[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]

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.

I'm not sure I get what you're saying here. I think you're suggesting that I can't apply my reasoning (of their decision being logical) to a hypothetical situation where they consider the decision of a Master of Logic to be illogical (which is the supposition on which your argument rests). That would not be correct as this hypothetical situation would go against one of the conditions of the game - i.e. that all decisions made are logical. Therefore, Masters of Logic would not consider hypothetical but impossible situations.

If what you are saying is correct then I could argue that, when asked what my score is likely to be if I roll two die, he would say zero. The reasoning would go along the lines of "I have to assess every single hypothetical scenario, regardless of whether it is possible or not. Therefore I must include the hypothetical scenarios where you roll 13, 42, -76 and every other possible number you can think of. As there is a hypothetical negative sum for each hypothetical positive, I can now conclude that your likely score is zero."

Of course this is nonsense and no Master of Logic would ever say this. They would discount the impossible scenarios and then work with what is left to come to an answer.

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.

OK, this and the shooting yourself thing is getting a bit deep and complicated. It also seems to be turning the original question into an absurd situation where sense, reason and logic does not actually come into it, because a "logical decision" does not exist.

If you consider a "logical decision" to be a decision based on logical reasoning, then this certainly does exist. Indeed that's what most logic problems are based on. If we assume that a Master of Logic is someone who will always be able to solve logic problems (as opposed to someone who will always get stuck and get them wrong) then I think it's safe to say that a Master of Logic can make a logical decision.

One other thing I have always assumed from Masters of Logic - that they will not get caught up in traps that defeat mere mortals. Most paradoxes are simply traps that have a logical explanation. For example, the trap in the Achilles and the tortoise paradox is that you are continually evaluating an infinite series that has a finite convergence in time and space, therefore instead of continually evaluating the infinite series, you should simply evaluate the convergence. I have always understood that a Master of Logic would not be taken in by such a trap and would instead approach it a different way.

TRYING TO PUT THIS ONE TO BED:

So far I believe I've found a problem with each approach or argument you have given. You have found a problem with my possible/impossible choices approach, which I acknowledge is not foolproof and I could find fault with myself. So far no one has posted a problem with my disproof of an assumption approach, which appears to be the only uncontested approach in here. Could you please take a look at it and either point out the flaw in my logic there or consider whether that could be the correct answer? Basically - as I've said above, I think a Master of Logic would avoid any approach that results in a trap of some sort, as that approach to a solution would be flawed. Instead they would seek an approach to a solution with no logical flaws, so that is what I think we need to look for here.

(By the way, I think that if a Master of Logic were to play anyone else, they would always defect, as in this case the other person's response is unpredictable and so defection is always the safest bet. However, when playing against another Master of Logic the other person's response becomes predictable and it is this that flips the problem on it's head and leads to the always cooperate approach.)

[Guest]

I think part of the problem at hand is that octopuppy is treating the Masters of Logic as Turing machines, ie. objects given a set of logical restraints that crunch the numbers and spit out a result. The problem with this approach relates to the Halting Problem and the reason why no Decision Engine can exist for First-Order logic.

I'll try to explain. We have three states that we can reach: defect, cooperate or contradiction (infinite regress). (In this particular instance, using octopuppy's logic causes the consideration of cooperation to be a contradiction.) We know that a MoL has to reach one of these three states, but the halting problem shows us that we can't ever know if the process will halt. We cannot be sure that a decision will be reached. Due to the nature of logic, some questions cannot be answered by a Turing machine (or equivalent) and I think that this question qualifies (at least when framed in octopuppy's terms).

If the MoL operate as Turing machines (as octopuppy seems to insist they must), then they will never get an answer because as soon as they consider the 'cooperation' option the machines with infinite regress and never stop. Looking from the outside, we could quickly realize that the process is trapped in an infinite loop, but that isn't from a process of the same order logic. Rather, we have to reason about the nature of the machine and that is something that the machine itself cannot do. We can see that the MoL will go back and forth on cooperation based on the rules that octopuppy set down and we can try to remedy them. Octopuppy decided that since no decision is possible from pure logic using cooperation, the only choice is to defect since that does not lead to contradiction.

What I think neida's approach leads to is creating a decision engine for the problem. The difficulty that octopuppy has with neida's conclusion is that the decision engine is not built from a MoL, since a master of logic is by definition incomplete* , meaning that no MoL could decide if a MoL would reach a decision about whether to defect or cooperate. Neida's reasoning engine looks at how the MoL operate and determines that since they will operate identically, then they don't have to consider the risk of an opponent defecting if they choose to cooperate and thus they can safely cooperate.

So I think that the fundamental problem between the two camps here is that octopuppy is trying to determine things using only a particular order of logic, while neida is recognizing the difficulty with such a restriction and is going above octopuppy's order of logic in order that a decision can be reached.

I may be wrong in some of the specifics of octopuppy's and neida's arguments, but I think that the underlying problem does exist and that is what leads to their opposing conclusions.

* The Wikipedia language seems kind of dense, but I don't want to try to define it in simpler terms right now.

[Guest]

I'm not sure I get what you're saying here. I think you're suggesting that I can't apply my reasoning (of their decision being logical) to a hypothetical situation where they consider the decision of a Master of Logic to be illogical (which is the supposition on which your argument rests). That would not be correct as this hypothetical situation would go against one of the conditions of the game - i.e. that all decisions made are logical. Therefore, Masters of Logic would not consider hypothetical but impossible situations.

Putting what I said back in original context, we were testing the assumption that mutual cooperation is the logical choice. Your view of the situation went along these lines:

1) Assume cooperation is logical

2) Would defection be a better choice for me? (if so it contradicts 1)

3) If I choose to defect the other player will too, because we will always play the same move

4) That makes my score less than it would be if we both cooperated

...so no contradiction there. Or is there? Suppose we expand on part (3) to include the reason why we always play the same move...

