The co-author model is introduced by Jackson and Wolinsky (1996, Journal of Economic Theory) as a typical example of the models of network formation. In this note, we study which network is pairwise stable and/or stochastically stable when the number of players is four.

Masaki Iimura

Seigo Uchida

Toru Hokari

English

Research Papers in Economics

Pairwise stable and stochastically stable networks in the
four-person co-author model 
Seigo Uchida
Graduate School of Humanities and Social Sciences, University of Tsukuba
Masaki Iimura Toru Hokari
Graduate School of Humanities and Social Sciences,
University of Tsukuba
Graduate School of Humanities and Social Sciences,
University of Tsukuba
Abstract
The co-author model is introduced by Jackson and Wolinsky (1996, Journal of Economic
Theory) as a typical example of the models of network formation. In this note, we study
which network is pairwise stable and/or stochastically stable when the number of players is
four.
Citation: Uchida, Seigo, Masaki Iimura, and Toru Hokari, (2006) "Pairwise stable and stochastically stable networks in the
four-person co-author model." Economics Bulletin, Vol. 3, No. 30 pp. 1-11
Submitted: October 30, 2006.  Accepted: November 14, 2006.
URL: http://economicsbulletin.vanderbilt.edu/2006/volume3/EB-06C70015A.pdf1 Introduction
The “co-author model” is introduced by Jackson and Wolinsky (1996) as a
typical example of the models of network formation. In this note, we study
which network is “pairwise stable” and/or “stochastically stable” when the
number of players is four.
2 The model
There are four researchers who get their payoﬀs by writing joint papers with
others. Each paper is written by a pair of researchers. By writing a paper
with researcher j, researcher i gets
a
ni
+
a
nj
+
b
ninj
;
where a > 0, b > 0, and ni and nj are the numbers of co-authors for i and
j, respectively. If i has no co-author, her payoﬀ is 0. See Figure 1 for the
payoﬀs to researchers in each network.1
A network is pairwise stable (Jackson and Wolinsky, 1996) if (i) no pair
of researchers want to form a new link between them, and (ii) no one wants
to sever any single direct link.
Whether a particular network is pairwise stable and/or Pareto optimal
depends on the values of a and b, as shown in Table 1.
3 Dynamic process
Let us consider the following discrete-time dynamic process. At each period,
a pair of researchers are chosen randomly. If they are already directly linked,
they can decide whether to keep the link or sever it. If they are not linked,
they can decide whether to form a new link between them. We assume that
each researcher is myopic. The resulting transition dynamics is described in
Figures 2, 3, and 4.
When there are more than one pairwise stable networks, we can use the
above dynamic process to check which one is more likely to be realized than
others.
1Jackson and Wolinsky (1996) assume a = b. So, our model is a slightly generalized
version of the original one.
1r
r r
r 0
0 0
0
(0)
r
r r
r 2a + b
2a + b 0
0
(1)
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
Figure 1: The payoﬀs to researchers in each network.
Conditions on a and b Pareto optimal pairwise stable
b · 5a
3 (2a), (2b), (3a), (3c), (4b), (5) (6)
5a
3 < b · 2a (2a), (2b), (3a), (3c), (4b) (6)
2a < b · 3a (2a), (2b), (3c), (4b) (2a), (6)
3a < b · 6a (2a), (2b), (3c) (2a), (6)
6a < b (2a), (2b), (3c) (2a), (4a), (6)
Table 1: Pareto optimal networks and pairwise stable networks
2Conditions on a and b pairwise stable stochastically stable
b · 2a (6) (6)
2a < b · 6a (2a), (6) (6)
6a < b (2a), (4a), (6) (2a), (4a), (6)
Table 2: Pairwise stable networks and stochastically stable networks
Next, we assume that researchers’ decisions are reversed with a small
probability ". This additional assumption generates a Markov chain which
has a unique stationary distribution, and the process converges to this dis-
tribution from any initial state. The transition dynamics with errors is de-
scribed in Figures 5, 6, and 7.
A network is stochastically stable (Jackson and Watts, 2002) if it has
a positive support in the limiting stationary distribution when " ! +0.
It is known that the set of stochastically stable networks is a subset of the
