The Nash Equilibrium and the traveller's dilemma

[Galwegian]

For anyone who thinks that the Nash equilibrium is the nuts - check the following excellent scientific american article

Traveller's Dilemma

or if you can't access that article you can look at the wikipedia page

wiki - traveller's dilemma

I'm curious - can anyone think of a poker example that would illustrate the same concept as the traveller's dilemma?

[djames]

This article doesn't argue that the Nash equilibrium isn't the nuts. In fact, it reiterates the fact that it is the nuts when playing against an opponent also playing optimally.

When opponents play suboptimally, it's widely accepted that playing an optimal strategy isn't necessarily a maximal strategy. Thus, strategies other than the NE choice of $2 in the traveller's dilemma certainly can have higher payoffs than the NE choice. However, such a strategy would be suboptimal and hence exploitable.

So gauging the strategy of your opponent is always paramount when playing a suboptimal (or maximal) strategy.

[Galwegian]

[ QUOTE ]
This article doesn't argue that the Nash equilibrium isn't the nuts. In fact, it reiterates the fact that it is the nuts when playing against an opponent also playing optimally.

When opponents play suboptimally, it's widely accepted that playing an optimal strategy isn't necessarily a maximal strategy. Thus, strategies other than the NE choice of $2 in the traveller's dilemma certainly can have higher payoffs than the NE choice. However, such a strategy would be suboptimal and hence exploitable.

So gauging the strategy of your opponent is always paramount when playing a suboptimal (or maximal) strategy.

[/ QUOTE ] I think it does argue that NE is not the nuts. It argues that there is difference between rational and optimal (in the game theoretic sense)

[djames]

Optimal has a mathematical meaning. Rational does not.

What may be "rational" to a game player who wants to play an unexploitable strategy would be the NE strategy (or other equilibrium strategies).

What may be rational to a game player who doesn't care if they can be exploited but rather cares to maximally exploit the weaknesses of their opponent may be a strategy that isn't the NE.

Either way, both players are playing "the nuts" when they have their goal in mind. I don't think rigid game theory accepts the term "rational" as it is not mathematical but rather subjective.

[Galwegian]

[ QUOTE ]
Optimal has a mathematical meaning. Rational does not.

[/ QUOTE ] And this is exactly the point. Indeed the author himself refers to the difficulty of formalizing the notion of rationality.

[djames]

[ QUOTE ]
[ QUOTE ]
Optimal has a mathematical meaning. Rational does not.

[/ QUOTE ] And this is exactly the point. Indeed the author himself refers to the difficulty of formalizing the notion of rationality.

[/ QUOTE ]

Good, then you can understand the difference between an optimal strategy and a maximal strategy. You can also understand that the idea that if we knew the second player's number, the optimal & maximal strategies are known and agree (e.g if their number is 100, ours is 99). But, when the other player's number is unknown, the optimal & maximal strategies (probably*) disagree. The author uses "rational" when speaking of a maximal strategy, not an optimal one. He lucks out that they are the same in the case where we know the other players action.

* I say probably because I haven't seen the derivation of the maximal strategy and I don't care to figure one out.

[PairTheBoard]

[ QUOTE ]
I don't think rigid game theory accepts the term "rational" as it is not mathematical but rather subjective.


[/ QUOTE ]

For me, this is the most interesting point here. The idea that what's "rational" is subjective. Such a notion puts the lie to the dominant attitude of many posters on this Forum. It's the arrogant attitude that what's rational can always be determined and that they have the ability to make that determination and dictate it to the rest of us.

PairTheBoard

[tolbiny]

[ QUOTE ]
[ QUOTE ]
This article doesn't argue that the Nash equilibrium isn't the nuts. In fact, it reiterates the fact that it is the nuts when playing against an opponent also playing optimally.

When opponents play suboptimally, it's widely accepted that playing an optimal strategy isn't necessarily a maximal strategy. Thus, strategies other than the NE choice of $2 in the traveller's dilemma certainly can have higher payoffs than the NE choice. However, such a strategy would be suboptimal and hence exploitable.

So gauging the strategy of your opponent is always paramount when playing a suboptimal (or maximal) strategy.

[/ QUOTE ] I think it does argue that NE is not the nuts. It argues that there is difference between rational and optimal (in the game theoretic sense)

[/ QUOTE ]

Basically it argues that optimal play can only occur with full knowledge of how player #2 will play. Optimal play if you know that you opponent will pick $100 is to pick 99$, them knowing this should pick 98$ all the way on down to 2. But when you don't know how your opponent is going to play this backwards induction doesn't work since you don't expect your opponent to work their way all the way back to 2$.

[Galwegian]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
This article doesn't argue that the Nash equilibrium isn't the nuts. In fact, it reiterates the fact that it is the nuts when playing against an opponent also playing optimally.

When opponents play suboptimally, it's widely accepted that playing an optimal strategy isn't necessarily a maximal strategy. Thus, strategies other than the NE choice of $2 in the traveller's dilemma certainly can have higher payoffs than the NE choice. However, such a strategy would be suboptimal and hence exploitable.

So gauging the strategy of your opponent is always paramount when playing a suboptimal (or maximal) strategy.

[/ QUOTE ] I think it does argue that NE is not the nuts. It argues that there is difference between rational and optimal (in the game theoretic sense)

[/ QUOTE ]

Basically it argues that optimal play can only occur with full knowledge of how player #2 will play. Optimal play if you know that you opponent will pick $100 is to pick 99$, them knowing this should pick 98$ all the way on down to 2. But when you don't know how your opponent is going to play this backwards induction doesn't work since you don't expect your opponent to work their way all the way back to 2$.

[/ QUOTE ] I understand the logic of the arguments presented in the article quite well. However, you are missing (I think) the deeper point of the article which is the dichotomy between optimal and rational - the interesting part of the article is towards the end

"What is interesting is that this rejection of formal rationality and logic has a kind of meta-rationality attached to it. If both players follow this meta-rational course, both will do well. The idea of behavior generated by rationally rejecting rational behavior is a hard one to formalize. But in it lies the step that will have to be taken in the future to solve the paradoxes of rationality that plague game theory and are codified in Traveler's Dilemma. "

I am just wondering if there are examples of this in poker - that is, situations where where two oppponents who understand optimal strategy might choose to reject it on the basis of this "meta-rationality" referred to in the article? This has nothing to do with exploitive play against a suboptimal opponent.

[Piers]

The confusion appears to be that the dilemma assumes (while stating it does not) that the travellers are playing against each other where in the story they are not playing a game but just both trying to get as much money as possible.

If your aim is to beat your opponent obviously you choose $2, if you just want a load of dosh and don’t care about the other guy you choose something around $100.

I don’t think it’s a paradox, just a case of the model not really fitting, anyone choosing $2 for real would be nuts.