1) Assume cooperation is logical

2) Would defection be a better choice for me? (if so it contradicts 1)

3) If I choose to defect, it will be because defection is the logical choice. The other player also makes the logical choice so they will also defect

4) That makes my score less than it would be if we both cooperated

This is no longer consistent. We went from assuming cooperation is logical to confirming that defection scores less, but stage 3 of that reasoning depends on defection being the logical choice! You cannot make this inference in a situation where we've already assumed cooperation to be the logical choice. The line of reasoning depends on a contradiction, but that contradiction was hidden within the generalisation that both players play the same move.

A more valid approach is:

1) Assume cooperation is logical

2) Would defection be a better choice for me? (if so it contradicts 1)

3) If I choose to defect instead, it would increase my score, with no other consequences. This contradicts (1).

However tempted you are to think "But there is a consequence! If you defect, the other player will too", remember that you cannot infer this without defection being the logical choice, contradicting assumption (1). Either way (1) is proved false.

OK, this and the shooting yourself thing is getting a bit deep and complicated. It also seems to be turning the original question into an absurd situation where sense, reason and logic does not actually come into it, because a "logical decision" does not exist.

If you consider a "logical decision" to be a decision based on logical reasoning, then this certainly does exist. Indeed that's what most logic problems are based on. If we assume that a Master of Logic is someone who will always be able to solve logic problems (as opposed to someone who will always get stuck and get them wrong) then I think it's safe to say that a Master of Logic can make a logical decision.

What I was saying, if correct, would render the question meaningless, as it would imply that a Master of Logic, if they existed, would perceive logic but be incapable of making even the most trivial decision. I thought that might explain why we can't agree on an answer. However, having given it more thought, I'm pretty sure I was wrong about that. You don't have to buy into the illusion of free will in order to make a perfectly rational decision. You just have to take an approach of trying to ascertain what you will do, instead of looking at various scenarios of what you might do. That's what I liked about plasmid's approach. Strictly speaking it wasn't all worded that way, but the structure was right, much more to the point than my ramblings tend to be. So you can safely ignore all that stuff.

One other thing I have always assumed from Masters of Logic - that they will not get caught up in traps that defeat mere mortals. Most paradoxes are simply traps that have a logical explanation. For example, the trap in the Achilles and the tortoise paradox is that you are continually evaluating an infinite series that has a finite convergence in time and space, therefore instead of continually evaluating the infinite series, you should simply evaluate the convergence. I have always understood that a Master of Logic would not be taken in by such a trap and would instead approach it a different way.

I'm inclined to agree, but not without doubts

EDIT: Just looked at Dawh's post above, much more doubts now. I've never got my head around all that stuff but maybe it's time to.

So far I believe I've found a problem with each approach or argument you have given.

Actually plasmid's argument will do for me. I know you've posted an answer to that, which I'll comment on if plasmid doesn't. But I don't want to hog the topic so I'll give him a day.

So far no one has posted a problem with my disproof of an assumption approach, which appears to be the only uncontested approach in here. Could you please take a look at it and either point out the flaw in my logic there or consider whether that could be the correct answer?

If it's you're referring to, stage (5) of that suffers from the problem I've been dealing with at the top of this post.

(By the way, I think that if a Master of Logic were to play anyone else, they would always defect, as in this case the other person's response is unpredictable and so defection is always the safest bet. However, when playing against another Master of Logic the other person's response becomes predictable and it is this that flips the problem on it's head and leads to the always cooperate approach.)

Now that's a really difficult question. Depends on what the Master of Logic thinks of the other person, but they could take a probabilistic approach here.

Assuming it's 50 rounds, if Tіt for Tat (on all rounds) and Always Defect were the only available strategies (obviously they aren't but it's just for illustration), by my calculation, in order for Always Defect to be a better strategy, the other player needs to be more than 48 times more likely to be a Defect player than a Tіt for Tat player. Hence the results you observed on your training course.

[unreality]

I haven't posted in a while but I've been reading every post, and thinking. At first I was agreeing with neida, but then when I thought about that kind of stuff myself I always ran into Godel's Incompleteness Theorem, over and over again. I realized that shouldn't be happening. Then I figured out a logical solution that stays first-order and never once makes an assumption about what the other person will do. This allows anyone to emulate the "logical decision" without having to be a Master of Logic, and gets rid of that paradox.

First we need to establish a lemma that each round is independent of all other rounds. I don't know how to formally prove this so I suppose this is the underlying assumption. We could use a proof of induction:

* the very first round does not depend on previous rounds

* if you made the logical decision last round, and last round did not depend on previous rounds, then this round you can make the logical decision without depending on previous rounds

And via induction, all rounds become independent.

Or you could show that whatever is done the last round will ricochet backwards. I have a few other inkling ideas of why each round of this game may be independent (FOR MASTERS OR LOGIC OR PEOPLE USING A STRATEGY THAT A MOL WOULD USE, obviously it's not independent in the general case). Let me know if you disagree with this assumption.

From there we can consider the game by just considering a single, arbitrary round.

Back to the basics:

I defect, s/he cooperates: +2

I defect, s/he defects: 0

I cooperate, s/he defects: -1

I cooperate, s/he cooperates: +1

Either way the opponent's move is locked in as the opponent's move. They are going to do what they are going to do. Therefore I should defect and thus get 1 more point than I would have if I had cooperated.

I realize that the "lemma" (that each round is independent) is probably under more scrutiny now since the "logical choice" follows so simply from that, but at the very least we can establish an if/then relationship:

IF the logical decision does not require input from the previous rounds, THEN defect.

However, if it can be shown that the logical decision requires the past moves to establish itself, then it would have to be proven in a different way...

My gut tells me that symmetry means everyone should cooperate all the time, and that's probably what I would do. But logic tells me I should defect

[Guest]

I haven't posted in a while but I've been reading every post, and thinking. At first I was agreeing with neida, but then when I thought about that kind of stuff myself I always ran into Godel's Incompleteness Theorem, over and over again. I realized that shouldn't be happening. Then I figured out a logical solution that stays first-order and never once makes an assumption about what the other person will do. This allows anyone to emulate the "logical decision" without having to be a Master of Logic, and gets rid of that paradox.

