Improving computation for hypergraphical game
When solving a hypergraphical game, it is possible for a given level k that an active player is not playing with any other players. When that is the case it means that the choice of the player isolated from the rest does not have any influence, even indirectly on the other players.
Given that we can considered the his variable x are not necessary for the polynomials of the other players, giving us the opportunity to have a simpler polynomials to solve. Considering that we can also considered this players as if he was at the level 0, we need only his best response, solving a pcp is not need so we can also consider "removing" or ignoring his variables from the pcp ("removing" them might avoid issue with dimension of the ideal).
Similarly, if a group of players are isolated from another, theirs variables are need only in the polynomials of the group they interact with (directly and inderectly).
Ignoring the arrow direction if I were to have the following graph
graph TD;
A-->B;
C;
The player C is isolated, meaning that his variables are not in the polynomials of A and B, and he could potentially be considered as a fixed player depending on his best response (if not degenerate)
graph TD;
A-->B;
B-->E;
C-->D;
There the polynomials of A include the variables of E, he interact with him indirectly. The polynomials of A,B and E do not include the variable of either C or D, they do not interact with each other, even indirectly.
graph TD;
A-->B;
C-->D;
A-->C;
B-->E;
There all the player interact with each other, directly or indirectly
A new implementation, using subgame, should take that in consideration when trying to create and solve a sub PCP of an hypergraphical game. It is also possible to consider "reordering" the players so that computation are keep independent as long as possible