Source code for quantecon.game_theory.game_generators.bimatrix_generators

"""
This module contains functions that generate NormalFormGame instances of
the 2-player games studied by Fearnley, Igwe, and Savani (2015):

* Colonel Blotto Games (`blotto_game`): A non-zero sum extension of the
  Blotto game as studied by Hortala-Vallve and Llorente-Saguer (2012),
  where opposing parties have asymmetric and heterogeneous battlefield
  valuations.

* Ranking Games (`ranking_game`): In these games, as studied by Goldberg
  et al. (2013), each player chooses an effort level associated with a
  cost and a score. The players are ranked according to their scores,
  and the player with the higher score wins the prize. Each player's
  payoff is given by the value of the prize minus the cost of the
  effort.

* SGC Games (`sgc_game`): These games were introduced by Sandholm,
  Gilpin, and Conitzer (2005) as a worst case scenario for support
  enumeration as it has a unique equilibrium where each player uses half
  of his actions in his support.

* Tournament Games (`tournament_game`): These games are constructed by
  Anbalagan et al. (2013) as games that do not have interim epsilon-Nash
  equilibria with constant cardinality supports for epsilon smaller than
  a certain threshold.

* Unit Vector Games (`unit_vector_game`): These games are games where
  the payoff matrix of one player consists of unit (column) vectors,
  used by Savani and von Stengel (2016) to construct instances that are
  hard, in terms of computational complexity, both for the Lemke-Howson
  and support enumeration algorithms.

Large part of the code here is based on the C code available at
https://github.com/bimatrix-games/bimatrix-generators distributed under
BSD 3-Clause License.

References
----------
* Y. Anbalagan, S. Norin, R. Savani, and A. Vetta, "Polylogarithmic
  Supports Are Required for Approximate Well-Supported Nash Equilibria
  below 2/3," WINE, 2013.

* J. Fearnley, T. P. Igwe, and R. Savani, "An Empirical Study of Finding
  Approximate Equilibria in Bimatrix Games," International Symposium on
  Experimental Algorithms (SEA), 2015.

* L. A. Goldberg, P. W. Goldberg, P. Krysta, and C. Ventre, "Ranking
  Games that have Competitiveness-based Strategies", Theoretical
  Computer Science, 2013.

* R. Hortala-Vallve and A. Llorente-Saguer, "Pure Strategy Nash
  Equilibria in Non-Zero Sum Colonel Blotto Games", International
  Journal of Game Theory, 2012.

* T. Sandholm, A. Gilpin, and V. Conitzer, "Mixed-Integer Programming
  Methods for Finding Nash Equilibria," AAAI, 2005.

* R. Savani and B. von Stengel, "Unit Vector Games," International
  Journal of Economic Theory, 2016.

"""

# BSD 3-Clause License
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# Copyright (c) 2015, John Fearnley, Tobenna P. Igwe, Rahul Savani
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import numpy as np
import scipy.special
from numba import jit
from ..normal_form_game import Player, NormalFormGame
from ...util import check_random_state
from ...gridtools import simplex_grid
from ...graph_tools import random_tournament_graph
from ...util.combinatorics import next_k_array, k_array_rank_jit