First we need to establish a lemma that each round is independent of all other rounds. I don't know how to formally prove this so I suppose this is the underlying assumption. We could use a proof of induction:

* the very first round does not depend on previous rounds

* if you made the logical decision last round, and last round did not depend on previous rounds, then this round you can make the logical decision without depending on previous rounds

And via induction, all rounds become independent.

Or you could show that whatever is done the last round will ricochet backwards. I have a few other inkling ideas of why each round of this game may be independent (FOR MASTERS OR LOGIC OR PEOPLE USING A STRATEGY THAT A MOL WOULD USE, obviously it's not independent in the general case). Let me know if you disagree with this assumption.

From there we can consider the game by just considering a single, arbitrary round.

Back to the basics:

I defect, s/he cooperates: +2

I defect, s/he defects: 0

I cooperate, s/he defects: -1

I cooperate, s/he cooperates: +1

Either way the opponent's move is locked in as the opponent's move. They are going to do what they are going to do. Therefore I should defect and thus get 1 more point than I would have if I had cooperated.

I realize that the "lemma" (that each round is independent) is probably under more scrutiny now since the "logical choice" follows so simply from that, but at the very least we can establish an if/then relationship:

IF the logical decision does not require input from the previous rounds, THEN defect.

However, if it can be shown that the logical decision requires the past moves to establish itself, then it would have to be proven in a different way...

My gut tells me that symmetry means everyone should cooperate all the time, and that's probably what I would do. But logic tells me I should defect

Sorry, unreality, but I think you're getting that a bit back to front. Past moves can be relevant if you're playing something like Tіt for Tat, where you offer an incentive to cooperate by copying the other player's last move, and in other circumstances that may be a more logical option. The reason it works is that there is potential to cooperate in future rounds.

But you can make an inductive argument from the last round backwards. The last round is special in that it has no possible future rounds, thus no potential for future cooperation. That removes any possible incentive for continuing to play cooperatively on this round, so regardless of what strategy we have been playing up to this point, it makes sense to defect then*. It's the lack of future rounds that makes past rounds irrelevant. I'll leave you to do the rest of the induction.

*The disagreement between Neida and I currently revolves around that last round. Or equivalently, what the Masters of Logic would do in a simple, single round game.

That may seem a trivial discussion but it's proving to have some interest. Your reasoning on that is that the other player's move is locked, and we must choose our own move. But bear in mind that your own move is also locked, even if you don't know it yet. You have to (or as a Master of Logic, you will) use the "right" method to determine what it is locked to. I'm not saying your method is wrong, only that that's what we're discussing.

[Guest]

I'm still not convinced - in fact I'm coming more around to believing that cooperation is almost certainly the right answer (but still willing to be proved wrong!) I'll try to address a few points here...

1) Assume cooperation is logical

2) Would defection be a better choice for me? (if so it contradicts 1)

...

A more valid approach is:

1) Assume cooperation is logical

2) Would defection be a better choice for me? (if so it contradicts 1)

3) If I choose to defect instead, it would increase my score, with no other consequences. This contradicts (1).

This isn't actually what I was saying, and your approach isn't valid either. The way you have put both of these is to make an assumption and then immediately make an assumption that goes against your first. If you do that then obviously you will come out with a contradiction, but that is only because you are making two contradicting assumptions.

To make it clear what I mean, if you say "assume cooperation is logical" then you can't say "would defection be a better choice?" - the answer is no by your first assumption. This would be like saying "assume a coin toss showed heads" and then saying "did it actually show tails?" If you're still unsure about the second assumption and think I'm just pointing to a question, then it should be clear to see that your next step is basically "assume I defect - what would happen" - so we have "assume cooperation, now assume defection." No approach can start with "assume cooperation/defection is the logical choice" as any deviation from that would, by definition, mean you make an illogical choice.

For the assumption approach to work, there should only be one assumption made - if you make more than one you also then don't know which one you are disproving! My approach does not assume cooperation. It states that, of the two possible scenarios of always cooperate and always defect we know that cooperation will yield more. Ordinarily I think you would just stop there, but as the question was raised "but surely if you change you will win more" we then go on to disprove that by saying "OK, let's assume that I were to defect" and see how that will change the currently known situation. There is only one assumption, so when we hit a contradiction we know it is that one that is disproved.

dawh, I know what you are saying about the Turing machine, but the more I think about it, the more I think there isn't an infinite loop here. The infinite loop is only introduced by flawed logic and considering an impossible scenario. The flaw is at the point you say "but what if I choose to defect and he doesn't - then I win more". The order of these three steps is wrong, as you are taking them in this order:

1. I choose to defect (and he doesn't)

2. I win more

3. Therefore he will choose to defect

What's being missed is the instantaneous step of the other person defecting. This is because of the statement "If I choose to defect then whatever line of reasoning I follow to make that choice will also be followed by the other person as we are both Masters of Logic in an identical situation." Note it does not matter what line of reasoning you take - i.e. think about "but if I do this, then that" all you like, but whatever would ultimately lead you to make the decision would simultaneously lead the other person to make the same decision. Therefore the order should actually be:

1. I choose to defect

2. He simultaneously chooses to defect

3. I win less

Note this simple (and logical) reasoning can apply from any angle. E.g. I know that if we both cooperate I will win one, if I then chose to defect, they would follow the same reasoning and defect so I will win nothing. Or, I know that if we both defect I will win nothing, if I then chose to cooperate, they would follow the same reasoning and cooperate and I will win one. Whichever way you approach this you end up with cooperation being the best outcome.

Now I know there was talk of hypothetical but impossible situations and I see again what dawh is saying about Turing machines and so not being able to ever figure out a solution, but I really don't think this is the case here and we seem to be overcomplicating (or dumbing down) Masters of Logic to fit the solution, rather than the other way around. I think that both of these thoughts stem from the difficulty in a Master of Logic determining the decision of another Master of Logic, but then consider the following:

I ask one Master of Logic to choose between accepting 0 or 1 beer tokens (let's take for granted that it is good to have more beer tokens that less, as per this problem). Before he responds I ask another Master of Logic to tell me what the first will say. According to your arguments, the second MoL would not be able to respond as he would reason that there are two choices available here, but a Master of Logic can not have two choices because there is only one logical answer, so I have now hit a contradiction and can't move forward. Either that or he would start considering the hypothetical situation of the first Master of Logic choosing 2 beer tokens. Now it's fairly obvious the Master of Logic would say 1, but I don't think we can really argue that another Master of Logic wouldn't be able to come to the same conclusion!

Finally, it's not part of our discussion which is why I haven't mentioned it before, but why does everyone keep bring up ti t for tat as giving the same result as cooperating. If you alternate cooperating and defecting (which is how I understand this approach to work) then, forgetting about the difficulty of deciding who would start, this would result in you winning 2, losing 1, winning 2, losing 1, etc. This results in a net win of 1 every two rounds, as opposed to 1 every round through cooperation. So why would anyone ever choose this approach?

[Guest]

From there we can consider the game by just considering a single, arbitrary round.

Back to the basics:

I defect, s/he cooperates: +2

I defect, s/he defects: 0

I cooperate, s/he defects: -1

I cooperate, s/he cooperates: +1

Either way the opponent's move is locked in as the opponent's move. They are going to do what they are going to do. Therefore I should defect and thus get 1 more point than I would have if I had cooperated.

This is what I think is wrong. Look at your last paragraph. You say that either way the opponent's move is locked in - they will do what they are going to do. What you are basically saying is that, although you are a Master of Logic and you know the other person is a slave to logic, you have no idea what they are going to do, so therefore you should just proceed with the approach that "beats" them regardless of what they do. This is like my example above of saying "will another Master of Logic choose 1 beer token or 0 beer tokens" and saying "I haven't got a clue!"

What if I was to say the following: If you both cooperate you both go free. If only one of you defects you go free with $1m and the other dies a slow and painful death. If both of you defect then you both die a quick and pain free death. Now apply your same reasoning: if they cooperate then {I cooperate = go free, I defect = go free with $1m} so I should defect; if they defect then {I cooperate = slow and painful death, I defect = quick and pain free death} so I should defect. Therefore both Masters of Logic defect and die. But this reasoning is based on having no idea what the other person will do. I argue that a Master of Logic can determine what the other will choose as they will both follow the same reasoning and so, based on this reasoning, they know that living is better than dying so they make that choice.

[Guest]

Actually plasmid's argument will do for me. I know you've posted an answer to that, which I'll comment on if plasmid doesn't. But I don't want to hog the topic so I'll give him a day.

Sorry, just one more thought for the moment which (I think) is a more straightforward counter to plasmid's argument.

Plasmid states that you know with absolute certaintywhat your opponent will do, so you can use that to inform your own decision. The flaw with this is that logically you can only know with absolute certainty what your opponent will do once you also know with absolute certainty what you will do. But if you say that you can apply that knowledge to change your mind then you didn't know with absolute certainty at all. Again we have a case of two contradicting assumptions "assume I've made up my mind, but then assume I change it."

Coin example again: "When I toss a coin will it be heads or tails? Well, let's assume it is heads. Now what if it is tails - if it is tails then it can't be heads and we have a contradiction, therefore it can't be heads. Turn that argument around and we can equally show it can't be tails. Therefore the tossed coin will show neither heads nor tails."

[Guest]

This isn't actually what I was saying, and your approach isn't valid either. The way you have put both of these is to make an assumption and then immediately make an assumption that goes against your first... To make it clear what I mean, if you say "assume cooperation is logical" then you can't say "would defection be a better choice?" - the answer is no by your first assumption.

Here you're merely confirming your assumption, on the basis that that's what you've assumed.

Example:

Master of Logic has to win as many beer tokens as possible. He is asked "Would you like a beer token?" and must answer "Yes" or "No". He only gets the beer token if he says "Yes".

Now consider whether it would be logical to reply "No".

Assuming it is, he will get zero beer tokens.

Would it be a better stategy to reply "Yes"?

It would not, as this would contradict our initial assumption that "No" is the logical answer and "Yes" is not.

Therefore our initial assumption is confirmed, and he will reply "No".

There's something not quite right with the above line of reasoning. We need to be able to consider the consequences of playing differently, to verify whether this would have been better. To be fair, the problem lies in the way the argument is structured, and the way I have expressed my own reasoning has been a bit too loose in that way. However, your own disproof of an assumption approach relies on the same language, and the main reason for what I was saying was to point out the flaw in that argument. The way is it structured is wrong, but that's not the main thing that's wrong with it, and I was trying to overlook the structural problems in order to highlight the underlying flaw in reasoning.

Like I said, plasmid came closer to wording it correctly, and the only reason I've reverted to this kind of language is to answer something we've been batting around for a few posts now. It might be better to drop it altogether and focus on plasmid's argument. There are parts of that which could be worded better, such as what you've just pointed out. But if you don't get too hung up on those, and look at the structure of the argument, it's pretty sound.

[Guest]

Here you're merely confirming your assumption, on the basis that that's what you've assumed.

Example:

Master of Logic has to win as many beer tokens as possible. He is asked "Would you like a beer token?" and must answer "Yes" or "No". He only gets the beer token if he says "Yes".

Now consider whether it would be logical to reply "No".

Assuming it is, he will get zero beer tokens.

Would it be a better stategy to reply "Yes"?

It would not, as this would contradict our initial assumption that "No" is the logical answer and "Yes" is not.

Therefore our initial assumption is confirmed, and he will reply "No".

There's something not quite right with the above line of reasoning.

I agree - I think this just further proves my point though. My point is that a proof can't consist of making two contradicting assumptions. As you've just shown, if you do this then you can come up with a "proof" for almost anything! Here you've proven it's best to say "No" using the same method that you've used to show that always defect is the right solution to the OP.

EDIT: I've just realised where the confusion might be - I am not trying to say that my argument against your approach therefore proves the opposite (or proves my approach). I am simply pointing out what I believe to be the flaw in your approach. If an approach is shown to be flawed that doesn't say anything in itself - you just simply need to look at a different approach.

Another thought I've had in the meantime is that another flaw with the "if I defect and he stays the same" argument is that you reach a conclusion and make an assertion before you have allowed the entire line of reasoning itself to reach a conclusion.

Here's a slightly different example to illustrate what I mean: The Grandmaster says to a Master of Logic "You currently have no beer tokens. Pick a number between 1 and 6 and I will give you that many beer tokens. At the end of this 'game', if you said 6 then I will take half of your beer tokens." According to your reasoning the Master of Logic would think "Well 5 sounds like the best answer, but what if I said 6 instead? In that case I would win 6 beer tokens which is better than 5, so I'm going to say 6. Yay!" At the end of the game the Master of Logic ends up with only 3 beer tokens and wonders where he went wrong. He then realises it was because he forgot to take into account the fact that his beer tokens would be halved if he said 6 and he wonders how he could have been so stupid to let himself get carried away half way through a line of reasoning and ignore such an obvious fact. He promptly resigns as a Master of Logic in shame.

This might seem simplistic, but I believe it's exactly the same principle that your argument is following for the single round game. I.e. "If we both cooperate then I get one beer token, but what if I defect? Then I get 2 beer tokens which is better than 1. Yay!" At the end of the game he gets no tokens and wonders where he went wrong. He then realises it was because he forgot to take into account the fact that his opponent would do exactly the same as him and he wonders how he could have been so stupid to let himself get carried away half way through a line of reasoning and ignore such an obvious fact. He promptly resigns as a Master of Logic in shame.

Edited January 14, 2010 by neida

[Guest]

This might seem simplistic, but I believe it's exactly the same principle that your argument is following for the single round game. I.e. "If we both cooperate then I get one beer token, but what if I defect? Then I get 2 beer tokens which is better than 1. Yay!" At the end of the game he gets no tokens and wonders where he went wrong. He then realises it was because he forgot to take into account the fact that his opponent would do exactly the same as him and he wonders how he could have been so stupid to let himself get carried away half way through a line of reasoning and ignore such an obvious fact. He promptly resigns as a Master of Logic in shame.

What you're not taking into account there is that by cooperating he would have lost a beer token. Granted, I use "would have" in a questionable sense since there's obviously no possible way he could have. But he would not resign on the basis that he didn't do as well as he would in a cooperate-cooperate scenario, since this would require him to believe that there was some way he might have not only chosen to cooperate, but also persuaded the other player to do the same.

[Guest]

Since I have said that plasmid's approach (being free-will-free) is essentially the right one, but not quite worded right, I've reworked it. I hope plasmid doesn't mind.

Bits in blue are redundant to the reasoning but left in for clarification.

Thought process of a Master of Logic (single round game or known to be on last round of game)

The other player will make the most logical move to maximise his/her points.

I do not yet know what that move is, but there is only one possibility and I cannot change it.¹

Since we both make decisions using perfect logic, it will be the same move that I decide to make at the end of this thought process.²

Since I do not yet know, I will consider both possibilities:

If s/he will play cooperate, the move that will get me the most points is defect.³

If s/he will play defect, the move that will get me the most points is defect.

In either case, the move that will get me the most points is defect, and since the acquisition of points is my objective, that is the most logical move.

¹ Strictly speaking a Master of Logic wouldn't think "I cannot change it " since free will is implied by the very idea that you might be able to. To him/her the inability goes without saying.

² I put this in for contrast with Neida's approach. It accepts the inevitability of the decision. Both players would be better off in the end if they both decide that cooperation is the logical choice, but using that to infer that cooperation is the logical choice is an appeal to consequences.

³ At this point I could infer a contradiction, as plasmid did. It would lead to the same conclusion.

[unreality]

This is what I think is wrong. Look at your last paragraph. You say that either way the opponent's move is locked in - they will do what they are going to do. What you are basically saying is that, although you are a Master of Logic and you know the other person is a slave to logic, you have no idea what they are going to do, so therefore you should just proceed with the approach that "beats" them regardless of what they do. This is like my example above of saying "will another Master of Logic choose 1 beer token or 0 beer tokens" and saying "I haven't got a clue!"

Thanks for pointing that out. That seems to be the core of our debate so I'll get back to that in a second.

First, regarding my inductive process from the first round up that octopuppy criticized, I included that I realized this was not valid against anyone other than a Master of Logic. Another proof is the back-to-front induction from plasmid that octopuppy gives in the above post, which is essentially the same as what I said. I think we are all in agreement that this problem condenses all rounds into one between the two Masters of Logic, and that is what's being debated: a single round. Does the MoL cooperate or does it defect?

So going back to what I quoted from neida, and with what octopuppy has been saying, this is my attempt at summarizing:

OCTOPUPPY: I will choose the logical choice, as will my opponent. Whatever they choose, I can do better by defecting than by cooperating. KEY LINE FROM PROOF: "Since I do not yet know, I will consider both possibilities"

NEIDA: Saying that you don't know what the opponent will do is illogical because, being a Master of Logic, you know what they will do. They will do the same as you, so the logical choice is for both to cooperate. KEY LINE FROM PROOF: "He forgot to take into account the fact that his opponent would do exactly the same as him"

I think that I would be on Neida's side if not for a flaw I see in his assumption of symmetry. It's that if the other MoL thinks that and decides to go with cooperate, you can one-up them by defecting. Then of course both will defect. etc.

The only stable non-looping solution seems to be when both players defect, even though the optimal seems to be when both players cooperate.

[Guest]

What you're not taking into account there is that by cooperating he would have lost a beer token. Granted, I use "would have" in a questionable sense since there's obviously no possible way he could have.

I think this is my point. You say there's obviously no possible way he could have, yet you also say a Master of Logic would have thought he would. That's not a very clever Master of Logic! You are saying his reasoning is "Possible scenario A - I win 1. But because of the possibility of impossible scenario B where I would lose 1, I should go with a different choice." Surely a Master of Logic would realise that the "possibility of an impossible scenario" is zero? Alternatively, and elsewhere, you are saying the reasoning is "Possible scenario A - I win 1. Impossible scenario B - I win 2. I will choose impossible scenario B. D'oh!"

Since we both make decisions using perfect logic, it will be the same move that I decide to make at the end of this thought process.

...

If s/he will play cooperate, the move that will get me the most points is defect.

