Optimal Stopping


  • Optimal Stopping Problems; One-Step-Look-Ahead Rule
  • The Secretary Problem.
  • Infinite Time Stopping


An Optimal Stopping Problem is an Markov Decision Process where there are two actions: a=0 meaning to stop, and a=1 meaning to continue. Here there are two types of costs

This defines a stopping problem.

Assuming that time is finite, the Bellman equation is

for s\in \mathbb N and C_0(x)=k(x).

Def 1. [OLSA rule] In the one step lookahead (OSLA) rule we stop when ever x\in S where

We call S the stopping set. In words, you stop whenever it is better stop now rather than continue one step further and then stop.

Def 2. [Closed Stopping Set] We say the set S\subset {\mathcal X} is closed, it once inside that said you cannot leave, i.e.

Ex 1. Given the set S is closed, argue that if C_{s-1}(x)=k(x) for x\in S then C_s(x)=k(x).

Ans 1. If x\in S then since S is closed \hat{X} \in S. In otherwords C_{s-1}(\hat{X}) =k(\hat{X}). Therefore, in this case, Bellman’s equation becomes

The last inequality above follows by the definition of x\in S.

Ex 2. [OS:Finite] If, for the finite time stopping problem, the set S given by the one step lookahead rule is closed then the one step lookahead rule is an optimal policy. (Hint: Induction on s.)

Ans 2.  The OSLA rule is optimal for s=1 steps, since OSLA is exactly the optimal policy for one step.

Suppose that the result is try for upto s-1 steps. Now consider the Optimal Stopping Problem with s steps.

If x\notin S, then clearly it’s better to continue.

Ex 3. [The Secretary Problem] There are N candidates for a secretary job. You interview candidates sequentially. After each interview, you must either accept or reject the candidate. We assume each candidate has the rank: 1,2,...,N And arrive for interview uniformly at random. Find the policy that maximises the probability that you hire the best candidate.

Ans 3.  All that matters at each time is if the current candidate is the best so far. At time t let

Since is uniform random where the best candidate is

Under the chosen policy, we let

be our reward function. Now

Thus the Bellman equation for the above problem is

Notice that R_t \geq R_{t+1}. Let t^* be the smallest t such that R_{t^*} \geq (t^*-1)/N . Starting from R_{N+1}=0 note that so long as R_{t+1} < \frac{t}{N} holds in second case in the above expression, we have that

Thus

Thus our condition for the optimal t^* is to take the smallest t^* such that

In other words, the optimal policy is to interview the first t^* candidates and then accept the next best candidate.

Ex 4. [The Secretary Problem, continued] Argue that as N\rightarrow \infty, the optimal policy is to interview e^{-1} of the candidates and then to accept the next best candidate.

Ans 4. From [3], the optimal condition is

We know that as N\rightarrow \infty

Thus \log N - \log t^* \geq 1 for t^*/N =e^{-1}.

Ex 5. You look for a parking space on street, each space is free with probability p=1-q. You can’t tell if space is free until you reach it. Once at space you must decide to stop or continue. From position s (s spaces from your destination), the cost of stopping is s. The cost of passing your destination without parking is D.

Ans 5. Let

Here stopping means take the next free parking space. The Bellman equation is

Consider the stopping set

where k(s) is the cost of taking the next available space. Note that

Recursion of this form have solution

Therefore

Since s is decreasing, this set if clearly closed. Therefore the optimal policy is to take the next available space once (Dp+1) q^s \geq 1 holds.

Ex 6. In a game show a contestant is asked a series of 10 questions. For each question q=1,...,10 there is a reward r_q for answering the question correctly. With probability p_q the contestant answers the question correctly. After correctly answering a question, the contestant can choose to stop and take their total winnings home or they can continue to the next question q+1. However, if the contestant answers a question incorrectly then the contestant looses all of their winnings. The probability of winning each round is decreasing and is such that the expected reward from each round, p_q r_q, is constant.

i) Write down the Bellman equation for this problem.

ii) Using the One-Step-Look-Ahead rule, or otherwise, find the optimal policy of the contestant.

