How to analyze this guessing game? I sometimes play a guessing game with a friend. So fx we might try to guess how many people live in Russia. So person A says 100 milion and then person B says either 'more' or 'less'. If B says 'more' and in fact more than 100 milion live in Russia, then B wins and conversely. Is this game fair? I have an intuition that B probably has an advantage, but I'm not sure how to formally analyse the situation.
This is an interesting question!

It would seem that the game is not fair: Player A is at a pretty serious disadvantage.

Let's imagine each player has their own probability distribution of beliefs that is a sampling distribution of some population distribution centered at the true value. Both players are aware of this, and know their own distribution but have no further information on the other's belief distribution.

The optimal strategies are for player A to guess at the center of their own distribution and for B to guess "More" if the center of B's distribution is above A's guess and "Less" if it's below.

Player A will win if their guess is between the true value and Player B's guess.

-Since there is a 50/50 shot for either guess to be above or below the true value, 50% of the time the guesses will lie on either side of the true value and player B will win, regardless of whose guess is closer.

Via symmetry:

-25% of the time their guesses will be on the same side of the true value and player B's will be closer. Player B will also win here (deservedly).

-25% of the time their guesses will be on the same side of the true value and player A's will be closer. Here player A will win (deservedly).

Hence, player A will win only 1/4 of the time.

However the game also becomes increasingly complicated once you allow each player to have first order beliefs about the other player's beliefs (ie. specific information about the shape and center of the other's belief distribution), because then there's the opportunity for say, A to exploit their belief that B's guess is below the true value by guessing slightly lower than they would otherwise. And with second-order beliefs things get even more complex, and so on and so on. I'm not sure whether player A's disadvantage is robust to these nuances, but it's still quite striking to me that a game that at first glance seems fair is actually under normal conditions pretty lopsided!