[docs]def blotto_game(h, t, rho, mu=0, random_state=None): """ Return a NormalFormGame instance of a 2-player non-zero sum Colonel Blotto game (Hortala-Vallve and Llorente-Saguer, 2012), where the players have an equal number `t` of troops to assign to `h` hills (so that the number of actions for each player is equal to (t+h-1) choose (h-1) = (t+h-1)!/(t!*(h-1)!)). Each player has a value for each hill that he receives if he assigns strictly more troops to the hill than his opponent (ties are broken uniformly at random), where the values are drawn from a multivariate normal distribution with covariance `rho`. Each player’s payoff is the sum of the values of the hills won by that player. Parameters ---------- h : scalar(int) Number of hills. t : scalar(int) Number of troops. rho : scalar(float) Covariance of the players' values of each hill. Must be in [-1, 1]. mu : scalar(float), optional(default=0) Mean of the players' values of each hill. random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. Returns ------- g : NormalFormGame Examples -------- >>> g = blotto_game(2, 3, 0.5, random_state=1234) >>> g.players[0] Player([[-0.44861083, -1.08443468, -1.08443468, -1.08443468], [ 0.18721302, -0.44861083, -1.08443468, -1.08443468], [ 0.18721302, 0.18721302, -0.44861083, -1.08443468], [ 0.18721302, 0.18721302, 0.18721302, -0.44861083]]) >>> g.players[1] Player([[-1.20042463, -1.39708658, -1.39708658, -1.39708658], [-1.00376268, -1.20042463, -1.39708658, -1.39708658], [-1.00376268, -1.00376268, -1.20042463, -1.39708658], [-1.00376268, -1.00376268, -1.00376268, -1.20042463]]) """ actions = simplex_grid(h, t) n = actions.shape[0] payoff_arrays = tuple(np.empty((n, n)) for i in range(2)) mean = np.array([mu, mu]) cov = np.array([[1, rho], [rho, 1]]) random_state = check_random_state(random_state) values = random_state.multivariate_normal(mean, cov, h) _populate_blotto_payoff_arrays(payoff_arrays, actions, values) g = NormalFormGame( [Player(payoff_array) for payoff_array in payoff_arrays] ) return g
@jit(nopython=True) def _populate_blotto_payoff_arrays(payoff_arrays, actions, values): """ Populate the ndarrays in `payoff_arrays` with the payoff values of the Blotto game with h hills and t troops. Parameters ---------- payoff_arrays : tuple(ndarray(float, ndim=2)) Tuple of 2 ndarrays of shape (n, n), where n = (t+h-1)!/ (t!*(h-1)!). Modified in place. actions : ndarray(int, ndim=2) ndarray of shape (n, h) containing all possible actions, i.e., h-part compositions of t. values : ndarray(float, ndim=2) ndarray of shape (h, 2), where `values[k, :]` contains the players' values of hill `k`. """ n, h = actions.shape payoffs = np.empty(2) for i in range(n): for j in range(n): payoffs[:] = 0 for k in range(h): if actions[i, k] == actions[j, k]: for p in range(2): payoffs[p] += values[k, p] / 2 else: winner = np.int(actions[i, k] < actions[j, k]) payoffs[winner] += values[k, winner] payoff_arrays[0][i, j], payoff_arrays[1][j, i] = payoffs
[docs]def ranking_game(n, steps=10, random_state=None): """ Return a NormalFormGame instance of (the 2-player version of) the "ranking game" studied by Goldberg et al. (2013), where each player chooses an effort level associated with a score and a cost which are both increasing functions with randomly generated step sizes. The player with the higher score wins the first prize, whose value is 1, and the other player obtains the "second prize" of value 0; in the case of a tie, the first prize is split and each player receives a value of 0.5. The payoff of a player is given by the value of the prize minus the cost of the effort. Parameters ---------- n : scalar(int) Number of actions, i.e, number of possible effort levels. steps : scalar(int), optional(default=10) Parameter