Multidimensional mutual choice problem Vladimir V. Mazalov Institute of Applied Mathematical Research Karelian Research Center of RAS Pushkinskaya str. 11 185910 Petrozavodsk, Russia e-mail:
[email protected] url: http://mathem.krc.karelia.ru Anna A. Falko Institute of Applied Mathematical Research Karelian Research Center of RAS Pushkinskaya str. 11 185910 Petrozavodsk, Russia e-mail:
[email protected] url: http://mathem.krc.karelia.ru Abstract Multidimensional mutual choice problem is considered. In the problem the individuals from three groups want to create a coalition from three individuals. We present the dynamic game with n periods where free individuals from different groups randomly meet each other each period. If they accept each other they create a coalition and leave the game. Suppose the initial distributions of the qualities are uniform on [0, 1]. If free individuals accept each other in the i-th period, i = 1, 2, ..., n, they leave the game and each receives the sum of the qualities of others as a payoff. In the last period n the individuals who don’t create the coalition receive zero. Each player aims to maximize her expected payoff. The strategies for players are derived and numerical results are given. Keywords: mutual choice, dynamic game, equilibrium
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Introduction
In this paper, we consider a multidimensional mutual choice problem. The individuals from three groups (e. g. businessmen) want to form a long-term relationship with members of others groups, i.e. to create a coalition. We present the dynamic game with n periods where free individuals from different groups randomly meet each other each period. If they accept each other they create a coalition and leave the game. Let (X, Y, Z) are the qualities of the members from respective group ( e.g. level of income). The initial distributions of qualities are known and equal F for first group, G for the second and H for the third. For simplicity, we suppose that F = G = H are uniform on [0, 1]. If free individuals accept each other in the i-th period, they leave the game and each receives as a payoff the sum of the qualities of others. In the last period n the individuals who don’t create the coalition receive zero. Each player aims to maximize her expected payoff. 1
The mutual choice problems are investigated for the mating model in [1] and for the job search model in [2]. Alpern and Reyniers [1] consider the twodimensional mutual choice game in which players create the couple with a highly ranked individual of other group. Our paper extends this problem for the case of several players. The optimal strategies for players are derived and numerical results are given. The work is supported by Russian Found for Basic Research, project 06-0100128-a and 08-01-98801-r-sever-a.
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The game with two periods
Consider the case n = 2. By symmetry the optimal behaviour of player from each group is similar. Therefore, in the first period the coalition is created if x + y ≥ w, y + z ≥ w and x + z ≥ w, where w < 1 is the threshold of the acceptance. The individuals with qualities (x, y, z) (in the shaded area of Figure 1) create the coalition in the first period.
Figure 1 The common number of individuals in each group in the second period (Figure 2) is equal to A2
=
w/2 R £
¤ ¤ Rw £ 1 − (1 − w + x)2 dx + 1 − (1 − w + x)2 + 12 (2x − w)2 dx
0
+
w/2
R1 £ 1 w
¤ 3 2 2 2 w dx = 4 w (2 − w).
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Figure 2 The density of the distribution of the qualities in the second period is following 1−(1−w+x)2 , 0 ≤ x < w/2; A2 1−(1−w+x)2 + 12 (2x−w)2 f (x) = , w/2 ≤ x < w; A2 12 w2 , w ≤ x ≤ 1. A2
Thus, the optimal threshold w for the player with quality z is obtained from the equation w = E(X + Y ); or w = 2E(X). Hence we obtain the following equation
w
· w/2 ¤ ¤ R £ Rw £ = x 1 − (1 − w + x)2 dx + x 1 − (1 − w + x)2 + 12 (2x − w)2 dx 0 w/2 ¸ R1 £ 1 2 ¤ + x 2 w dx 2 A2
w
or 11w2 − 40w + 24 = 0. The optimal w is equal to w=
√ 2 (10 − 34) = 0.758. 11
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In the paper [1] the threshold for two periods and two groups is equal to 0.382, but the value 0.758 is less then doubled threshold of the two-dimensional problem. In the problem with two periods and one decision maker who aims to maximize the sum of the two uniform distributed variables the threshold of the acceptance is equal to 1. But in our statement the individuals become less choosy in the first period during the mutual choice.
