def no_arbitage_wagering(forecasts, wagers, f_norm=f_norm_Brier_subgradient):
# forecasters - the predicted probability of a binary random variable
# by each forecasters
# f_norm - f_norm_function()
# !!! requires normalizing the range of score rule to [0,1]
if np.ndim(forecasts) > 1:
print("No_arbitage_wagering(): forecasters dimension error!")
return 0
n = np.shape(forecasts)[0]
w = wagers
sum_w = np.sum(w)
p = forecasts
P_matrix = np.column_stack((p, 1 - p))
coefficient = np.multiply(w, sum_w - w) / sum_w
p_bar = np.zeros(n)
for i in range(n):
w_zero_i = np.array(w)
w_zero_i[i] = 0
p_bar[i] = f_norm(p, w_zero_i)
P_bar_matrix = np.column_stack((p_bar, 1 - p_bar))
scores = (sc.Brier_score(P_matrix, b=0.5, a=-0.5) -
sc.Brier_score(P_bar_matrix, b=0.5, a=-0.5))
return coefficient[:, None] * scores
def surrogate_NAWM(forecasts, wagers, f_norm=f_norm_Brier_subgradient, e1=0.25, e0=0.25):
# forecasters - the predicted probability of a binary random variable
# by each forecasters
# f_norm - f_norm_function()
# !!! requires normalizing the range of score rule to [0,1]
# e1, e0 - outcome-flipped rate under outcome 1 and 0
if np.ndim(forecasts) > 1:
print("No_arbitage_wagering(): forecasters dimension error!")
return 0
n = np.shape(forecasts)[0]
w = wagers
sum_w = np.sum(w)
p = forecasts
P_matrix = np.column_stack((p, 1 - p))
coefficient = np.multiply(w, sum_w - w) / sum_w
p_bar = np.zeros(n)
for i in range(n):
w_zero_i = np.array(w)
w_zero_i[i] = 0
p_bar[i] = f_norm(p, w_zero_i)
P_bar_matrix = np.column_stack((p_bar, 1 - p_bar))
# if the outcome is 1:
score1 = np.zeros(n)
for i in range(n):
tmp1= sc.Brier_score(P_matrix[i, :].reshape(1, -1), b=0.5, a=-0.5)
tmp2 = sc.Brier_score(P_bar_matrix[i, :].reshape(1, -1), b=0.5, a=-0.5)
s1 = tmp1[0, 0]
s0 = tmp1[0, 1]
avg_s1 = tmp2[0, 0]
avg_s0 = tmp2[0, 1]
if (np.random.rand() < e1): # outcome flipped to 0
score1[i] = (surrogate_score(s1=s1, s0=s0, e1=e1, e0=e0, outcome=0)
- surrogate_score(s1=avg_s1, s0=avg_s0, e1=e1, e0=e0, outcome=0)
)
else:
score1[i] = (surrogate_score(s1=s1, s0=s0, e1=e1, e0=e0, outcome=1)
- surrogate_score(s1=avg_s1, s0=avg_s0, e1=e1, e0=e0, outcome=1)
)
# if the outcome is 0:
score0 = np.zeros(n)
for i in range(n):
tmp1= sc.Brier_score(P_matrix[i, :].reshape(1, -1), b=0.5, a=-0.5)
tmp2 = sc.Brier_score(P_bar_matrix[i, :].reshape(1, -1), b=0.5, a=-0.5)
s1 = tmp1[0, 0]
s0 = tmp1[0, 1]
avg_s1 = tmp2[0, 0]
avg_s0 = tmp2[0, 1]
if (np.random.rand() < e1): # outcome flipped to 1
score0[i] = (surrogate_score(s1=s1, s0=s0, e1=e1, e0=e0, outcome=1)
- surrogate_score(s1=avg_s1, s0=avg_s0, e1=e1, e0=e0, outcome=1)
)
else:
score0[i] = (surrogate_score(s1=s1, s0=s0, e1=e1, e0=e0, outcome=0)
- surrogate_score(s1=avg_s1, s0=avg_s0, e1=e1, e0=e0, outcome=0)
)
return coefficient[:, None] * np.vstack((score1, score0)).T
def no_arbitrage_wagering(forecasts, wagers, f_norm=f_norm_Brier_subgradient):
# forecasts - the predicted probability of a random variable
# by each forecasters
# f_norm - f_norm_function()
# !!! requires normalizing the range of score rule to [0,1]
# return value
n, m = np.shape(forecasts)
w = wagers
sum_w = np.sum(w)
coefficient = np.multiply(w, sum_w - w) / sum_w
p_bar = np.zeros((n, m))
for i in range(n):
w_zero_i = np.array(w)
w_zero_i[i] = 0
p_bar[i, :] = f_norm(forecasts, w_zero_i)
scores = (sc.Brier_score(forecasts, b=1, a=-0.5) -
