I encountered the following 2-player game in a TV show and am very curious if there is an optimal strategy. Suppose there are two players, A and B. They first choose an even number between 0 and 30. The judge then calculate the average, say $a$, of this two number. After that, two players will take turn to speak out some numbers. They need to follow the rules below:
- whoever choose to start first need to begin with 1, and one needs to start from the number next to the last number spoken out by the previous player.
- each turn, the numbers a player speaks out have to be in increasing order, with one adjacent to another.
- each turn, at least one number needs to be spoke out.
For example, player A begin by speaking out '1,2,3', player B then speaks out '4,5,6,7,8,9', then player A will start from 10 and continue... Once the number $a$ is spoke out, the game ends, and the player who speaks out number $a$ loses.
If you are playing this simple game with someone else, what number will you pick? Will you start to speak first or not? And What would be your strategy? Is there an optimal strategy(maximizing the probability of winning)?
I am not very sure how to define a strategy in this game. It could be a triple $(x,b,f)$, where $x$ is the number you choose, $b$ indicates whether to start first or not, and $f$ is a multi-function such that $f(n)$ represent the numbers one will speak out given the last number spoke out by your opponent is $n$. But this can of course be over-simplified.
Any comment are welcomed!
I am assuming the numbers are picked by the players and do not include $0$ or $30$ because if both players picked $0$ the game will not end. A naive assumption is that you should never call more than one number, as that just gives you more chances to lose. At the start you can know that you will not lose because if you chose $20$ you know the average is at least $11$, but you can worry about giving your opponent information about what you picked. If we assume that each player will only say one number each turn, all that matters is whether the numbers match $\bmod 4$ and who starts. If the numbers match $\bmod 4$ the average will be even and you want to start. If the numbers do not match $\bmod 4$ the average will be odd and you do not want to start. If you think your opponent will chose numbers that are $0 \bmod 4$ half the time you are guessing.
I think if I knew nothing about the opponent I would assume s/he is more likely to choose a number that is $0 \bmod 4$ because they are "rounder". I would then choose a number that is also $0 \bmod 4$ and try to start with $1$. I have little enough faith in the advantage of this strategy that I would not name two numbers to restore the parity if my opponent did so.