This is the final exam for Ergodicity & the Time Average. It pulls the whole course together: the two averages — the ensemble average taken across a crowd of players at a single instant, and the time average taken along one player’s own path — and the crucial fact that they only agree in an ergodic world. The flagship coin promises a cheerful per round on the ensemble mean yet quietly delivers ruin to the typical player, because multiplying shrinks a lone bankroll every pair of flips. The fix is not to chase the average across a crowd you will never be, but to optimise the average you actually live — take the log, grow at the arithmetic mean of log-returns, and above all survive the absorbing barrier first, because after zero there is nothing left to compound. Several questions look easy until you spot the trap: an EV that silently assumes you never hit zero, a “safe” percentage bet that is really multiplicative, or a slogan dressed up as a mechanism.
How this exam works
Read carefully — this exam is final. Each question appears one at a time. Once you submit an answer it is locked for good: there’s no going back, no retry, and no restart. Your score is hidden until the end, where you’ll see a pass/fail verdict. The pass mark is 70%. A few questions ask you to select all correct answers.
The whole course rests on two different averages. Which pair of definitions is correct?
Select an answer to continue.
Course Recap
Big picture
Ergodicity and the time average, in one picture
- Ergodicity and the time average
- Ergodic vs non-ergodic
- The ensemble average runs across a crowd at one instant while the time average runs along one path over many rounds, ergodic means they agree so one path samples the crowd, non-ergodic means they diverge so your path is not the crowd, and Boltzmann coined the word in statistical mechanics
- The flagship coin flip
- A fair coin that pays times one-and-a-half on heads and times zero-point-six on tails has an ensemble expected value of one-point-zero-five or plus 5 percent per round, yet a head-tail pair multiplies to one-and-a-half times zero-point-six equals zero-point-nine so the time average is the square root of zero-point-nine near zero-point-nine-four-nine or minus 5 percent, the crowd swells while the typical player drifts to zero as a right-skewed lognormal mean races past the median
- Additive vs multiplicative
- Adding fixed amounts is path-independent and ergodic so the ensemble expected value is your fate, multiplying by percentages of current wealth compounds and is non-ergodic, plus 50 percent then minus 40 percent is really times zero-point-nine or minus 10 percent, and taking the log turns products into sums so growth is the arithmetic mean of log-returns
- The Kelly fix
- Kelly maximises expected log-wealth E ln W which is the time-average growth rate, from Bernoulli in 1738 to Kelly in 1956, the stake is f-star equals bp minus q over b so b equals 1 and p equals zero-point-six gives 20 percent, the growth curve is hump-shaped so overbetting is strictly worse, ruin is refused for free because ln 0 is minus infinity, and half-Kelly keeps most growth for far less variance
- Ruin and the real world
- Zero is an absorbing barrier that nothing rebounds from and the ensemble expected value silently assumes you never touch it, a tiny per-period ruin chance compounds so one minus zero-point-nine-nine-nine to the thousandth is about zero-point-six-three, leverage lifts the ensemble mean but lowers growth and raises ruin, and insurance buys back ergodicity by paying a small certain premium to remove a ruinous tail even at negative expected value
- Where the model lies
- Not everything is non-ergodic since small bounded repeatable non-ruinous bets can be judged on the ensemble mean, non-ergodicity is about which average is decision-relevant and bites even a risk-neutral agent with Peters versus expected-utility a live debate, and to avoid an empty slogan demand the mechanism of multiplicativity a reachable absorbing barrier and diverging averages
- Ergodic vs non-ergodic
Key takeaways
Ergodicity is the difference between two averages: the ensemble average across a crowd at one instant, and the time average along your own path. When they agree the world is ergodic and the ensemble EV is your fate; when they diverge it is non-ergodic and your path is not a sample of the crowd. The flagship coin — fair, heads / tails — makes this vivid: the ensemble mean climbs at a round while the typical player bleeds toward zero, because and the time-average factor is . The engine is multiplicativity: adding fixed amounts is additive and ergodic, but multiplying by percentages compounds and is non-ergodic — which is why then is really , and why a loss needs a gain to recover. Take the log and products become sums, so the growth you live is the arithmetic mean of log-returns — exactly what Kelly (, maximising ) optimises, refusing ruin for free because , punishing overbetting, and rewarding half-Kelly caution. Above all, respect the absorbing barrier at zero: small per-period ruin odds compound, leverage trades growth for a higher ensemble mean, and insurance buys back ergodicity — a small certain premium removing a ruinous tail, rational even at negative EV. But do not over-apply it: small, bounded, non-ruinous bets are fine on the mean, non-ergodicity is about which average matters (not risk-aversion), and “it’s non-ergodic” is only real if you can show the mechanism — multiplicativity, a reachable barrier, and diverging averages.