A non-zero-sum no-information best-choice game

摘要

A given number of n applicants are to be interviewed for a position of secretary. They present themselves one-by-one in random order, all n! permutations being equally likely. Two players I and II jointly interview the i-th applicant and observe that his (or her) relative rank is y for I and z for II, relative to i - I applicants that have already seen (rank I is for the best). Each player chooses one of the two choices Accept or Reject. If choice-pair is R-R, then the i-th is rejected, and the players face the next i + 1-th applicant. If A-A is chosen, then the game ends with payoff to I's (II), the expected absolute rank under the condition that the i-th has the relative rank y (z). If players choose different choices, then arbitration comes in, and forces players to take the same choices as I's (II's) with probability p (<(p)over bar> = 1-p). Arbitration is fair if p = 1/2. If all applicants except the last have been rejected, then A-A should be chosen for the last. Each player aims to minimize the expected payoff he can get. Explicit solution is derived to this it stage game, and numerical results are given for some n and p. The possibility of an interactive approach in this selection problem is analyzed.