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The Job-search Problem with Incomplete Information
Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 55 (2015) 159 – 164
Information Technology and Quantitative Management (ITQM 2015)
The job-search problem with incomplete information Vladimir Mazalova , Elena Konovalchikovab a Institute
of Applied Mathematical Research, Karelian Research Center of RAS, ul. Pushkinskaya, 11, Petrozavodsk, 185910, Russia b Transbaikal State University, ul. Alekzavodskaya, 30, Chita, 672039, Russia
Abstract This paper considers a best-choice game with incomplete information associated with the job search problem. Players (experts) observe a sequence of independent identically distributed random variables (xi , yi ), i = 1 . . . , n, which represent the quality of incoming objects. The first component is announced to the players and the other one is hidden. The players choose an object based on known information about it. The winner is the player having a higher sum of the quality components (total quality) than the opponent. And finally, the optimal strategies of the players are derived. c 2015 2015 Published The Authors. Published Elsevier B.V.access article under the CC BY-NC-ND license © by Elsevier B.V.by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of ITQM2015. Peer-review under responsibility of the Organizing Committee of ITQM 2015
Keywords: best-choice game, incomplete information, equilibrium, threshold strategy
1. Introduction In this paper, we study a best-choice game associated with the job search problem. Here m employers (experts) have to make a single choice from a number of items (objects) presented sequentially. The job search problem possesses a long history and a series of alternative names including the secretary problem, the marriage problem and the asset-selling problem. As the secretary problem [1], it has many statements and variations. For instance, the number of items is finite or infinite, a decision-maker knows the actual value of each item as it is presented or its relative rank among presented items only. In some models, items are random variables with a known probability distribution function. Several authors considered the case of a given probability distribution with unknown parameters [4]. The game-theoretic setting of this problem was developed in [8]. Different generalizations of bestchoice games were introduced in the papers [6, 7, 10, 12-14]. There are few models devoted to mutual best-choice games, where both groups of game participants (experts and candidates) make their choice [3, 9, 11]. In what follows, we explore a two-player best-choice game with incomplete information. Assume that two experts observe a sequence of independent identically distributed random variables (xi , yi ), i = 1 . . . , n, which represent the quality of incoming objects. The first component is announced to the players and the other component is hidden. In a practical interpretation, the first component reflects professional skills of a candidate and the second one describes its computing skills. Each expert has to maximize the sum xi + yi of the quality components (the so-called total quality) of the selected candidate. Within the game-theoretic framework, a player strives to choose a candidate with higher total quality than the candidates selected by other players. ∗ Vladimir Mazalov. Tel.: +7-814-278-1108; fax: +7-814-276-6313. E-mail address: [email protected].
1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ITQM 2015 doi:10.1016/j.procs.2015.07.025
160
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
