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In the statistics, the game theory and the decision theory the secretary problem (also admits as marriage problem) designates the goal to select from a row successively regarded candidates of different quality the best. A refusal is irrevocable. Because of the contained coincidence elements the problem is usually formulated in such a way to determine the largest probability to select the best offer.

Problem

The usually-quoted example is that an organization, which would like to adjust a secretary. The applicants call successively; in the investigation an order of rank can to be set up and the qualities of each applicant be able to be held. However a rejected applicant separates finally and is not available in the further process any longer, a premise, which contradicts the actual personnel occupation reality. Another formulation of the problem goes from the choice of a marriage partner out of a set of candidate out this problem coating is close-to-reality. The problem definition includes also that the probability is to be maximized to select the best in each case applicant. Is instead the expectancy value related to all whom are applicable candidates to be maximized, a deviating strategy would be necessary.

Strategy

The problem has a very simple strategy, which is also still optimal in addition: regard the first r candidates with 1 \ leq r<n - and reject it. Select from the remaining NR applicants first, which is better than each of the first R. it leaves itself to show that for large n the optimal value for r results from r \ approx n/e, whereby e is the basis of the Naperian logarithm (Euler number). With this strategy the probability is appropriate for the best candidates to select with 1/e. For smaller values of n this probability is ever higher.

Applicability in practice

The practical applicability of the problem might be smaller, than one will first assume. Problematic it is first that, in order the optimal "stop number" determined to be able, from the outset admits be must, how high n is. This may be still possible, if a being certain number of application discussions is agreed upon, in the coating of the marriage problem is however difficult this prognosis.

Further the secretary problem presupposes that the quality of the candidates is coincidental and independent of the place number. Also this condition may be perhaps given in the case of application. A counter example is however the marriage problem like exemplary under taking as a basis traditional, outdated role pictures is shown: With increasing age of the woman the quality of the potentially interested partners will tendentious sink, on the other hand with an increasing prosperity of the man and/or an improvement of its social position the quality of its potentially interested will partner inside possibly rise. Corresponding would be to be guessed/advised concerning this case of the woman to a weighted decrease of the stop number, to the man rather to an increase of the stop number.

A further problem is however that alone due to the out-hesitated decision phase a longer time without problem solution must be overcome, which can lead to welfare losses.

A further difficulty is the fact that the solution of the secretary problem thereupon is optimized to select the very best solution for which a comparatively high probability is taken in purchase that finally the place-number-moderately last, possibly clearly worse solution must be taken in purchase. In practice, in which a such optimization is often unnecessary for best solution, a risikoaversere strategy might be often more favorable. If thus before reaching a solution appears even objectively good about already to the stop number as good compromise or, then this solution in deviation from the strategy should be selected immediately.

Variants

The problem was examined in different variants, under it:

  1. The choice may fall on two candidates, instead of only on one
  2. The number of the applicants is unknown
  3. Equivalent candidates are of importance
  4. Rejected applicants separated not finally
  5. The choice may fall also on the secondarybest applicant

See also

Used topics, with which one can make the optimal decision of the remainder problem from partial information:

  • Odds strategy
  • Conversion paradox
  • Goat problem
  • Two-note play

Literature

  • F. Thomas Bruss: Strategies of the best choice. In: Spektrum der Wissenschaft. 05/2004. Spektrum der Wissenschaft publishing house company, P. 102-104

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