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Technologos comments on Fundamentally Flawed, or Fast and Frugal? - Less Wrong
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Comments (74)
(Not directly related, but may be interesting to someone. )
In a certain technical sense, "satisficing" is formally equivalent to expected utility maximization. Specifically, consider an interval on a real line (e.g. the amount of money that could be made), and a continuous and monotonous utility function on that interval. Expected utility maximization for that utility function u (i.e. the choice of a random variable X with codomain in the amounts of money) is then equivalent to maximization of probability Pr(X>V), where V is a random variable that depends only on u. Pr(X>V) looks quite like satisficing: you are trying to choose the outcome X so that it's better than a threshold V, except you are uncertain about what the "correct" threshold is.
In detail, an appropriately scaled utility function u(x) is a cumulative distribution function for a random variable V, so expected utility can be written as
Reference: E. Castagnoli & M. Licalzi (1996). `Expected utility without utility'. Theory and Decision 41(3):281-301.
That utility function would have a very interesting second derivative, though...
Also, the example appears to depend on simultaneous consideration of the options; with sequential consideration, might not a small standard deviation for V induce a situation where many options will have high Pr(X>V) and only EU maximization would support rejection of early options?
Could you say that more explicitly? What sequential consideration? Where do Pr(X>V) and EU(X) disagree, given that they are equal?
To be clear, I am not disagreeing with your analysis of the model you presented; I am arguing that satisficing and EU maximization are not equivalent in general, but rather only when certain conditions are satisfied. Imagine, for instance, that there was no uncertainty in V; then two distributions of X could both have Pr(X>V) = 1 with different EUs.
I was thinking of sequential consideration as essentially introducing uncertainty about the set of possible X distributions, but on reflection it's clear that this would be inadequate by itself to change C&L's result. The above modification--or any variant where satisficing includes a threshold requirement for Pr(X>V) rather than trying to maximize that quantity--would have to be integrated to make sequential consideration matter.
Finally, if V depends only on money, rather than utility, then having a utility function with positive second derivative could make EU maximizers pick an X distribution with higher mean and standard deviation than satisficers might.
I can't make sense of this statement.
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).