Stochastic Processes · 免费预览
赌博策略与停时
在不利的博弈中，财富会逐渐缩水；翻倍策略之所以失败，是因为其期望损失等于期望盈利；可预测的下注系统无法把亏钱的博弈变得公平；而在恰当选择的时刻停止，则能以停时形式保持鞅性质。
1. Expectations drift the wrong way
Expectations drift the wrong way
2. The doubling strategy, an autopsy
The doubling strategy, an autopsy
3. You cannot bet your way out
You cannot bet your way out
4. Stopping honestly
Stopping honestly
5. Which way the drift goes
What happens to the expected fortune over time in a supermartingale?
6. The doubling trick
In the doubling strategy, why does every won round show a net profit of exactly one dollar?
7. Where the doubling strategy hides its cost
Over ten fair plays, why do the doubling strategy's expected losses equal its expected winnings of five dollars?
8. Predictable means no peeking
What makes a betting strategy H n predictable or admissible?
9. The betting-system theorem
What does the theorem say about applying a bounded nonnegative predictable strategy to a supermartingale?
10. Quitting without cheating
Which quitting rule is a legitimate stopping time?
11. Four losses then a win
A doubling-strategy gambler loses the first 4 plays and then wins the fifth. Compute the size of the fifth bet in dollars and write the number.
12. The winnings side of the ledger
Over 10 independent fair plays, the number of rounds the doubling gambler wins is binomial with parameters 10 and one half. Compute the expected winnings in dollars and write the number.
13. The disaster scenario
If the doubling gambler loses all 10 plays, the total loss is 2 to the 10th minus 1 dollars. Write this loss in dollars as a number.