This oriented-probability-interval stuff does seem to perform as advertised. But I just want to point to another, in my opinion simpler, way to rationally refuse to play Savage-style expected utility games. The simple way contests the axioms dealing with preference, rather than those dealing with probability. If some options are incomparable, Savage’s argument fails. (Of course if an agent is forced to choose between incomparable options, it will choose, but that doesn’t mean it considers one of the options “better” in a straightforward way, nor that a classical utility function can be derived.)
What if Principle of Indifference-inspired probability theories can actually be made to work? Would you end your defiance of classical utility theory?
Here’s an alternate interpretation of this method:
If two events have probability intervals that don’t overlap, or they overlap but they have the same orientation and neither contains the other, then I’ll say that one event is unambiguously more likely than the other. If two events have the exact same probability intervals (including orientation), then I’ll say that are equally likely. Otherwise they are incomparable.
Under this interpretation, I claim that I do obey rule 2 (see prev post): if A is unambiguously more likely than B, then (A but not B) is unambiguously more likely than (B but not A), and conversely. I still obey rule 3: (A or B) is either unambiguously more likely than A, or they are equally likely. I also claim I still obey rule 4. Finally, I claim “unambiguously more likely” is transitive, but it is not total: there are incomparable events. So I break that part of rule 1.
Passing to utility, I’ll also have “unambiguously better”, “equally good”, and “incomparable”.
Of course if an agent is forced to choose between incomparable options, it will choose, but that doesn’t mean it
considers one of the options “better” in a straightforward way,
Exactly. But there’s a major catch: unlike with equal choices, an agent cannot choose arbitrarily between incomparable choices. This is because incomparability is intransitive. If the agent doesn’t resolve ambiguities coherently, it can get money pumped. For instance, an 18U bet on green and a 15U bet on red are incomparable. Say it picks red. A 15U bet on green and a 15U bet on red are also incomparable. Say it picks green. But then, an 18U bet on green is unambiguously better than a 15U bet on green.
The rest of the post is then about one method to resolve imcomparability coherently.
I personally think this interpretation is more natural. I also think it will be even less palatable to most LW readers.
This oriented-probability-interval stuff does seem to perform as advertised. But I just want to point to another, in my opinion simpler, way to rationally refuse to play Savage-style expected utility games. The simple way contests the axioms dealing with preference, rather than those dealing with probability. If some options are incomparable, Savage’s argument fails. (Of course if an agent is forced to choose between incomparable options, it will choose, but that doesn’t mean it considers one of the options “better” in a straightforward way, nor that a classical utility function can be derived.)
What if Principle of Indifference-inspired probability theories can actually be made to work? Would you end your defiance of classical utility theory?
Upvoted, first and foremost for appendix B.
Here’s an alternate interpretation of this method:
If two events have probability intervals that don’t overlap, or they overlap but they have the same orientation and neither contains the other, then I’ll say that one event is unambiguously more likely than the other. If two events have the exact same probability intervals (including orientation), then I’ll say that are equally likely. Otherwise they are incomparable.
Under this interpretation, I claim that I do obey rule 2 (see prev post): if A is unambiguously more likely than B, then (A but not B) is unambiguously more likely than (B but not A), and conversely. I still obey rule 3: (A or B) is either unambiguously more likely than A, or they are equally likely. I also claim I still obey rule 4. Finally, I claim “unambiguously more likely” is transitive, but it is not total: there are incomparable events. So I break that part of rule 1.
Passing to utility, I’ll also have “unambiguously better”, “equally good”, and “incomparable”.
Exactly. But there’s a major catch: unlike with equal choices, an agent cannot choose arbitrarily between incomparable choices. This is because incomparability is intransitive. If the agent doesn’t resolve ambiguities coherently, it can get money pumped. For instance, an 18U bet on green and a 15U bet on red are incomparable. Say it picks red. A 15U bet on green and a 15U bet on red are also incomparable. Say it picks green. But then, an 18U bet on green is unambiguously better than a 15U bet on green.
The rest of the post is then about one method to resolve imcomparability coherently.
I personally think this interpretation is more natural. I also think it will be even less palatable to most LW readers.