## Tuesday, November 1, 2011

Consider two different methods for what to do with the opinion of someone more expert than yourself, on a matter where both you and the expert have an opinion.

Adopt: When the expert's opinion differs from yours, adopt the expert's opinion.

Caution: When the expert's opinion differs from yours, suspend judgment.

To model the situation, we need to assign some epistemic utilities.  The following are reasonable given that the disvalue of a false opinion is significantly worse than the value of a true belief, at least by a factor of ~2.346 in the case of confidence level 0.95, according to the hate-love ratio inequality.
• Utility of having a true opinion: +1
• Utility of having a false opinion: approximately -2.346
• Utility of suspending judgment: 0
Given these epistemic utilities, we can do some quick calculations.  Suppose for simplicity that you're perfect at identifying the expert as an expert (surprisingly, replacing this by a 0.95 confidence level makes almost no difference).  Suppose the expert's level of expertise is 0.95, i.e., the expert has probability 0.95 of getting the right answer.  Then it turns out that Adopt is the better method when your level of expertise is below 0.89, while Caution is the better method when your level of expertise is above 0.89.  Approximately speaking, Adopt is the better method when you're more than about twice as likely to be wrong as the expert; otherwise, Caution is the better method.

In general, Adopt is the better method when your level of expertise is less than e/(D-e(D-1)), where e is the expert's level of expertise and D is the disutility of having a false opinion (which should be at least 2.346 for opinions at confidence level 0.95).  If your level of expertise is higher than that, Caution is the better method.

Here is a graph (from Wolfram Alpha) of the level of expertise you need to have (y-axis), versus the expert's level of expertise (x-axis), in order for adopting Caution rather than Adopt to be epistemic-decision-theory rational, where D=2.346.

Here is a further interesting result. If you set the utility of a false opinion to -1, which makes things more symmetric but leads to an improper scoring rule (with undesirable results like here), then it turns out that Adopt is better than Caution whenever your level of expertise is lower than the expert's. But for any utility of false opinion that's smaller than -1, it will be better to adopt Caution when the gap in level of expertise is sufficiently small.
If you want to play with this stuff, I have a Derive worksheet with this. But I suspect that there aren't many Derive users any more.