First, consider the model, where player 1 has a priority to make a choice. Suppose that random variables (xi , yi ), i = 1 . . . , n, are uniformly distributed on the segment [0, 1]. At step 1, expert 1 observes the value x1 from the pair (x1 , y1 ) describing candidate 1. This expert may accept it and finish the game. The goal of expert 2 consists in choosing a candidate (xi , yi ) such that xi + yi > x1 + y1 to win the game. Otherwise, the winner is expert 1. Assume that expert 1 rejects candidate 1. Then expert 2 makes its choice, accepting or rejecting this candidate. In the former case, player 1 has to choose a better candidate than the accepted one. If expert 2 also rejects candidate 1, the game Γn proceeds to step 2 (actually, to the game Γn−1 ) and the described procedure repeats. An expert selecting a candidate with higher total quality wins in this game. 2. Game Γ2 Let n = 2, i.e., there are two candidates for a position. At step 1, expert 1 observes x1 . Define its selection rule via a threshold u. In other words, if x1 u, then the expert accepts candidate 1 and leaves the game. Otherwise, this candidate is accepted by expert 2 and expert 1 chooses candidate 2. The optimal value of the threshold u can be established based on the following considerations. Assume that x1 = u is the known quality of the candidate at step 1. If expert 1 accepts the candidate, its payoff makes up Ha (u) = 1 · P{u + y1 x2 + y2 } − 1 · P{x2 + y2 u + y1 } = 2 · P{u + y1 x2 + y2 } − 1. On the other hand, in the case of rejection its payoff is Hr (u) = −Ha (u). The optimal value u1 satisfies the equation Ha (u) = Hr (u) or 2 · P{u + y1 x2 + y2 } − 1 = 0. It can be equivalently rewritten as 1−u
(y1 + u)2 dy1 + 2
0
1 1−
(2 − (y1 + u))2 1 dy1 = . 2 2
1−u
It follows that the optimal value is u1 =
1 , yielding the payoff 2
1
2 H1 = 1 +
[1 · P{x1 + y1 < x2 + y2 } − 1 · P{x1 + y1 x2 + y2 }] dx1 + 0
[1 · P{x1 + y1 x2 + y2 } − 1 · P{x1 + y1 < x2 + y2 }] dx1 = 0.354. 1 2
3. Game Γ3 Let n = 3 and denote by x1 is the first value observed. Expert 1 moves first, accepting or rejecting it. Suppose that its selection rule is determined by a threshold u: expert 1 accepts the candidate under x1 u and rejects otherwise. Expert 2 uses a threshold v. According to a natural assumption, v < u, as far as expert 1 has a priority in the game. Imagine that x1 ≤ v. In this case, expert 1 rejects candidate 1. Next, the decision is made by expert 2. If expert 2 also rejects the candidate, then the game Γ3 proceeds to step 2 (the game Γ2 ) with the known payoff H1 . The optimal value v can be found from the following condition: the payoffs of expert 2 in the cases of candidate acceptance and rejection do coincide. Suppose that x1 = v2 , and expert 2 accepts the candidate. Then expert 1 can select any from the two available candidates. We emphasize that expert 1 knows the complete information m1 = v2 + y1 . Therefore, its goal is to
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
select a candidate i with xi + yi m1 . Expert 1 sets a new threshold s1 = s(m1 ): if x2 s1 , then it accepts the candidate (and rejects otherwise). The optimal value s1 depends on m1 < 1 or not. By comparing the payoffs of expert 1 for the cases when it accepts or rejects the candidate, we evaluate the optimal threshold ⎧ ⎪ m21 ⎪ ⎪ ⎪ , i f m1 < 1, − m ⎪ 1 ⎪ ⎨ 2 s1 = ⎪ ⎪ ⎪ m2 ⎪ ⎪ ⎪ ⎩ 1 − m1 + 1 , i f m1 1. 2 Consequently, the payoff of expert 2 owing to accepting the candidate with the quality x1 = v2 is ⎤ 1−v2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢ ⎥ ⎢⎢⎢ 1 − 2 · P{x3 + y3 m1 } dx2 + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 + Ha(II) (v2 ) = ⎢⎣ ⎦ s1 0 0 ⎡ ⎤ 1 ⎢⎢s1 1 ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 , 1 − 2 · P{x3 + y3 m1 } dx2 + ⎣ ⎦ 1−v2
s1
0
