The Nash Equilibrium and the traveller's dilemma

[wazz]

It gets much more interesting when you read this far.

'Game theorists have made a number of attempts to explain why a lot of players do not choose the Nash equilibrium in TD experiments. Some analysts have argued that many people are unable to do the necessary deductive reasoning and therefore make irrational choices unwittingly. This explanation must be true in some cases, but it does not account for all the results, such as those obtained in 2002 by Tilman Becker, Michael Carter and Jörg Naeve, all then at the University of Hohenheim in Germany. In their experiment, 51 members of the Game Theory Society, virtually all of whom are professional game theorists, played the original 2-to-100 version of TD. They played against each of their 50 opponents by selecting a strategy and sending it to the researchers. The strategy could be a single number to use in every game or a selection of numbers and how often to use each of them. The game had a real-money reward system: the experimenters would select one player at random to win $20 multiplied by that player's average payoff in the game. As it turned out, the winner, who had an average payoff of $85, earned $1,700.'

[wazz]

'Many of us do not feel like letting down our fellow traveler to try to earn only an additional dollar, and so we choose 100 even though we fully understand that, rationally, 99 is a better choice for us as individuals.'

But this must be where game theory goes wrong; the aim of choosing 100 or 99 is nothing to do with 'letting down our fellow traveller' but in assuming your fellow traveller is rational and wants to give himself as much of a chance of making money as possible. If you assume the other traveller is rational, it is decidedly +EV to choose 99 or 100 because it makes no sense for the traveller to give up on the actual reward of 98-101 just to get one over on the other guy and collect $4 or even $2.

[vhawk01]

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I hate to point this out, but if the objective is to 'make as much money as possible' you should choose $99 or $100. The solution can only be $2 if the objective is to make more money than the other person.

On further thinking, $2 couldn't be the right answer either - assuming the other guy picks $2 as well, which is a 'safe' assumption given all this reasoning, you can make $3 by choosing $1.

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No. Explain how 99 makes you the most money. Pick 99 and pray he picks 100?

[CallMeIshmael]

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Two observations. Tournament poker is not zero sum. There are many situations (for example) where there are just two players left in a hand, and where their strategies will result in changes of equity for the players not in the hand. Easy example - say there are 4 players left in a standard STT (3 places paid) and it is folded to the small blind. Now we have a heads up between the blinds that is not a zero sum game. In fact, virtually every tournament situation where some players have folded their hands results in a non zero sum game between the remaining players.

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Well, for starters, the equity doesnt change, so, yes, this is still a constant sum game, which is a modified form of a zero sum game. Lets say that 10 players start a 10 dollar SNG (ignore rake for simplicities sake). No matter what the chip stacks, the total equity for all players must be 100. So, it might start off as {10,10,10...10}, and later become {60,40,0,0...0}, but, no matter what the equity doesnt change.

Now, I cant remember the name of the theorem, but there is a theorem that states that you can subtract/add a constant to every payoff and not change the nature of the game, so, subtracting 10 from all payoffs in this game transforms the above example payoffs into {0,0,0..,0} and {50,30,-10,-10..-10}, and we have a zero sum game.

Beyond that, I guess this all depends on how closely you want the situations to be before they are analogous, but, the sitution you just described has more than 2 players, whereas the TD does not. If you want to make it a 3 player game, where two players can collude to get all the payoff out of a third player, then you can, perhaps, find a good equivalent situation to concept of the TD. But, that is changing things around quite a bit, imo.



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Second, it is not clear to me at all that the non zero sum nature of TD has anything to do with the surprising Nash equilibrium

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It has everythhing to do with it.

The whole point of the TD is that two players can cooperate to achieve a much higher payment than they would at the Nash.

In any two player zero sum game, the payoffs are in the form {X,-X}, and there is no way for players to collude such that BOTH payoffs are higher than all NEs, since, by definition, if one is higher, the other must be lower.

Beyond that, this can be extended for all N-player 0 sum games, since if (a1,a2,a3...aN) are the payoffs at NE, and sum to 0, it is impossible for all payoffs to be increased and still sum to 0.

[wazz]

Well I'm pretty sure I'm right, but obviously that 2nd paragraph was wrong.

Aside from 'mathematics of poker' i couldn't really say I'm in any way versed or educated in game theory, so I'm not saying I am right in a game theoretical sense here. I'm saying that it doesn't make sense to choose $2 - the fact that $100 is dominated by $99 is an issue, but to resolve that you just take $99 instead; taking $98, via that logic, is equivalent to choosing $100 in the first place as that's the maximum payout you can hope to achieve. $99 'dominates' (in hyphens because it's possible I'm not getting some of the subtleties of that word) both these choices.

[CallMeIshmael]

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I hate to point this out, but if the objective is to 'make as much money as possible' you should choose $99 or $100. The solution can only be $2 if the objective is to make more money than the other person.

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Again, this is just mistaken.


Also, if you think picking 100 will give you the 'most money possible', can you tell me the number that the opponent played that made 100 such a good choice?

[CallMeIshmael]

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Well I'm pretty sure I'm right, but obviously that 2nd paragraph was wrong.

Aside from 'mathematics of poker' i couldn't really say I'm in any way versed or educated in game theory, so I'm not saying I am right in a game theoretical sense here. I'm saying that it doesn't make sense to choose $2 - the fact that $100 is dominated by $99 is an issue, but to resolve that you just take $99 instead; taking $98, via that logic, is equivalent to choosing $100 in the first place as that's the maximum payout you can hope to achieve. $99 'dominates' (in hyphens because it's possible I'm not getting some of the subtleties of that word) both these choices.

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99 does not dominate 98, since, IF my opponent chooses 99, 98 is better.

And, since you just said that you would resolve the issue of 100 being doiminated by playing 99 instead, 98 looks pretty attractive, no?

[CallMeIshmael]

All,

instead of the possible numbers being $2-$100, imagine instead that you are playing the game with a magic genie, who is going to let you write down $200 billion - $10000 billion, in increments of $100 billion (ie, same game, but with much larger payoffs), what do you write?

EDIT: I just saw piers' post where he brought up the same thing

[vhawk01]

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All,

instead of the possible numbers being $2-$100, imagine instead that you are playing the game with a magic genie, who is going to let you write down $200 billion - $10000 billion, in increments of $100 billion (ie, same game, but with much larger payoffs), what do you write?

EDIT: I just saw piers' post where he brought up the same thing

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No doubt that the paltry $2 payoff emotionally skews peoples knee-jerk impressions of the problem. 2 is practically the same as 0 to most adults who have jobs, so it cant possibly be correct...ANYTHING is better than 0!

[CallMeIshmael]

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All,

instead of the possible numbers being $2-$100, imagine instead that you are playing the game with a magic genie, who is going to let you write down $200 billion - $10000 billion, in increments of $100 billion (ie, same game, but with much larger payoffs), what do you write?

EDIT: I just saw piers' post where he brought up the same thing

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No doubt that the paltry $2 payoff emotionally skews peoples knee-jerk impressions of the problem. 2 is practically the same as 0 to most adults who have jobs, so it cant possibly be correct...ANYTHING is better than 0!

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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.