What would be the best strategy for the following game?
One of the two players starts by transferring $x$ points to the other player. The second player then either keeps the $x$ points or gives back $2x$ points. The game continues, each time doubling the amount of points, until one player decides to keep all the points. This is repeated until a limit of $n$ rounds is reached. The objective is to finish the game with the largest point difference possible.
The game would also be equivalent to "Two players each give a natural number, the lowest number, $a$, wins. The winner receives $2^a$ points."
How would the best strategy change if:
- There are more than two players
- The amount of points increases linearly instead of exponentially (each player gives back $x + 1$ points, or $a$ points are received instead of $2^a$ in the equivalent version of the game)
- Each player can choose how many points to give back (something other than $2x$)
- The game ends once a player reaches a certain number of points rather than a round limit
- In the equivalent game of naming a number $a$, the players can also name a positive real number $a$, rather than just a natural number