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In the framework of maximization of expected utility (MEU), is it somehow optimal or justifiable to make choices based on the subjective probability distribution when we know it may be inaccurate (hence, almost always), even if it is the best we have?
Is there a justification for such a choice in Savage's theory?

Richard Hardy
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  • Could you elaborate on how we are intended to interpret "inaccurate ... [yet] the best we have"? I'm hoping for a quantitative statement, for otherwise it looks impossible to provide any theoretical justification as you ask. – whuber Sep 10 '19 at 12:50
  • @whuber, I must say upfront that I might be inadvertently mixing different definitions of probability here, so some of the confusion might arise from that. What I mean here can be illustrated by an example. Let us say, the agent believes the probability of rain tomorrow is 50% but actually it is 90%. The agent does not have the relevant information or cannot process it well enough to make a better judgment. – Richard Hardy Sep 10 '19 at 12:54
  • I still don't get it: if the agent is wrong, all bets are off (literally!). In what sense could one possibly pursue any kind of argument that it is "optimal or justifiable" to act on incorrect information or belief? – whuber Sep 10 '19 at 14:00
  • @whuber, It is obvious that in most realistic situations the agent is wrong, to a larger or smaller degree. At the same time, there is a whole theory (Savage's) about decision under uncertainty with subjective probabilities. It would be unfortunate if the whole theory were made redundant by such a simple fact of life. So I expect something smart can be said about such situations in light of the theory, and this is what I am asking about. – Richard Hardy Sep 10 '19 at 14:11
  • And that's precisely what I'm getting at, too: until you can explain how you might quantify that degree of wrongness, it's difficult to see what anyone could say generally. You seem to be asking a question that is too vague and broad for us to cope with in this forum. – whuber Sep 10 '19 at 14:13
  • @whuber, what about the following example: we have a perfectly balanced die and each of {1}, ..., {6} has a probability of 1/6 to occur. Meanwhile, the agent believes wrongly that the die is loaded and the probabilities are 1/3, 1/3, 1/3, 0, 0, 0. Does the fact that the beliefs are wrong make the framework of MEU w.r.t. subjective probabilities useless? Or can we say something about optimality of or justification for using it in such a setup? – Richard Hardy Sep 10 '19 at 14:22
  • In such a specific situation, the existing theory tells you how to evaluate the misbelief: compare its risk to the risk associated with acting on a correct belief. In general the action based on a misbelief will not be optimal--that's almost a tautology derived from the definition of optimality--but how unoptimal that might be will depend on the circumstances. Applications in finance (no-arbitrage arguments, change of measure, etc.) show how to develop procedures that don't depend on knowing the true probabilities at all: that's another approach. – whuber Sep 10 '19 at 14:32
  • @whuber, thank you, I think I get that. I am not primarily concerned with evaluating performance per se but rather with the question of whether we must dump Savage's theory based on a simple observation that our beliefs are hardly ever perfect. As a decision maker facing uncertainty, I need to make a choice, and I need to make it before I will be able to measure how well I did. Knowing that my beliefs are not perfect, what are my options? Can I justify basing my choices on Savage's theory? – Richard Hardy Sep 10 '19 at 14:38
  • As a general proposition, no theory has ever been abandoned because it failed to be a fully correct and perfectly precise model or predictor of reality. Thoughtful users of any theory are always weighing the extent to which it might depart from the idealization on which it is based and considering whether that should cause them to modify the theory's predictions. This is such a broad, general issue that I cannot help asking what in particular about Savage's theory you are hoping might be addressed here. – whuber Sep 10 '19 at 20:00
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    @whuber, Hmm, that is a helpful perspective. I will think about it. Thank you! – Richard Hardy Sep 11 '19 at 11:27

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