egttools.analytical.StochDynamics¶

class StochDynamics(nb_strategies, payoffs, pop_size, group_size=2, mu=0)[source]¶

Bases: object

A class containing methods to calculate the stochastic evolutionary dynamics of a population.

Defines a class that contains methods to compute the stationary distribution for the limit of small mutation (only the monomorphic states) and the full transition matrix.

Parameters

nb_strategies (int) – number of strategies in the population
payoffs (numpy.ndarray[numpy.float64[m,m]]) –
Payoff matrix indicating the payoff of each strategy (rows) against each other (columns). When analyzing an N-player game (group_size > 2) the structure of the matrix is a bit more involved, and we can have 2 options for structuring the payoff matrix:

1) If we consider a simplified version of the system with a reduced Markov Chain which only contains the states at the edges of the simplex (the Small Mutation Limit - SML), then, we can assume that, at most, there will be 2 strategies in a group at any given moment. In this case, StochDynamics expects a square matrix of size nb_strategies x nb_strategies, in which each entry is a function that takes 2 positional arguments k and group_size, and an optional *args argument, and will return the expected payoff of the row strategy A in a group with k A strategists and group_size - k B strategists (the column strategy). For all the elements in the diagonal, only 1 strategy should be present in the group, thus, this function should always return the same value, i.e., the payoff of a row strategy when all individuals in the group adopt the same strategy. See below for an example.

2) If we want to consider the full Markov Chain composed of all possible states in the simplex, then the payoff matrix should be of the shape nb_strategies x nb_group_configurations, where the number of group configurations can be calculated using egttools.calculate_nb_states(group_size, nb_strategies). Moreover, the mapping between group configurations and integer indexes must be done using egttools.sample_simplex(index, group_size, nb_strategies). See below for an example
pop_size (int) – population size
group_size (int) – group size
mu (float) – mutation probability

Notes

We recommend that instead of`StochDynamics`, you use PairwiseComparison because the latter is implemented in C++, runs faster and supports more precise types.

Examples

Example of the payoff matrix for case 1) mu = 0:

>>> def get_payoff_a_vs_b(k, group_size, \*args):
...     pre_computed_payoffs = [4, 5, 2, ..., 4] # the size of this list should be group_size + 1
...     return pre_computed_payoffs[k]
>>> def get_payoff_b_vs_a(k, group_size, \*args):
...     pre_computed_payoffs = [0, 2, 1, ..., 0] # the size of this list should be group_size + 1
...     return pre_computed_payoffs[k]
>>> def get_payoff_a_vs_a(k, group_size, \*args):
...     pre_computed_payoffs = [1, 1, 1, ..., 1] # the size of this list should be group_size + 1
...     return pre_computed_payoffs[k]
>>> def get_payoff_b_vs_b(k, group_size, \*args):
...     pre_computed_payoffs = [0, 0, 0, ..., 0] # the size of this list should be group_size + 1
...     return pre_computed_payoffs[k]
>>> payoff_matrix = np.array([
...     [get_payoff_A_vs_A, get_payoff_A_vs_B],
...     [get_payoff_B_vs_A, get_payoff_B_vs_B]
...     ])

Example of payoff matrix for case 2) full markov chain (mu > 0):

>>> import egttools
>>> nb_group_combinations = egttools.calculate_nb_states(group_size, nb_strategies)
>>> payoff_matrix = np.zeros(shape=(nb_strategies, nb_group_combinations))
>>> for group_configuration_index in range(nb_group_combinations):
...     for strategy in range(nb_strategies):
...         group_configuration = egttools.sample_simplex(group_configuration_index, group_size, nb_strategies)
...         payoff_matrix[strategy, group_configuration_index] = get_payoff(strategy, group_configuration)

