It isn’t a matter of agreement. It is the logical conclusion to make by every “perfect logician”, if such had been specified. The Nash Equilibrium requires that no player CAN benefit by any other choice and thus does nothing. That doesn’t apply to this scenario because merely by every driver knowing the situation, they all know that by flipping a coin, they all benefit.
Adding the noise of flipping a coin, or any ordered understanding/agreement to establish who changes and who doesn’t, immediately improves the situation.
It is like Zeno’s paradox, because that is dealing with infinities and in a sense making the contrast to our reality not being like that. Start at 1 on an infinite line, then where do you draw 2?.. you have to travel from point a – b which with an infinite line you cannot arrive at the next cardinal point without taking an infinity to do so. If there were no movement we can imagine as in calculus that we can just jump to an infinite amount of finite divisions [which is imho a contradiction] along an infinite line. The integers there are in meta[physical] position ~ there would be nothing to denote fixed points/cardinality. Start moving and then your car has to actually traverse between two infinities at finite speed!
As pure thought experiment and mathematically, your solution is fine though.
It seems that you do not understand calculus. Calculus is about comparing infinities. When you compare two infinities (the infinite number of points and the infinitesimal time it takes to traverse each of them), you can have a finite ratio, a finite velocity. Zeno has no actual paradoxes.
No i do understand the philosophical basis of calculus, but understanding how the universe can be infinite and finite is to me the most important question of our time. I dont think calculus answers it so i am taking it as something [and combining the meaning in this thread] to build off of ~ to ask further questions.
Consider that in real terms we get to relativity and qm before we can think of potential particles in infinite metaposition via calculus. So already the cardinality has broken down beyond maths as we understand it. It seems a tad strange or illogical even, to then denote singularity to the cardinality of what lies beyond qm/r. The answer wont be points and spaces, but more like an elastic band which can be stretched infinitely - probably to the point of transparecy.
Ergo I am attempting to further the debate rather than simply reiterating the given/‘known’.
There is no assumption that the drivers are perfect logicians.
No car benefits itself by changing lanes, it only benefits the cars behind it.
Each driver knows that if the drivers in front of them flip coins, they benefit. No driver has to participate in the coin flipping in order to benefit. Indeed, they have no reason to participate, as their participation does not in any way affect their own outcome.
You are adding noise by adding coin flipping, which is exactly my point: if we add noise, the outcome improves.
I’m not interested in talking about the Blue Eye Problem. If you want to, please take it up in one of the threads where we’ve discussed that problem, I’m happy to continue our conversation in an appropriate thread. Your objection to Nash equilibria on the basis that all Nash equilibria are self-defeating because they rely on common knowledge is also beyond the scope of this discussion; if it helps, I’m happy to acknowledge that my argument here is dependent on the assumption that Nash equilibria are not a self-defeating concept.
Amorphos, I believe the outcome I describe still results if we restrict ourselves to discrete states and discrete math. Say the cars are composed of pixels, and time ticks, and the cars jump ahead Y pixels per tick, but would like to jump ahead Z>Y pixels per tick. I think the result should be the same: if the cars randomly jump to lane 2 with some probability, the cars will be able to jump ahead Z pixels per tick.
The Nash Equilibrium requires it by specifying that there is no better decision possible and that every member knows it. Only perfect logicians could know that.
That is in a quasi-“yes and no” area. No car detracts from itself by changing lanes. It would be a zero-loss, zero-gain except by choosing to make the decision by a random method, all cars that are behind every car benefits. Logically, that inherently includes every driver.
What do you call it when there is zero risk of loss and zero prospect of gain … unless some choose to do X at which time it becomes zero risk of loss and 100% chance of gain?
By more than one understanding that game scenario, they know to give themselves the chance of gain because there is no risk and the alternative is zero gain.
The Nash Equilibrium requires that they all already know that. And to me, that means that they would all choose to flip the coin, or something similar.
For the above reason, that isn’t really true.
By “participating”, every knows that everyone gains.
It is similar to the situation of the solder. In an army, every solder shares a risk and thus allows for every solder to win the war. From the perspective of each individual, there is great risk and no benefit by fighting in a war. There is potential maximum loss (death) by fighting and no direct gain other than a paycheck that could have been gained by other means. By every solder being an altruist and participating in the risk, the risk is reduced, potentially to zero, and the gain is the shared booty (whatever the war was about) acquired by team work.
I agree. And I stated that … twice. Why are you arguing with me?
If the question is “will a driver benefit by changing lanes?”, the answer is a solid “no” where doing so is zero-gain.
And again, even in your hypothesized solution, the expected gain for an individual driver in changing lanes is zero. For driver D, the difference between everyone flipping a coin and everyone-but-D flipping a coin is zero.
It’s true for any driver D that, if all the drivers ahead of D changed their strategy, D would benefit. But that does not entail that D benefits by changing strategies. We know that D does not.
If we assume that D wants to both go fast and be altruistic, we’re just changing the meaning of ‘benefit’: we’re adding a new value that drivers are seeking to maximize, and so it isn’t surprising that we’re destroying the equilibrium.
They are all required to be the same. They all know the same scenario and all know that they all know it. There is no “D” acting differently than the others. So whatever D does in the way of decision making, the others also do.
This is not a given and does not follow from the givens. Nothing about the construction of the scenario requires that if one car has a thought, they all have the thought. They aren’t clones or computer programs (they aren’t people either, they’re cars). They’re players in a game-theoretic scenario, and their motivations are fully specified. It’s perfectly consistent for them to be different in innumerable ways so long as they otherwise satisfy the criteria in the scenario.
It is an inherent condition of the Nash Equilibrium. It is a prerequisite that no one can make a better decision. And that means that everyone is making the best decision. And since their situation is identical, it means that they are all making the same decision.
In general, it isn’t even a condition of a Nash equilibrium that all players have the same strategy or motivation, let alone that their strategy and motivation be so the same that they are literally identical.
And it’s not a condition of this Nash equilibrium because the problem doesn’t specify certain things. Lets say some drivers are slight misanthropes (but still care first and foremost about their own speed reaching z), some oblivious to line of cars behind them, some are oblivious to the cars in front of the car in front of them, some are robots, and some have never seen a coin. None of these conditions will affect the equilibrium or violate the setup, but they will break any solution that depends on the line of reasoning, “they’re perfect, so they’re the same, so that think of whatever I think of first which is flipping a coin.”
The stipulation is that they CANNOT make a better decision. So regardless of what color shoes they are wearing or who they voted for or where they wish to go and at what speed, they ARE ALL making the “best decision” that they CAN, else it isn’t a Nash Equilibrium.
I agree. That includes flipping a coin themselves (because it cannot improve their outcome), nor everyone in the line flipping a coin (because that is not a decision that they can make).
The only decision that cannot be bettered is to flip the coin, thus they ALL flip the coin.
I agreed that such introduces the noise of which you spoke. In reality, all that is needed is the introduction of something outside of the incumbent system. Sometimes noise would help sometimes not.
Hmmm… Are you taking ‘do nothing’ as the default strategy, i.e. any other strategy is a change from the strategy of ‘do nothing’?
I think if you don’t take ‘do nothing’ to be the default strategy, then ‘pick randomly’ is just as valid. But if ‘do nothing’ is the default strategy, and there’s no expected gain in changing to ‘pick randomly’, there’s an equilibrium in everyone sticking with the default.
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