Robbins' problem

In probability theory, Robbins' problem of optimal stopping, named after Herbert Robbins, is sometimes referred to as the fourth secretary problem or the problem of minimizing the expected rank with full information. Its statement is as follows.

Let X₁, ... , X_n be independent, identically distributed random variables, uniform on [0, 1]. We observe the X_k's sequentially and must stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value?

The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not only on simpler sufficient statistics of these. Only bounds are known for the limiting value v as n goes to infinity, namely 1.908 < v < 2.329. It is known that there is some room to improve the lower bound by further computations for a truncated version of the problem. It is still not known how to improve on the upper bound which stems from the subclass of memoryless threshold rules.

One of the motivations to study Robbins' Problem is that with its solution all classical (four) secretary problems would be solved. But the major reason is to understand how to cope with full history dependence in a (deceptively easy-looking) problem. On the Ester's Book International Conference in Israel (2006) Robbins' Problem was accordingly named one of the four most important problems in the field of optimal stopping and sequential analysis.

Herbert Robbins presented the above described problem at the International Conference on Search and Selection in Real Time in Amherst, 1990. He concluded his address with the words I should like to see this problem solved before I die. Scientists working in the field of optimal stopping have since called this problem Robbins' Problem.

If the problem to the outcome has a very simple equation, all probable outcomes lead to Infinity the Ultimatum can be defined as the end of the life cycle.

Attributes consist of (Xi), (Xn) where (Xi is greater then Xn)

Within an equation you have an X amount of (Xi)s and an X amount of (Xn)s

The Probability (P) known and then there is the UnKnown (K) both outcomes attribute each other during the lifecycle

The Probability (P) stands for the probable outcome P(A) then the probable outcome can either have -0≤(P)≥1

The probabililty of (P)≥(PA)

Based on the formula of probability we can identify relevance (R)

Another factor that can be added into the equation unknown (K)

(V)≥(R - K)

then there are the variables (V) that define relevance (R)

attributes = Xi ≤(A)≥ Xn Ultimatum= (U) Infinity = ∞

The lifecycle ends with the path impacted most by either (Xi),(Xn)

The path is defined by the attributes that influence the path

Xi = attributes (Negative) Xn = attributes (Positive)

then you have the ultimatum or the final outcome (U) and you have Infinity (I) the direction can be calculated through the impact of (X1),(Xn) on various touch points through the lifecycle

Event (A) taking place at the time will take into account (Xi),(Xn) as what defines the direction to the final outcome

Direction (D) can be understood over the lifecycle based on the attributes that have taken place during the direction

Based on the attributes of (P),(K) The direction can be defined leading to the final outcome.

the direction can be defined through variables that equal to (R)

Once the lifecycle reaches its end we can calculate its impact through (P)-(K)+Xi/Xn

The optimal equation:

so the journey leading up to the collection of information will define:

(Optimal stoping)= An-Ci / Xn-Xi = (U) = (I) (Optimal) = ((An)+(Xn)-(K)-(Ci)

The ultimatum can be defined at the end of the lifecycle.

(CiK) = Compromise the Unknown

(An) = Absolute (Ci)= Compromise

Optimal = (Xi)-(Xn)+(Ci) Optimal stopping = (Xn)+(Xn)-(CiK)= (XnCi) + (PA)+ (Relevance) Fully Optimal = (Xn)+(Xn)+(K)= (An)

Relavance on K must be ignored completely.

We can then use this formula to rank the lifecycle up until the lifecycle comes to an end.

By spending more on (Xi) the outcome will be negative

However when focusing on (Xn) the outcome will be positive

(U) = ∞ (Information) Optimal, Optimal stopping, Fully Optimal

(U) where (U) merges with information

The outcome can be broken down into percentages based on True Value

Outcome (OC) = (P)+(Xn)-(Xi)+(Time and Relevance)-(K)

The Outcome factors in the law of probability, UnKnown causes Negatives, Positives, Time, Relevance

The direction of the outcome will be determined (Xn), (Xi) over through the lifecycle by removing (K) out of the equation

(U) = change

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