[API]: Naming and structure of enumpoly

[tturocy]

The method we currently call enumpoly is somewhat mis-named. It actually conflates several ideas.

At the core of the method is actually support enumeration - generating the full set of supports which have the property that there are no dominated strategies in the game projected to that support.

Given such a support, it is then possible to formulate the equations required for an equilibrium in which each player is indifferent between all of their strategies.

The "poly" in the name refers to "polynomial" - insofar as these equations are indeed polynomial equations (indeed for strategies and sequences they're multinomials). This reflected the intention to solve these equations using methods from e.g. algebraic geometry which exploit the fact that these are polynomials.

Except.... that's not actually what Gambit's implementation does. It is actually a branch-and-bound Newton solver - which does take some advantage of the fact that the system is formed of polynomials, but it is not based on ideas from algebraic geometry.

In contrast, the method also offers the ability to use external solvers (currently only PHCpack) which are truly "polynomial system solvers".

So these aren't actually the same thing (compared to, say, using Gambit's LP solver versus an external LP solver, which can be thought of more as different implementations solving the same problem formulation).

Gambit's implementation actually points out that there's nothing inherent about the "polynomial" aspect - support enumeration simply creates projections of games (possibly onto small supports) .

So, a case can be made that we need a better name and organisation of these methods, which separate out the concept of "support enumeration" (and express precisely what support enumeration does) versus what is actually done to find equilibria on the projected game (and to check whether the equilibria on the projected game extend to Nash equilibria on the full game).