FracESSA

Fractional ESS Analyzer

FracESSA finds evolutionarily stable strategies (ESS) in symmetric games with rational payoffs. Equivalently, it finds the strict local maximizers of a rational quadratic form over the probability simplex. It provides a C++ command-line program and a Python API.

FracESSA logo showing a simplex in three-dimensional coordinates

Mathematical problem

FracESSA analyzes the standard quadratic optimization problem

\[\max_{x\in\Delta^n} x^\mathsf{T} A x,\]

where \(A\) is a symmetric matrix with rational entries and \(\Delta^n\) is the probability simplex. In the corresponding symmetric partnership game, the ESS are exactly the strict local maximizers of this quadratic form.

Numerical contract

Matrix entries, returned probability vectors, payoffs, candidate decisions in safe mode, and every final stability decision are exact. FracESSA offers two explicitly selected search methods:

Method

Intended use

Guarantee

safe

Complete correctness-first analysis

Uses exact arithmetic for every candidate and stability decision.

fast

Faster exploratory analysis

Uses floating-point candidate filtering followed by exact checks. It can miss candidates and is not a completeness certificate.

The nonempty support space can contain 2^n - 1 supports. FracESSA generates supports one at a time, prunes strict supersets of exact equilibria, and exploits circular symmetry when present. Candidate output is therefore an intermediate record of the ESS search, not a census of every algebraic Nash equilibrium. One matrix uses one CPU core; Python can analyze independent matrices in parallel.

Quick start

Install the Python package:

python -m pip install pyfracessa

Run one exact analysis:

from pyfracessa import Matrix, RunConfig, run

result = run(
    "safe",
    Matrix(1, "3#-1,0,0,-2,0,-3"),
    config=RunConfig(include_candidates=True),
)

print(result["ess_count"])
print(result["candidates"])

Prebuilt CLI binaries are available from the GitHub releases page. Continue with Getting Started for matrix encoding, command-line use, result interpretation, and multiprocessing.

Documentation

Documentation