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guslThe claim I refuted in class last week: if the number of turns is known, then the optimal strategy is to always compete/defect.
Refutation:
Claim: if the above is true, then when you have an unknown number of turns, the optimal strategy is to always compete. This is absurd (though I won't prove that it is).
Proof: There will be a finite number of turns n.
Since for every n, the optimal strategy is to always compete, then the optimal strategy is to always compete.
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1$, 2$ paradox
There are 10 dollars in a bowl, and two players sitting opposite each other. At each move, a player can either take $1 and pass his turn, or he can take $2 and end the game right there.
Jacques suggested this should be equivalent to the Prisoner's Dilemma (since it's a cooperative competitive game), but I argued it wasn't since the number of turns is under the players' control in this game. He suggested the existence of an isomorphism, but didn't prove it. I didn't disprove it either. If this is isomorphic to the Prisoner's Dilemma, then we're in trouble.
Paradox: common knowledge of rationality makes both players compete at the first move, and thus get one of the worst possible outcomes. This is because at the penultimate move, the player's optimal strategy is to take $2. If this is foreseen by the other player, he's going to take $2 the turn before. But if that is foreseen, then the first player will take $2 the turn before, and so on... (presented by Wiebe van der Hoek, ESSLLI2002).
The paradox is that the most rational pair of people will do worse than a less rational pair people, provided that the players know about the other player's level of rationality. Or perhaps it's that assuming that the other player is dumb can be a good thing.
I am no longer troubled by The Prisoner's Dilemma (an old problem to me), but this paradox intrigues me.
I will add this to my collection of paradoxes once I get a Wiki running.
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The Balls Puzzle (Human Irrationality)
You have 100 balls, in three colors: red, green and blue. All you know is
that exactly 30 are red.
The game is to guess a color and draw a ball. If it matches you win; if it
doesn't, you lose.
(1) What is the optimal color to pick?
(2) Now suppose you could pick two colors. What is the optimal pick?
Most people will answer "red" to the first example, and "green & blue" to
the second, which is inconsistent. Apparently, they are operating under the
assumption that their choice will turn out to be worse than expected. e.g.
they act as if: in case they pick "green", then there will be fewer greens
than reds (though this will never actually be observed), and that if they
pick "blue" there will be fewer blues than reds. But even if it were
expected that one of these will be true, randomizing the choice 50/50 is
still better than picking "red".
(told by Joe Halpern)
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On Paradoxes (from an email conversation)
> Do you know of Martin Garnder, who used to
> write a Mathematical Games column for (I think) Scientific American? He
> has published a couple of books (_Aha! Gotcha_ and _Aha! Insight_) filled
> with real paradoxes and "false paradoxes" like that one, where our
> intuitions often give us exactly the wrong answer :)
Interesting, though I tend to believe there are no "true paradoxes", so I'd like to see what he means. A paradox to me is just a counterintuitive result. Do you know what he means by that distinction?
Refutation:
Claim: if the above is true, then when you have an unknown number of turns, the optimal strategy is to always compete. This is absurd (though I won't prove that it is).
Proof: There will be a finite number of turns n.
Since for every n, the optimal strategy is to always compete, then the optimal strategy is to always compete.
---------------------------------------------
1$, 2$ paradox
There are 10 dollars in a bowl, and two players sitting opposite each other. At each move, a player can either take $1 and pass his turn, or he can take $2 and end the game right there.
Jacques suggested this should be equivalent to the Prisoner's Dilemma (since it's a cooperative competitive game), but I argued it wasn't since the number of turns is under the players' control in this game. He suggested the existence of an isomorphism, but didn't prove it. I didn't disprove it either. If this is isomorphic to the Prisoner's Dilemma, then we're in trouble.
Paradox: common knowledge of rationality makes both players compete at the first move, and thus get one of the worst possible outcomes. This is because at the penultimate move, the player's optimal strategy is to take $2. If this is foreseen by the other player, he's going to take $2 the turn before. But if that is foreseen, then the first player will take $2 the turn before, and so on... (presented by Wiebe van der Hoek, ESSLLI2002).
The paradox is that the most rational pair of people will do worse than a less rational pair people, provided that the players know about the other player's level of rationality. Or perhaps it's that assuming that the other player is dumb can be a good thing.
I am no longer troubled by The Prisoner's Dilemma (an old problem to me), but this paradox intrigues me.
I will add this to my collection of paradoxes once I get a Wiki running.
----------------------------------------------
The Balls Puzzle (Human Irrationality)
You have 100 balls, in three colors: red, green and blue. All you know is
that exactly 30 are red.
The game is to guess a color and draw a ball. If it matches you win; if it
doesn't, you lose.
(1) What is the optimal color to pick?
(2) Now suppose you could pick two colors. What is the optimal pick?
Most people will answer "red" to the first example, and "green & blue" to
the second, which is inconsistent. Apparently, they are operating under the
assumption that their choice will turn out to be worse than expected. e.g.
they act as if: in case they pick "green", then there will be fewer greens
than reds (though this will never actually be observed), and that if they
pick "blue" there will be fewer blues than reds. But even if it were
expected that one of these will be true, randomizing the choice 50/50 is
still better than picking "red".
(told by Joe Halpern)
------------------------------------------
On Paradoxes (from an email conversation)
> Do you know of Martin Garnder, who used to
> write a Mathematical Games column for (I think) Scientific American? He
> has published a couple of books (_Aha! Gotcha_ and _Aha! Insight_) filled
> with real paradoxes and "false paradoxes" like that one, where our
> intuitions often give us exactly the wrong answer :)
Interesting, though I tend to believe there are no "true paradoxes", so I'd like to see what he means. A paradox to me is just a counterintuitive result. Do you know what he means by that distinction?
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(no subject)
Date: 2003-08-09 07:01 pm (UTC)By the way, my difficulty with guaging how much rationality my opponent has built into their stategy is the real reason I never had a clue with chess. *And* Go, though I haven't given up on that permanently (I haven't been to a game since Christmas though. bah.)
On paradoxes; I tend also to believe there're no "true paradoxes", but on closer reflection, this is a bias (or intuition, or religous belief). In practice, I'm not so sure. Indeed, practice (experience) doesn't really help here, since you never know if there's something you don't understand that'll render it paradox a counterintuitive result.
I swear, mathematical reasoning really does release endorphins in my brain,
A glossary
Date: 2003-08-10 10:03 pm (UTC)arguement: argument
finsish: finish
effects: affects
hwow: who
guaging: gauging (is that right?)
religous: religious
it paradox: a paradox
bah.
(no subject)
Date: 2003-08-09 07:09 pm (UTC)To be technical, the "paradox" is that the subgame perfect nash equilibrium ("always defect") isn't pareto optimal. While this might be aesthetically unpleasant, I can't think of any sense in which it's a "paradox."
--
I am about to go out of town, but chat with me about the finite PD when I get back. :)