Bayes ’ Model of the Best-Choice Problem with Disorder

We consider the best-choice problemwith disorder and imperfect observation. The decision-maker observes sequentially a known number of i.i.d random variables from a known distribution with the object of choosing the largest. At the random time the distribution law of observations is changed. The random variables cannot be perfectly observed. Each time a random variable is sampled the decision-maker is informed only whether it is greater than or less than some level specified by him. The decision-maker can choose at most one of the observation. The optimal rule is derived in the class of Bayes’ strategies.


Introduction
In the papers we consider the following best-choice problem with disorder and imperfect observations.A decision-maker observes sequentially n iid random variables ξ 1 , . . ., ξ θ−1 , ξ θ , . . ., ξ n .The observations ξ 1 , . . ., ξ θ−1 are from a continuous distribution law F 1 x state S 1 .At the random time θ, the distribution law of observations is changed to continuous distribution function F 2 x i.e., the disorder happen-state S 2 .The moment of the disorder has a geometric distribution with parameter 1 − α.The observer knows parameters α, F 1 x , and F 2 x , but the exact moment θ is unknown.
At each time in which a random variable is sampled, the observer has to make a decision to accept and stop the observation process or reject the observation and continue the observation process .If the decision-maker decided to accept at step k 1 ≤ k ≤ n , she receives as the payoff the value of the random variable discounted by the factor λ k−1 , where 0 < λ < 1.The random variables cannot be perfectly observed.The decision-maker is only informed whether the observation is greater than or less than some level specified by her.

International Journal of Stochastic Analysis
The aim of the decision-maker is to maximize the expected value of the accepted discounted observation.
We find the solution in the class of the following strategies.At each moment k 1 ≤ k ≤ n , the observer estimates the a posterior probability of the current state and specifies the threshold s s n−k .The decision-maker accepts the observation x k if and only if it is greater than the corresponding threshold s.
This problem is the generalization of the best-choice problem 1, 2 and the quickest determination of the change-point disorder problem 3-5 .The best-choice problems with imperfect information were treated in 6-8 .Only few papers related to the combined best-choice and disorder problem are published 9-11 .Yoshida 9 considered the fullinformation case and found the optimal stopping rule which maximizes the probability that accepted value is the largest of all θ m − 1 random variables for a given integer m.
Closely related work to this study is Sakaguchi 10 where the optimality equation for the optimal expected reward is derived for the full-information model.In 11 , we constructed the solution of the combined best-choice and disorder problem in the class of single-level strategies, and, in this paper, we search the Bayes' strategy which maximizes the expected reward in the model with imperfect observation.

Optimal Strategy
According to the problem the observer does not know the current state S 1 or S 2 .But she can estimate the state using the Bayes' formula: Here, s s i is the threshold specified by the decision-maker within i steps until the end i.e., at the step n − i , π is the a prior probability of the state S 1 i.e., before getting the information that x ≤ s , F π s πF 1 s πF 2 s , and π 1 − π.We use the dynamic programming approach to derive the optimal strategy.Let v i π be the payoff that the observer expects to receive using the optimal strategy within steps until the end.The optimality equation is as follows: Simplifying 2.2 , we get Here, The following theorem gives the presentation of the expected payoff in linear form on π.

Theorem 2.1. For any i the function v i π can be written in the form
where Proof.Using the formula 2.3 , one can show that where Assume the theorem is correct for certain i k.Then, for i k 1 where

2.9
The theorem is proved.
The following lemma takes place.
Proof.It is obvious that the sequence v i π is increasing by i.

International Journal of Stochastic Analysis
Now, we prove that the sequence of the expected payoffs has an upper bound.

