(1) How we define the term ‘coherence theorems’ doesn’t matter. What matters is that Premise 1 (striking out the word ‘coherence’, if you like) is false.
(2) The way I define the term ‘coherence theorems’ seems standard.
Now making point (1) in more detail:
Reserve the term ‘coherence theorems’ for whatever you like. Premise 1 is false: there are no theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. The VNM Theorem doesn’t say that, nor does Savage’s Theorem, nor does Bolker-Jeffrey, nor do Dutch Books, nor does Cox’s Theorem, nor does the Complete Class Theorem. That is the error in coherence arguments. Premise 1 is false.
Now for point (2):
I take the Appendix to make plausible enough that my use of the term ‘coherence theorems’ is standard, at least in online discussions. Here are some quotations.
1.
Now, by the general idea behind coherence theorems, since we can’t view this behavior as corresponding to expected utilities, we ought to be able to show that it corresponds to a dominated strategy somehow
2.
Roughly, the general claim these theorems make is that any system either (a) acts like an expected utility maximizer under some probabilistic model, or (b) throws away resources in a pareto-suboptimal manner.
3.
Summary: Violations of coherence constraints in probability theory and decision theory correspond to qualitatively destructive or dominated behaviors.
Again, we see a manifestation of a powerful family of theorems showing that agents which cannot be seen as corresponding to any coherent probabilities and consistent utility function will exhibit qualitatively destructive behavior
4.
One of the most pleasing things about probability and expected utility theory is that there are many coherence arguments that suggest that these are the “correct” ways to reason. If you deviate from what the theory prescribes, then you must be executing a dominated strategy.
5.
‘Coherence arguments’ mean that if you don’t maximize ‘expected utility’ (EU)—that is, if you don’t make every choice in accordance with what gets the highest average score, given consistent preferability scores that you assign to all outcomes—then you will make strictly worse choices by your own lights than if you followed some alternate EU-maximizing strategy (at least in some situations, though they may not arise). For instance, you’ll be vulnerable to ‘money-pumping’—being predictably parted from your money for nothing.
6.
The overall message here is that there is a set of qualitative behaviors and as long you do not engage in these qualitatively destructive behaviors, you will be behaving as if you have a utility function.
7.
I think that to contain the concept of Utility as it exists in me, you would have to do homework exercises I don’t know how to prescribe. Maybe one set of homework exercises like that would be showing you an agent, including a human, making some set of choices that allegedly couldn’t obey expected utility, and having you figure out how to pump money from that agent (or present it with money that it would pass up).
8.
The view that utility maximizers are inevitable is supported by a number of coherence theories developed early on in game theory which show that any agent without a consistent utility function is exploitable in some sense.
Maybe the term ‘coherence theorems’ gets used differently elsewhere. That is okay. See point (1).
Two points, made in order of importance:
(1) How we define the term ‘coherence theorems’ doesn’t matter. What matters is that Premise 1 (striking out the word ‘coherence’, if you like) is false.
(2) The way I define the term ‘coherence theorems’ seems standard.
Now making point (1) in more detail:
Reserve the term ‘coherence theorems’ for whatever you like. Premise 1 is false: there are no theorems which state that, unless an agent can be represented as maximizing expected utility, that agent is liable to pursue strategies that are dominated by some other available strategy. The VNM Theorem doesn’t say that, nor does Savage’s Theorem, nor does Bolker-Jeffrey, nor do Dutch Books, nor does Cox’s Theorem, nor does the Complete Class Theorem. That is the error in coherence arguments. Premise 1 is false.
Now for point (2):
I take the Appendix to make plausible enough that my use of the term ‘coherence theorems’ is standard, at least in online discussions. Here are some quotations.
1.
2.
3.
4.
5.
6.
7.
8.
Maybe the term ‘coherence theorems’ gets used differently elsewhere. That is okay. See point (1).