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The game with three periods
Consider the case n = 3. Denote w2 , w3 are the thresholds of the acceptance (0 < w3 < w2 < 1). In the first period the coalition is created if x + y ≥ w2 , y + z ≥ w2 and x + z ≥ w2 , where w2 is the threshold of the acceptance in the first period. Denote following functions S1 (x, wi ) = 1 − (1 − wi + x)2 ; S2 (x, wi ) = 1 − (1 − wi + x)2 + 12 (2x − wi )2 ; S3 (x, wi ) = 12 wi2 , i = 2, 3. Then the common number of individuals in each group in the second period is equal to A2
=
wR 2 /2
S1 (x, w2 )dx +
0
w R2
S2 (x, w2 )dx +
R1 w2
w2 /2
S3 (x, w2 )dx = 43 w22 (2 − w2 ).
The density of the distribution of the qualities in the second period is following S (x,w ) 1 2 , 0 ≤ x < w2 /2; A2 S2 (x,w2 ) f2 (x) = , w2 /2 ≤ x < w2 ; A2 S3 (x,w2 ) , w2 ≤ x ≤ 1. A2 In the second period the coalition is created if x + y ≥ w3 , y + z ≥ w3 and x + z ≥ w3 , where w3 is the threshold of the ecceptance in the third period and x, y and z have the probability distribution f2 (x), f2 (y) and f2 (z) respectively. The common number of individuals in each group in the third period is equal to · w3 /2 wR 2 /2 R S1 (x, w3 )S1 (x, w2 )dx + A3 = A12 S2 (x, w3 )S1 (x, w2 )dx 0
+
w R3 w2 /2
S2 (x, w3 )S2 (x, w2 )dx +
w R2
w3 /2
S3 (x, w3 )S2 (x, w2 )dx +
w3
R1
¸ S3 (x, w3 )S3 (x, w2 )dx .
w2
The density of the distribution of the qualities in the second period is fol-
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lowing
f3 (x) =
S1 (x,w3 )S1 (x,w2 ) , A2 A3
0 ≤ x < w3 /2;
S2 (x,w3 )S1 (x,w2 ) , A2 A3
w3 /2 ≤ x < w2 /2;
S2 (x,w3 )S2 (x,w2 ) , A2 A3
w2 /2 ≤ x < w3 ;
S3 (x,w3 )S2 (x,w2 ) , A2 A3
w3 ≤ x < w2 ;
S3 (x,w3 )S3 (x,w2 ) , A2 A3
w2 ≤ x ≤ 1.
The optimal w2 and w3 are obtained from the equations w3
=2
w2
= w3
R1 0
xf3 (x)dx; w R3 0
f2X+Y (z)dz +
R1 w3
xf2X+Y (z)dz,
f2X+Y
where (z) is the density of the distribution of the variable Z (Z = X + Y ), X and Y have the distribution f2 (x). From the system we obtain the optimal thresholds in the second and third periods: w3 = 0.823; w2 = 0.449. In the paper [1] the threshold for three periods and two groups is equal to 0.321 and 0.482, but obtained values are less then double thresholds of the two-dimensional problem.
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The game with n periods
Denote wi is the threshold of the acceptance in the i-th period (i = 2, 3, ..., n). In each period the coalition is created if x + y ≥ wi , y + z ≥ wi and x + z ≥ wi , where wi is the threshold of the ecceptance in each period. Denote following function for wi < 1 (Figire 2, w = wi ) S1 (x, wi ) = 1 − (1 − wi + x)2 ; S2 (x, wi ) = 1 − (1 − wi + x)2 + 12 (2x − wi )2 ; S3 (x, wi ) = 12 wi2 , and for wi ≥ 1 (Figure 3) S1 (x, wi ) = 1; S2 (x, wi ) = 1 − (1 − wi + x)2 ; S3 (x, wi ) = 1 − (1 − wi + x)2 + 12 (2x − wi )2 , i = 2, 3, ..., n.
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Figure 3 Then the common number of individuals in each group in the i-th period denote Ai and the density of the distribution of the qualities in the second period is fi (x). The optimal wi are obtained from the equations wn
=2
wi
= wi+1
R1
xfn (x)dx;
0
wRi+1 0
fiX+Y (z)dz +
R1 wi+1
xfiX+Y (z)dz, i = 2, 3, ..., n,
where fiX+Y (z) is the density of the distribution of the variable Z (Z = X + Y ), X and Y have the distribution fi (x), i = 2, 3, ..., n.
References [1] Alpern S., Reyniers D. Strategic mating with common preferences. Journal of Theoretical Biology, 2005, 237, pp. 337-354. [2] McNamara J., Collins E. The job search problem as an employer-candidate game. J.Appl. Prob., 1990, 28, pp.815–827.
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