The following market concerns a paradox in which it is possible to guess whether a randomly selected number out of two distinct secret numbers is the larger one or not, with >50% probability, no matter what the two secret numbers are.
If 29 is in fact the larger number, this market resolves YES if the average probability of the linked market is >50% and NO otherwise. If 29 is the smaller number, it resolves YES if the linked market's average probability is <50% and NO otherwise. I will use exact probabilities from the API.
I will not bet in this market, since my bets could reveal information about the secret numbers that I am not allowed to hint at.
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@JosephNoonan If the way to win is to get in between the numbers, then one should pick their random number in the range of something like 28.9 - 29.1. Sounds reasonable to say that if you pick small enough range, you have close to 50% chance to get in between the numbers. So you’ll have 50% chance to be 100% correct + a 50% chance to be 50% correct. So something like 2/3 sounds alright.
But I think there’s a error in the reasoning in the video that describes the paradox, and the strategy doesn’t work.
@Roma You have to pick your number before the first number is shown to you. If you make your number depend on the revealed first number, the reasoning doesn't work anymore.
@FlorisvanDoorn The error I think of still there: when you don’t get in between, and then answer according to the strategy, the probability of getting it right is lower than 50%. Because you’re betting on the hidden number getting in between of your number and the shown number, and that range is smaller than the remaining range it could be in.
But I’m probably wrong, if this is a real paradox :)
@Roma Nah the 50% comes from the game master choosing one of the two, not from your actions.
If you pick 80, then the game master draws the numbers 2 and 3, then if he randomly (50%) decides to show "2" then you win, and if "3" you lose.
@Mqrius Hm, right, I think game can be re-defined to make it more obvious:
GM chooses 2 numbers, and doesn’t show you any.
You get a chance to guess a number between the two. If you guessed it, game ends, you won.
If you didn’t guess, GM flips a coin to determine whether you win.
Did I miss something, is this the same game?
@FlorisvanDoorn You don't have to pick it before the first number is shown to you, but you do have to pick it independently of the first number. I wonder if seeing the first number does have some effect on what probability distribution people using the strategy will use.
@Roma And yes, it should be noted that the probability of guessing right is only >50% before I choose which secret number to reveal first, even if you don't choose your random number until after that point. And that is for exactly the reason above: If your randomly chosen number isn't between the two secret numbers, then you have a 50-50 shot at being right, since the correctness of your guess depends on which number is chosen, and both have a 50% chance of being chosen. But if your random number is between them, you are right no matter what.
So depending on what distribution your using, you may or may not have a >50% chance of being right (given the value that the other number has) now that I've revealed one number.
@Roma This is equivalent if you use the Cover's paradox strategy. It's not totally equivalent because some people might just not use the strategy.