4.1. No Fault Case
First, we consider the case that there is no possibility that any node will be destroyed. Note that the network at each period
t must be subgame perfect due to the prescribed sequential nature of a stage game. By a subgame perfect network, we mean a network that is realized in a subgame perfect equilibrium in the
t-th stage game. Thus, under the requirement that a network be connected, i.e.,
, a dynamically
-stable network at period
t is defined by
such that (i)
is a subgame perfect network and (ii)
,
where
. If
is a dynamically
-stable network at period
t for all
t, we call
dynamically
-stable. We will use dynamic
-stability as our main solution concept. Additionally, note that a sequence of
-stable networks at period
t is strictly increasing in the sense that
for any
, because the assumption that the cost of linking in a period becomes sunk in the next period thereafter eliminates the possibility that an existing link is severed. This concept of
-stability is based on pairwise stability by Jackson and Wolinsky [
2] that requires a new link to be formed by mutual consent of the linking agents.
5It is worthwhile to mention the relations of our concept to other concepts in literature. Our definition of dynamic
-stability is distinguished from that of Watts [
9] and Jackson and Watts [
10] who assume that players are myopic. In our definition, players are farsighted in the sense that they make linking decisions by comparing
with
that will be completed at the end of period
t. In other words, player
i does not compare
with
because he is farsighted enough to expect other links to follow in period
t. However, they are not perfectly farsighted in the sense that they do not compare
with
,
and ⋯. Thus, we call our definition of dynamic stability intraperiod-farsighted stability to distinguish it from farsighted stability by Chwe [
11] and Page
et al. [
12].
To characterize dynamic -stable networks, we need several lemmas. Lemma 1 tells us whether two extreme players (the most senior player and the most junior player) in a line network have an incentive to link between them.
Lemma 1. For any given t, define by Then, (i) , (ii) for , and (iii) .
This lemma says that is a partition of that consists of a monotonically increasing sequence of intervals. The interval is essential to characterizing the stable network, because and have the interpretations as the benefits from node 1 linking with node and node t, respectively. Intuitively, if , node 1 will have no incentive to link with node at period because but he will have an incentive to link with node t at period t because , assuming that is a line.
To understand the intuition, let
be the benefit of node
k from linking with the new node
t at period
t. Then, the benefit is the difference in his delay status. Assuming that
is a line, the incentive of node 1 to link with node
t is exactly identical to the incentive of node
to link with node
t. Hence,
implies
.
6 Thus,
where
. If
is odd,
and if
is even,
Similarly,
can be computed by replacing
t by
in (
1) and (
2).
It is intuitively clear that node 1 has more incentive to link with the entrant as the length of the line network is longer. Thus, Lemma 2 follows.
Lemma 2. Assume that is a line; i.e., . Then, is strictly decreasing in k, for .
This lemma implies that if node 1 has no incentive to link with node t, neither does node k for . The following lemma is helpful for understanding the incentive of node to link with node t.
Lemma 3. Assume that is a line. Then, implies .
This lemma implies that the incentive of node to link with the entrant t in a line network is congruent with the incentive of node 1.
Lemma 4. For given t, if , i.e., , is a line.
The main insight behind this lemma is that the most senior player enjoys the first mover advantage. He can save the connecting cost by taking advantage of juniors who must link with the newcomers. We can call this the deferral principle.
7 The first mover (senior) advantage is mainly due to the requirement that the network must be connected.
To illustrate -stable networks, consider . It is clear that , since . At , node 1 would not link with node 3 if , because he knows that if he does not, node 2 must link to node 3. Accordingly, for any . This is contrasted with the case that in which node 1 will link to the new node 3 even if he knows that a direct link between node 2 and node 3 will follow his linking with node 3 because the benefit of reducing a delay by the direct link with node 3 exceeds the linking cost (). At , node 3 must link with the new node 4 (i.e., ) if for all . If , node 3 will not link with node 4 (i.e., ), as far as . If , if because the linking cost c is smaller than the benefit of reducing delay which is 2. If , would lead to and . Therefore, node 1 would prefer not to link to node 4, because in that case, node 3 must link with node 4 and this is better for node 1. Thus, if and only if . The resulting stable network at is a circle if and a line if . This argument can be generalized for by the following proposition.
Proposition 1. If , the complete graph is the only dynamically -stable network. For any , there exists a finite time such that, for all , is a line, and is a circle if . In particular, is determined by t such that , and is strictly increasing in c.
In
Appendix B, we will provide an algorithm which we call optimal deferral algorithm to find the
-stable network for any
t.
By varying c instead of varying t and fixing c, Proposition 1 can be interpreted as follows.
Corollary 1. Suppose that for some . Then, is a circle if (), and is a line if .
We can compare the result in this dynamic model of network formation with that of a static model that was obtained in Jun and Kim [
1]. First, the complete network is uniquely
-stable if
in both the dynamic model and the static model. Second, the line network is statically
-stable if
for even
k and
for odd
k. Proposition 1 shows that the line network is dynamically
-stable in the same range of
c. Third, the circle network is dynamically
-stable if
. Let
be the range of
c in which a circle is
-stable in a static model. Then, we have
by Proposition 1 of Jun and Kim [
1]. Since
for all
, the dynamic
-stability of a circle implies static
-stability, but not vice versa.
4.2. Fault Case
Now, we consider the possibility that
nodes are destroyed due to exogenous shocks, for example, an enemy’s attack.
8 Thus, in this section, besides connectivity, we impose a further requirement that a graph remains connected even after the deletion of any at most
q nodes and their direct links. Then a graph
at period
t is dynamically
-stable if and only if (i)
is subgame perfect and (ii)
,
.
Proposition 2. If , the complete graph is the only -stable network. For any , if , the unique dynamically -stable network is the complete graph. If , it must be a graph such that there are centers called hubs forming a complete subnetwork among them and the rest of the nodes have links to the centers.
Without the possibility of node extinction, hubs would not be necessary for stability, but hubs are essential for stability in the presence of a possibility of attacks on nodes, and furthermore, more hubs are necessary in the presence of attacks on more nodes (larger q).
It is easy to see that a dynamically
-stable network is, in general, not
-stable in a static model. Dynamic
-stability requires
centers that form a clique.
9 If a pair of centers delete the link between them, it is profitable if
, because the delay cost per each node increases by one and
. Therefore, if
, no dynamically
-stable network is
-stable in a static model. This disparity mainly comes from our assumption that
c is sunk. Once a link is formed in a previous period, it is never severed in our dynamic model, even if it may become redundant by subsequently formed links. Thus, the resulting dynamically
-stable network may not be
-stable in a static sense because such a link may not be formed in a static model. To elaborate, note that all nodes must have at least
degrees in a
q-connected network (see Lemma 2 of Jun and Kim [
1]). In our dynamic model,
centers form
links with all the other center nodes and then the
-th node form links with
center nodes. Let
be the remaining center node; i.e.,
. Similarly, if
-th node forms links with center nodes except node
, it is dynamically
-stable. It is, however, not statically
-stable, because two other center nodes
for
could benefit from severing their direct link if
, while the remaining graph is still
q-connected, although severing the link is not permissible in the dynamic model in which the linking cost is sunk.
Clearly, there is a coordination problem,
10 because no player would want to be a hub if hubs are more likely to be attacked.
11 In our dynamic model, however, such a coordination problem is easily resolved by seniority. The most senior players must form the hubs to maintain
q-connectivity. In this sense, the possibility of external attacks gives seniors a disadvantage to becoming a hub.