The way you set up the model, V was a threshold of utility. Thus, anything that increased one’s expected utility also increased one’s expected probability of being above that threshold.
If, however, V was a threshold of money (distributed, say, as N($100,$10)), then look at these two X-distributions, given a utility function U(x) = x (the case of a function with positive second derivative just makes the following more extreme):
1) 100% probability of $200
2) 90% probability of $100 and 10% probability of $2100
Expected utilities:
1) 200
2) 300
Probabilities of meeting threshold:
1) 1- the probability of being 10 standard deviations above the mean, or “very damn close to 1”
2) 90% 50% + 10% “even closer to 1 than the above” = 55%
So EU-maxers will take the latter choice, where satisficers will take the former.
Note that if U(x) = x^2, then the disparity is even stronger.
I believe you are confused, but can’t pinpoint on the first look in what way exactly. V is not “threshold of utility”, it is a random variable of the same kind as X. I don’t see what you mean by setting V to be normally distributed and U(x)=x, given that by construction they determine each other by the rule Pr(x>V)=u(x). If you redefine the concepts, you should do so more clearly.
If the decision agent is trying to maximize the probability of its utility being greater than a draw from the random variable V (where V is specified in utility) then it is trying to maximize the probability of being above some (yet-unknown) threshold value, no?
The departure from your model that I was clarifying (unsuccessfully) in the last comment was for V to be a random variable not of utility but of money, distributed normally in this example. U(x) = x is the utility function for the EU-maxing agent, because when V is specified in money, the satisficing agent no longer needs to worry about utility.
The rule you gave is only true when the satisficer defines the threshold level in terms of utility.
No luck. You should write everything in math, specifying types (domains/codomains) of all functions/random variables. It’ll really be easier, and the confusion (mine or yours) will be instantly resolved.
Ah, I think I found it. I took V to have a codomain in utilons in your example (that was my interpretation of “V is a random variable that depends only on u”).
Reinterpreting the subsequent comments in that context, I can see that I was responding to “formally equivalent” in the original comment as if it meant “expected utility maximization of the traditional sort, where each outcome x is itself assigned a value by a function on x that does not involve V, will produce the same decisions as satisficing of the type described under these conditions.”
Interestingly, the latter may be true if V did have a codomain in utilons (or at least, I was unable to come up with a consistent counterexample).
The way you set up the model, V was a threshold of utility. Thus, anything that increased one’s expected utility also increased one’s expected probability of being above that threshold.
If, however, V was a threshold of money (distributed, say, as N($100,$10)), then look at these two X-distributions, given a utility function U(x) = x (the case of a function with positive second derivative just makes the following more extreme):
1) 100% probability of $200
2) 90% probability of $100 and 10% probability of $2100
Expected utilities:
1) 200
2) 300
Probabilities of meeting threshold:
1) 1- the probability of being 10 standard deviations above the mean, or “very damn close to 1”
2) 90% 50% + 10% “even closer to 1 than the above” = 55%
So EU-maxers will take the latter choice, where satisficers will take the former.
Note that if U(x) = x^2, then the disparity is even stronger.
I believe you are confused, but can’t pinpoint on the first look in what way exactly. V is not “threshold of utility”, it is a random variable of the same kind as X. I don’t see what you mean by setting V to be normally distributed and U(x)=x, given that by construction they determine each other by the rule Pr(x>V)=u(x). If you redefine the concepts, you should do so more clearly.
If the decision agent is trying to maximize the probability of its utility being greater than a draw from the random variable V (where V is specified in utility) then it is trying to maximize the probability of being above some (yet-unknown) threshold value, no?
The departure from your model that I was clarifying (unsuccessfully) in the last comment was for V to be a random variable not of utility but of money, distributed normally in this example. U(x) = x is the utility function for the EU-maxing agent, because when V is specified in money, the satisficing agent no longer needs to worry about utility.
The rule you gave is only true when the satisficer defines the threshold level in terms of utility.
No luck. You should write everything in math, specifying types (domains/codomains) of all functions/random variables. It’ll really be easier, and the confusion (mine or yours) will be instantly resolved.
Also, thanks for posting the original comment—it’s actually useful to some research I’m doing, now that I actually understand it!
Ah, I think I found it. I took V to have a codomain in utilons in your example (that was my interpretation of “V is a random variable that depends only on u”).
Reinterpreting the subsequent comments in that context, I can see that I was responding to “formally equivalent” in the original comment as if it meant “expected utility maximization of the traditional sort, where each outcome x is itself assigned a value by a function on x that does not involve V, will produce the same decisions as satisficing of the type described under these conditions.”
Interestingly, the latter may be true if V did have a codomain in utilons (or at least, I was unable to come up with a consistent counterexample).