determining the upper bound for the size of the random steps for the scores and costs for each player: The step sizes for the scores are drawn from `1`, ..., `steps`, while those for the costs are multiples of `1/(n*steps)`, where the cost of effort level `0` is 0, and the maximum possible cost of effort level `n-1` is less than or equal to 1. random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. Returns ------- g : NormalFormGame Examples -------- >>> g = ranking_game(5, random_state=1234) >>> g.players[0] Player([[ 0. , 0. , 0. , 0. , 0. ], [ 0.82, -0.18, -0.18, -0.18, -0.18], [ 0.8 , 0.8 , -0.2 , -0.2 , -0.2 ], [ 0.68, 0.68, 0.68, -0.32, -0.32], [ 0.66, 0.66, 0.66, 0.66, -0.34]]) >>> g.players[1] Player([[ 1. , 0. , 0. , 0. , 0. ], [ 0.8 , 0.8 , -0.2 , -0.2 , -0.2 ], [ 0.66, 0.66, 0.66, -0.34, -0.34], [ 0.6 , 0.6 , 0.6 , 0.6 , -0.4 ], [ 0.58, 0.58, 0.58, 0.58, 0.58]]) """ payoff_arrays = tuple(np.empty((n, n)) for i in range(2)) random_state = check_random_state(random_state) scores = random_state.randint(1, steps+1, size=(2, n)) scores.cumsum(axis=1, out=scores) costs = np.empty((2, n-1)) costs[:] = random_state.randint(1, steps+1, size=(2, n-1)) costs.cumsum(axis=1, out=costs) costs[:] /= (n * steps) _populate_ranking_payoff_arrays(payoff_arrays, scores, costs) g = NormalFormGame( [Player(payoff_array) for payoff_array in payoff_arrays] ) return g
@jit(nopython=True) def _populate_ranking_payoff_arrays(payoff_arrays, scores, costs): """ Populate the ndarrays in `payoff_arrays` with the payoff values of the ranking game given `scores` and `costs`. Parameters ---------- payoff_arrays : tuple(ndarray(float, ndim=2)) Tuple of 2 ndarrays of shape (n, n). Modified in place. scores : ndarray(int, ndim=2) ndarray of shape (2, n) containing score values corresponding to the effort levels for the two players. costs : ndarray(float, ndim=2) ndarray of shape (2, n-1) containing cost values corresponding to the n-1 positive effort levels for the two players, with the assumption that the cost of the zero effort action is zero. """ n = payoff_arrays[0].shape[0] for p, payoff_array in enumerate(payoff_arrays): payoff_array[0, :] = 0 for i in range(1, n): for j in range(n): payoff_array[i, j] = -costs[p, i-1] prize = 1. for i in range(n): for j in range(n): if scores[0, i] > scores[1, j]: payoff_arrays[0][i, j] += prize elif scores[0, i] < scores[1, j]: payoff_arrays[1][j, i] += prize else: payoff_arrays[0][i, j] += prize / 2 payoff_arrays[1][j, i] += prize / 2
[docs]def sgc_game(k): """ Return a NormalFormGame instance of the 2-player game introduced by Sandholm, Gilpin, and Conitzer (2005), which has a unique Nash equilibrium, where each player plays half of the actions with positive probabilities. Payoffs are normalized so that the minimum and the maximum payoffs are 0 and 1, respectively. Parameters ---------- k : scalar(int) Positive integer determining the number of actions. The returned game will have `4*k-1` actions for each player. Returns ------- g : NormalFormGame Examples -------- >>> g = sgc_game(2) >>> g.players[0] Player([[ 0.75, 0.5 , 1. , 0.5 , 0.5 , 0.5 , 0.5 ], [ 1. , 0.75, 0.5 , 0.5 , 0.5 , 0.5 , 0.5 ], [ 0.5 , 1. , 0.75, 0.5 , 0.5 , 0.5 , 0.5 ], [ 0. , 0. , 0. , 0.75, 0. , 0. , 0. ], [ 0. , 0. , 0. , 0. , 0.75, 0. , 0. ], [ 0. , 0. , 0. , 0. , 0. , 0.75, 0. ], [ 0. , 0. , 0. , 0. , 0. , 0. , 0.75]]) >>> g.players[1] Player([[ 0.75, 0.5 , 1. , 0.5 , 0.5 , 0.5 , 0.5 ], [ 1. , 0.75, 0.5 , 0.5 , 0.5 , 0.5 , 0.5 ], [ 0.5 , 1. , 0.75, 0.5 , 0.5 , 0.5 , 0.5 ], [ 0. , 0. , 0. , 0. , 0.75, 0. , 0. ], [ 0. , 0. , 0. , 0.75, 0. , 0. , 0. ], [ 0. , 0. , 0. , 0. , 0. , 0. , 0.75], [ 0. , 0. , 0. , 0. , 0. , 0.75, 0. ]]) """ payoff_arrays = tuple(np.empty((4*k-1, 4*k-1)) for i in range(2)) _populate_sgc_payoff_arrays(payoff_arrays) g = NormalFormGame( [Player(payoff_array) for payoff_array in payoff_arrays] ) return g
