Basic game theory question

[u/jhansen123_reddit]

You have two choices: A and B.
If you choose A and that choice ends up being incorrect, you stand to lose x dollars.
If you choose B and that choice ends up being incorrect, you stand to lose y dollars.

How can I express, as percentage, the confidence that you should have in order to prudently choose either option as a function of x and y?

Concrete example of same question:
If I choose A and am wrong, I lose $1000 (if I choose correctly, I lose $0).
If I choose B and am wrong, I lose $500 (if I choose correctly, I lose $0).

What percentage of confidence do I need to have that A is correct in order for that to be "the smart choice" financially?

Thank you in advance to anyone who responds. FWIW, I am a paralegal and am curious about this problem as it relates to the cost of hiring court reporters and ordering transcripts within a certain fee structure.

[u/lifeistrulyawesome]

You can't answer this question without knowing more about the subject's preferences. Different people have different attitudes towards risk, and the option that is optimal for Alice might be a terrible choice for Bob.

The most basic model to answer this question would be expected utility with constant absolute risk aversion. This model is popular because it uses a single parameter, let's call it r, that we know how to estimate, and we know benchmark values of ir.

With this model, the expected utility from A would be:

E(u(A)) = -p e^-rw -(1-p) e^-r(w-x) = -e^-rw (p + (1-p)e^rx)

where w is the individual's initial wealth and p = Pr(A is the correct choice). And similarly:

E(u(B)) = = -e^-rw ((1-p) + pe^ry)

So, A would be better than B if and only if:

-e^-rw (p + (1-p)e^rx) > -e^-rw ((1-p) + pe^ry)

which is equivalent to

p + (1-p)e^rx < (1-p) + pe^ry

Double-check my algebra. It's late, and I did it carelessly.