Yup, you need more dimensions to include utility of certainty at a time, utility of impact on future games, emotional cost of negotiation, and other factors that aren't mentioned in your simplistic 2D payout matrix. And each player's utility function is the projection of this many-dimensional space onto a line for that decision. Your simpler fix is insufficient - the uncertainty cost is not necessarily smooth, and is not the only factor missing.
This more complete modeling, in theory, will just make sure the points are in the right place on your 2-d projection, and you still get a convex hull over them. Most classes teaching this will mention that the utility is "all inclusive", but don't spend much time on defining that, or noting how weak it makes the theory. Note that the costs of uncertainty can vary with the probability distribution, so you can't necessarily pick anything in between without re-projecting the points (or treating each distribution as a new projection, and you can only pick actual intersecting points).
In practice, humans don't have a utility function, don't know how to introspect what preferences they do have, and have inconsistencies that make this fail for almost all real decisions.
I agree my "fix" is insufficient - in fact I'd go so far as agreeing with JBlack below that including it was net negative to the question.
I'd like to pin down what you mean by your description of a more complete model, I hope you don't mind.
Let me flesh out the restaurant story. The actors are (me) and (my friend). The restaurants are and . There are two events we care about: the first is me and my friend choosing the lottery parameter , and the second is actually running the lottery.
After picking ...
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 , then my mental model of the expected utility boundary as 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 is away from or 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.
It's almost a rule that as soon as you have a "utility of possible outcomes" plot like this:
You must then say "and by randomly choosing between the outcomes, we can achieve any intermediate outcome in terms of utility within the convex hull of these points" resulting in a plot like this:
Cool, I've done a linear interpolation before, seems reasonable. Plus, convex hulls are super nice to work with. But all models are imperfect - how accurate is this convex hull idea in practice?
Three stories
My point is that in practice the mapping from lottery probability p and outcome utilities U1,U2 to lottery utility Up is probably not Up=pU1+(1−p)U2. I wonder if it's occasionally not even close. I would expect the "lottery closure" of the three outcomes above to look something like this:
I'm pretty darn sure I'm not the first person to think about this, so: how big an issue is this when thinking about things like bargaining solutions, or stochastic Nash equilibria?