Ten thousand players and one player

The coin game of the preface can be worked out exactly. Each round, wealth is multiplied by 1.5 or by 0.6, each with probability one half.

The ensemble average is the average over many parallel players at the same moment, that is, the mathematical expectation: the expected factor per round is 0.5 × 1.5 + 0.5 × 0.6 = 1.05, so expected wealth grows by 5 % a round, and after a hundred rounds it is 1.05¹⁰⁰ ≈ 132 times the start.

The time average is the average along one player’s time line. In a hundred rounds there will be roughly fifty heads and fifty tails, leaving wealth at about 1.5⁵⁰ × 0.6⁵⁰ = 0.9⁵⁰ ≈ 0.005, five thousandths of the start. The typical factor per round is the geometric mean √(1.5 × 0.6) ≈ 0.9487, or −5.13 % a round.

# The multiplicative coin game: each round wealth is multiplied by 1.5 (heads) or 0.6
# (tails), each with probability 1/2. Compare the ensemble average (the expectation, an
# average over many parallel players) with the time average (the growth one player
# experiences by playing on).
import math, random, statistics

UP, DOWN = 1.5, 0.6
ens = 0.5 * UP + 0.5 * DOWN                  # expected factor per round
tim = math.sqrt(UP * DOWN)                   # geometric mean factor per round
print(f"expected factor per round   {ens:.4f} -> ensemble growth {100 * (ens - 1):+.2f} %")
print(f"time-average factor         {tim:.4f} -> time-average growth"
      f" {100 * (tim - 1):+.2f} %")
print(f"expected wealth after 100 rounds          {ens ** 100:,.0f} x start")
print(f"typical (median) wealth after 100 rounds  {tim ** 100:.4f} x start")

random.seed(1)
N, T = 10_000, 100
finals = []
for _ in range(N):
    w = 1.0
    for _ in range(T):
        w *= UP if random.random() < 0.5 else DOWN
    finals.append(w)
finals.sort()
print(f"\nsimulation: {N:,} players, {T} rounds each")
print(f"  sample mean of final wealth       {statistics.fmean(finals):8.2f}")
print(f"  median final wealth               {statistics.median(finals):8.4f}")
print(f"  share ending below the start      {sum(w < 1 for w in finals) / N:8.1%}")
print(f"  share ending below 1 % of start   {sum(w < 0.01 for w in finals) / N:8.1%}")
top = sum(finals[-N // 100:]) / sum(finals)
print(f"  wealth held by the richest 1 %    {top:8.0%}")
# With N players the sample mean tracks the expectation only while some player is likely
# to have had an exceptional run; for long games it falls far below the expectation, is
# set by the one or two luckiest players, and jumps around between simulations.
for T2 in (10, 100, 1000):
    random.seed(2)
    m = statistics.fmean(math.prod(UP if random.random() < 0.5 else DOWN
                                   for _ in range(T2)) for _ in range(N))
    print(f"  T = {T2:4d}: expectation {ens ** T2:9.3g}, sample mean of {N:,} {m:9.3g}")
# expected factor per round   1.0500 -> ensemble growth +5.00 %
# time-average factor         0.9487 -> time-average growth -5.13 %
# expected wealth after 100 rounds          132 x start
# typical (median) wealth after 100 rounds  0.0052 x start
#
# simulation: 10,000 players, 100 rounds each
#   sample mean of final wealth         109.53
#   median final wealth                 0.0052
#   share ending below the start         86.4%
#   share ending below 1 % of start      54.0%
#   wealth held by the richest 1 %         98%
#   T =   10: expectation      1.63, sample mean of 10,000      1.62
#   T =  100: expectation       132, sample mean of 10,000      41.6
#   T = 1000: expectation  1.55e+21, sample mean of 10,000      68.8

The simulation shows every feature of the game:

• After a hundred rounds about 86 % of players are below where they started, and more than half have less than 1 % of their starting wealth; the median is about 0.5 % of the start, as the geometric mean predicts.
• The sample mean of the ten thousand players is still far above the start, but it comes almost entirely from a very few exceptionally lucky people: the richest 1 % of players hold about 98 % of all the wealth.
• The longer the game, the further the sample mean falls behind the expectation. At a hundred rounds the expectation is 132 times the start while the sample mean of ten thousand players is a few dozen and varies a lot from one simulation to the next; at a thousand rounds the expectation is of order 10²¹, while the sample mean is set by the one or two luckiest players. The expectation describes an ensemble of infinitely many players, and any finite group drifts ever further from it.

So “the expected value is positive” is true here, and “almost everyone loses” is also true. Mathematically, each player’s wealth tends to zero with probability one: ever closer to zero, though never exactly zero.

Ergodicity

A quantity is ergodic if its average over time equals its average over the ensemble: the average one person obtains by playing long enough is the same number as the average obtained by countless people each playing once.

