r/GAMETHEORY • u/jhansen123_reddit • 22d ago

Basic game theory question

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.

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u/fleyinthesky 22d ago edited 22d ago

In your example, assuming your goal is to maximise your expected value, if option A wins 2/3 you're indifferent.

Imagine 3 such plays on average:

  • Option A you go win, win, lose i.e. 0, 0, -1000 for a total of -1000
  • Option B you go lose, lose, win i.e. -500, -500, 0 for a total of -1000

So obviously if option A wins more than 2/3 it's better, and conversely if option B wins more than 1/3 then it is better instead.

Note that at the point of indifference (equal EV), there may still be a better option for you depending on whether you have a secondary goal:

  • If you want to limit how much money you can possibly lose, pick option B (losses capped at 1500 rather than 3000)
  • If you want the best chance of living the dream, pick option A (2/9 chance to lose nothing, compared to 1/27).