m2 m2 where s1 = m1 − 1 in the first integral and s1 = 1 − m1 + 1 in the second integral. 2 2 If expert 2 rejects the candidate, then the game proceeds to the game Γ2 with the payoff −H1 for expert 2. Thus, the optimal value v2 meets the equation ⎤ 1−v2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢⎢⎢ ⎥⎥ + y m } dx + y m } dx + 1 − 2 · P{x 1 − 2 · P{x ⎢⎢⎣ ⎥⎥⎦ dy1 + 2 2 1 2⎥ 3 3 1 2 s1 0 ⎤ ⎡ 0 1 ⎢⎢s1 1 ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 = −H1 . 1 − 2 · P{x3 + y3 m1 } dx2 + ⎦ ⎣ 1−v2
s1
0
The above equation possesses the solution v2 = 0.402. Now, find the optimal threshold u2 of expert 1 at step 1. If expert 1 accepts the candidate with the quality x1 = u2 , then its payoff is defined by formula (1): Ha(I) (u2 ) = Ha(II) (u2 ). In the case of rejection by expert 1, expert 2 accept the candidate (so long as u2 > v2 ). And so, the payoff of expert 1 constitutes −Ha(II) (u2 ). Consequently, the optimal threshold u2 satisfies the equation ⎤ 1−u2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢⎢⎢ ⎥ 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 + ⎢⎢⎣ ⎦ s1 0 ⎤ ⎡ 0s1 1 1 ⎢⎢ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 = 0. ⎣ ⎦ 1−u2
0
s1
The solution is u2 = 0.637. And finally, we can calculate the value of the game Γ3 :
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Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
⎡ 1−x1⎡ s1 ⎤ u2 ⎢⎢ 1 ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢
⎢ ⎥ ⎢ ⎢⎢⎢ 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 + 2 · P{x3 + y3 m1 } − 1 dx2 + = H1 dx1 + ⎢⎢⎢ ⎣ ⎣ ⎦ s1 0 ⎤ ⎡ s1 v2 0 0 ⎤ 1 1 ⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎥ ⎢⎢⎢ + 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 + ⎣ ⎦ ⎦ s1 1−x1 0 ⎡ 1−x1⎡ s1 ⎤ 1 1 ⎢⎢ ⎢⎢ ⎥⎥⎥ ⎢⎢ ⎢⎢⎢
⎥ ⎢⎢ + ⎢⎢⎢⎢ 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 + 1 − 2 · P{x3 + y3 m1 } dx2 + ⎢ ⎣ ⎣ ⎦ s1 ⎤ ⎡u2 s1 0 0 ⎤ 1 1 ⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎥ + 1 − 2 · P{x3 + y3 m1 } dx2 + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 = 0.237. ⎢⎢⎢⎣ ⎦ ⎦ v2
H
(I)
1−x1
s1
0
4. Equally weighted players In this section, we assume that the experts have equal weights. If both experts accept a candidate, the latter 1 selects each of them with equal probability . 2 Let n = 1, i.e., the experts consider only one candidate. Obviously, both experts accept the candidate. This leads to the optimal thresholds u∗ = v∗ = 0 and the game value H1 = 0. Suppose that n > 1. Due to the symmetry of the problems, the game value is Hn = 0 and the optimal thresholds do coincide. To find them, study the following optimization problem. Imagine that expert 2 makes a choice and its selected candidate has the total quality m. But expert 1 still continues selection and considers n candidates. Designate by Hn (m) its optimal payoff. Find the optimal strategy of expert 1 at step 1. Candidate 1 with quality components (x1 , y1 ) is presented. Let s indicate some value of the threshold of expert 1. If x1 < s, expert 1 rejects the candidate and moves to the next step with the optimal payoff Hn−1 (m). Otherwise, under x1 s, it accepts the candidate and wins the game only if x1 + y1 m. Consequently, the optimality equation acquires the form 1
s Hn−1 (m) dx1 +
Hn (m) = 0
(P{x1 + y1 m} − P{x1 + y1 < m}) dx1 . s
Consider the case of m < 1. The optimal threshold is s m and the optimality equation is defined by m Hn (m) = Hn−1 (m)s +
m
1 dx1 = Hn−1 (m)s +
(2 · P{x1 + y1 m} − 1) dx1 + s
m
(1 − 2m + 2x1 ) dx1 + 1 − m. s
Under m ≥ 1, the optimality equation becomes s Hn (m) =
1 Hn−1 (m) dx1 +
0
1 (2 · P{x1 + y1 m} − 1) dx1 = Hn−1 (m) · s +
s
(2 · P{x1 + y1 m} − 1) dx1 = s
1 = Hn−1 (m)s +
(1 − 2m + 2x1 ) dx1 . s
Expert 1 seeks to maximize the payoff Hn (m). The optimal threshold s can be evaluated from the following condition. The payoffs of the expert gained by acceptance or rejecting a candidate with quality x1 = s do coincide: 2 · P{sn + y1 m} − 1 = Hn−1 (m).
