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 A (me) and B (my friend). The restaurants are X and Y. There are two events we care about: the first is me and my friend choosing the lottery parameter p=P(X), and the second is actually running the lottery.
After picking p but before the lottery, my friend and I have (for simplicity) fixed costs C(p)=(CA(p),CB(p)) and outcome-dependent utilities UX=(UA(X),UB(X)) and UY. Our expected utilities are indeed exactly what you’d expect: pUX+(1−p)UY−C(p). Is this what you mean by eventually projecting to a straight line?
The “standard model”/convex hull isn’t describing the space of outcomes of the lottery, it’s describing the space of lotteries by summarizing them as expected utilities. However, as p varies C(p) can draw any number of weird and wonderful (and as you say, sharp and discontinuous) shapes. Once we fix C and hence {C(p)|0≤p≤1}, we get a specific shape/space of (expected) lottery outcomes. Is this what you mean by having the utility function be “all inclusive”?
Now that we’ve got a nice, fixed set of outcomes with an associated utility per outcome, we can take a hyperprior over p to get a distribution over that space of outcomes, and we’re back in standard-utility-theory land.
I think I’ve identified my confusion: we should distinguish between the distribution choice parameterized by p, and the prior distribution over expected outcomes which we can get with a distribution over p. If we were playing a game where we made a choice about that distribution over p, we’d have the same problem: our utilities could depend on the prior and so the outcome space would again be an arbitrary shape.
So, summary: it’s invalid, as a design choice when formulating e.g. a bargaining solution or a game equilibrium, to do the following:
Start from a space of outcomes.
Say “and now the players choose a distribution over the outcomes”.
Conclude “our new space of outcomes is the convex hull of the old space of outcomes”.
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 A (me) and B (my friend). The restaurants are X and Y. There are two events we care about: the first is me and my friend choosing the lottery parameter p=P(X), and the second is actually running the lottery.
After picking p but before the lottery, my friend and I have (for simplicity) fixed costs C(p)=(CA(p),CB(p)) and outcome-dependent utilities UX=(UA(X),UB(X)) and UY. Our expected utilities are indeed exactly what you’d expect: pUX+(1−p)UY−C(p). Is this what you mean by eventually projecting to a straight line?
The “standard model”/convex hull isn’t describing the space of outcomes of the lottery, it’s describing the space of lotteries by summarizing them as expected utilities. However, as p varies C(p) can draw any number of weird and wonderful (and as you say, sharp and discontinuous) shapes. Once we fix C and hence {C(p)|0≤p≤1}, we get a specific shape/space of (expected) lottery outcomes. Is this what you mean by having the utility function be “all inclusive”?
Now that we’ve got a nice, fixed set of outcomes with an associated utility per outcome, we can take a hyperprior over p to get a distribution over that space of outcomes, and we’re back in standard-utility-theory land.
I think I’ve identified my confusion: we should distinguish between the distribution choice parameterized by p, and the prior distribution over expected outcomes which we can get with a distribution over p. If we were playing a game where we made a choice about that distribution over p, we’d have the same problem: our utilities could depend on the prior and so the outcome space would again be an arbitrary shape.
So, summary: it’s invalid, as a design choice when formulating e.g. a bargaining solution or a game equilibrium, to do the following:
Start from a space of outcomes.
Say “and now the players choose a distribution over the outcomes”.
Conclude “our new space of outcomes is the convex hull of the old space of outcomes”.
Does that sound right?