I don't know if I'm just losing this altogether, but the flaws in your reasoning seem to be becoming more obvious. Look at the two sentences I've quoted above. You said yourself that the reasoning would occur in this order. The first one says we have to make the same choice. The second one then says that you should make a different choice. But you can't - because you've already determined that you have to make the same choice!

Would it make any difference to you if we said at the start that a rule of the game is that both players must make the same choice? From what you're saying I think it might make a difference to you, but I don't think it would make a difference to the game - a Master of Logic would simply reason "well that's obvious - it will happen anyway, it does not need to be a rule, it is superfluous." The very fact that we are dealing with two Masters of Logic means that we have an unwritten rule that they will make the same choice - and being Masters of Logic they both know this - everyone has agreed with this so far (it's been used in the argument from both sides!)

What if I reworded my "pick any number from 1 to 6" problem earlier to "pick any number from 1 to 6 as long as it's odd". By your reasoning the Master of Logic then thinks "I know that my number must be odd" but then "I know that the number that gives me most is 6" so he therefore chooses 6! Your argument here is exactly the same. "I know, by definitions given in the question/situation, that our answers must be the same" and then "I know that if I choose a different answer to them I win more." I'm sorry, but I'm really failing to see any sense in this at all! And I'm also failing to see how any Master of Logic, by any definition, would think the same.

I do think it's interesting that we can passionately argue about whether a theoretical but impossible person, in a theoretical but impossible situation, can think about a theoretical but impossible outcome!!! No wonder it's making my head hurt!!!

By the way, although I stick to my argument that always cooperate would work, I was thinking about your line of inductive reasoning and whether that would apply in any case. Generally I think I agree that if defect in the last round is right then it should be right throughout, but then I thought that it might not... In the final round, your reasoning relies on the fact that, worst case, you definitely don't lose one and end up with zero instead. That's fine for 1 round, but over 50 rounds would you apply the same reasoning? If you were to go on the basis that as soon as one of you defect, trust is lost, so all future rounds would be defect, then would it really be logical to give up everything on the basis that you weren't prepared to lose 1 token on the last round? At the very least, would it not be more logical (even if playing against someone you have no idea about what they would do) to risk losing 1 token in the first round in the hope of getting at least close to 50? That's pretty good odds and, even if the other person did defect at the end, surely 47 or more would be better than nothing? (I say 47 because I think the option of defecting comes in at round 48 with the rock/paper/scissors scenario discussed earlier) Anyway, I don't want to get sidetracked (as I don't think this is something up for consideration, except maybe in a "two random people play the game" scenario), but thought it might be something for you to think about!

[Guest]

I think that I would be on Neida's side if not for a flaw I see in his assumption of symmetry. It's that if the other MoL thinks that and decides to go with cooperate, you can one-up them by defecting. Then of course both will defect. etc.

OK, I think we are thinking more along similar lines here.

Note that I've said previously that if the objective of the game was to beat your opponent (or even more clearly, if it was not to lose against your opponent) then I agree that defection is the better choice. What you've said here is that you can "one-up" your opponent by defecting. Now you could reword this by saying that you definitely won't get "one-upped" by defecting. Either way, it is about beating your opponent. But with the knowledge that whatever line of reasoning you follow, they will do the same you can never reach a conclusion that you will win more (or even the same amount of) beer tokens (which is more important) - and also I don't believe it is cyclic in any case.

I have mentioned this before, but to go into further detail: if you make any assumption, you must first apply everything that you know will happen before you can determine the result of that assumption. I.e. you can't only apply half of what you know will happen, which is what everyone seems to be doing to get the cyclic situation. It goes along the lines of "If we are both currently thinking of cooperating and winning 1, then if I were to make a decision to defect then I know that the other person would also defect. Now that I have applied everything that I know, I can determine the result of this decision (assumption) and see that I win zero. Therefore it is not a good decision." What you have actually done here is say "If we are both currently thinking of cooperating and winning 1, then what if I were to defect? Well, if I ignore everything that I know would happen in that scenario, then I would win 2." That is not a very logical approach.

The only stable non-looping solution seems to be when both players defect, even though the optimal seems to be when both players cooperate.

I still have a problem seeing why people think defection is a stable non-looping solution. Consider the following: "If we are both currently thinking of defecting and winning nothing, then if I were to cooperate then I know that the same reasoning would lead them to cooperate so I would win 1." Alternatively, reverting to the 50 round game, consider the following "If we are both currently thinking of always defecting and winning nothing, then we both know that there exist alternative strategies that would result in each of us winning more. As our joint objective is to maximise our personal winnings, we must therefore have chosen the wrong strategy." (This also actually applies in the single round game, but is easier to see in a 50 round game).

Edited January 15, 2010 by neida

[unreality]

I still have a problem seeing why people think defection is a stable non-looping solution. Consider the following: "If we are both currently thinking of defecting and winning nothing, then if I were to cooperate then I know that the same reasoning would lead them to cooperate so I would win 1."

The recursion kicks in when you reason about the other person's reasoning. That's what I've been trying to avoid because it leads to infinite regress.

[Guest]

The recursion kicks in when you reason about the other person's reasoning. That's what I've been trying to avoid because it leads to infinite regress.

Mmmm, it only leads to infinite regress on the always defect side of the argument (I've shown that it doesn't on the always cooperate side).

I think what you are saying here is that if you reason about the other person's reasoning then always defect becomes an infinite regress and always cooperate doesn't. Alternatively if you don't consider this then always cooperate becomes an infinite regress and always defect doesn't. You've chosen to ignore it so that the always defect argument works. Whereas I argue that a Master of Logic is unable to ignore it - it would be illogical not to think about what the other person would do as it has a direct impact on your result.

[Guest]

I don't know if I'm just losing this altogether, but the flaws in your reasoning seem to be becoming more obvious. Look at the two sentences I've quoted above. You said yourself that the reasoning would occur in this order. The first one says we have to make the same choice. The second one then says that you should make a different choice. But you can't - because you've already determined that you have to make the same choice!