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
This equation yields
1 − Hn−1 (m) . 2 Thus, the optimal payoff of expert 1 can be expressed by ⎧ m ⎪ ⎪ ⎪ ⎪ ⎪ Hn−1 (m) · sn + (1 − 2m − 2x1 ) dx1 + 1 − m, m < 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ sn Hn (m) = ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hn−1 (m) · sn + (1 − 2m − 2x1 ) dx1 , m 1, ⎪ ⎪ ⎪ ⎩ sn = m −
sn
where the optimal threshold is
1 − Hn−1 (m) . 2 Now, we can calculate the equilibrium thresholds of the experts at the beginning of the game. Assume that the experts use the thresholds u and v, respectively, and u v. Then the payoff of expert 1 acquire the form ⎡ 1−x ⎡ ⎤ 1⎢ u ⎢⎢⎢ v m ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ ⎥ ⎢ ⎢ 0 · dx1 + ⎢⎢⎢ Hn (u, v) = ⎢⎢⎢Hn−1 (m) · sn + (1 − 2m + 2x2 ) dx2 + 1 − m⎥⎥⎥⎥⎥ dy1 + ⎣ ⎣ ⎦ sn = m −
0
v
sn
0
⎡ ⎤ ⎤ 1 ⎢⎢⎢ 1 1 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥⎥⎥ + ⎢⎢⎢Hn−1 (m) · sn + (1 − 2m + 2x2 ) dx2 ⎥⎥⎥ dy1 ⎥⎥⎥ dx1 + 0 · dx1 = ⎣ ⎦ ⎦ 1−x1
u = v
sn
u
⎡ 1−x ⎡ ⎤ x 1 +y1 ⎢⎢⎢ 1⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥ ⎢⎢⎢⎢ (x + y ) · s + (1 − 2(x + y ) + 2x ) dx + 1 − (x + y ) H ⎢ ⎥⎥⎥ dy1 + 1 n 1 1 2 2 1 1 ⎥ ⎢⎢⎣ n−1 1 ⎢⎢⎣ ⎦ 0
sn
⎡ ⎤ ⎤ 1 ⎢⎢⎢ 1 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥ + ⎢⎢⎢Hn−1 (x1 + y1 ) · sn + (1 − 2(x1 + y1 ) + 2x2 ) dx2 ⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 . ⎣ ⎦ ⎦ sn
1−x1
The payoff function H(u, v) is concave in u and convex in v. Consequently, the game admits a unique equilibrium. The optimal thresholds of the experts equal each other and meet the equations ⎡ ⎤ 1−u∗⎢ m ⎥⎥ ⎢⎢⎢ ⎢⎢⎢H (u∗ + y ) · s + (1 − 2(u∗ + y ) + 2x ) dx + 1 − (u∗ + y )⎥⎥⎥⎥⎥ dy + n−1 1 n 1 2 2 1 ⎢⎢⎣ ⎥⎥⎦ 1 sn
0
⎡ ⎤ 1 ⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢ ⎥ + ⎢⎢⎢Hn−1 (u∗ + y1 ) · sn + (1 − 2(u∗ + y1 ) + 2x2 ) dx2 ⎥⎥⎥⎥⎥ dy1 = 0. ⎣ ⎦ 1−u∗
sn
The obtained equations allow optimal thresholds evaluation for any n. For instance, the optimal thresholds make up 0.5 (the game with 2 candidates), 0.637 (the game with 3 candidates) and 0.711 (the game with 4 candidates). Acknowledgements This work was supported by the Russian Foundation for Basic Research (project no. 13-01-00033-a), Russian Humanitarian Science Foundation (project no. 15-02-00352) and by the Division of Mathematical Sciences of Russian Academy of Sciences.
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References [1] Dynkin EB. The optimal choice of the stopping of a Markov process. Reports of the Academy of Sciences of the USSR 1963;150(2): 238-40. [2] Mazalov VV. Mathematical Game Theory and Applications. New York: Wiley; 2014. [3] Alpern S, Reyniers D. Strategic mating common preferences. Journal of Theoretical Biology 2005; 237: 337-54. [4] Ano K. On a partial information multiple selection problem. Games Theory and Application 1998; 4: 1-10. [5] Enns E. Selecting the maximum of a sequence with imperfect information. Journal of the American Statistical Association 1975; 70(351): 640-3. [6] Enns ES, Ferenstein EZ. The horse game. J. Oper. Res. Soc. Japan. 1985; 28: 51-62. [7] Fushimi M. The secretary problem in a competitive situation. J. Oper. Res. Soc. Japan. 1981; 24: 350-8. [8] Gilbert J, Mosteller F. Recognizing the maximum of a sequence. J. Amer. Statist. Ass. 1966; 61: 35-73. [9] McNamara J, Collins E. The job search problem as an employer-candidate game. J. Oper. Res. Soc. Japan. 1990; 28: 815-27. [10] Mazalov VV. Game related to optimal stopping of two sequences of independent random variables having different distributions. Mathematica Japonica 1996; 43(1): 121-8. [11] Mazalov VV, Falko AA. Nash equilibrium in two-sided mate choice problem. International Game Theory Review 2008; 10(4): 421-35. [12] Kurano M, Nakagami J, Yasuda M. Multi-variate stopping problem with a majority rule. J. Oper. Res. Soc. Japan. 1980;23: 205-23. [13] Sakaguchi M. Non-zero-sum games related to the secretary problem. J. Oper. Res. Soc. Japan. 1980; 23(3): 287-93. [14] Sakaguchi M. Non-zero-sum best-choce games where two stops are required. Scientiae Mathematicae Japonicae 2003; 58(1): 137-176.
