Plots for computing optimal strategy in winner-take-all games with many opponents

monte_carlo.py

from __future__ import division
from matplotlib import pyplot as plt
import numpy as np

sigma_mine = 2.0
sigma_they = 1.0
mean_mine = 0.3
mean_they = 0.0

N_other_players = 4
N_samples = 100000

if __name__ == "__main__":
    win_count = 0
    for i in range(N_samples):
        other_player_scores = np.random.normal(mean_they, sigma_they, (N_other_players))
        my_score = np.random.normal(mean_mine, sigma_mine)
        if my_score >= np.max(other_player_scores):
            win_count += 1

    win_pct = win_count / N_samples
    print(win_pct)

n_breakeven

from __future__ import division
from scipy import integrate
from scipy.stats import norm
import numpy as np
import plotly.graph_objects as go

def myfunc(x, mu, sigma, N):
    pdf = norm.pdf(x, loc=mu, scale=sigma)
    dpdf_dsigma = (x-mu)**2 / sigma **3 * pdf - pdf / sigma
    f = norm.cdf(x)**N * dpdf_dsigma
    return f


if __name__ == "__main__":
    mu = 0.3
    sigma = 2.0
    N = 1
    mus = np.linspace(-1.0, 3.0, 21)
    Ns = np.linspace(1.0, 200.0, 20)
    probs = []
    for i in range(21):
        probs.append([])
        for j in range(20):
            args = (mus[i], 1.0, Ns[j])
            probs[i].append(integrate.quad(myfunc, -np.inf, np.inf, args=args)[0])

    colorscale = [[0, 'black'], [1, 'black']]

    fig = go.Figure(data =
        go.Contour(
            z=probs,
            x=Ns, # horizontal axis
            y=mus, # vertical axis
            contours=dict(start=0,end=0,size=1),
            contours_coloring='lines',
            colorbar=None,
            colorscale=colorscale,
            line_width=5,

        ))

    fig.update_layout(
        title="High/low-variance strategy breakeven",
        xaxis_title=r"$\text{Opponent count} \, (N)$",
        yaxis_title=r"$\text{Player normalized skill} \, (\frac{\mu_0-\mu}{\sigma})$",
        showlegend=False,
        font=dict(
        size=18,
        color="#7f7f7f")
    )

    fig.show()

win_prob_contour.py

from __future__ import division
from scipy import integrate
from scipy.stats import norm
import numpy as np
import plotly.graph_objects as go

def myfunc(x, mu, sigma, N):
    f = norm.cdf(x)**N * norm.pdf(x, loc=mu, scale=sigma)
    return f


if __name__ == "__main__":
    mu = 0.3
    sigma = 2.0
    N = 5
    mus = np.linspace(-2.0, 2.0, 20)
    sigmas = np.linspace(0.1, 3.0, 20)
    probs = []
    for i in range(20):
        probs.append([])
        for j in range(20):
            args = (mus[i], sigmas[j], N)
            probs[i].append(integrate.quad(myfunc, -np.inf, np.inf, args=args)[0])


    fig = go.Figure(data =
        go.Contour(
            z=probs,
            x=sigmas, # horizontal axis
            y=mus # vertical axis
        ))

    fig.update_layout(
        title="Win Probability with N="+str(N)+" opponents",
        xaxis_title=r"$\text{Player variability} \, (\sigma_0)$",
        yaxis_title=r"$\text{Player normalized skill} \, (\frac{\mu_0-\mu}{\sigma})$",
        font=dict(
        size=18,
        color="#7f7f7f"))

    fig.show()