A gambler plays a fair coin-flip game, winning or losing $1 each round. Before each round they may look at the entire history of outcomes and apply any stopping strategy they choose. According to the optional stopping theorem, what are their expected total winnings when they stop?
APositive — a sufficiently clever strategy can exploit patterns in the history to generate positive expected value
BZero — regardless of the stopping strategy, expected wealth at any bounded stopping time equals the initial wealth
CNegative — variance accumulates over time, reducing expected returns
DIndeterminate — expected winnings depend on which specific stopping strategy is used
The optional stopping theorem states E[M_T] = E[M₀] for a bounded stopping time T. Since the gambler's wealth is a martingale (fair game), no stopping strategy — however clever — can change the expected value. The 'martingale betting strategy' (doubling bets after each loss) is the most famous misconception here: it seems to guarantee profit but requires unbounded wealth and time. With realistic constraints, expected winnings remain zero.
Question 2 Multiple Choice
The squared wealth process M²ₙ for a symmetric random walk (win/lose $1) — is it a martingale?
AYes — squaring preserves the fairness of the game
BYes, but only if the walk starts at zero
CNo — M²ₙ is a submartingale (tends to increase), but M²ₙ − n is a martingale after compensation
DNo — M²ₙ is a supermartingale because variance growth makes future values tend to be smaller than present values
E[M²_{n+1} | ℱₙ] = E[(Mₙ ± 1)² | ℱₙ] = M²ₙ + 1 > M²ₙ. So M²ₙ has a positive expected increment — it is a submartingale (satisfies the ≥ condition). Subtracting the compensating term n gives M²ₙ − n, which has zero expected increment and is therefore a true martingale. This illustrates the general pattern: important processes often become martingales only after an appropriate centering or compensation.
Question 3 True / False
A martingale's defining property is that, given the current state and all past history, the best prediction for the next value is simply the current value.
TTrue
FFalse
Answer: True
This is exactly the martingale condition: E[M_{n+1} | ℱₙ] = Mₙ. The filtration ℱₙ encodes all information up to time n. Note that this does not mean future values are deterministic — they can be highly variable. It means the expected change is zero. In a symmetric random walk, you genuinely don't know where you'll be next, but your best single guess is where you are now.
Question 4 True / False
A martingale bounded in L¹ converges in L¹ (in mean) to a limit.
TTrue
FFalse
Answer: False
The martingale convergence theorem guarantees almost sure convergence for a martingale bounded in L¹ — but almost sure convergence does not imply L¹ convergence. L¹ convergence additionally requires uniform integrability. A classic counterexample involves the Doob martingale for a branching process that becomes extinct: it converges almost surely to 0 but not necessarily in L¹. This is a subtle but important distinction in probability theory.
Question 5 Short Answer
In your own words, explain why the optional stopping theorem implies that no betting strategy can yield a positive expected return in a fair game.
Think about your answer, then reveal below.
Model answer: A fair game is modeled as a martingale: E[M_{n+1} | ℱₙ] = Mₙ, meaning expected future wealth always equals current wealth, regardless of history. A betting strategy with a stopping rule chooses when to quit based on observed history — this is a stopping time T. The optional stopping theorem says that for a bounded stopping time, E[M_T] = E[M₀]. Since E[M₀] is your initial wealth, the expected wealth when you stop equals what you started with. No strategy, however cleverly designed, can change this expectation.
The key insight is that stopping a martingale at a cleverly chosen time doesn't change its expected value. You cannot exploit a fair process by choosing when to observe it. This is also why casino games, once they are unfavorable (supermartingales), cannot be beaten by stopping strategies alone — the expected loss is baked into the process itself.