egttools.analytical.utils.find_roots_and_stability

find_roots_and_stability(gradient_function, nb_strategies, nb_initial_random_points=3, atol=1e-07, atol_neg=0.0001, atol_pos=0.0001, atol_zero=0.0001, tol_close_points=0.0001, method='hybr')[source]

Searches for the roots of the differential equation gradient_function and calculates the stability based on an estimate of the Jacobian. This estimate is often imprecise which leads to wrong results.

Parameters

  • gradient_function (Callable[[np.ndarray], np.ndarray]) – function that returns a numpy.ndarray with the gradient of every strategy/type given a current population state.

  • nb_strategies (int) – number of strategies/types present in the population.

  • nb_initial_random_points (int) – number of random points to use as initial states for the root function. These are additional to the vertex of the simplex.

  • atol (float) – tolerance for considering that a point is in the simplex.

  • atol_neg (float) – tolerance to consider a value negative.

  • atol_pos (float) – tolerance to consider a value positive.

  • atol_zero (float) – tolerance to determine if a value is zero.

  • tol_close_points (float) – tolerance for considering that two points are equal.

  • method (str) – one of the options described in scipy.optimize.root (see https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html)

Returns

A tuple containing the list of roots and a list with 1 indicating stable points, 0 saddle points and -1 unstable points.

Return type

Tuple[List[np.array], List[int]]