Let the marginal utility of player i in network g be µl
i(g) := ui(g ∪ l) − ui(g \ l). While
linking actions are made by individual players, they might affect the utility of any agent in the network. Consider a network g and and a network g0 = g∪ ij. Recall that w(g0)≥ w(g) ⇐⇒
P
h∈Nuh(g∪ ij) −Ph∈Nuh(g) ≥ 0 ⇐⇒ Ph∈Nµijh(g) ≥ 0; a network g0 has a higher welfare
than a network g if the sum of marginal utilities (of a change from g to g0) is positive. To
emphasize the role of the active and passive players we can also write
w(g0)≥ w(g) ⇐⇒ µi(g) + µj(g) +
X
k∈N \{i,j}
µk(g)≥ 0. (4.12)
To analyze how different actions affect the utility of the focal players (i, j) and of the players not involved in a link (k ∈ N \ {i, j}), we have to analyze the fundamentals of utility in our model: ui(g) = (1− λ)CLOSEi(g) + λBET Wi(g)− cli(g). Clearly, two agents forming a link
Centrality 92
following Lemma shows that individuals establishing a link, strictly increase their closeness and weakly increase their betweenness. If the link is critical, it also weakly increases the closeness and betweenness of all other players. If the link is non-critical, it also weakly increases the closeness of all other players, but not so for betweenness.
Lemma 4.4.1. By the definition of closeness and betweenness the following holds:
(i) For all g, and ij 6∈ g, CLOSEi(g∪ij) > CLOSEi(g) and CLOSEj(g∪ij) > CLOSEj(g)
and for all k ∈ N \ {i, j} it holds that CLOSEk(g∪ ij) ≥ CLOSEk(g).
(ii) For all g, and i, j : dij(g) = M , BET Wi(g ∪ ij) ≥ BET Wi(g) and BET Wj(g∪ ij) ≥
BET Wj(g) and for all k ∈ N \ {i, j} it holds that BET Wk(g∪ ij) ≥ BET Wk(g).
(iii) For all g, and i, j : 1 < dij(g) < M, BET Wi(g∪ ij) ≥ BET Wi(g) and BET Wj(g∪ ij) ≥
BET Wj(g); and
P
k∈N \{i,j}BET Wk(g∪ ij) <
P
k∈N \{i,j}BET Wk(g).
Part (i) is based on a fundamental property that additional links can only reduce distances between players but never increase them. For part (ii) and (iii) w.r.t. i and j, we use that any new path in g∪ ij (that was not present in g) uses link ij, such that the brokerage of i and j cannot decrease. Part (ii) and (iii) w.r.t k∈ N \ {i, j} distinguishes between critical and non- critical links. Critical links establish new connections and can lead to additional brokerage. Non-critical links reduce aggregate betweenness by reducing the sum of distances (without changing the number of unconnected pairs, see Eq. 4.2). Since the betweenness of the focal players cannot decrease, there must be other players k who lose betweenness benefits. This is plausible, because the link ij first of all takes away the brokerage benefits for all agents that were on their geodesics before. Moreover i, j can now be on shortest paths were others were before.
Effects on Welfare
When individuals alter the network structure, they do not consider the consequences for other players. First of all, there might be diverging interests between two agents (i and j) about forming a link. While one of them might want to form, the other one can hinder him; respectively, if the link is already present (g0), one agent can cut the link, without considering
the harm it does to the other player involved. Such a problem can be relaxed by allowing agents to pay transfers to other players whom they form a link with. In Chapter 5 we define the concept
of pairwise stability with transfers. Our focus here will be on a more robust problem: The effect of an action (of one or two players, i, j) on the players not involved in a link k ∈ N \ {i, j}.72
Suppose two agents i and j agree to form a link. With use of Lemma 4.4.1, we can partially characterize whether this agreement increases or decreases welfare:
Proposition 4.5. In the centrality model for any g ∈ G and i, j ∈ N : ui(g∪ ij) > ui(g) and
uj(g∪ ij) ≥ uj(g), the following holds (according to the utilitarian welfare function):
(i) If dij(g) = M, then w(g∪ ij) > w(g).
(ii) If dij(g) < M and λ≥ ˆλ, then w(g ∪ ij) < w(g).
(iii) If λ = 0, then w(g∪ ij) > w(g).
Part (iii) obviously follows from Lemma 4.4.1 (i): by assumption, player i strictly increases his utility, player j weakly increases his utility, while for all other players the benefits weakly increase (based on closeness only), but the costs (based on degree) do not change. Similarly, part (i) follows from Lemma 4.4.1 (i) and (ii). In contrast, part (ii) is directly shown by using W2. It is based on the increase of total costs, while the total benefits decrease. In fact, (ii) does not require the condition that ui(g∪ ij) > ui(g) and uj(g∪ ij) ≥ uj(g), that is: it holds
for any g ∈ G and i, j ∈ N : 1 < dij(g) < M. In words Prop. 4.5 shows the following effect
of a new link: if the link is critical, it increases welfare. If the link is non-critical and λ ≥ ˆλ, then it decreases welfare. For λ = 0 the link increases welfare, whether it is critical or not. The only case that is not covered by Prop. 4.5 is when ij is not critical (for g) and 0 < λ < ˆλ. Then welfare may increase or decrease (because aggregate benefits increase, while aggregate costs decrease).
Note that the statements are formulated for the addition of links. For non-severance the proposition must be adapted by ui(g∪ ij) ≥ ui(g). Then, statement (i) and (iii) are not strict
anymore (w(g∪ ij) ≥ w(g)), since it might happen that everybody is indifferent.73
Remark 4.4.1. This proposition does not make statements about severance of links! Consider a network g and a player i : ui(g\ ij) > ui(g). If the link is critical (i), we can show that for
all players k ∈ N \ {i, j} it holds that uk(g\ ij) ≤ uk(g), but the effect for j is not clear. In the
third case (iii) (λ = 0), it can be shown that for all k∈ N \{i, j} it holds that uk(g\ij) ≤ uk(g),
but the effect for j is not clear (since his closeness decreases and his costs decrease). In the
72
We will informally call this effect “spillovers” throughout this section, while in the next chapter we formally define externalities and analyze their role in strategic network formation.
73
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discussion of integrating an isolate in Subsection 3.3.3, we have already observed that marginal utilities of a new link can be very different. Only in the second case (ii) (if the link is non- critical and λ ≥ ˆλ), the effect of cutting a link is clear: welfare increases. Considering W2, one can see that total benefits do not decrease, while total costs strictly decrease. Agent j then might have negative marginal utility, but he cannot block i0s decision.
Let us study how the discrepancy between individual and collective interests drives ineffi- ciency in three examples, one for each case of Prop. 4.5.