The Secretary Problem

This note reanalyzes a non·zero-sum game version of the secretary problem recently treated by Kurano, Nakagami and Yasuda [3], under a modified formulation of the problem. The equilibrium derived differs from the former one and has an interesting assymptotic behavior which reconfirms a main theorem formerly proved by Presman and Sonin [4]. Th" equilibrium value in the limit is a positive number which is a unique root in (0, 1/2) of a transcendental equation. 1_ Introduction and Purpose In their paper [3 J, Kurano, Nakag8lTIi and Yasuda analysed a two-person game version of thE· so-called secretary problem in which the situation can be represented aE follows: There are two competitors and two identical sets of N applicants, and each player considers his set in the conditions of the "ellknown secretary problem, taking decisions independently of the other player_ As soon as one of the players stops, the other player is informed of this information and he drops out of the g8lTIe _ The goal of each player is to maximize the probability of win, Le. stopping at the best one in his set of objeets. In [3J the authors give a non-zero-sum game formulation and shows that the unique Nash-equibilium strategy for each player has a threshold character, i.e. each player must stop at the arrival of the first candidate (the obj ect which is best among those preceding it) positive integer satisfying m*NIl j=r.* after the m*-th, where m* is the smalle:,t -2 j <1. This result i<; clearly a non-interesting one in opportunity analysis since both of the threshold point and the equilibrium value tend to zero in the limit as N -+ 00 The purpose of this note is to rec.nalyze the above problem, showing first 287 © 1980 The Operations Research Society of Japan