Extend support enumeration to N-player games

[oyamad]

By using a nonlinear equation/complementarity problem solver or a polynomial equation solver (if any).

What packages are available in Julia (and in Python)?

[oyamad]

Well-known polynomial equation solvers (not in Julia):

• Bertini 2.0
• PHCpack

Some discussion: https://groups.google.com/forum/?fromgroups=#!topic/julia-users/HTbw39F0ejc

[oyamad]

Are these applicable?

• HomotopyContinuation.jl
• SemialgebraicSets.jl

[blegat]

The plan is to make SemialgebraicSets.jl independent on the algorithm/library used a bit like Polyhedra.jl. In Polyhedra, you build your polyhedron with polyhedron(::HRepresentation, ::somelibrary). In SemialgebraicSets, you will do algebraicset(::Vector{<:AbstractPolynomial}, ::somelibrary). There is a solver implemented in pure Julia in SemialgebraicSets but we are planning to also make HomotopyContinuation an option. There is also this solver: https://github.com/wbhart/Singular.jl which works in exact arithmetic. Ideally it would be nice if it could be used from the algebraicset function. We could even make it so that the default library if the polynomials are Nemo polynomials then the default library is Singular.

It would of course also be nice to add wrappers to C/C++ libraries like Bertini to be used through the same interface :-P

[oyamad]

@blegat Thanks. Sounds great!

[oyamad]

https://github.com/QuantEcon/GameTheory.jl/pull/174#issuecomment-1367809331