Ans 6. The Bellman equation for this problem is

The stopping set for this problem is

Since by assumption r_t p_t = c and p_t \searrow 0 (therefore c/1-p_t) is decreasing) and x_t \nearrow the set S is closed. Thus the OSLA rule is optimal for this problem.

Ex 7. [Whittle’s Burglar] A burglar robs houses over N nights. At any night the burglar may choose to retire and thus take home his total earnings. On the tth night house he robs has a reward r_t where r_t is an iidrv with mean \bar{r}. Each night the probability that he is caught is p and if caught he looses all his money. Find the optimal policy for the burglar’s retirement. (Hint: OLSA)

Ex 7. [Bruss’ Odds Algorithm] You sequentially treat patients t=1,...,T with a new trail treatment. The probability of success is p_t=1-q_t. We must minimize the number of unsuccessful treatments while treating all patients for which the trail is will be successful. (i.e. if we label 1 for success and 0 for failure, we want to stop on the last 1). Argue, using the One-Step-Look-Ahead rule that the optimal policy is the stop treating at t^* the largest integer such that

This procedure is called Bruss’ Odds Algorithm.


Optimal stopping in infinite time

We now give conditions for the one step look ahead rule to be optimal for infinite time stopping problems.

Ex 8. [OS:Converge] If the following two conditions hold

  • K = \max_{x} k(x) < \infty
  • C= \min_{x} c(x) > 0

then

Ans 8. An optimal policy \pi exists by Thrm [IDP:NegBellman]. Suppose that the optimal policy \pi stops at time \tau then

Therefore if we follow optimal policy \pi but for the s time horizon problem and stop at s if \tau \geq s then

Thus L_s(x) \rightarrow L(x) as required.

Ex 9. [continued] Suppose that from the one step lookahead that

Then it is optimal to stop if and only if x\in S.

Ans 9. As before (for the finite time problem), it is no optimal to stop if x\notin S and for the finite time problem L_s(x) = k(x) for all x\in S. Therefore by [[OS:Converge]] , since L_s(x) \rightarrow L(x) we have that L(x)=k(x) for all x\in S and there for it is optimal to stop for x\in S.

Ex 10. You own a “toxic” asset its value, x_t at time t, belongs to \{1,2,3,...\}. The daily cost of holding the asset is x_t. Every day the value moves up to x+1 with probability 1/2 or otherwise remains the same at x. Further the cost of terminating the asset after holding it for t days is C(1-\alpha)^t. Find the optimal policy for terminating the asset.


The one step lookahead rule is not always the correct solution to an optimal stopping problem.

Def 3. [Concave Majorant] For a function r:\{0,...,N\}\rightarrow \mathbb R a concave majorant is a function G such that

  • G(x)\geq \frac{1}{2} G(x-1) + \frac{1}{2} G(x+1)
  • G(x) \geq r(x).

Ex 11. [Stopping a Random Walk] Let X_t be a symmetric random walk on \{0,...,N\} where the process is automatically stopped at 0 and N. For each x\in \{ 0,...,N\}, there is a positive reward of r(x) for stopping. We are asked to maximize

where T is our chosen stopping time. Show that the optimal value function is a concave majorant.

Ans 11. The Bellman equation is

with R(x)=r(0) and R(N)=r(N). Thus the optimal value function is a concave majorant.

Ex 12. Show that is R_{s}(x) is the optimal reward starting from x and stopping before s steps (here R_0(x)=0). Then R_s(x) \leq G(x) for any concave majorant.

Ans 12. We will show that the optimal policy is the minimal concave majorant of r(x). We do so by, essentially applying induction on value iteration.

First R_0(x)=0 \leq G(x) for any concave majorant of r(x). Now suppose that R_{s-1}, the function reached after s-1 value iterations, satisfies R_{s-1}(x)\leq G(x) for all x, then

Ex 13. Show that the optimal value function V(x) is the minimal concave majorant, and that it is optimal to stop whenever V(x)=r(x).

Ans 13. Since value iteration converges R_s(x) \nearrow V(x), where V(x) satisfies V(x) \leq G(x), as required. Finally observe that the optimal stopping rule is to stop whenever V(x)=r(x) for the minimal concave majorant.

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