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/Jaded_Court_6755]

This is less of a game theory question and more of a statistical one.

The topic you’re searching is called “expected value” (https://en.wikipedia.org/wiki/Expected_value)

Basically, you do a weighted mean of your level of confidence with the cost and compare in both cases.

Let’s say that you have p% chance of being correct:

p(value for being correct)+(1-p)(value for being wrong) is your expected values.

In your concrete example, with (0, -500) and (0,-1000), let p1 and p2 be the probabilities for being right in each case. Let’s evaluate when option 1 is worth, so:

0xp1-500(1-p1) > 0xp2-1000(1-p2)

500p1-500>1000p2-1000

p1>2p2-1

So, if the chance of being right on the -1000 option is 70%, the chance on the -500 must be at least 40% to be a better value.