ScienceDirect Procedia Computer Science 55 (2015) 159 – 164
Information Technology and Quantitative Management (ITQM 2015)
The job-search problem with incomplete information Vladimir Mazalova , Elena Konovalchikovab a Institute
of Applied Mathematical Research, Karelian Research Center of RAS, ul. Pushkinskaya, 11, Petrozavodsk, 185910, Russia b Transbaikal State University, ul. Alekzavodskaya, 30, Chita, 672039, Russia
Abstract This paper considers a best-choice game with incomplete information associated with the job search problem. Players (experts) observe a sequence of independent identically distributed random variables (xi , yi ), i = 1 . . . , n, which represent the quality of incoming objects. The first component is announced to the players and the other one is hidden. The players choose an object based on known information about it. The winner is the player having a higher sum of the quality components (total quality) than the opponent. And finally, the optimal strategies of the players are derived. c 2015 2015 Published The Authors. Published Elsevier B.V.access article under the CC BY-NC-ND license © by Elsevier B.V.by This is an open (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of ITQM2015. Peer-review under responsibility of the Organizing Committee of ITQM 2015
Keywords: best-choice game, incomplete information, equilibrium, threshold strategy
1. Introduction In this paper, we study a best-choice game associated with the job search problem. Here m employers (experts) have to make a single choice from a number of items (objects) presented sequentially. The job search problem possesses a long history and a series of alternative names including the secretary problem, the marriage problem and the asset-selling problem. As the secretary problem [1], it has many statements and variations. For instance, the number of items is finite or infinite, a decision-maker knows the actual value of each item as it is presented or its relative rank among presented items only. In some models, items are random variables with a known probability distribution function. Several authors considered the case of a given probability distribution with unknown parameters [4]. The game-theoretic setting of this problem was developed in [8]. Different generalizations of bestchoice games were introduced in the papers [6, 7, 10, 12-14]. There are few models devoted to mutual best-choice games, where both groups of game participants (experts and candidates) make their choice [3, 9, 11]. In what follows, we explore a two-player best-choice game with incomplete information. Assume that two experts observe a sequence of independent identically distributed random variables (xi , yi ), i = 1 . . . , n, which represent the quality of incoming objects. The first component is announced to the players and the other component is hidden. In a practical interpretation, the first component reflects professional skills of a candidate and the second one describes its computing skills. Each expert has to maximize the sum xi + yi of the quality components (the so-called total quality) of the selected candidate. Within the game-theoretic framework, a player strives to choose a candidate with higher total quality than the candidates selected by other players. ∗ Vladimir Mazalov. Tel.: +7-814-278-1108; fax: +7-814-276-6313. E-mail address: [email protected].
