Source code for nashpy.game

"""A class for a normal form game"""

import numpy as np
import numpy.typing as npt
from typing import Optional, Any
from .algorithms.lemke_howson import lemke_howson
from .algorithms.support_enumeration import support_enumeration
from .algorithms.vertex_enumeration import vertex_enumeration
from .linalg.minimax import linear_program
from .egt.moran_process import moran_process, fixation_probabilities
from .learning.fictitious_play import fictitious_play
from .learning.replicator_dynamics import (
    asymmetric_replicator_dynamics,
    replicator_dynamics,
)
from .learning.stochastic_fictitious_play import stochastic_fictitious_play
from .utils.is_best_response import is_best_response


[docs]class Game: """ A class for a normal form game. Parameters ---------- - A, B: 2 dimensional list/arrays representing the payoff matrices for non zero sum games. - A: 2 dimensional list/array representing the payoff matrix for a zero sum game. """ def __init__(self, *args: Any) -> None: if len(args) == 2: if (not len(args[0]) == len(args[1])) or ( not len(args[0][0]) == len(args[1][0]) ): raise ValueError("Unequal dimensions for matrices A and B") self.payoff_matrices = tuple([np.asarray(m) for m in args]) if len(args) == 1: self.payoff_matrices = np.asarray(args[0]), -np.asarray(args[0]) self.zero_sum = np.array_equal( self.payoff_matrices[0], -self.payoff_matrices[1] ) def __repr__(self) -> str: if self.zero_sum: tpe = "Zero sum" else: tpe = "Bi matrix" return """{} game with payoff matrices: Row player: {} Column player: {}""".format( tpe, *self.payoff_matrices ) def __getitem__(self, key: Any) -> npt.NDArray: row_strategy, column_strategy = key return np.array( [ np.dot(row_strategy, np.dot(m, column_strategy)) for m in self.payoff_matrices ] )
[docs] def vertex_enumeration(self): """ Obtain the Nash equilibria using enumeration of the vertices of the best response polytopes. Algorithm implemented here is Algorithm 3.5 of [Nisan2007]_. 1. Build best responses polytopes of both players 2. For each vertex pair of both polytopes 3. Check if pair is fully labelled 4. Return the normalised pair Returns ------- generator The equilibria. """ return vertex_enumeration(*self.payoff_matrices)
[docs] def support_enumeration(self, non_degenerate=False, tol=10**-16): """ Obtain the Nash equilibria using support enumeration. Algorithm implemented here is Algorithm 3.4 of [Nisan2007]_. 1. For each k in 1...min(size of strategy sets) 2. For each I,J supports of size k 3. Solve indifference conditions 4. Check that have Nash Equilibrium. Parameters ---------- non_degenerate : bool Whether or not to consider supports of equal size. By default (False) only considers supports of equal size. tol : float A tolerance parameter for equality. Returns ------- generator The equilibria. """ return support_enumeration( *self.payoff_matrices, non_degenerate=non_degenerate, tol=tol )
[docs] def lemke_howson_enumeration(self): """ Obtain Nash equilibria for all possible starting dropped labels using the lemke howson algorithm. See `Game.lemke_howson` for more information. Note: this is not guaranteed to find all equilibria. Yields ------ Tuple An equilibria """ for label in range(sum(self.payoff_matrices[0].shape)): yield self.lemke_howson(initial_dropped_label=label)
[docs] def lemke_howson(self, initial_dropped_label): """ Obtain the Nash equilibria using the Lemke Howson algorithm implemented using integer pivoting. Algorithm implemented here is Algorithm 3.6 of [Nisan2007]_. 1. Start at the artificial equilibrium (which is fully labeled) 2. Choose an initial label to drop and move in the polytope for which the vertex has that label to the edge that does not share that label. (This is implemented using integer pivoting) 3. A label will now be duplicated in the other polytope, drop it in a similar way. 4. Repeat steps 2 and 3 until have Nash Equilibrium. Parameters ---------- initial_dropped_label: int The initial dropped label. Returns ------- Tuple An equilibria """ return lemke_howson( *self.payoff_matrices, initial_dropped_label=initial_dropped_label )
[docs] def fictitious_play(self, iterations, play_counts=None): """ Return a given sequence of actions through fictitious play. The implementation corresponds to the description of chapter 2 of [Fudenberg1998]_. 1. Players have a belief of the strategy of the other player: a vector representing the number of times the player has chosen a given strategy. 2. Players choose a best response to the belief. 3. Players update their belief based on the latest choice of the opponent. Parameters ---------- iterations : int The number of iterations of the algorithm. play_counts : array The play counts. Returns ------- Generator The play counts """ return fictitious_play( *self.payoff_matrices, iterations=iterations, play_counts=play_counts )
[docs] def stochastic_fictitious_play( self, iterations, play_counts=None, etha=10**-1, epsilon_bar=10**-2 ): """Return a given sequence of actions and mixed strategies through stochastic fictitious play. The implementation corresponds to the description given in [Hofbauer2002]_. Parameters ---------- iterations : int The number of iterations of the algorithm. play_counts : array The play counts. etha : float The noise parameter for the logit choice function. epsilon_bar : float The maximum stochastic perturbation. Returns ------- Generator The play counts """ return stochastic_fictitious_play( *self.payoff_matrices, iterations=iterations, play_counts=play_counts, etha=etha, epsilon_bar=epsilon_bar )
