Dynamic Process: Base Case (5.2.1, 5.2.4, 5.3, C.8, C.9)
[1]:
from matplotlib import pyplot as plt
from poisson_approval import *
[2]:
N_SAMPLES = 10000
N_MAX_EPISODES = 1000
[3]:
rand_profile = RandProfileHistogramUniform(n_bins=1)
[4]:
results = monte_carlo_fictitious_play(
factory=rand_profile,
n_samples=N_SAMPLES,
n_max_episodes=N_MAX_EPISODES,
voting_rules=VOTING_RULES,
init='random_tau',
monte_carlo_settings=[
MCS_CONVERGES,
MCS_WELFARE_LOSSES,
MCS_UTILITY_THRESHOLDS,
MCS_BALLOT_STATISTICS,
MCS_DECREASING_SCORES
],
file_save='sav/base_case.sav',
)
Convergence (5.2.1)
[5]:
results[APPROVAL]['mean_converges']
[5]:
0.952
[6]:
results[PLURALITY]['mean_converges']
[6]:
1.0
[7]:
results[ANTI_PLURALITY]['mean_converges']
[7]:
0.0
Welfare Losses (5.2.4)
[8]:
results[APPROVAL]['mean_utilitarian_welfare_loss']
[8]:
0.006778897505995395
[9]:
results[PLURALITY]['mean_utilitarian_welfare_loss']
[9]:
0.060140989673448934
[10]:
results[ANTI_PLURALITY]['mean_utilitarian_welfare_loss']
[10]:
0.09870457398113966
[11]:
plot_welfare_losses(results, 'utilitarian_welfare_losses')
plt.savefig('img/fUWelfare.png', dpi=600, bbox_inches="tight")
_13_0.png)
[12]:
plot_welfare_losses(results, 'plurality_welfare_losses')
plt.savefig('img/fPLWelfare.png', dpi=600, bbox_inches="tight")
_14_0.png)
[13]:
plot_welfare_losses(results, 'anti_plurality_welfare_losses')
plt.savefig('img/fAPLWelfare.png', dpi=600, bbox_inches="tight")
_15_0.png)
Utility Thresholds (5.3, C.8)
[14]:
plot_utility_thresholds(results, APPROVAL)
plt.savefig('img/fThresholds.png', dpi=600, bbox_inches="tight")
_17_0.png)
[15]:
results[APPROVAL]['p_utility_threshold_0']
[15]:
0.1847010518611007
[16]:
results[APPROVAL]['p_utility_threshold_1']
[16]:
0.4722538949283068
[17]:
results[APPROVAL]['p_utility_threshold_not_0_or_1']
[17]:
0.3430450532105924
Ballot Statistics (5.3)
[18]:
results[APPROVAL]['mean_share_single_votes']
[18]:
0.6814864927666361
[19]:
results[APPROVAL]['mean_share_double_votes']
[19]:
0.3185135072333639
Scores of the Candidates (5.3, C.9)
[20]:
plot_distribution_scores(results, APPROVAL)
plt.savefig('img/fScores.png', dpi=600, bbox_inches="tight")
_25_0.png)
A counter-example where the challenger may have more than 50% of approval:
[21]:
profile = ProfileHistogram({('abc', (1., )): .33, ('acb', (1., )): .01,
('bac', (1., )): .08, ('bca', (1., )): .08,
('cab', (1., )): .02, ('cba', (1., )): .48})
profile
[21]:
<abc: 0.33 [1.], acb: 0.01 [1.], bac: 0.08 [1.], bca: 0.08 [1.], cab: 0.02 [1.], cba: 0.48 [1.]> (Condorcet winner: c)
Let us try some random taus at initialization, until one gives the desired counter-example behavior:
[22]:
i = 0
while True:
i += 1
print()
print('Tau attempt number %s' % i)
results = profile.fictitious_play(
init='random_tau', n_max_episodes=1000,
perception_update_ratio=one_over_log_t_plus_one,
ballot_update_ratio=one_over_log_t_plus_one,
winning_frequency_update_ratio=one_over_log_t_plus_one
)
if results['tau'] is None:
print('No convergence')
continue
scores = results['tau'].scores.values()
print('Results: %s' % results)
print('Scores: %s' % results['tau'].scores)
if len([s for s in scores if s > .5]) > 1:
break
Tau attempt number 1
Results: {'converges': True, 'tau': TauVector({'a': 0.33, 'ac': 0.03, 'b': 0.16, 'bc': 0.48}), 'strategy': StrategyThreshold({'abc': 1.0, 'acb': 0, 'bac': 1, 'bca': 1, 'cab': 0, 'cba': 0}), 'tau_init': TauVector({'a': 0.08413489495889137, 'ab': 0.47035846921310287, 'ac': 0.031199702607497493, 'b': 0.003972010294148842, 'bc': 0.12120334467892513, 'c': 0.2891315782474343}), 'n_episodes': 41, 'd_candidate_winning_frequency': {'b': Fraction(1, 1)}}
Scores: {a: 0.36, b: 0.64, c: 0.51}