1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ITQM 2015 doi:10.1016/j.procs.2015.07.025
160
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
First, consider the model, where player 1 has a priority to make a choice. Suppose that random variables (xi , yi ), i = 1 . . . , n, are uniformly distributed on the segment [0, 1]. At step 1, expert 1 observes the value x1 from the pair (x1 , y1 ) describing candidate 1. This expert may accept it and finish the game. The goal of expert 2 consists in choosing a candidate (xi , yi ) such that xi + yi > x1 + y1 to win the game. Otherwise, the winner is expert 1. Assume that expert 1 rejects candidate 1. Then expert 2 makes its choice, accepting or rejecting this candidate. In the former case, player 1 has to choose a better candidate than the accepted one. If expert 2 also rejects candidate 1, the game Γn proceeds to step 2 (actually, to the game Γn−1 ) and the described procedure repeats. An expert selecting a candidate with higher total quality wins in this game. 2. Game Γ2 Let n = 2, i.e., there are two candidates for a position. At step 1, expert 1 observes x1 . Define its selection rule via a threshold u. In other words, if x1 u, then the expert accepts candidate 1 and leaves the game. Otherwise, this candidate is accepted by expert 2 and expert 1 chooses candidate 2. The optimal value of the threshold u can be established based on the following considerations. Assume that x1 = u is the known quality of the candidate at step 1. If expert 1 accepts the candidate, its payoff makes up Ha (u) = 1 · P{u + y1 x2 + y2 } − 1 · P{x2 + y2 u + y1 } = 2 · P{u + y1 x2 + y2 } − 1. On the other hand, in the case of rejection its payoff is Hr (u) = −Ha (u). The optimal value u1 satisfies the equation Ha (u) = Hr (u) or 2 · P{u + y1 x2 + y2 } − 1 = 0. It can be equivalently rewritten as 1−u
(y1 + u)2 dy1 + 2
0
1 1−
(2 − (y1 + u))2 1 dy1 = . 2 2
1−u
It follows that the optimal value is u1 =
1 , yielding the payoff 2
1
2 H1 = 1 +
[1 · P{x1 + y1 < x2 + y2 } − 1 · P{x1 + y1 x2 + y2 }] dx1 + 0
[1 · P{x1 + y1 x2 + y2 } − 1 · P{x1 + y1 < x2 + y2 }] dx1 = 0.354. 1 2
3. Game Γ3 Let n = 3 and denote by x1 is the first value observed. Expert 1 moves first, accepting or rejecting it. Suppose that its selection rule is determined by a threshold u: expert 1 accepts the candidate under x1 u and rejects otherwise. Expert 2 uses a threshold v. According to a natural assumption, v < u, as far as expert 1 has a priority in the game. Imagine that x1 ≤ v. In this case, expert 1 rejects candidate 1. Next, the decision is made by expert 2. If expert 2 also rejects the candidate, then the game Γ3 proceeds to step 2 (the game Γ2 ) with the known payoff H1 . The optimal value v can be found from the following condition: the payoffs of expert 2 in the cases of candidate acceptance and rejection do coincide. Suppose that x1 = v2 , and expert 2 accepts the candidate. Then expert 1 can select any from the two available candidates. We emphasize that expert 1 knows the complete information m1 = v2 + y1 . Therefore, its goal is to
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
select a candidate i with xi + yi m1 . Expert 1 sets a new threshold s1 = s(m1 ): if x2 s1 , then it accepts the candidate (and rejects otherwise). The optimal value s1 depends on m1 < 1 or not. By comparing the payoffs of expert 1 for the cases when it accepts or rejects the candidate, we evaluate the optimal threshold ⎧ ⎪ m21 ⎪ ⎪ ⎪ , i f m1 < 1, − m ⎪ 1 ⎪ ⎨ 2 s1 = ⎪ ⎪ ⎪ m2 ⎪ ⎪ ⎪ ⎩ 1 − m1 + 1 , i f m1 1. 2 Consequently, the payoff of expert 2 owing to accepting the candidate with the quality x1 = v2 is ⎤ 1−v2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢ ⎥ ⎢⎢⎢ 1 − 2 · P{x3 + y3 m1 } dx2 + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 + Ha(II) (v2 ) = ⎢⎣ ⎦ s1 0 0 ⎡ ⎤ 1 ⎢⎢s1 1 ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 , 1 − 2 · P{x3 + y3 m1 } dx2 + ⎣ ⎦ 1−v2
s1
0
