A (difficult) example (B.4)

[1]:

from fractions import Fraction
import sympy as sp
from poisson_approval import *

[2]:

tau = TauVector({'a': Fraction(3, 20), 'b': Fraction(9, 20), 'ac': Fraction(1, 20), 'bc': Fraction(7, 20)},
                symbolic=True)

[3]:

tau.pivot_weak_ab.mu

[3]:

$\displaystyle - \frac{1}{5}$

[4]:

tau.pivot_weak_bc.mu

[4]:

$\displaystyle - \frac{1}{5}$

[5]:

tau.pivot_weak_ac.mu

[5]:

$\displaystyle - \frac{7}{10} + \frac{\sqrt{21}}{10}$

[6]:

tau.trio.mu

[6]:

$\displaystyle - \frac{7}{10} + \frac{\sqrt{21}}{10}$

[7]:

float(tau.trio.mu)

[7]:

-0.241742430504416

$\delta_{ab}(\tau)$:

[8]:

tau.score_ab_in_duo_ab - tau.score_c_in_duo_ab

[8]:

$\displaystyle \frac{1}{8}$

$\delta_{bc}(\tau)$:

[9]:

tau.score_bc_in_duo_bc - tau.score_a_in_duo_bc

[9]:

$\displaystyle \frac{1}{5}$

[10]:

tau.pivot_weak_ab.psi_ac

[10]:

$\displaystyle 2$

[11]:

tau.pivot_weak_bc.psi_a

[11]:

$\displaystyle 1$

[12]:

tau.pivot_weak_bc.psi_b

[12]:

$\displaystyle \frac{1}{3}$

[13]:

tau.duo_ab.asymptotic

[13]:

exp(- n/5 - log(n)/2 - log(8*pi/5)/2 + o(1))

[14]:

tau.duo_bc.asymptotic

[14]:

exp(- n/5 - log(n)/2 - log(3*pi/5)/2 + o(1))

[15]:

utility_threshold_program = tau.d_ranking_best_response['abc'].utility_threshold
utility_threshold_program

[15]:

$\displaystyle \frac{27 \sqrt{2}}{16 \sqrt{3} + 27 \sqrt{2}}$

[16]:

utility_threshold_theoretical = 9 * sp.sqrt(3) / (9 * sp.sqrt(3) + 8 * sp.sqrt(2))
utility_threshold_theoretical

[16]:

$\displaystyle \frac{9 \sqrt{3}}{8 \sqrt{2} + 9 \sqrt{3}}$

Check that the value found by the program is actually the same as the theoretical value:

[17]:

sp.simplify(utility_threshold_program - utility_threshold_theoretical)

[17]:

$\displaystyle 0$