The Nash Equilibrium and the traveller's dilemma

[wazz]

I say again, 99 is optimal.

The problem here is that game theory is butting its head in where it's not needed. This is a simple maths/logic question.

Every number is dominated by the number 1 lower. However, it dominates every choice 2 numbers lower; this is because the aim is to make money rather than make more money than the other guy. You don't even need to appeal to your sense of marginal utility to get this, I feel.

Maybe this can prove my point. If both agents were fully rational, they need not be able to communicate, but would have the same inner dialogue with each other, i.e. there's no need to screw each other over when we can both take $100 guaranteed. The extra $1 is not worth it when you take into account the possibility the other guy will take this strategy down to the wire.

Consider the paragraph about the game theorists I quoted from the example above; if the reward were directly proportional to the amount of money you earned, the result would have nothing to do with game theory. As it is, one is rewarded for how much more than everyone else you earned rather than absolute amounts.

I agree that game theory is needed when the aim is to beat the other guy; as it is, if you're just trying to make +EV decisions, take $99 or $100.

[wazz]

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this is EXACTLY the point of why I asked the question

Inherent in a lot of answers in the assumption that $2 negligible.


Which, dont get me wrong, for most intents and purposes, it is. But it is a very slight irrationality that is exploited in this problem.

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Ever heard of marginal utility? Nothing irrational about it.

[CallMeIshmael]

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this is EXACTLY the point of why I asked the question

Inherent in a lot of answers in the assumption that $2 negligible.


Which, dont get me wrong, for most intents and purposes, it is. But it is a very slight irrationality that is exploited in this problem.

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Ever heard of marginal utility? Nothing irrational about it.

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Of course I have. But this isnt it.

Do you agree that the assumption that $2 is the same as $0 is irrational, if only slightly?

[wazz]

I agree slightly. Put £5 in the changebox of a coke machine and I pick it up. Put 5 pence on the floor of a train station and I don't.

[vhawk01]

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I agree slightly. Put £5 in the changebox of a coke machine and I pick it up. Put 5 pence on the floor of a train station and I don't.

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As long as it is even SLIGHTLY irrational, then this problem does exactly what it is trying to do. If it isn't irrational at all, then this is the dumbest problem ever.

[vhawk01]

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I say again, 99 is optimal.

The problem here is that game theory is butting its head in where it's not needed. This is a simple maths/logic question.

Every number is dominated by the number 1 lower. However, it dominates every choice 2 numbers lower; this is because the aim is to make money rather than make more money than the other guy. You don't even need to appeal to your sense of marginal utility to get this, I feel.

Maybe this can prove my point. If both agents were fully rational, they need not be able to communicate, but would have the same inner dialogue with each other, i.e. there's no need to screw each other over when we can both take $100 guaranteed. The extra $1 is not worth it when you take into account the possibility the other guy will take this strategy down to the wire.

Consider the paragraph about the game theorists I quoted from the example above; if the reward were directly proportional to the amount of money you earned, the result would have nothing to do with game theory. As it is, one is rewarded for how much more than everyone else you earned rather than absolute amounts.

I agree that game theory is needed when the aim is to beat the other guy; as it is, if you're just trying to make +EV decisions, take $99 or $100.

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A better way to look at it, as opposed to your inner monologue idea (which I do like) is to say that, if you are both perfectly rational, then it is essentially the same as saying "Have the first guy write his answer down on a sheet, and let the other guy look at it and then write whatever he wants. First guy cannot change."

Now, in this scenario, do you think the first guy is writing...what? 99? And so now if you are the second guy, are you seriously contending that 99 gets you the most money? Pretend you are either player and you can see that you will NEVER be writing 99.

[wazz]

If the first guy can't change his choice, the first guy writes 100, the second guy writes 99. That's clearly not the same as having a pretend discussion with the other guy in your head, though. Whatever the first guy chooses, he can expect to earn $2 less, unless he chooses $2. Thus he maximizes expectation by choosing $100.

[vhawk01]

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If the first guy can't change his choice, the first guy writes 100, the second guy writes 99. That's clearly not the same as having a pretend discussion with the other guy in your head, though. Whatever the first guy chooses, he can expect to earn $2 less, unless he chooses $2. Thus he maximizes expectation by choosing $100.

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If you are the first guy, you write 100? God, why? You suck at this game! Didn't you read the rules!?

[CallMeIshmael]

That is the crux of the issue

By chossing something like 100, you risk of the situation of (100,99) --> (97,101), where you "only" get $97.

The difference between 97, and the payoff of switching to 98 (100), is so small that people dont think of these as different numbers. But, in the GT world, they are.



Now, when you say things like "I say again, 99 is optimal." This is just flat out wrong. Now, whether or not this is a maximal strategy, depends on the opponent. And, for the record, Id NEVER pick 2 in real world situations. But, the term "optimal" has a rigorous definition, and 2 is the optimal strategy.


In the theoretic world, it doesnt matter if the increments in the game are $2 or $200 billion, the solution is the same.

It is only when we introduce the slight human irrationalities like "$2 ~= $0" and "$200 billion ~= $10000 billion" that we are able to come to "real world" solutions to the problem.

Now, as you mentioned, the marginal utility of 0 and 2 dollars are very very close, as are the marginal utilities for 200 billion and 10000 billion, but they ARE different. 2 is slightly better than 0, and 10000 billion is slightly better than 200 billion.

The fact that the utility of these values are so close as to be negligible in the real world, doesnt make them equal in the theoretic world.

[CallMeIshmael]

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If the first guy can't change his choice, the first guy writes 100, the second guy writes 99. That's clearly not the same as having a pretend discussion with the other guy in your head, though. Whatever the first guy chooses, he can expect to earn $2 less, unless he chooses $2. Thus he maximizes expectation by choosing $100.

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If you are the first guy, you write 100? God, why? You suck at this game! Didn't you read the rules!?

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100 is correct, methinks