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Stopping RuleIn decision theory, a stopping rule is a mechanism for deciding whether to continue or stop a process on the basis of the present position and past events, and which will almost always lead to a decision to stop at some time, known as a stopping time. As an example, consider a gambler playing roulette, starting with $100: - Playing until she either runs out of money or has played 500 games is a stopping rule.
- Playing until she doubles her money (borrowing if necessary if she goes into debt) is not a stopping rule, as there is a positive probability that she will never double her money.
- Playing until she either doubles her money or runs out of money is a stopping rule, even though there is potentially no limit to the number of games she plays, since the probability that she stops in a finite time is 1.
- Playing until she is the maximum amount ahead is not a stopping rule, as it requires information about the future as well as the present and past.
The theory of stopping rules and stopping times can be analysed in probability and statistics, notably in the optional stopping theorem. This says that under certain conditions, the expected value of a martingale at a stopping time is equal to its initial value.
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