set of pairwise stable networks. It is also known that, when there are more
than one pairwise stable networks, there is a method to ﬁnd out which one
is stochastically stable. Basically, this method uses a picture such as Figures
2, 3, and 4 and counts the number of “mutations” which is required to go
from one pairwise stable network to others.2 The results are presented in
Table 2. When 2a < b · 5a, although both (2a) and (6) are pairwise stable,
only (6) is stochastically stable. When 6a < b, however, the set of pairwise
stable networks coincides with that of stochastically stable networks. In such
a case, one way to ﬁnd out which network is more likely to be realized than
others is to calculate the limiting stationary distribution explicitly, and this
is what we do in the rest of the note.
Suppose that 6a < b. Given " > 0, the stationary distribution of the
corresponding Markov chain is deﬁned by the following equations:
p0 = "p0 +
"
6
¢ p1;
p1 = (1 ¡ ")p0 +
1 + 4"
6
¢ p1 +
2"
6
¢ p2a +
2"
6
¢ p2b;
p2a =
1 ¡ "
6
¢ p1 + (1 ¡ ")p2a +
1 ¡ "
6
¢ p3a;
2See Jackson and Watts (2002) or Young (1998) for the details of this method.
3p2b =
4(1 ¡ ")
6
¢ p1 +
2(1 + ")
6
¢ p2b +
2"
6
¢ p3a +
3"
6
¢ p3b +
3"
6
¢ p3c;
p3a =
4"
6
¢ p2a +
2(1 ¡ ")
6
¢ p2b +
4 ¡ 2"
6
¢ p3a +
4"
6
¢ p4a +
2(1 ¡ ")
6
¢ p4b;
p3b =
1 ¡ "
6
¢ p2b +
3
6
¢ p3b +
"
6
¢ p4b;
p3c =
1 ¡ "
6
¢ p2b +
3
6
¢ p3c +
"
6
¢ p4b;
p4a =
1 ¡ "
6
¢ p3a + (1 ¡ ")p4a +
1 ¡ "
6
¢ p5;
p4b =
2"
6
¢ p3a +
3(1 ¡ ")
6
¢ p3b +
3(1 ¡ ")
6
¢ p3c +
2(1 + ")
6
¢ p4b +
4"
6
¢ p5;
p5 =
2"
6
¢ p4a +
2(1 ¡ ")
6
¢ p4b +
4 ¡ 2"
6
¢ p5 + "p6;
p6 =
1 ¡ "
6
¢ p5 + (1 ¡ ")p6;
1 = p0 + p1 + p2a + p2b + p3a + p3b + p3c + p4a + p4b + p5 + p6:
By solving these equations, we have
p0 =
4"2 ¡ 4"3 + 3"4 + 36"5 ¡ 25"6 + 58"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p1 =
24" ¡ 48"2 + 42"3 + 198"4 + 366"5 + 498"6 ¡ 348"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p2a =
105"2 ¡ 204" + 594"3 ¡ 1374"4 + 1380"5 ¡ 735"6 + 174"7 + 60
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p2b =
24" + 120"2 ¡ 204"3 ¡ 36"4 + 780"5 ¡ 1380"6 + 696"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p3a =
336" ¡ 816"2 ¡ 276"3 + 3132"4 ¡ 4548"5 + 2868"6 ¡ 696"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p3b =
8" + 36"2 + 32"3 ¡ 136"4 ¡ 240"5 + 532"6 ¡ 232"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p3c =
8" + 36"2 + 32"3 ¡ 136"4 ¡ 240"5 + 532"6 ¡ 232"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p4a =
9"2 ¡ 288" + 1230"3 ¡ 2388"4 + 2124"5 ¡ 951"6 + 174"7 + 90
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p4b =
12" + 420"2 ¡ 576"3 ¡ 1536"4 + 3756"5 ¡ 2772"6 + 696"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
4p5 =
204" ¡ 372"2 ¡ 858"3 + 3114"4 ¡ 3534"5 + 1794"6 ¡ 348"7
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
;
p6 =
662"3 ¡ 81"2 ¡ 96" ¡ 1108"4 + 888"5 ¡ 357"6 + 58"7 + 34
28" ¡ 587"2 + 674"3 ¡ 267"4 + 36"5 + 4"6 + 184
:
Thus, in the limiting distribution, we have p2a = 60
184 ¼ 0:326, p4a = 90
184 ¼
0:489, p6 = 34
184 ¼ 0:185, and all other probabilities are zero. So, one can say
that among these three stochastically stable networks, network (4a) is more
likely to be realized than others.
References
[1] Matthew O. Jackson and Alison Watts. “The evolution of social and
economic networks.” Journal of Economic Theory, 106:265–295, 2002.
[2] Matthew O. Jackson and Asher Wolinsky. “A strategic model of social
and economic networks.” Journal of Economic Theory, 71:44–74, 1996.
[3] H. P. Young. Individual Strategy and Social Structure: An Evolutionary
Theory of Institutions. Princeton UP, 1998.
5r
r r
r 0
0 0
0
(0)
- 1
r
r r
r 2a + b
2a + b 0
0
(1)
¾
1
6
1
6
À ?
4
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2
6
?
1
6
?
1
6
)
2
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
2
6
6
1
6
?
1
6
q
2
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
?
3
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
?
3
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
-
4
6
?
2
6
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
4
6
À
2
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
5
6
-
1
