One catch is that in the examples, the state spaces being compared aren’t probability mixtures at all.
In the 6.59pm restaurant lottery example, the outcomes at 7pm are not just “you eat at restaurant X” for two possible values of X. They also include “you had to use extra resources to cover both contingencies”, “your mood was affected by the late decision” and possibly even “your friend’s option was drawn but was too far away for you to get to by 7pm so you had to go home and eat microwaved ramen instead”.
That is, none of these outcomes are the same as any of the outcomes from a 7am lottery (or a nonrandom restaurant choice), and it doesn’t matter what cost function you assign to entropy of the distribution. There are real physical differences that mean that the utilities will generally be different.
Sometimes utility may even be higher for the more uncertain outcomes. For example some people value anticipation and revelation of potential gifts more than receiving the same gift with knowledge in advance of what it will be.
Hmm, I’m not sure what I should be taking away from that. You’ve pointed out that the morning and evening lotteries are materially different, but that’s not contentious to me: if uncertainty has costs then those costs have to show up as differences in the world compared to a world without that uncertainty.
I guess the restaurant story failed to focus on the-bit-that’s-weird-to-me, which is that if my friend and I were negotiating over the lottery parameter p, then my mental model of the expected utility boundary as p varies is not a straight line.
To be explicit, the “standard model” of my friend and I having a lottery looks like this, whereas once you account for the costs of increasing uncertainty when p is away from 0 or 1 it ends up looking like this.
Yes, I’m not contending against your fundamental point. In fact, I think that the curve from 0 to 1 can be even stranger than that with discontinuities in it, and that under some circumstances it can even have parts that go above the straight line. Focusing on a specific formula based on entropy doesn’t really match reality and detracts from the main point.
One catch is that in the examples, the state spaces being compared aren’t probability mixtures at all.
In the 6.59pm restaurant lottery example, the outcomes at 7pm are not just “you eat at restaurant X” for two possible values of X. They also include “you had to use extra resources to cover both contingencies”, “your mood was affected by the late decision” and possibly even “your friend’s option was drawn but was too far away for you to get to by 7pm so you had to go home and eat microwaved ramen instead”.
That is, none of these outcomes are the same as any of the outcomes from a 7am lottery (or a nonrandom restaurant choice), and it doesn’t matter what cost function you assign to entropy of the distribution. There are real physical differences that mean that the utilities will generally be different.
Sometimes utility may even be higher for the more uncertain outcomes. For example some people value anticipation and revelation of potential gifts more than receiving the same gift with knowledge in advance of what it will be.
Hmm, I’m not sure what I should be taking away from that. You’ve pointed out that the morning and evening lotteries are materially different, but that’s not contentious to me: if uncertainty has costs then those costs have to show up as differences in the world compared to a world without that uncertainty.
I guess the restaurant story failed to focus on the-bit-that’s-weird-to-me, which is that if my friend and I were negotiating over the lottery parameter p, then my mental model of the expected utility boundary as p varies is not a straight line.
To be explicit, the “standard model” of my friend and I having a lottery looks like this, whereas once you account for the costs of increasing uncertainty when p is away from 0 or 1 it ends up looking like this.
Yes, I’m not contending against your fundamental point. In fact, I think that the curve from 0 to 1 can be even stranger than that with discontinuities in it, and that under some circumstances it can even have parts that go above the straight line. Focusing on a specific formula based on entropy doesn’t really match reality and detracts from the main point.
Fair point re. focusing on a specific formula, I’ll remove that from the post.