A year or so back, on the verge of falling asleep, I thought up this question:
You have come to me ready to gamble. I have two envelopes on the table, one containing the amount of my bet, and one containing the probability I will win the bet (possibly in the form of some sports game result to check, etc. Something whose result is unknown to both of us, but whose odds have been estimated more or less exactly). There are no ties (the unknown probability is from a Bernoulli distribution). Clearly, I know what is inside both of these envelopes, but the only one I let you open is the one showing how much money I have bet, not the one containing the probability I win.
What is the optimal bet for you, given that you do not intend to refuse the bet?
If we play $n$ times, is there a winning strategy? (How about in the limit?)
I haven't really come up with a satisfying answer yet (if there is one), and I don't think I'll get around to it, so I'm sharing.
- The winner of the bet receives the full amount of the other person's bet (for that round, if the game is repeated)
- Both players have a finite amount of money, that has a minimum increment.
- In the infinite game version, players may choose to repeat the bet with new parameters (odds selected by me, your bet amount selected by you) until one player is bankrupted.