You and an opponent are playing the following game:
- Your opponent picks two different real numbers, however he wants to, and writes them on two cards, which he presents to you face down
- You choose one of the cards, and look at its value
- You say "the card I picked is higher" or "the card I picked is lower"
- Your opponent reveals his card, and if you were correct, you score a point. Otherwise, your opponent scores a point.
Your aim is to pick a strategy for this game such that your probability of winning each point is greater than 50%. To make it more difficult, there is an extra rule:
- You must tell your opponent your strategy before he picks his numbers, and stick to that strategy. He keeps his owns strategy secret.
Your opponent is completely unlimited as to which numbers he picks, he can pick any real numbers, with any strategy he wants. Obviously this is a bit unrealistic, because some strategies would in reality be impossible to actually compute (e.g. if your opponent wanted to randomly pick two numbers from a flat probability distribution over all real numbers).
So to be clear, both you and your opponent have the magic ability to do arbitrarily precise mathematics in a short length of time, and are able to mentally generate random numbers from a probability distribution in a way that's not constrained by whether a real computer could do the same thing.
The main rule is: No cheating or loopholes. This is a mathematical logic puzzle, not an attempt to find a way around the rules. I'll try to close some of the more obvious ones here, but if any more come up that I haven't thought of, I'll add them here too.
- Any number can fit on a card, even if in reality it would take an infinite amount of space.
- Likewise, any number can be written onto or read from a card in a reasonable length of time, unrelated to how large that number is or how many digits it has.
- The "magical" power for mental maths you and your opponent have is just about being able to do calculations or generate numbers from probability distributions that wouldn't normally be possible either because of arbitrarily precise numbers or arbitrarily large numbers. You're not psychic, or anything similar. So, for example, your opponent can't have the strategy "I'm going to write 5 on this card if my opponent is thinking about a cat, otherwise I'm going to write 6.2"