Optimal stopping

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The theory of optimal stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of Statistics. A key example of an optimal stopping problem is the Secretary problem.

[edit] Definition

Stopping rule problems are associated with two objects:

A sequence of random variables $X 1, X 2,...$ , whose joint distribution is something assumed to be known
A sequence of 'reward' functions $(y_i)_{i\ge 1}$ which depend on the observed values of the random variables in 1.:
$y i = y i (x 1,.., x i)$

Given those objects, the problem is as follows:

You are observing the sequence of random variables, and at each step i, you can choose to either stop observing or continue
If you stop observing, you will receive reward yi
You want to choose a stopping rule to maximise your expected reward (or equivalently, minimise your expected loss)

[edit] Examples

Coin tossing ( $(y i)$ converges)

You have a fair coin and are repeatedly tossing it. Each time, before it is tossed, you can choose to stop tossing it and get paid (in dollars, say) the average number of heads observed.

You wish to maximise the amount you get paid by choosing a stopping rule. If X_i (for i ≥ 1) forms a sequence of independent, identically distributed random variables with distribution

$Binomial\left(1,\frac{1}{2}\right),$

and if

$y_i = E((\sum_{k=1}^{i} X_k)/i),$

then the sequences $(X_i)_{i\geq 1}$ , and $(y_i)_{i\geq 1}$ are the objects associated with this problem.

House selling ( $(y i)$ does not necessarily converge)
You have a house and wish to sell it. Each day you are offered $X_n for your house, and pay $k to continue advertising it. If you sell your house on day n, you will earn $ $y n$ , where $y n = (X n - n k)$ .
You wish to maximise the amount you earn by choosing a stopping rule.
In this example, the sequence (X_i) is the sequence of offers for your house, and the sequence of reward functions is how much you will earn.

Secretary problem ( $(X i)$ is a finite sequence)

Main article: Secretary problem

You are observing a sequence of objects which can be ranked from best to worst. You wish to choose a stopping rule which maximises your chance of picking the best object.
Here, if $R 1,..., R n$ (n is some large number, perhaps) are the ranks of the objects, and $y i$ is the chance you pick the best object if you stop intentionally rejecting objects at step i, then $(R i)$ and $(y i)$ are the sequences associated with this problem.