutely sure, 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. But here's the rub: They are still Masters of Logic. The reasoning plasmid put forth still applies! I'm inclined to think that even though anyone with a grain of sense would cooperate in this situation, a Master of Logic would defect.

This is starting to look pretty paradoxical now. I think the problem is to do with decisions.

Example of a decision: "I have a gun in my hand. I could shoot myself in the head. But if I did I would almost certainly die. I don't want to die. So I won't shoot myself in the head."

Bit of a no-brainer, that one (prrrp-tch!). What we are doing is imagining a hypothetical scenario and evaluating its desirability. But how much sense does that make? Take the statement "I could shoot myself in the head". Could you really? There is something that will stop you from doing that, namely the current thought process. If we say "I could shoot myself in the head, current thought process notwithstanding", it still doesn't make much sense since you're considering a scenario in which your thought processes can result in anything. The hypothetical scenario is an unrealistic fantasy. You wouldn't shoot yourself in the head. Therefore you couldn't. The physical systems in your brain will see to it that you don't. In order to make any decisions, we have to buy into the illusion of free will.

I should stop there and acknowledge that the many-worlds view of quantum physics causes considerable problems for what I just claimed. In a very small minority of possible futures the thought process will go more along the lines of:

"I have a gun in my hand. I could shoot myself in the head. But if I did... umm" . . . [brain-fart] . . . "wibble wibble" . . . [shoots self in head]

However, this does not apply to Masters of Logic (phew! saved).

What we're disagreeing about is exactly what kinds of unrealistic fantasies the Masters of Logic may envisage in their reasoning. Normally this wouldn't be an issue but because their decision-making is self-referential, they are simultaneously depending on the illusion of free will (can't perform the reasoning to make a logical decision without it), while accepting the inevitability of their decisions*. Our disagreement arises over whether we should first draw a generalisation about "potential" final outcomes, then choose one, or use the more standard method of decision making, which acknowledges that what you are deciding is your own action, nothing more. Either way it's flawed because it has at its heart the concept of choice, in a situation where the outcome is inevitable.

So I'm inclined to conclude that a "logical decision" is a self-contradicting phrase. The contradiction isn't normally a problem, but here it is. Which is one reason why Masters of Logic don't exist.

Come to think of it, that has implications for artificial intelligence since there seem to be limitations on what a decision-making engine can aspire to.

"Forget your perfect offering

There is a crack, a crack in everything

That's how the light gets in. "

Leonard Cohen

By the way, thanks for this! I'm enjoying the stimulating debate and challenging my own thoughts - reminds me why I joined the forum in the first place!

Likewise! You've made me really doubt my position.

It would be interesting to see people's thoughts on that one!

Intriguing, reminds me of nim. I'll definitely have a closer look later today.

EDIT: Misread the question. Nothing like nim at all!

* Actually on second thoughts I'm not sure about that. Plasmid's line of reasoning determines what the logical choice will be, simply by demonstrating that it can not be cooperation. No free will is assumed.