@jit(nopython=True) def _populate_sgc_payoff_arrays(payoff_arrays): """ Populate the ndarrays in `payoff_arrays` with the payoff values of the SGC game. Parameters ---------- payoff_arrays : tuple(ndarray(float, ndim=2)) Tuple of 2 ndarrays of shape (4*k-1, 4*k-1). Modified in place. """ n = payoff_arrays[0].shape[0] # 4*k-1 m = (n+1)//2 - 1 # 2*k-1 for payoff_array in payoff_arrays: for i in range(m): for j in range(m): payoff_array[i, j] = 0.75 for j in range(m, n): payoff_array[i, j] = 0.5 for i in range(m, n): for j in range(n): payoff_array[i, j] = 0 payoff_array[0, m-1] = 1 payoff_array[0, 1] = 0.5 for i in range(1, m-1): payoff_array[i, i-1] = 1 payoff_array[i, i+1] = 0.5 payoff_array[m-1, m-2] = 1 payoff_array[m-1, 0] = 0.5 k = (m+1)//2 for h in range(k): i, j = m + 2*h, m + 2*h payoff_arrays[0][i, j] = 0.75 payoff_arrays[0][i+1, j+1] = 0.75 payoff_arrays[1][j, i+1] = 0.75 payoff_arrays[1][j+1, i] = 0.75
[docs]def tournament_game(n, k, random_state=None): """ Return a NormalFormGame instance of the 2-player win-lose game, whose payoffs are either 0 or 1, introduced by Anbalagan et al. (2013). Player 0 has n actions, which constitute the set of nodes {0, ..., n-1}, while player 1 has n choose k actions, each corresponding to a subset of k elements of the set of n nodes. Given a randomly generated tournament graph on the n nodes, the payoff for player 0 is 1 if, in the tournament, the node chosen by player 0 dominates all the nodes in the k-subset chosen by player 1. The payoff for player 1 is 1 if player 1's k-subset contains player 0's chosen node. Parameters ---------- n : scalar(int) Number of nodes in the tournament graph. k : scalar(int) Size of subsets of nodes in the tournament graph. random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. Returns ------- g : NormalFormGame Notes ----- The actions of player 1 are ordered according to the combinatorial number system [1]_, which is different from the order used in the original library in C. Examples -------- >>> g = tournament_game(5, 2, random_state=1234) >>> g.players[0] Player([[ 0., 0., 0., 0., 1., 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0., 0., 0., 0., 0., 1.], [ 1., 0., 0., 0., 0., 0., 0., 0., 0., 0.], [ 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.], [ 0., 1., 0., 1., 0., 1., 0., 0., 0., 0.]]) >>> g.players[1] Player([[ 1., 1., 0., 0., 0.], [ 1., 0., 1., 0., 0.], [ 0., 1., 1., 0., 0.], [ 1., 0., 0., 1., 0.], [ 0., 1., 0., 1., 0.], [ 0., 0., 1., 1., 0.], [ 1., 0., 0., 0., 1.], [ 0., 1., 0., 0., 1.], [ 0., 0., 1., 0., 1.], [ 0., 0., 0., 1., 1.]]) References ---------- .. [1] `Combinatorial number system <https://en.wikipedia.org/wiki/Combinatorial_number_system>`_, Wikipedia. """ m = scipy.special.comb(n, k, exact=True) if m > np.iinfo(np.intp).max: raise ValueError('Maximum allowed size exceeded') payoff_arrays = tuple(np.zeros(shape) for shape in [(n, m), (m, n)]) tourn = random_tournament_graph(n, random_state=random_state) indices, indptr = tourn.csgraph.indices, tourn.csgraph.indptr _populate_tournament_payoff_array0(payoff_arrays[0], k, indices, indptr) _populate_tournament_payoff_array1(payoff_arrays[1], k) g = NormalFormGame( [Player(payoff_array) for payoff_array in payoff_arrays] ) return g
