For instance, someone is providing you with information about where the princess is… but they secretly prefer that you be eaten rather than wed another!
The theorem proved below says that before you make an observation, you cannot expect it to decrease your utility, but you can sometimes expect it to increase your utility. I’m ignoring the cost of obtaining the additional data, and any losses consequential on the time it takes.
As you indicated, the information assumed in the proof is not assumed in your gloss.
Perhaps it should read something like, “the expected difference in the expected value of a choice upon learning information about the choice, when you are aware of the reliability of the information, is non-negative,” but pithier?
Because it seems that if I have a lottery ticket with a 1-in-1000000 chance of paying out $1000000, before I check whether I won, going to redeem it has an expected value of $1, but I expect that if I check whether I have won, this value will decrease.
For instance, someone is providing you with information about where the princess is… but they secretly prefer that you be eaten rather than wed another!
It is explicit in the hypotheses that you know how reliable your observations are, i.e. you know P_c(u|o).
It is explicitly stated in the hypotheses that you know how reliable your observations are.
Where? I see
It’s always a good idea to read below the fold before commenting, an example of more information being a good thing.
(BTW, my deleted comment was a draft I had second thoughts about, then decided was right anyway and reposted here.)
P_c(u|o) is assumed to be known to the agent.
No need to be snide. I think the description of your theorem, as written above, is false. What conditions need to hold before it becomes true?
I think it is true. I don’t see whatever problem you see.
As you indicated, the information assumed in the proof is not assumed in your gloss.
Perhaps it should read something like, “the expected difference in the expected value of a choice upon learning information about the choice, when you are aware of the reliability of the information, is non-negative,” but pithier?
Because it seems that if I have a lottery ticket with a 1-in-1000000 chance of paying out $1000000, before I check whether I won, going to redeem it has an expected value of $1, but I expect that if I check whether I have won, this value will decrease.
“The prior expected value of new information is non-negative.”
But summaries leave out details. That is what makes them summaries.