[docs] def replicator_dynamics(self, y0=None, timepoints=None, mutation_matrix=None): """ Implement replicator dynamics Return an array showing probability of each strategy being played over time. The total population is constant. Strategies can either stay constant if equilibria is achieved, replicate or die. Parameters ---------- y0 : array The initial population distribution. timepoints: array The iterable of timepoints. mutation_matrix : array The mutation rate matrix. Element [i, j] gives the probability of an individual of type i mutating to an individual of type j. Default behaviour is to be the identity matrix which corresponds to no mutation. Returns ------- array The population distributions over time. """ A, _ = self.payoff_matrices return replicator_dynamics( A=A, y0=y0, timepoints=timepoints, mutation_matrix=mutation_matrix )
[docs] def asymmetric_replicator_dynamics(self, x0=None, y0=None, timepoints=None): """ Returns two arrays, corresponding to the two players, showing the probability of each strategy being played over time using the asymmetric replicator dynamics algorithm. Parameters ---------- x0 : array The initial population distribution of the row player. y0 : array The initial population distribution of the column player. timepoints: array The iterable of timepoints. Returns ------- Tuple The 2 population distributions over time. """ A, B = self.payoff_matrices return asymmetric_replicator_dynamics( A=A, B=B, x0=x0, y0=y0, timepoints=timepoints )
[docs] def is_best_response(self, sigma_r, sigma_c): """ Checks if sigma_r is a best response to sigma_c and vice versa. Parameters ---------- sigma_r : array The row player strategy sigma_c : array The column player strategy Returns ------- tuple A pair of booleans, the first indicates if sigma_r is a best response to sigma_c. The second indicates if sigma_c is a best response to sigma_r. """ A, B = self.payoff_matrices is_row_strategy_best_response = is_best_response( A=A, sigma_c=sigma_c, sigma_r=sigma_r, ) is_column_strategy_best_response = is_best_response( A=B.T, sigma_c=sigma_r, sigma_r=sigma_c, ) return (is_row_strategy_best_response, is_column_strategy_best_response)
[docs] def moran_process( self, initial_population, mutation_probability=0, replacement_stochastic_matrix: Optional[npt.NDArray] = None, interaction_graph_adjacency_matrix: Optional[npt.NDArray] = None, ): """ Return a generator of population across the Moran process. The last population is when only a single type of individual is present in the population. Parameters ---------- initial_population : array the initial population mutation_probability : float the probability of an individual selected to be copied mutates to another individual from the original set of strategies (even if they are no longer present in the population). replacement_stochastic_matrix: array Individual i chosen for replacement will replace individual j with probability P_{ij}. Default is None: this is equivalent to P_{ij} = 1 / N for all i, j. interaction_graph_adjacency_matrix : array the adjacency matrix for the interaction graph G: individuals of type i interact with individuals of type j count towards fitness iff G_{ij} = 1. Default is None: if so a complete graph is used -- this corresponds to all individuals interacting with each other (with no self interactions) Returns ------- Generator The generations. """ A, _ = self.payoff_matrices return moran_process( A=A, initial_population=initial_population, mutation_probability=mutation_probability, interaction_graph_adjacency_matrix=interaction_graph_adjacency_matrix, replacement_stochastic_matrix=replacement_stochastic_matrix, )
[docs] def fixation_probabilities( self, initial_population, repetitions, replacement_stochastic_matrix: Optional[npt.NDArray] = None, interaction_graph_adjacency_matrix: Optional[npt.NDArray] = None, ): """ Return the fixation probabilities for all types of individuals. The returned array will have the same dimension as the number of rows or columns as the payoff matrix A. The ith element of the returned array corresponds to the probability that the ith strategy becomes fixed given the initial population. This is a stochastic algorithm and the probabilities are estimated over a number of repetitions. Parameters ---------- initial_population : array the initial population repetitions : int The number of iterations of the algorithm. replacement_stochastic_matrix: array Individual i chosen for replacement will replace individual j with probability P_{ij}. Default is None: this is equivalent to P_{ij} = 1 / N for all i, j. interaction_graph_adjacency_matrix : array the adjacency matrix for the interaction graph G: individuals of type i interact with individuals of type j count towards fitness iff G_{ij} = 1. Default is None: if so a complete graph is used -- this corresponds to all individuals interacting with each other (with no self interactions) Returns ------- array The fixation probability of each type. """ A, _ = self.payoff_matrices return fixation_probabilities( A=A, initial_population=initial_population, repetitions=repetitions, interaction_graph_adjacency_matrix=interaction_graph_adjacency_matrix, replacement_stochastic_matrix=replacement_stochastic_matrix, )
[docs] def linear_program(self): """ Returns the Nash Equilibrium for a zero sum game by solving the Linear Program that corresponds to the minimax theorem. Returns ------- tuple The Nash equilibria Raises ------ ValueError A value error is raised if the game is not zero sum """ if self.zero_sum is False: raise ValueError( "The Linear Program corresponding to the minimax theorem is defined only for Zero Sum games." ) A, B = self.payoff_matrices row_strategy = linear_program(row_player_payoff_matrix=A) column_strategy = linear_program(row_player_payoff_matrix=B.T) return row_strategy, column_strategy