Use symbolic payoffs for a 2-player game

general_four_state_fitness_function returns fully symbolic payoffs for a 2-player, 2-action game. You can build and simplify the transition and absorption matrices algebraically, then substitute specific payoffs at the end.

The four states and their symbolic labels are:

State	Label
`[0, 0]`	\(a\)
`[0, 1]`	\(b\)
`[1, 0]`	\(c\)
`[1, 1]`	\(d\)

Player \(i\)'s fitness in state \(x\) is written \(f_i(x)\).

Evaluate the fitness function

>>> import ludics.fitness_functions
>>> import numpy as np

>>> ludics.fitness_functions.general_four_state_fitness_function(
...     np.array([0, 1])
... )
array([f_1(b), f_2(b)], dtype=object)

>>> ludics.fitness_functions.general_four_state_fitness_function(
...     np.array([1, 0])
... )
array([f_1(c), f_2(c)], dtype=object)

Build a symbolic transition matrix

>>> import ludics
>>> import ludics.fitness_functions
>>> import numpy as np

>>> state_space = ludics.get_state_space(N=2, k=2)
>>> transition_matrix = ludics.generate_transition_matrix(
...     state_space=state_space,
...     fitness_function=ludics.fitness_functions.general_four_state_fitness_function,
...     compute_transition_probability=ludics.compute_moran_transition_probability,
...     selection_intensity=1,
... )

Compute and simplify the symbolic absorption matrix

Use calculate_absorption_matrix for symbolic matrices, then sympy.cancel to reduce the expressions:

>>> import sympy as sym

>>> absorption = ludics.calculate_absorption_matrix(transition_matrix)
>>> sym.cancel(absorption)
Matrix([
[(1.0*f_1(b) + 1.0)/(1.0*f_1(b) + 1.0*f_2(b) + 2.0), (1.0*f_2(b) + 1.0)/(1.0*f_1(b) + 1.0*f_2(b) + 2.0)],
[(1.0*f_2(c) + 1.0)/(1.0*f_1(c) + 1.0*f_2(c) + 2.0), (1.0*f_1(c) + 1.0)/(1.0*f_1(c) + 1.0*f_2(c) + 2.0)]])

The columns correspond to fixation in [0, 0] and [1, 1] respectively. This is the standard Moran fixation probability formula for a 2-player game. From state \(b = [0, 1]\), the fixation probability into all-zeros is:

\[\rho_{b \to [0,0]} = \frac{f_1(b) + 1}{f_1(b) + f_2(b) + 2}\]

The fixation probability is proportional to the fitness of the invading strategy relative to the total fitness in the mixed state.

Substitute specific payoffs

To evaluate for a particular game, substitute numerical values for the symbolic fitness terms. For a Prisoner's Dilemma with \(T=5\), \(S=0\):

State \(b = [0, 1]\): player 0 defects, player 1 cooperates, so \(f_1(b) = T = 5\), \(f_2(b) = S = 0\)
State \(c = [1, 0]\): player 0 cooperates, player 1 defects, so \(f_1(c) = S = 0\), \(f_2(c) = T = 5\)

>>> simplified = sym.cancel(absorption)
>>> subs = {
...     sym.Function("f_1")(sym.Symbol("b")): 5,
...     sym.Function("f_2")(sym.Symbol("b")): 0,
...     sym.Function("f_1")(sym.Symbol("c")): 0,
...     sym.Function("f_2")(sym.Symbol("c")): 5,
... }
>>> simplified.subs(subs)
Matrix([
[0.857142857142857, 0.142857142857143],
[0.857142857142857, 0.142857142857143]])

Both transient states fix on defection with probability ~86%, as expected when defection is the dominant strategy.