The Nash Equilibrium and the traveller's dilemma

[GMontag]

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So in other words, one of the starting assumptions of the problem is nonsensical.

There's only two possibilities here, either the problem is self-contradictory and meaningless, or both players will *always* come up with the same number. Because that's what mutual knowledge of perfect rationality implies. Since you are implicitly rejecting that, the only thing left is a self-contradictory situation.

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Read my post a few up. Of course the two players will come up with the same number. But they dont get there by one person putting whatever the heck he wants and the other one putting the same via telepathy. There are assumptions that lead to the inevitable outcome that they both pick the same number, and those assumptions have additional implications. One of them happens to be that they both pick $2.

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The assumption of mutual knowledge directly implies that both pick the same number without any of the additional implications. The argument that leads to (2,2) only works if you ignore the mutual knowledge. In fact the argument for (2,2) has both players assuming that they are "more rational", or can think on at least one additional level, than the other player, in direct contradiction to one of the premises of the problem. When you recognize that fallacy for what it is, the argument breaks down completely.

EDIT: removed some of the quotes for readability.

[Freyalise]

Didn't read the whole thread, but personally I pick $99. I expect this performs the best against the field of 'random people picking'.

I gain $0 instead of $2 against perfectly rational Nash-equilibrists, but gain $101 against GMontag and others. This seems like a good strategy as I expect there are more of the latter than the former.

Surely no-one can justify picking $100, a number which *always* does worse than picking $99 - no matter what number the opponent picked. I just cannot imagine any credible explanation for $100. Every other number makes sense in that it can do well against opponents with a certain range, but $100 can never be correct.

I also assume that everyone who picked $100 would also choose 'keep silent' in the prisoner's dilemma, even though it's a dominated strategy?

Would be interesting to hear the justification of anyone who would pick $100 here, but choose 'Betray' in prisoner's dilemma.

[GMontag]

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Didn't read the whole thread, but personally I pick $99. I expect this performs the best against the field of 'random people picking'.

I gain $0 instead of $2 against perfectly rational Nash-equilibrists, but gain $101 against GMontag and others. This seems like a good strategy as I expect there are more of the latter than the former.

Surely no-one can justify picking $100, a number which *always* does worse than picking $99 - no matter what number the opponent picked. I just cannot imagine any credible explanation for $100. Every other number makes sense in that it can do well against opponents with a certain range, but $100 can never be correct.

I also assume that everyone who picked $100 would also choose 'keep silent' in the prisoner's dilemma, even though it's a dominated strategy?

Would be interesting to hear the justification of anyone who would pick $100 here, but choose 'Betray' in prisoner's dilemma.

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The justification depends on the opponent. In this situation, the problem stipulates "infinite" rationality (i.e. I know he's rational, he knows I know he's rational, I know he knows I know he's rational, ad infinitum) for both players. The normal formulation of the Prisioner's Dilemma doesn't make such a stipulation. If it did, the correct answer there would also be to keep silent.

[wazz]

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Surely no-one can justify picking $100, a number which *always* does worse than picking $99 - no matter what number the opponent picked. I just cannot imagine any credible explanation for $100. Every other number makes sense in that it can do well against opponents with a certain range, but $100 can never be correct.

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Incorrect! See if you can work out why.

[CallMeIshmael]

PTB, GM:

Imagine a situation in which YOU are playing this game, against someone who is not only perfectly rational, but assumes you are, assumes that you know that he is...

Bascially, you have somehow convinced him/her that you are perfectly rational. Perhaps its the shirt you are wearing.

What do you write?

Im assuming 99, since you seem to be under the impression that a perfectly rational being writes 100.

Is it odd that a less than perfect being does better than a perfect being against the same opponent in this game?

[GMontag]

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PTB, GM:

Imagine a situation in which YOU are playing this game, against someone who is not only perfectly rational, but assumes you are, assumes that you know that he is...

Bascially, you have somehow convinced him/her that you are perfectly rational. Perhaps its the shirt you are wearing.

What do you write?

Im assuming 99, since you seem to be under the impression that a perfectly rational being writes 100.

Is it odd that a less than perfect being does better than a perfect being against the same opponent in this game?

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Why would it be odd that a person with faulty knowledge does worse than a person with correct knowledge?

[CallMeIshmael]

They both have correct knowledge.



Lets say there are 3 players, A, B and C.


Lets say A is perfectly rational, and so is C, but B is not. However, C assumes they are both perfectly rational.


So, according to you, when the game is AC, they both play 100.

And, when the game is BC, C also plays 100.


Now, assuming you are B, you play 99, yes?


And, to note, NEITHER A nor B has faulty knowledge. They both, correctly, know that their opponent is perfectly rational. Yet, the less than perfect B does better than A.


It seems odd that the perfectly rational A doesnt, just this once, forget that he is perfectly rational, and switches to 99.

[GMontag]

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They both have correct knowledge.

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No, you said I had managed to *convince* my opponent that I was perfectly rational and had knowledge of his infinite inductive chain. As that is not true, he was operating on faulty assumptions.

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Lets say there are 3 players, A, B and C.


Lets say A is perfectly rational, and so is C, but B is not. However, C assumes they are both perfectly rational.


So, according to you, when the game is AC, they both play 100.

And, when the game is BC, C also plays 100.


Now, assuming you are B, you play 99, yes?


And, to note, NEITHER A nor B has faulty knowledge. They both, correctly, know that their opponent is perfectly rational. Yet, the less than perfect B does better than A.

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Because of the faulty knowledge of C. Not because of any "better than perfect"ness on B's part.


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It seems odd that the perfectly rational A doesnt, just this once, forget that he is perfectly rational, and switches to 99.

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The only reason that strategy would work better would be because then C's knowledge about A would also be faulty. If C's knowledge is not faulty, then 100 is better than 99.

The whole problem with this setup is you start out by saying "Assume you can't outthink your opponent", then you go about making an argument where you assume you can outthink your opponent. It's nonsense.

[CallMeIshmael]

"The only reason that strategy would work better would be because then C's knowledge about A would also be faulty."

So you admit that it is in a perfectly rational players interest to deviate, essentially pretending they are not rational, but also believe they would opt to not do this.

OK.

[GMontag]

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"The only reason that strategy would work better would be because then C's knowledge about A would also be faulty."

So you admit that it is in a perfectly rational players interest to deviate, essentially pretending they are not rational, but also believe they would opt to not do this.

OK.

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I admit no such thing. If C's knowledge about A is faulty, then A will benefit from that, but A cannot "pretend" to be irrational while still being rational, as that would be predicted by C, being perfectly rational.