m2 m2 where s1 = m1 − 1 in the first integral and s1 = 1 − m1 + 1 in the second integral. 2 2 If expert 2 rejects the candidate, then the game proceeds to the game Γ2 with the payoff −H1 for expert 2. Thus, the optimal value v2 meets the equation ⎤ 1−v2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢⎢⎢ ⎥⎥ + y m } dx + y m } dx + 1 − 2 · P{x 1 − 2 · P{x ⎢⎢⎣ ⎥⎥⎦ dy1 + 2 2 1 2⎥ 3 3 1 2 s1 0 ⎤ ⎡ 0 1 ⎢⎢s1 1 ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 = −H1 . 1 − 2 · P{x3 + y3 m1 } dx2 + ⎦ ⎣ 1−v2
s1
0
The above equation possesses the solution v2 = 0.402. Now, find the optimal threshold u2 of expert 1 at step 1. If expert 1 accepts the candidate with the quality x1 = u2 , then its payoff is defined by formula (1): Ha(I) (u2 ) = Ha(II) (u2 ). In the case of rejection by expert 1, expert 2 accept the candidate (so long as u2 > v2 ). And so, the payoff of expert 1 constitutes −Ha(II) (u2 ). Consequently, the optimal threshold u2 satisfies the equation ⎤ 1−u2⎡ 1 ⎢⎢⎢s1 ⎥⎥⎥
⎢⎢⎢ ⎥ 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 + ⎢⎢⎣ ⎦ s1 0 ⎤ ⎡ 0s1 1 1 ⎢⎢ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎢⎢⎢ + 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 = 0. ⎣ ⎦ 1−u2
0
s1
The solution is u2 = 0.637. And finally, we can calculate the value of the game Γ3 :
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⎡ 1−x1⎡ s1 ⎤ u2 ⎢⎢ 1 ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢
⎢ ⎥ ⎢ ⎢⎢⎢ 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 + 2 · P{x3 + y3 m1 } − 1 dx2 + = H1 dx1 + ⎢⎢⎢ ⎣ ⎣ ⎦ s1 0 ⎤ ⎡ s1 v2 0 0 ⎤ 1 1 ⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎥ ⎢⎢⎢ + 2 · P{x3 + y3 m1 } − 1 dx2 + 2 · P{x2 + y2 m1 } − 1 dx2 ⎥⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 + ⎣ ⎦ ⎦ s1 1−x1 0 ⎡ 1−x1⎡ s1 ⎤ 1 1 ⎢⎢ ⎢⎢ ⎥⎥⎥ ⎢⎢ ⎢⎢⎢
⎥ ⎢⎢ + ⎢⎢⎢⎢ 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 + 1 − 2 · P{x3 + y3 m1 } dx2 + ⎢ ⎣ ⎣ ⎦ s1 ⎤ ⎡u2 s1 0 0 ⎤ 1 1 ⎢⎢ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢
⎥ ⎥ + 1 − 2 · P{x3 + y3 m1 } dx2 + 1 − 2 · P{x2 + y2 m1 } dx2 ⎥⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 = 0.237. ⎢⎢⎢⎣ ⎦ ⎦ v2
H
(I)
1−x1
s1
0
4. Equally weighted players In this section, we assume that the experts have equal weights. If both experts accept a candidate, the latter 1 selects each of them with equal probability . 2 Let n = 1, i.e., the experts consider only one candidate. Obviously, both experts accept the candidate. This leads to the optimal thresholds u∗ = v∗ = 0 and the game value H1 = 0. Suppose that n > 1. Due to the symmetry of the problems, the game value is Hn = 0 and the optimal thresholds do coincide. To find them, study the following optimization problem. Imagine that expert 2 makes a choice and its selected candidate has the total quality m. But expert 1 still continues selection and considers n candidates. Designate by Hn (m) its optimal payoff. Find the optimal strategy of expert 1 at step 1. Candidate 1 with quality components (x1 , y1 ) is presented. Let s indicate some value of the threshold of expert 1. If x1 < s, expert 1 rejects the candidate and moves to the next step with the optimal payoff Hn−1 (m). Otherwise, under x1 s, it accepts the candidate and wins the game only if x1 + y1 m. Consequently, the optimality equation acquires the form 1
s Hn−1 (m) dx1 +
Hn (m) = 0
(P{x1 + y1 m} − P{x1 + y1 < m}) dx1 . s
Consider the case of m < 1. The optimal threshold is s m and the optimality equation is defined by m Hn (m) = Hn−1 (m)s +
m
1 dx1 = Hn−1 (m)s +
(2 · P{x1 + y1 m} − 1) dx1 + s
m
(1 − 2m + 2x1 ) dx1 + 1 − m. s
Under m ≥ 1, the optimality equation becomes s Hn (m) =
1 Hn−1 (m) dx1 +
0
1 (2 · P{x1 + y1 m} − 1) dx1 = Hn−1 (m) · s +
s
(2 · P{x1 + y1 m} − 1) dx1 = s
1 = Hn−1 (m)s +
(1 − 2m + 2x1 ) dx1 . s
Expert 1 seeks to maximize the payoff Hn (m). The optimal threshold s can be evaluated from the following condition. The payoffs of the expert gained by acceptance or rejecting a candidate with quality x1 = s do coincide: 2 · P{sn + y1 m} − 1 = Hn−1 (m).