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1
Figure 2: The dynamics in the case of 2a < b · 4a.
6r
r r
r 0
0 0
0
(0)
- 1
r
r r
r 2a + b
2a + b 0
0
(1)
¾
1
6
1
6
À ?
4
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2
6
?
1
6
?
1
6
)
2
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
4
6
6
1
6
?
1
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
?
3
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
?
3
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
-
4
6
?
2
6
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
2
6
À
2
6
i
2
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
5
6
-
1
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1
Figure 3: The dynamics in the case of 4a < b · 6a.
7r
r r
r 0
0 0
0
(0)
- 1
r
r r
r 2a + b
2a + b 0
0
(1)
¾
1
6
1
6
À ?
4
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2
6
?
1
6
?
1
6
)
2
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
4
6
6
1
6
?
1
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
?
3
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
?
3
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
- 1
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
2
6
À
2
6
i
2
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
4
6
-
1
6
6
1
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1
Figure 4: The dynamics in the case of 6a < b.
8r
r r
r 0
0 0
0
(0)
- 1 ¡ " - "
r
r r
r 2a + b
2a + b 0
0
(1)
¾
"
6
¾
1+4"
6
À
1¡"
6
?
4(1¡")
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1 ¡ "
Á
2"
6
?
4"
6
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2(1+")
6
?
1¡"
6
6
2"
6
?
1¡"
6
)
2(1¡")
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
2(1+")
6
6
1¡"
6
?
1¡"
6
1
2"
6
q
2(1¡")
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
-
4¡2"
6
6
4"
6
?
2(1¡")
6
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
4¡2"
6
À
2(1¡")
6
6
"
6
6
"
6
i
2"
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
5¡4"
6
-
1¡"
6
Á
4"
6
6
"
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1 ¡ "
¾
"
Figure 5: The dynamics with errors in the case of 2a < b · 4a.
9r
r r
r 0
0 0
0
(0)
- 1 ¡ " - "
r
r r
r 2a + b
2a + b 0
0
(1)
¾
"
6
¾
1+4"
6
À
1¡"
6
?
4(1¡")
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1 ¡ "
Á
2"
6
?
4"
6
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2(1+")
6
?
1¡"
6
6
2"
6
?
1¡"
6
)
2(1¡")
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
4¡2"
6
6
1¡"
6
?
1¡"
6
1
2"
6
q
2"
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
-
4¡2"
6
6
4"
6
?
2(1¡")
6
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
2(1+")
6
À
2(1¡")
6
6
"
6
6
"
6
i
2(1¡")
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
5¡4"
6
-
1¡"
6
Á
4"
6
6
"
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1 ¡ "
¾
"
Figure 6: The dynamics with errors in the case of 4a < b · 6a.
10r
r r
r 0
0 0
0
(0)
- 1 ¡ " - "
r
r r
r 2a + b
2a + b 0
0
(1)
¾
"
6
¾
1+4"
6
À
1¡"
6
?
4(1¡")
6
r
r r
r 2a + b
2a + b 2a + b
2a + b
(2a)
- 1 ¡ "
Á
2"
6
?
4"
6
r
r r
r 3a+b
2
3a + b 3a+b
2
0
(2b)
¾
2(1+")
6
?
1¡"
6
6
2"
6
?
1¡"
6
)
2(1¡")
6
r
r r
r 3a+b
2
5a
2 + 3b
4
5a
2 + 3b
4
3a+b
2
(3a)
-
4¡2"
6
6
1¡"
6
?
1¡"
6
1
2"
6
q
2"
6
r
r r
r ¡
¡
¡
2a + b
2
2a + b
2 2a + b
2
0
(3b)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r
@
@
@ 4a+b
3
4a + b 4a+b
3
4a+b
3
(3c)
¾
3
6
6
3"
6
?
3(1¡")
6
r
r r
r 2a + b
2
2a + b
2 2a + b
2
2a + b
2
(4a)
- 1 ¡ "
6
4"
6
?
2"
6
r
r r
r
@
@
@ 11a
6 + 5b
12
3a + 2b
3
4a+b
3
11a
6 + 5b
12
(4b)
¾
2(1+")
6
À
2(1¡")
6
6
"
6
6
"
6
i
2(1¡")
6
r
r r
r
@
@
@ 5a+b
3
7a
3 + 4b
9
5a+b
3
7a
3 + 4b
9
(5)
-
4¡2"
6
-
1¡"
6
Á
4"
6
6
1¡"
6
r
r r
r
@
@
@ ¡
¡
¡
2a + b
3
2a + b
3 2a + b
3
2a + b
3
(6)
¾1 ¡ "
¾
"
Figure 7: The dynamics with errors in the case of 6a < b.
11

A strategic model of social and economic networks.”

Individual Strategy and Social Structure: An Evolutionary Theory of Institutions. Princeton UP,

The evolution of social and economic networks.”

Pairwise stable and stochastically stable networks in the four-person co-author model

http://www.accessecon.com/pubs/EB/2006/Volume3/EB-06C70015A.pdf

Pairwise stable and stochastically stable networks in the four-person co-author model

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics