One way to do it to get to the desired outcome is to replace U(x) with U(x,p) (with x being the money reward and p the probability to get it), and define U(x,p)=2x if p=1 and U(x,p)=x, otherwise.
The problem with this is that dealing with p=1 is iffy. Ideally, our certainty response would be triggered, if not as strongly, when dealing with 99.99% certainty—for one thing, because we can only ever be, say, 99.99% certain that we read p=1 correctly and it wasn’t actually p=.1 or something! Ideally, we’d have a decaying factor of some sort that depends on the probabilities being close to 1 or 0.
The reason I asked is that it’s very possible that a correct model of “attaching a utility to certainty” would be equivalent to a model with diminishing utility of money. If that were the case, we would be arguing over nothing. If not, we’d at least stand a chance of formulating gambles clarifying our intuitions if we knew what the alternatives are.
Comparing a .33 with a .34 chance makes me think that there’s gotta be a lot of guesswork involved, and that, with error margins and confidence intervals and such, there’s usually a sizeable chance that the underlying probabilities might be equal or reversed, so going for the higher reward makes sense.
If the 33% and 34% chances are in the middle of their error margins, which they should be, our uncertainty about the chances cancels out and the expected utility is still the same. Going for the higher expected value makes sense.
I brought up Settlers of Catan because, if I imagine a tile on the board with $24K and 34 dots under it, and another tile with $27K and 33 dots, suddenly I feel a lot better about comparing the probabilities. :) Does this help you, or am I atypical in this way?
Imagine you are a mathematical advisor to a king who asks you to advise him of a course of action and to predict the outcome.
Obviously with the advisor situation, you have to take your advisee’s biases into account. The one most relevant to risk avoidance is, I think, the status quo bias: rather than taking into account the utility of the outcomes in general, the king might be angry at you if the utility becomes worse, and not as picky if the utility becomes better (than it is now). You have to take your own utility into account, which depends not on the outcome but on your king’s satisfaction with it.
The problem with this is that dealing with p=1 is iffy. Ideally, our certainty response would be triggered, if not as strongly, when dealing with 99.99% certainty—for one thing, because we can only ever be, say, 99.99% certain that we read p=1 correctly and it wasn’t actually p=.1 or something! Ideally, we’d have a decaying factor of some sort that depends on the probabilities being close to 1 or 0.
The reason I asked is that it’s very possible that a correct model of “attaching a utility to certainty” would be equivalent to a model with diminishing utility of money. If that were the case, we would be arguing over nothing. If not, we’d at least stand a chance of formulating gambles clarifying our intuitions if we knew what the alternatives are.
If the 33% and 34% chances are in the middle of their error margins, which they should be, our uncertainty about the chances cancels out and the expected utility is still the same. Going for the higher expected value makes sense.
I brought up Settlers of Catan because, if I imagine a tile on the board with $24K and 34 dots under it, and another tile with $27K and 33 dots, suddenly I feel a lot better about comparing the probabilities. :) Does this help you, or am I atypical in this way?
Obviously with the advisor situation, you have to take your advisee’s biases into account. The one most relevant to risk avoidance is, I think, the status quo bias: rather than taking into account the utility of the outcomes in general, the king might be angry at you if the utility becomes worse, and not as picky if the utility becomes better (than it is now). You have to take your own utility into account, which depends not on the outcome but on your king’s satisfaction with it.