Source code for quantecon.game_theory.support_enumeration

"""
Compute all mixed Nash equilibria of a 2-player (non-degenerate) normal
form game by support enumeration.

References
----------
B. von Stengel, "Equilibrium Computation for Two-Player Games in
Strategic and Extensive Form," Chapter 3, N. Nisan, T. Roughgarden, E.
Tardos, and V. Vazirani eds., Algorithmic Game Theory, 2007.

"""
import numpy as np
from numba import jit
from ..util.numba import _numba_linalg_solve
from ..util.combinatorics import next_k_array


[docs]def support_enumeration(g): """ Compute mixed-action Nash equilibria with equal support size for a 2-player normal form game by support enumeration. For a non-degenerate game input, these are all the Nash equilibria. The algorithm checks all the equal-size support pairs; if the players have the same number n of actions, there are 2n choose n minus 1 such pairs. This should thus be used only for small games. Parameters ---------- g : NormalFormGame NormalFormGame instance with 2 players. Returns ------- list(tuple(ndarray(float, ndim=1))) List containing tuples of Nash equilibrium mixed actions. """ return list(support_enumeration_gen(g))
[docs]def support_enumeration_gen(g): """ Generator version of `support_enumeration`. Parameters ---------- g : NormalFormGame NormalFormGame instance with 2 players. Yields ------- tuple(ndarray(float, ndim=1)) Tuple of Nash equilibrium mixed actions. """ try: N = g.N except: raise TypeError('input must be a 2-player NormalFormGame') if N != 2: raise NotImplementedError('Implemented only for 2-player games') return _support_enumeration_gen(g.payoff_arrays)
@jit(nopython=True) # cache=True raises _pickle.PicklingError def _support_enumeration_gen(payoff_matrices): """ Main body of `support_enumeration_gen`. Parameters ---------- payoff_matrices : tuple(ndarray(float, ndim=2)) Tuple of payoff matrices, of shapes (m, n) and (n, m), respectively. Yields ------ out : tuple(ndarray(float, ndim=1)) Tuple of Nash equilibrium mixed actions, of lengths m and n, respectively. """ nums_actions = payoff_matrices[0].shape n_min = min(nums_actions) for k in range(1, n_min+1): supps = (np.arange(0, k, 1, np.int_), np.empty(k, np.int_)) actions = (np.empty(k+1), np.empty(k+1)) A = np.empty((k+1, k+1)) while supps[0][-1] < nums_actions[0]: supps[1][:] = np.arange(k) while supps[1][-1] < nums_actions[1]: if _indiff_mixed_action( payoff_matrices[0], supps[0], supps[1], A, actions[1] ): if _indiff_mixed_action( payoff_matrices[1], supps[1], supps[0], A, actions[0] ): out = (np.zeros(nums_actions[0]), np.zeros(nums_actions[1])) for p, (supp, action) in enumerate(zip(supps, actions)): out[p][supp] = action[:-1] yield out next_k_array(supps[1]) next_k_array(supps[0]) @jit(nopython=True, cache=True) def _indiff_mixed_action(payoff_matrix, own_supp, opp_supp, A, out): """ Given a player's payoff matrix `payoff_matrix`, an array `own_supp` of this player's actions, and an array `opp_supp` of the opponent's actions, each of length k, compute the opponent's mixed action whose support equals `opp_supp` and for which the player is indifferent among the actions in `own_supp`, if any such exists. Return `True` if such a mixed action exists and actions in `own_supp` are indeed best responses to it, in which case the outcome is stored in `out`; `False` otherwise. Array `A` is used in intermediate steps. Parameters ---------- payoff_matrix : ndarray(ndim=2) The player's payoff matrix, of shape (m, n). own_supp : ndarray(int, ndim=1) Array containing the player's action indices, of length k. opp_supp : ndarray(int, ndim=1) Array containing the opponent's action indices, of length k. A : ndarray(float, ndim=2) Array used in intermediate steps, of shape (k+1, k+1). out : ndarray(float, ndim=1) Array of length k+1 to store the k nonzero values of the desired mixed action in `out[:-1]` (and the payoff value in `out[-1]`). Returns ------- bool `True` if a desired mixed action exists and `False` otherwise. """ m = payoff_matrix.shape[0] k = len(own_supp) for i in range(k): for j in range(k): A[j, i] = payoff_matrix[own_supp[i], opp_supp[j]] # transpose A[:-1, -1] = 1 A[-1, :-1] = -1 A[-1, -1] = 0 out[:-1] = 0 out[-1] = 1 r = _numba_linalg_solve(A, out) if r != 0: # A: singular return False for i in range(k): if out[i] <= 0: return False val = out[-1] if k == m: return True own_supp_flags = np.zeros(m, np.bool_) own_supp_flags[own_supp] = True for i in range(m): if not own_supp_flags[i]: payoff = 0 for j in range(k): payoff += payoff_matrix[i, opp_supp[j]] * out[j] if payoff > val: return False return True