Methods

`calculate_full_transition_matrix`	Returns the full transition matrix in sparse representation.
`calculate_stationary_distribution`	Calculates the stationary distribution of the monomorphic states is mu = 0 (SML).
`fermi`	The fermi function determines the probability that the first type imitates the second.
`fitness_group`	In a population of x i-strategists and (pop_size-x) j strategists, where players interact in group of 'group_size' participants this function returns the average payoff of strategies i and j.
`fitness_pair`	Calculates the fitness of strategy i versus strategy j, in a population of x i-strategists and (pop_size-x) j strategists, considering a 2-player game.
`fixation_probability`	Function for calculating the fixation_probability probability of the invader in a population of residents.
`full_fitness_difference_group`	Calculate the fitness difference between strategies :param i and :param j assuming that player interacts in groups of size group_size > 2 (n-player games).
`full_fitness_difference_pairwise`	Calculates the fitness of strategy i in a population with state :param population_state, assuming pairwise interactions (2-player game).
`full_gradient_selection`	Calculates the gradient of selection for an invading strategy, given a population state.
`full_gradient_selection_without_mutation`	Calculates the gradient of selection for an invading strategy, given a population state.
`gradient_selection`	Calculates the gradient of selection given an invader and a resident strategy.
`prob_increase_decrease`	This function calculates for a given number of invaders the probability that the number increases or decreases with one.
`prob_increase_decrease_with_mutation`	This function calculates for a given number of invaders the probability that the number increases or decreases with taking into account a mutation rate.
`transition_and_fixation_matrix`	Calculates the transition matrix (only for the monomorphic states) and the fixation_probability probabilities.
`update_group_size`	Updates the groups size of the game (and the methods used to compute the fitness)
`update_payoffs`	Updates the payoff matrix
`update_population_size`	Updates the size of the population and the number of possible population states.

__init__(nb_strategies, payoffs, pop_size, group_size=2, mu=0)[source]¶

calculate_full_transition_matrix(beta, *args)[source]¶

Returns the full transition matrix in sparse representation.

Parameters

beta (float) – Intensity of selection.
args (Optional[list]) – Other arguments. Can be used to pass extra arguments to functions contained in the payoff matrix.

Returns

The full transition matrix between the two strategies in sparse format.

Return type

scipy.sparse.csr_matrix

calculate_stationary_distribution(beta, *args)[source]¶

Calculates the stationary distribution of the monomorphic states is mu = 0 (SML). Otherwise, it calculates the stationary distribution including all possible population states.

This function is recommended only for Hermitian transition matrices.

Parameters

beta (float) – intensity of selection.
args (Optional[list]) – extra arguments for calculating payoffs.

Returns

A vector containing the stationary distribution

Return type

numpy.ndarray

static fermi(beta, fitness_diff)[source]¶

The fermi function determines the probability that the first type imitates the second.

Parameters

beta (float) – intensity of selection
fitness_diff (float) – Difference in fitness between the strategies (f_a - f_b).

Returns

the probability of imitation

Return type

numpy.typing.ArrayLike

fitness_group(x, i, j, *args)[source]¶

In a population of x i-strategists and (pop_size-x) j strategists, where players interact in group of ‘group_size’ participants this function returns the average payoff of strategies i and j. This function expects that

\[x \in [1,pop_size-1]\]

Parameters

x (int) – number of individuals adopting strategy i in the population
i (int) – index of strategy i
j (int) – index of strategy j
args (Optional[list]) – Other Parameters. This can be used to pass extra parameters to functions stored in the payoff matrix

Return type

float

Returns

float
Returns the difference in fitness between strategy i and j

fitness_pair(x, i, j, *args)[source]¶

Calculates the fitness of strategy i versus strategy j, in a population of x i-strategists and (pop_size-x) j strategists, considering a 2-player game.

Parameters

x (int) – number of i-strategists in the population
i (int) – index of strategy i
j (int) – index of strategy j
args (Optional[list]) –

Return type

float

Returns

float
the fitness difference among the strategies

fixation_probability(invader, resident, beta, *args)[source]¶

Function for calculating the fixation_probability probability of the invader in a population of residents.

TODO: Requires more testing!

Parameters

invader (int) – index of the invading strategy
resident (int) – index of the resident strategy
beta (float) – intensity of selection
args (Optional[list]) – Other arguments. Can be used to pass extra arguments to functions contained in the payoff matrix.

Returns

The fixation_probability probability.

Return type

float