Please correct me if any of my assumptions are innacurate, and I apologize if this comment comes off as completely tautological.
Expected utility is explicity defined as the statistic
U(x)})where X is the set of all possible outcomes associated with a particular gamble, p(x) is the proportion of times that outcome x occurs within the gamble, and U(x) is the utility of outcome x, a function that must be strictly increasing with respect to the monetary value of outcome x.
To reduce ambiguity:
1A, 1B, 2A, and 2B are instances of gambles.
For 1B, the possible outcomes are $27000 and $0.
For 1B, the expected utility is p($27000) * U($27000) + p($0) * U($0) = 33⁄34 * U($27000) + 1⁄34 * U($0).
If you choose 1A over 1B and 2B over 2A, what can we conclude?
that you are not using the rule “maximize expected utility” to make your decisions. Thus you do not fit the definition, as given by the Axiom of Independence, of consistent decision making.
If you choose 1A over 1B and 2B over 2A, what can we not conclude?
that your decision rule changes arbitrarily. You could, for example, always follow the rule, “Maximize minimum net utility. In the case of a tie, maximize expected utility.” In this case, you would choose 1A and 2B.
that you would be wrong or stupid for using a different decision rule when you only get to play one time, than the rule you would use when you get to play 100 times.
Hello. I’m William. I am a thirty year-old undergraduate student in the University of Wisconsin—Madison’s Industrial and Systems Engineering department, with some additional study in Computer Science.
The study of logic and rational thought have always been hobbies of mine. My interest in mathematical optimization techniques has also been developing for decades, but this interest in these dark arts started taking steroids when I realized simple ways to apply the techniques to video games and Poker.
I originally stumbled upon this site two years ago, while Google searching various paradoxes. The Allais Paradox was the first post that I read here. I was outraged upon reading it. I checked a few other sources to see what they had to say about Allais. The worst was a video posted by some ponytail-having MBA, because of his poor choice of words in describing the paradox (e.g. “wrong” instead of “inconsistent”).
I have reread the Allais post about a dozen times since then. It no longer outrages me.
The interesting subject matter brought me here originally.
The intuitive and elegant explanations of the subject matter brought me back.
The shockingly high level of class and quality in comments and discussion has made this site one of my favorites.