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 outcomex occurs within the gamble, and U(x) is the utility of outcomex, a function that must be strictly increasing with respect to the monetary value of outcomex.
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.
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
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.
That all seems pretty uncontroversial.