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lukeprog comments on Stupid Questions Open Thread Round 4 - Less Wrong Discussion
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I can see how the money pump argument demonstrates the irrationality of an agent with cyclic preferences. Is there a more general argument that demonstrates the irrationality of an agent with intransitive preferences of any kind (not merely one with cyclic preferences)?
I don't understand what you mean. Can you give me an example of preferences that are intransitive but not cyclic?
A little bit of googling turned up this paper by Gustafsson (2010) on the topic, which says that indifference allows for intransitive preferences that do not create a strict cycle. For instance, A>B, B>C, and C=A.
The obvious solution is to add epsilon to break the indifference. If A>B, then there exists e>0 such that A>B+e. And if e>0 and C=A, then C+e>A. So A>B+e, B+e>C+e, and C+e>A, which gives you a strict cycle that allows for money pumping. Gustafsson calls this the small-bonus approach.
Gustafsson suggests an alternative, using lotteries and applying the principle of dominance. Consider the 4 lotteries:
Lottery 1: heads you get A, tails you get B
Lottery 2: heads you get A, tails you get C
Lottery 3: heads you get B, tails you get A
Lottery 4: heads you get C, tails you get A
Lottery 1 > Lottery 2, because if it comes up tails you prefer Lottery 1 (B>C) and if it comes up heads you are indifferent (A=A).
Lottery 2 > Lottery 3, because if it comes up heads you prefer Lottery 2 (A>B) and if it comes up tails you are indifferent (C=A)
Lottery 3 > Lottery 4, because if it comes up heads you prefer Lottery 3 (B>C) and if it comes up tails you are indifferent (A=A)
Lottery 4 > Lottery 1, because if it comes up tails you prefer Lottery 4 (A>B) and if it comes up heads you are indifferent (C=A)
This is the kind of thing I was looking for; thanks!
Just in case - synchronising the definitions.
I usually consider something transitive if "X≥Y, Y≥Z then X≥Z" holds for all X,Y,Z.
If this holds, preferences are transitive. Otherwise, there are some X,Y,Z: X≥Y, Y≥Z, Z>X. I would call that cyclical.