sc.Brier_score(p_bar, b=1, a=-0.5))
return coefficient[:, None] * scores
import numpy as np import scores as sc # source: scores.py import matplotlib.pyplot as plt # Set the outcome and participant d = 2 # # of realizations of outcome n = 4 # # of participants # Generate the ground truth distribution of the outcome p_truth = np.random.dirichlet(np.ones((d, ))) # Generate participants' true believes and each row represents one belief forecasts = np.random.dirichlet(np.ones((d, )), size=n) forecasts = np.vstack([[1, 0], [1, 0], [0, 1], [0, 1]]) print(forecasts) Brier_scores = sc.Brier_score(forecasts) competitive_scores = sc.competitive_score_rule(sc.Brier_score(forecasts)) np.set_printoptions(precision=3, suppress=True) # print((quadratic_scores[:,0],competitive_quadratic_scores[:,0])) print(np.column_stack((Brier_scores[:, 0], competitive_scores[:, 0]))) # print(np.reshape(quadratic_scores[:,0]-competitive_quadratic_scores[:,0]-(quadratic_scores[0,0]-competitive_quadratic_scores[0,0]),(-1,1))) ''' # Set the number of realizations of a future event d = 2 # # of realizations of a future event running_times = 1000 range_n = range(5,101,5) average_average = [] average_max = []
for t in range(running_times):
# Generate forecasters' true believes and each row represents one belief
forecasts = np.random.dirichlet(np.ones((d,)), size=n)
# forecasts = np.random.dirichlet([1, 0.2], size=n)
# forecasts = np.random.dirichlet([0.3, 0.3], size=n)
forecasts[0, :] = [1, 0]
# print(forecasts)
# Generate forecasters' wagers
wagers = np.ones((n,))
# wagers = np.random.beta(100, 100, size=n)
# wagers = (np.random.pareto(1.16, size=n) + 1) * 1
total_wager = np.sum(wagers)
# Compute the outcome of no-arbitage wagering mechanism under each realization
net_pays_CSR = sc.competitive_score_rule(sc.Brier_score(forecasts, 2, -1)) / 2
net_pays_NAW = wg.no_arbitage_wagering(forecasts[:, 0], wagers)
net_pays_PPM = wg.net_payoff_PPM(forecasts, wagers)
# value duplication for further modification
MnEx_CSR_PS, MnEx_NAW_PS, MnEx_PPM_PS = (np.array(net_pays_CSR),
np.array(net_pays_NAW),
np.array(net_pays_PPM))
MnEx_CSR_NG, MnEx_NAW_NG, MnEx_PPM_NG = (np.array(net_pays_CSR),
np.array(net_pays_NAW),
np.array(net_pays_PPM))
# print(net_pays_CSR)
# print(net_pays_NAW)
# print(net_pays_PPM)
max_complete_right_CSR = max(net_pays_CSR[0, 0], max_complete_right_CSR)
import numpy as np
import scores as sc #source: scores.py
import wager as wg #source: wagers.py
import matplotlib.pyplot as plt
# Set the outcome vand participant
d = 2 # # of realizations of outcome
n = 100 # # of participants
# Generate the ground truth distribution of the outcome
p_truth = np.random.dirichlet(np.ones((d, )))
# print(p_truth)
# Generate participants' true believes and each row represents one belief
forecasts = np.random.dirichlet(np.ones((d, )), size=n)
quadratic_scores = -sc.Brier_score(forecasts)
competitive_quadratic_scores = sc.competitive_quadratic_score_rule(forecasts)
np.set_printoptions(precision=3, suppress=True)
# print((quadratic_scores[:,0],competitive_quadratic_scores[:,0]))
print(
np.column_stack((quadratic_scores[:, 0], competitive_quadratic_scores[:,
0])))
print(
np.reshape(
quadratic_scores[:, 0] - competitive_quadratic_scores[:, 0] -
(quadratic_scores[0, 0] - competitive_quadratic_scores[0, 0]),
(-1, 1)))
'''
# Set the number of realizations of a future event