The second one says "If s/he will play cooperate, the move that will get me the most points is defect." As I said in the footnotes, and plasmid mentioned, the defect result contradicts the knowledge that we will play the same move. Yet it is the best move, based on the initial assumption (s/he will play cooperate). An assumption that implies a contradiction is false. That's the whole basis of proof by contradiction, which wouldn't really work so well if you weren't allowed to infer contradictions! I've been careful about how I worded things this time.

Would it make any difference to you if we said at the start that a rule of the game is that both players must make the same choice?

Yes, although it's a bizarre situation because each player has to make a decision for the other player as well as themselves. Normally people only decide their own actions. You could probably envisage a variant on the rules so one player wanted to do one thing and the other wanted to do another. Who would get their way?

From what you're saying I think it might make a difference to you, but I don't think it would make a difference to the game - a Master of Logic would simply reason "well that's obvious - it will happen anyway, it does not need to be a rule, it is superfluous." The very fact that we are dealing with two Masters of Logic means that we have an unwritten rule that they will make the same choice - and being Masters of Logic they both know this - everyone has agreed with this so far (it's been used in the argument from both sides!)

But it's so important that this is only a consequence of them both making the logical choice. It doesn't mean that each gets to decide what the other does. And that's what you need for cooperation, because cooperation is of no benefit to the player who is cooperating, only to the other player.

"I know, by definitions given in the question/situation, that our answers must be the same" and then "I know that if I choose a different answer to them I win more." I'm sorry, but I'm really failing to see any sense in this at all! And I'm also failing to see how any Master of Logic, by any definition, would think the same.

Look carefully at the wording and you'll see that I have avoided any statements like "if I do this then that will happen". That's free will talking. There is no illusion of choice in the reasoning, since this muddies the waters enormously. It is just an evaluation of the best move.

I do think it's interesting that we can passionately argue about whether a theoretical but impossible person, in a theoretical but impossible situation, can think about a theoretical but impossible outcome!!! No wonder it's making my head hurt!!!

I have a spanner or two up my sleeve to throw in the works if we even agree on this much. Trust me, it gets worse.

By the way, although I stick to my argument that always cooperate would work, I was thinking about your line of inductive reasoning and whether that would apply in any case. Generally I think I agree that if defect in the last round is right then it should be right throughout, but then I thought that it might not... In the final round, your reasoning relies on the fact that, worst case, you definitely don't lose one and end up with zero instead. That's fine for 1 round, but over 50 rounds would you apply the same reasoning? If you were to go on the basis that as soon as one of you defect, trust is lost

Whoa, I'll have to stop you there. This is Masters of Logic we're talking about. There was never any trust, except insofar as players can be trusted to do what is best for themselves.

, so all future rounds would be defect, then would it really be logical to give up everything on the basis that you weren't prepared to lose 1 token on the last round? At the very least, would it not be more logical (even if playing against someone you have no idea about what they would do) to risk losing 1 token in the first round in the hope of getting at least close to 50? That's pretty good odds and, even if the other person did defect at the end, surely 47 or more would be better than nothing? (I say 47 because I think the option of defecting comes in at round 48 with the rock/paper/scissors scenario discussed earlier) Anyway, I don't want to get sidetracked (as I don't think this is something up for consideration, except maybe in a "two random people play the game" scenario), but thought it might be something for you to think about!

The average of the 4 scores attainable is 0.5, so two random people would actually do considerably better than Masters of Logic. A Master of Logic playing against an unknown player would probably do much better. The problem with two Masters of Logic is that neither is ever in any doubt about what the other will do.

[unreality]

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

[Guest]

I think that the two Masters of Logic would choose to cooperate until round 50, at which point I'm not really sure what would happen. My reasoning for this is that both Masters of Logic know that their goal is to maximize points and that by cooperating all 50 rounds they will get 50 points. Therefore, 50 points is what they would consider as their minimum attainable points, and would only be looking to gain from there.

On round 48, if they had cooperated on all previous rounds, they would each have 47 points, defecting when your opponent cooperates would lead to only 49, which is below the potential 50 that could be gained from cooperating every round.

Even on round 49. If one of the Masters decided to defect at this point when the other cooperates, they would then have 50 points, but the other Master would then defect on the last round, resulting in 50 points still being the maximum any one of them could gain.

However on round 50 this reasoning breaks down, as defecting when your opponent cooperates results in 51 points, which is better than cooperating all 50 rounds time. However, both Masters would know this, and so THEN there would be a conflict of what to do.

Like I said before, this problem only arises on the last round, as the goal is to maximize points, which can only be done by cooperating until the final round. To do anything different at a previous round would not be maximizing profits. Even though you could gain more points than your opponent, that was not the stated goal.

[Guest]

I think that the two Masters of Logic would choose to cooperate until round 50, at which point I'm not really sure what would happen. My reasoning for this is that both Masters of Logic know that their goal is to maximize points and that by cooperating all 50 rounds they will get 50 points. Therefore, 50 points is what they would consider as their minimum attainable points, and would only be looking to gain from there.

On round 48, if they had cooperated on all previous rounds, they would each have 47 points, defecting when your opponent cooperates would lead to only 49, which is below the potential 50 that could be gained from cooperating every round.

Even on round 49. If one of the Masters decided to defect at this point when the other cooperates, they would then have 50 points, but the other Master would then defect on the last round, resulting in 50 points still being the maximum any one of them could gain.

However on round 50 this reasoning breaks down, as defecting when your opponent cooperates results in 51 points, which is better than cooperating all 50 rounds time. However, both Masters would know this, and so THEN there would be a conflict of what to do.

Like I said before, this problem only arises on the last round, as the goal is to maximize points, which can only be done by cooperating until the final round. To do anything different at a previous round would not be maximizing profits. Even though you could gain more points than your opponent, that was not the stated goal.

So far so good, and this is why we are discussing what will happen on the final round. Consider what the situation would be if both Masters of Logic knew for certain that the other will defect on the 50th round. How does that affect play in the 49th round?

(I don't want to get into a long discussion about the inductive reasoning right now, but that's just a nudge to give you an idea of the significance of the final round)

[Guest]

I'm starting to think that we're going to have to agree to disagree on this one, as we seem to be going around in circles. However, I have one last line of reasoning which I'll try out first. In order to set it up, I'll need to counter some of your latest arguments first...

