Mathematics of a Simple Counting Game I wonder how can one think mathematically about the following game:

People sit in a circle. One of them says "One!". Then somebody (no
matter who - he/she can even be the former person) says "Two!" and so
on. If two persons say the number in the same time, they will both be
elliminated. (This is a version of this game.)
The game ends either when counting reaches 20 or just one person has remained.
Everyone who says a number is given one point. The winner is the
person with the highest point.

 A: If we use continuous time, the strategy space could be very large and make the game intractable. So let's say the game takes place in discrete time: $t=1,2,3,\ldots$ and let $I$ represent the set of players.
Let the state vector include the current number (the last number called before round $t$ was reached), $n_t$, the list of players that were not eliminated yet $J\subset I$ and their current scores (number of points), $\mathbf{p}=(p_j)_{j\in J}$. A typical state or history is $h_t=(n_t,J,\mathbf(p))$
A (Markov) strategy for player $i$, is a mapping $s_i$ from the set of all possible values of the state vector to the interval $[0,1]$ which we interpret as saying  $s_i(h_t)$ is the probability that player $i$ calls at round $t$ when the last called number is $n_t$, the set of active players is $J$ and their respective scores are $\mathbf{p}$. 
It simplifies the analysis a lot if we restricted attention to Markov strategies (i.e. strategies that depend only on the state and not on the calendar time), so $s_i$ would be a function only on the state vector $h_t$ but not of $t$.
The question is not clear if being eliminated or finishing in 2nd  or 3rd is the same or not. Does a player only cares about winning or not? A player that is behind and has no chances of winning may eliminate the front-runner. Does a player prefers that no one else wins to the outcome someone else wins or he does not care? Do players care about finishing the game quickly or they don't care? Anyway, we need to make assumptions to have payoffs clearly defined.
Given a strategy for each player we can compute the probabilities the state vector is going to transition to its new value. If we now how players rank the final state, we can compute the expected payoffs.
Of course the game is still too complex for us to obtain a solution here. But one should be able to solve the sub-game that happens when only two players remain in the game and one has $10$ points while the other has $9$ points. Perhaps (or perhaps not) we may need to make the players impatient: winning at round $t$ is worth $\delta^t$ and losing is worth always zero, where $\delta\in(0,1)$. 
