The Secretary Game

Ball 1 Ball 2 Ball 3 Ball 4 Ball 5 Ball 6 Ball 7 Ball 8 Ball 9 Ball 10 Ball 11 Ball 12 Ball 13 Ball 14 Ball 15 Ball 16 Ball 17 Ball 18 Ball 19 Ball 20 Ball 21 Ball 22 Ball 23 Ball 24 Ball 25
Ball 1 Ball 2 Ball 3 Ball 4 Ball 5 Ball 6 Ball 7 Ball 8 Ball 9 Ball 10 Ball 11 Ball 12 Ball 13 Ball 14 Ball 15 Ball 16 Ball 17 Ball 18 Ball 19 Ball 20 Ball 21 Ball 22 Ball 23 Ball 24 Ball 25
Distribution graph

Description

In the secretary problem, there are \(n\) candidates, totally ranked from best to worst, with no ties. The candidates arrive sequentially, in random order. We can not observe the absolute ranks of the candidates as they arrive, only the relative ranks. Our goal is to choose the best candidate; any other outcome is failure.

In the secretary game, the candidates are represented as balls. The app has buttons for starting a new game, rejecting a candidate, and accepting a candidate. The labels on the balls show the relative ranks of the candidates (1 is best). Candidates with relative ranks greater than 1 (non-optimal) are colored red; the candidate with relative rank 1 (the best so far) is colored green. Once a candidate is accepted, the game is over and the second row of balls shows the absolute ranks, again with non-optimal candidates red and the best candidate green. The data table and data graph show the relative frequency of successes and failures. The number (arrival order) of the selected candidate \( X \), the number of the best candidate \( Y \), and the outcome (success or failure) \( W \) are recorded in the record table for each game. The number of candidates can be varied with the input control.