2.10
Further one can show using the induction that for any i ≥ 1 and any 0 ≤ π ≤ 1 the expected payoff at the step i has the upper bound The lemma is proved.
Corollary 2.3.Theorem 2.1 and the lemma yield that there are such A and B that

2.12
As i → ∞ the expected payoff satisfies the following equation:

2.13
To find the components of the expected payoff for a case of huge number of observation we should solve the following equation:

2.15
The solution of the system is as follows

2.16
The expected payoff is v π max s πA B 2.17 and the optimal threshold is s s π arg max s πA B .

2.18
The above results are summarized in the following theorem.
Theorem 2.4.For i → ∞, the solution of 2.3 is defined as where 2.20

Examples
Consider the examples of using the Bayes' strategy B defined by the formula 2.18 comparing with two strategies with constant thresholds that do not depend on π.

Normal Distribution
Consider the example of the normal distribution of the random variables where functions F 1 x and F 2 x have the variance σ 2 1 and the expectation μ 1 10 and μ 2 9, respectively.Strategies A 1 and A 2 with constant thresholds defined by the following formula: where F s ≡ F 1 s and E s ≡ E 1 s for the strategy A 1 ; F s ≡ F 2 s and E s ≡ E 2 s for the strategy A 2 .
The values of the thresholds of strategies A 1 and A 2 depending on discount rate are tabulated in Table 1.
Table 1 shows how much the discount rate is affect on the thresholds.Figure 1 shows the graphics of the optimal thresholds for strategies A 1 and A 2 s 1 and s 2 , resp.and strategy B s opt depending on π.As the figure shows, the strategy B depends   on the a posterior probability of the state S 1 π .As π tends to zero, the optimal threshold of the strategy B tends to threshold s 2 .
We compare the payoffs that the observer expects to receive using different strategies.Define V α as the expected payoff for π 1 and depending on probability of disorder α.
Figure 2 shows the numerical results of the expected payoffs of the observer who uses the strategies A 1 , A 2 , and B thresholds s 1 , s 2 , and s opt , resp. .The expected payoff of the observer who uses the Bayes' strategy B is greater if she uses one of the strategies A 1 or A 2 .The difference is significant for α ∈ 0.75, 0.98 , because of uncertainty of the current state of the system.Table 2 shows the numerical results of the main characteristics of the best-choice process.
For the small probability of the disorder 1 − α 0.1 , the expected payoff according to the strategy A 2 is greater 10.429 than according to the strategy A 1 10.035 .But the Bayes' strategy B that depends on π gives the largest expected payoff 10.500 .
Table 2 shows that the average time of accepting the observation is increasing with respect to the value of the threshold.Note that the strategy A 1 does not depend on the disorder and this leads to a high value of the average time of accepting the observation.Both strategies A 2 and B have a small average time of accepting the observation.

Exponential Distribution
Consider the example of the exponential distribution of the observations.Let F 1 x and F 2 x have the exponential distribution with parameters λ 1 0.5 and λ 2 1, respectively.As in the previous example, consider the strategies A 1 and A 2 comparing with the Bayes' strategy B, s E s 1 − λF s , 3.2 where F s ≡ F 1 s and E s ≡ E 1 s for the strategy A 1 ; F s ≡ F 2 s and E s ≡ E 2 s for the strategy A 2 .Table 3 shows the values of the thresholds for the strategies A 1 and A 2 depending on the discount rate.
The value of the optimal threshold of the strategy B as in the case of the normal distribution of the observations is increasing by π and equal to the threshold of the strategy A 2 at π 0. The graphics of the expected payoffs have the same view as in Figure 2. Table 4 shows the main characteristics of the best-choice process for different strategies.As in the previous example, the Bayes' strategy gives better payoff than the strategy A 2 , but it has bigger average time of accepting the observation.The strategy A 1 is the worst for all the parameters.

Figure 2 :
Figure 2:Expected payoffs of the observer who uses the strategies A 1 , A 2 and B for α 0.9, λ 0.99.

Table 1 :
The values of the thresholds of strategies A 1 and A 2 .

Table 2 :
Main characteristics of the best-choice process for α 0.9, λ 0.99.

Table 3 :
The values of the thresholds of strategies A 1 and A 2 .

Table 4 :
Main characteristics of the best-choice process for α 0.9, λ 0.99.