Let's move to the formal analysis of the game.
First of all, I will describe the game form (see:
https://www.matematicamente.it/f/viewtopic.php?t=20428, sorry for the Italian...)
We have six players. The strategy space for each one is ${0,1,2,\ldots,20}$.
For sake of convenience, I will denote by $X_i$ the strategy space of player $i$. Of course, all of the $X_i$ coincide with ${0,1,2,\ldots,20}$.
As $E$ we can take $RR^6$, even if we could make more parsimonious choices.
The function $h$ is defined as follows. Since $h$ take values in $RR^6$, it will be convenient to describe it as $(h_1,\ldots,h_6)$, where $h_i$ are real-valued functions.
We have:
$h_i(x_1,\ldots,x_6) = 20 - x_i + \frac{1}{2} \cdot \sum_{j=1}^6 x_j$.
Notice that the game form is completely independent of any assumption about the preferences of the players. So, in particular, it is independent of what I said in my previous post about the "meaning" of the game we are discussing.
In the next post I will discuss the preferences of the players, so that we shall be able to say something more interesting, perhaps.
