[mrow84]

I was interested in seeing the results myself, so here is some python:

    import numpy as np
    import itertools
    from matplotlib import pyplot as plt

    def ensemble_mean(outcomes):
        # Assume we are given a (K, T) array of outcomes, and compute the ensemble average
        # for T+1 time steps, starting with 1 wealth.
        K, T = outcomes.shape
        X = np.ones((K, T+1), dtype=np.float64)
        X[:, 1:] = np.where(outcomes, 1.5, 0.6)
        Z = np.cumprod(X, axis=1)
        return Z.mean(axis=0)

    time_steps = 20

    all_outcomes = np.array(list(itertools.product([0, 1], repeat=time_steps-1)))
    exhaustive_mean = ensemble_mean(all_outcomes)

    ensemble_size = 100
    ensemble_samples = 10000

    ensemble_means = np.zeros((time_steps, ensemble_samples))
    for i in range(ensemble_samples):
        print(i)
        # generate ensembles as though we were sampling (i.e. with replacement)
        J = np.random.choice(all_outcomes.shape[0], size=ensemble_size, replace=True)
        ensemble_means[:, i] = ensemble_mean(all_outcomes[J, :])

    plt.hist(ensemble_means[-1], bins=1000, histtype='step')
    plt.axvline(exhaustive_mean[-1])
    plt.title("Modal sampled ensemble mean is below true ensemble mean")
    plt.show()