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Unnamed comments on Model of unlosing agents - Less Wrong Discussion
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Suppose you have A>B>C>A, with at least a $1 gap at each step of the preference ordering. Consider these 3 options:
Option 1: I randomly assign you to get A, B, or C
Option 2: I randomly assign you to get A, B, or C, then I give you the option of paying $1 to switch from A to C (or C to B, or B to A), and then I give you the option of paying $1 to switch again
Option 3: I take $2 from you and randomly assign you to get A, B, or C
Under standard utility theory Option 2 dominates Option 1, which in turn strictly dominates Option 3. But for an unlosing agent which initially has cyclic preferences, Option 2 winds up being equivalent to Option 3.
Incidentally, if given the choice, the agent would choose option 1 over option 3. When making choices, unlosing agents are indistinguishable from vNM expected utility maximisers.
Or another way of seeing it, the unlosing agent could have three utility functions remaining: A>B>C, B>C>A, and C>A>B, and all of these would prefer option 1 to option 3.
What's more interesting about your example is that it shows that certain ways of breaking transitivities are better than others.
Which is a good argument to break circles early, rather than late.