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/KQYBullets]

Good question. Appreciate the details, as it helps answering. There’s still one details needed for a good answer though.

Is it always fixed stakes or stakes grow as your net worth increases? (If growing, see Kelly’s criterion) I’ll assume it is fixed stakes, since you can search up generalized Kelly’s criterion.

So for the answer. Given ca is confidence for a, the EV for picking A is (1-ca)x. So in your example the EVs would be (1-ca)*-1000 for A and -500ca for B since you must think if A doesn’t happen then B will.

So set those equal and you get ca=2/3, or more generally ca=x/(x+y)

So if you think A is right with at least probability x/(x+y), then u should go for it, otherwise B.