Vladimir Mazalov and Elena Konovalchikova / Procedia Computer Science 55 (2015) 159 – 164
This equation yields
1 − Hn−1 (m) . 2 Thus, the optimal payoff of expert 1 can be expressed by ⎧ m ⎪ ⎪ ⎪ ⎪ ⎪ Hn−1 (m) · sn + (1 − 2m − 2x1 ) dx1 + 1 − m, m < 1, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ sn Hn (m) = ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Hn−1 (m) · sn + (1 − 2m − 2x1 ) dx1 , m 1, ⎪ ⎪ ⎪ ⎩ sn = m −
sn
where the optimal threshold is
1 − Hn−1 (m) . 2 Now, we can calculate the equilibrium thresholds of the experts at the beginning of the game. Assume that the experts use the thresholds u and v, respectively, and u v. Then the payoff of expert 1 acquire the form ⎡ 1−x ⎡ ⎤ 1⎢ u ⎢⎢⎢ v m ⎥⎥⎥ ⎢⎢⎢ ⎢⎢⎢ ⎥ ⎢ ⎢ 0 · dx1 + ⎢⎢⎢ Hn (u, v) = ⎢⎢⎢Hn−1 (m) · sn + (1 − 2m + 2x2 ) dx2 + 1 − m⎥⎥⎥⎥⎥ dy1 + ⎣ ⎣ ⎦ sn = m −
0
v
sn
0
⎡ ⎤ ⎤ 1 ⎢⎢⎢ 1 1 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥⎥⎥ + ⎢⎢⎢Hn−1 (m) · sn + (1 − 2m + 2x2 ) dx2 ⎥⎥⎥ dy1 ⎥⎥⎥ dx1 + 0 · dx1 = ⎣ ⎦ ⎦ 1−x1
u = v
sn
u
⎡ 1−x ⎡ ⎤ x 1 +y1 ⎢⎢⎢ 1⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥ ⎢⎢⎢⎢ (x + y ) · s + (1 − 2(x + y ) + 2x ) dx + 1 − (x + y ) H ⎢ ⎥⎥⎥ dy1 + 1 n 1 1 2 2 1 1 ⎥ ⎢⎢⎣ n−1 1 ⎢⎢⎣ ⎦ 0
sn
⎡ ⎤ ⎤ 1 ⎢⎢⎢ 1 ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎥ + ⎢⎢⎢Hn−1 (x1 + y1 ) · sn + (1 − 2(x1 + y1 ) + 2x2 ) dx2 ⎥⎥⎥ dy1 ⎥⎥⎥⎥⎥ dx1 . ⎣ ⎦ ⎦ sn
1−x1
The payoff function H(u, v) is concave in u and convex in v. Consequently, the game admits a unique equilibrium. The optimal thresholds of the experts equal each other and meet the equations ⎡ ⎤ 1−u∗⎢ m ⎥⎥ ⎢⎢⎢ ⎢⎢⎢H (u∗ + y ) · s + (1 − 2(u∗ + y ) + 2x ) dx + 1 − (u∗ + y )⎥⎥⎥⎥⎥ dy + n−1 1 n 1 2 2 1 ⎢⎢⎣ ⎥⎥⎦ 1 sn
0
⎡ ⎤ 1 ⎢⎢⎢ 1 ⎥⎥⎥ ⎢⎢⎢ ⎥ + ⎢⎢⎢Hn−1 (u∗ + y1 ) · sn + (1 − 2(u∗ + y1 ) + 2x2 ) dx2 ⎥⎥⎥⎥⎥ dy1 = 0. ⎣ ⎦ 1−u∗
sn
The obtained equations allow optimal thresholds evaluation for any n. For instance, the optimal thresholds make up 0.5 (the game with 2 candidates), 0.637 (the game with 3 candidates) and 0.711 (the game with 4 candidates). Acknowledgements This work was supported by the Russian Foundation for Basic Research (project no. 13-01-00033-a), Russian Humanitarian Science Foundation (project no. 15-02-00352) and by the Division of Mathematical Sciences of Russian Academy of Sciences.
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