imagine the following scenario. You are competing in a game with $2$ other players, throwing a $21$-faced dice (labeled $1-21$). All three of you get to choose a number and then roll the dice. Whoever chose the number closest to the outcome wins. If all players tie, there is no winner. What is your strategy?
And an additional question: what if all three of you can communicate?
I know generally what to do with the case of just two players. But I'm a bit unsure when there are three. Could it be just a generalisation of the previous case? That is, if player $1$ chooses $x$, player $2$ chooses $y=x+1$ or $y=x-1$ then player $3$ chooses $z=x+2$ or $z=x-2$. I believe there was a similar problem posted here; I would appreciate a solution with perhaps extended, explicit calculations.
Thanks!