I have a question in the secretary problem.
The problem is "Suppose that an employer seeks to fill a vacancy for which there are N candidates. Candidates are interviewed sequentially and following each interview, the employer must decide whether or not to offer the position to the current candidate. If the employer does not offer them the position, then that individual is removed from the pool of potential candidates and the next candidate is interviewed. There are many variations on this problem, but here we will assume that the candidates can be unambiguously ranked from best to worst and that the employer’s objective is to maximize the probability of offering the job to the most-preferred candidate. We can reformulate the problem as follows. Suppose that a collection of N objects is ranked from 1 to N and that the lower the ranking the more preferred the object is. An observer encounters these objects sequentially, one at a time, in a random order, where each of the N! possible orders is equally likely to occur. Although the absolute rank of each object is unknown to the observer, the relative ranks are known without error, e.g., the observer can tell whether the second object encountered is more or less preferred than the first object, but the observer cannot determine at that stage whether the first or second object is the best in the entire group of N if N > 2."
In this kind of case, how can I describe the transition probabilities p_t(j|s)?