The Secretary Problem with a Selection Committee: Do Conformist Committees Hire Better Secretaries?

This paper analyses a variation of the secretary problem in which two selectors, with different fields of interest, each want to appoint one of the n candidates with as much expertise as possible in their field. Selectors simultaneously vote to accept or reject: Unanimous decisions are respected and candidates with a split decision are hired with probability p. Each candidate arrives with expertises x and y in the two fields, uniformly and independently distributed on [0,1] and observable to both selectors. If a candidate with expertise pair (x, y) is hired by unanimous decision, the payoffs to the selectors are simply x and y. However in order to model the level of conformity in the firm, we deduct a positive ’consensus cost’ c from the utility of a selector who has rejected a candidate who is nevertheless hired. We show (Theorem 1) that each stage game has a unique equilibrium in which there are two thresholds, z < v, and say selector I accepts candidate (x, y) if x > v or x > z and y > v. We show that for p and c sufficiently large, decisions are unanimous and that as the number n of candidates goes to infinity the equilibrium value of the game goes to the golden mean. We show that as the consensus cost c increases from 0, this hurts the selectors (Theorem 4) but helps the firm (Theorem 6), whose utility from hiring candidate (x, y) is a weighted average of x and y. Thus a little conformity is good for the firm.