Wealth itself is not ergodic in the coin game: its expectation grows by 5 % a round while its time average falls by 5.13 % a round. But the change in the logarithm of wealth is ergodic: each round’s Δln x is an independent, identically distributed random variable, and its time average equals its expectation, E[Δln x] = 0.5 ln 1.5 + 0.5 ln 0.6 ≈ −0.0527. For a multiplicative process, the right question about what will happen to one person in the long run concerns the expectation of the logarithm, not the expectation of wealth.

Bernoulli’s logarithm, and the economists’ reply

In 1738, answering the “St Petersburg paradox” (a gamble with infinite expected value that nobody will pay much to play), Daniel Bernoulli proposed that people maximise not expected wealth but the expected “utility” of wealth, and suggested the logarithm of wealth as that utility (Bernoulli 1954). This is where expected utility theory begins.

For multiplicative gambles, Bernoulli’s logarithmic utility and Peters’ time average lead to the same decisions: both require maximising E[ln x]. What differs is the reason. Bernoulli’s logarithm is an assumption about preferences (an extra coin is worth more to the poor than to the rich); Peters’ logarithm comes from the dynamics themselves (in a multiplicative world, only the log growth rate is ergodic).

This book takes a practical position. Peters’ questions (“what dynamics does my wealth follow? do I care about the ensemble or about my own single path?”) are useful to any person or firm that lives only once; whether his view “replaces” expected utility theory is a theoretical question still under discussion, and the book does not rule on it.

Stake only part, and the outcome changes

The problem with the coin game is not that the gamble is bad, but that it stakes all of one’s wealth every round. If each round only a fraction f of wealth is staked, wealth is multiplied by 1 + 0.5f on heads and by 1 − 0.4f on tails.

# Two ways out of the coin game's paradox. (1) Dynamics: for ADDITIVE gains (win or lose
# fixed amounts) the time average equals the expectation; for MULTIPLICATIVE gains the
# time-average growth rate is the expected change in log wealth. (2) Stake only a
# fraction f of wealth each round: the factor becomes 1 + 0.5 f or 1 - 0.4 f.
import math

def g(f):          # time-average (log) growth rate per round when staking fraction f
    return 0.5 * math.log(1 + 0.5 * f) + 0.5 * math.log(1 - 0.4 * f)

print("additive version (+50 or -40 units a round): expected and time-average change +5")
print("multiplicative version: time-average growth per round = E[ln factor]")
print(f"{'fraction staked f':>18} {'growth per round':>18}")
for f in (0.0, 0.1, 0.25, 0.5, 0.75, 1.0):
    print(f"{f:18.2f} {100 * (math.exp(g(f)) - 1):+17.3f} %")
# maximise g: 0.25/(1+0.5f) = 0.2/(1-0.4f)  ->  f* = 0.25
fs = [i / 1000 for i in range(1000)]
best = max(fs, key=g)
zero = min((f for f in fs if f > 0.05), key=lambda f: abs(g(f)))
gb = 100 * (math.exp(g(best)) - 1)
print(f"best fraction {best:.3f} (exactly 1/4): {gb:+.3f} % a round")
print(f"growth falls back to zero near f = {zero:.3f}, twice the best fraction")
# A Bernoulli (log-utility) decision maker declines the whole-wealth game, because
# E[ln factor] < 0, but accepts staking a quarter of wealth.
print(f"E[ln factor]: whole wealth {g(1.0):+.4f}; a quarter of wealth {g(0.25):+.4f}")
# additive version (+50 or -40 units a round): expected and time-average change +5
# multiplicative version: time-average growth per round = E[ln factor]
#  fraction staked f   growth per round
#               0.00            +0.000 %
#               0.10            +0.399 %
#               0.25            +0.623 %
#               0.50            +0.000 %
#               0.75            -1.893 %
#               1.00            -5.132 %
# best fraction 0.250 (exactly 1/4): +0.623 % a round
# growth falls back to zero near f = 0.500, twice the best fraction
# E[ln factor]: whole wealth -0.0527; a quarter of wealth +0.0062

Staking a quarter of wealth each round turns the long-run growth rate from −5.13 % into +0.62 %; staking half makes it exactly zero, and staking more makes it negative. The best fraction, f = 1/4, is exactly what the Kelly criterion of Chapter 3 gives. It also shows that a decision maker with logarithmic utility (or one who cares about the time average) will refuse the “stake everything” game and accept the “stake a quarter” game: the same positive-expectation opportunity, depending on how one takes part.

Summary

In a multiplicative world the expected value describes the average of infinitely many parallel players, and the time average describes your own single path. The coin game’s expectation rises by 5 % a round while almost every player’s wealth shrinks by about 5.13 % a round. The log growth rate is ergodic and links the two; Bernoulli’s logarithmic utility and Peters’ time average give the same decisions for multiplicative gambles, and their theoretical status is still debated. Staking only part of one’s wealth turns the same gamble from certain decline into steady growth. The next chapter moves to continuous time and uses Itô’s lemma to show how volatility itself levies a tax on the growth rate.