The second one says "If s/he will play cooperate, the move that will get me the most points is defect." As I said in the footnotes, and plasmid mentioned, the defect result contradicts the knowledge that we will play the same move. Yet it follows from the initial assumption (s/he will play cooperate). An assumption that implies a contradiction is false. That's the whole basis of proof by contradiction, which wouldn't really work so well if you weren't allowed to infer contradictions! I've been careful about how I worded things this time.

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.

Anyway, maybe we're getting bogged down in this one, so moving on...

But it's so important that this is only a consequence of them both making the logical choice. It doesn't mean that each gets to decide what the other does.

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!"

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. However, even with your argument, it is only paradoxical at a micro level - i.e. if they consider discrete decisions and the direct outcome of each discrete decision.

If I move back to the Achilles and the Tortoise paradox again, think about what the actual question is: Will Achilles catch and ultimately pass the tortoise? When looking at the situation as a whole the obvious (and logical) answer is yes. After all Achilles is travelling faster, so it is trivial to show that there exists a point in time where he is past the tortoise. However, when you look at a micro level and consider a set of discrete situations, you can get tied into an infinite cycle - "by the time Achilles reaches where the tortoise is now, the tortoise will have moved forward; by the time Achilles reaches that new position, the tortoise will have moved forward again;" and so on. This is the paradox. It does not prove that Achilles will never pass the tortoise. However, it also does not prove that he won't. It simply proves nothing.

Coming back to this problem, let's consider what the overall question really is: How do I maximise my gains across this entire game? Now we already know that if we look at discrete decisions we end up in all sorts of problems (we're already on page 8 on the forum!), so how about taking a step back and approach it like this:

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.

Alternatively (and possibly more simply) the Master of Logic could think "I will consider every possible way for two people to play this game (not Masters of Logic) and determine the net result of each. As I am playing a Master of Logic with the same objective as me, I must then rule out any non-symmetric strategies, as we will both play the same strategy. The best remaining strategy is the one that yields the maximum winnings and so is the logical solution." It is trivial to show that the symmetric strategy that yields the most is always cooperate.

I think that what I am saying here (especially with the first approach) is that always cooperate is a very delicate equilibrium, but it is nonetheless an equilibrium. Any deviation whatsoever quickly results in it falling apart and turning into an always defect equilibrium, which admittedly is a far more stable equilibrium. Think of it like walking along the very fine crest of a hill with steep sides on both sides. Any deviation from the crest will lead you tumbling to the bottom where you will stay, but provided you stick to it rigidly you can still get to the other side. My point is that, whilst mere mortals may deviate (because of lack of trust, etc.), two Masters of Logic would recognise that there are only two equilibrium states at play here - one at the top of the hill and one at the bottom. The top is much better (both cooperatively and individually) than the bottom so, although it is far less stable, they know that a Master of Logic would not deviate from it and step onto that slippery slope.

[Guest]

I think that what I am saying here (especially with the first approach) is that always cooperate is a very delicate equilibrium, but it is nonetheless an equilibrium. Any deviation whatsoever quickly results in it falling apart and turning into an always defect equilibrium, which admittedly is a far more stable equilibrium. Think of it like walking along the very fine crest of a hill with steep sides on both sides. Any deviation from the crest will lead you tumbling to the bottom where you will stay, but provided you stick to it rigidly you can still get to the other side. My point is that, whilst mere mortals may deviate (because of lack of trust, etc.), two Masters of Logic would recognise that there are only two equilibrium states at play here - one at the top of the hill and one at the bottom. The top is much better (both cooperatively and individually) than the bottom so, although it is far less stable, they know that a Master of Logic would not deviate from it and step onto that slippery slope.

Actually, this post leads perfectly into what I was planning to say. If I recollect my Game Theory (and I'm a little rusty on that), the Prisoner's Dilemma has an unstable Pareto Optima solution on mutual cooperation and it has a stable Nash Equilibrium on mutual Defection. So it occurs to me that you are both finding a valid solution (in a sense) and merely arguing over which would be a better solution. I tend to agree with neida in this case that the MoL would recognize the mutual cooperation as the pareto optima (where no one can change a decision without hurting the opponent) and realize that they could not get better than that when faced against another MoL. However, like neida said, if anything breaks that delicate balance, both sides need to 'fall' to the stable equilibrium to avoid further losses (since any deviation from defection by one party would be to his/her own detriment at that point).

I think if someone created a learning algorithm that repeatedly tried this 50 round thing, it would eventually reach the mutual cooperation when faced against itself since it would find that no other approach led to more wins.

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. (Not to drag Turing machines back into the discussion, but it does seem that octopuppy's insistence on the determinism of the MoL choice could be easily modeled by a Turing machine following a set of simple axiomatic instructions. If you go the route of following such instructions blindly, you inevitably reach the defect solution since you ignore the larger picture success of cooperating for the individual 'win' of defecting each round maximizing the potential without examining the actual.)

* The Wikipedia article has a different name for it, but it's the same paradox.

[Guest]

to me it seems, if there is only 1 round, defecting is your best option, as there is no after consequence for doing so, and whether your opponent defects or cooperates, you get the most points you can. when you play multiple rounds however, always cooperate (unless opponent defects of course) is the clear way to go, as any defection will quickly result in retaliation. it's only the last round the questionable. here I would make the argument for cooperation though, for two reasons. 1) you've already built a system of trust between the two players, there's no need to break it. 2) even assuming you wanted to score more, defecting won't garentee that you would do so, as if it did your opponent would also defect.

[Guest]

now here's a real puzzle. say a moderator offers you and another random person the same game. and he says it will be 1 round. you defect, but say your opponent cooperates. then the moderator says "okay now we'll play rounds until someone scores 10 points. that will be the winner, if its a tie, we'll split the winnings."

obviously your opponent will want some vengeance for defecting, so you probably should defect the next round, but how many rounds should you wait before cooperating? say you opponent defects that round, should you go back to defection?