@jit(nopython=True) def _populate_tournament_payoff_array0(payoff_array, k, indices, indptr): """ Populate `payoff_array` with the payoff values for player 0 in the tournament game given a random tournament graph in CSR format. Parameters ---------- payoff_array : ndarray(float, ndim=2) ndarray of shape (n, m), where m = n choose k, prefilled with zeros. Modified in place. k : scalar(int) Size of the subsets of nodes. indices : ndarray(int, ndim=1) CSR format index array of the adjacency matrix of the tournament graph. indptr : ndarray(int, ndim=1) CSR format index pointer array of the adjacency matrix of the tournament graph. """ n = payoff_array.shape[0] X = np.empty(k, dtype=np.int_) a = np.empty(k, dtype=np.int_) for i in range(n): d = indptr[i+1] - indptr[i] if d >= k: for j in range(k): a[j] = j while a[-1] < d: for j in range(k): X[j] = indices[indptr[i]+a[j]] payoff_array[i, k_array_rank_jit(X)] = 1 a = next_k_array(a) @jit(nopython=True) def _populate_tournament_payoff_array1(payoff_array, k): """ Populate `payoff_array` with the payoff values for player 1 in the tournament game. Parameters ---------- payoff_array : ndarray(float, ndim=2) ndarray of shape (m, n), where m = n choose k, prefilled with zeros. Modified in place. k : scalar(int) Size of the subsets of nodes. """ m = payoff_array.shape[0] X = np.arange(k) for j in range(m): for i in range(k): payoff_array[j, X[i]] = 1 X = next_k_array(X)
[docs]def unit_vector_game(n, avoid_pure_nash=False, random_state=None): """ Return a NormalFormGame instance of the 2-player game "unit vector game" (Savani and von Stengel, 2016). Payoffs for player 1 are chosen randomly from the [0, 1) range. For player 0, each column contains exactly one 1 payoff and the rest is 0. Parameters ---------- n : scalar(int) Number of actions. avoid_pure_nash : bool, optional(default=False) If True, player 0's payoffs will be placed in order to avoid pure Nash equilibria. (If necessary, the payoffs for player 1 are redrawn so as not to have a dominant action.) random_state : int or np.random.RandomState, optional Random seed (integer) or np.random.RandomState instance to set the initial state of the random number generator for reproducibility. If None, a randomly initialized RandomState is used. Returns ------- g : NormalFormGame Examples -------- >>> g = unit_vector_game(4, random_state=1234) >>> g.players[0] Player([[ 1., 0., 1., 0.], [ 0., 0., 0., 1.], [ 0., 0., 0., 0.], [ 0., 1., 0., 0.]]) >>> g.players[1] Player([[ 0.19151945, 0.62210877, 0.43772774, 0.78535858], [ 0.77997581, 0.27259261, 0.27646426, 0.80187218], [ 0.95813935, 0.87593263, 0.35781727, 0.50099513], [ 0.68346294, 0.71270203, 0.37025075, 0.56119619]]) With `avoid_pure_nash=True`: >>> g = unit_vector_game(4, avoid_pure_nash=True, random_state=1234) >>> g.players[0] Player([[ 1., 1., 0., 0.], [ 0., 0., 0., 0.], [ 0., 0., 1., 1.], [ 0., 0., 0., 0.]]) >>> g.players[1] Player([[ 0.19151945, 0.62210877, 0.43772774, 0.78535858], [ 0.77997581, 0.27259261, 0.27646426, 0.80187218], [ 0.95813935, 0.87593263, 0.35781727, 0.50099513], [ 0.68346294, 0.71270203, 0.37025075, 0.56119619]]) >>> pure_nash_brute(g) [] """ random_state = check_random_state(random_state) payoff_arrays = (np.zeros((n, n)), random_state.random_sample((n, n))) if not avoid_pure_nash: ones_ind = random_state.randint(n, size=n) payoff_arrays[0][ones_ind, np.arange(n)] = 1 else: if n == 1: raise ValueError('Cannot avoid pure Nash with n=1') maxes = payoff_arrays[1].max(axis=0) is_suboptimal = payoff_arrays[1] < maxes nums_suboptimal = is_suboptimal.sum(axis=1) while (nums_suboptimal==0).any(): payoff_arrays[1][:] = random_state.random_sample((n, n)) payoff_arrays[1].max(axis=0, out=maxes) np.less(payoff_arrays[1], maxes, out=is_suboptimal) is_suboptimal.sum(axis=1, out=nums_suboptimal) for i in range(n): one_ind = random_state.randint(n) while not is_suboptimal[i, one_ind]: one_ind = random_state.randint(n) payoff_arrays[0][one_ind, i] = 1 g = NormalFormGame( [Player(payoff_array) for payoff_array in payoff_arrays] ) return g