Example C: Very High Costs

Consider the setting (λ, c) = (0.1, ˜c) with 1+0.1

n−1 < ˜c < T 2 (=

(1−λ)(M n−2n+2) 2(M −1)(n−1) +

λ

2(n−1)), where

the star network is efficient but not stable.

For this setting of c, the sequence of link formation will lead to sparse networks (any method showed a strong pattern that density is decreasing with c). For c = hi already non-critical links only rarely persist in the long run. However, networks that only consist of critical links, contain loose ends and cannot emerge for c > 1+0.1_n−1 by Prop. 3.3 (ii). In the enumeration for M = n = 8, 97.5 percent of the stable networks are not stable for c > 1+0.1

n−1 . There are only three candidates

for emerging networks: the circle network, a network consisting of a 7-circle and one isolate, and the empty network. We have not run any simulation for such high costs, c = v_hi, but our conjecture is that in the large majority of simulation runs the empty network would emerge.

Consider two players willing to form a link in the empty network. They would increase their benefits by 1−0.1

n−1 , which is not worth the costs c. Forming such a link would not have any

spillovers to other players, thus individual interest and collective interest in this link coincide. However, consider the network g =_{{12, 34} (for n ≥ 5). Forming a link between player 2 and} 3, would change their benefits by β := (1−λ)[2M −3]_{(M −1)(n−1)} +_{(n−1)(n−2)}2∗2λ . Now consider that ˜c = β, then 2 and 3 are indifferent about forming the link. For lower costs ˜c < β the link 23 will be formed, while also agent 1 and 4 benefit from that (their closeness increases by (1−λ)[2M −5]_{(M −1)(n−1)}, their degree and betweenness stays constant). By Prop. 4.5 (i) welfare increased. If, however, ˜c is slightly greater than β, then players 2 and 3 do not form the link, although it would still be socially beneficial (as long as 2c < 2β + 2(1−λ)[2M −5]_{(M −1)(n−1)}).

spillovers or no spillovers. In fact, we can generally show that there are no negative spillovers: for any g and ∀ij : dij(g) = M (ij is critical for g), it holds that for any k ∈ N \ {i, j},

uk(g∪ ij) ≥ uk(g). 79

Moreover, it can be can shown that if λ < 1 and Ni(g)∪ Nj(g)6= ∅ (not

both are isolates), it holds that P

k∈N \i,juk(g∪ ij) >P_{k∈N \i,j}uk(g).

While in the network discussed above (g =_{{12, 34}) both players agree not to form a critical} link, although it would be socially desirable, it is generally sufficient that one player involved in the link does not accept it. For example, in the efficient network, the star network, the central player of the star is not willing to keep his links, since the costs ˜c exceed the benefit of a neighbor that does not lead to indirect connections 1+0.1

n−1 . This central player does not consider

the harm he does to the peripheral player, who is willing to keep his link. Neither he considers the negative spillovers to the other peripheral players.

We argue that the issues discussed in this example can be generalized to any setting of λ, for sufficiently high c (c high enough that most non-critical links are not formed and loose ends are not stable, but below the thresholds where the empty network is efficient). First, it may happen that critical links typically have positive spillovers that are not internalized by the two players involved. At the same time a different issue seems to be predominant: A critical link that would increase welfare, is only accepted by one of the two players involved.80

Thus, inefficient outcomes (empty network emerges, while trees, i.e. the star network and the line network, are efficient) are rather explained by an asymmetry of link formation – it takes two players to form a link but one to sever it – than by positive spillovers of critical links.

In the three examples of this section, we identified some basic effects leading to inefficient networks. Example C presents agents that decrease the welfare of a network by unilaterally severing critical links. Those links exhibit positive spillovers either for the second player involved or for other players or for both. A similar issue is illustrated in Example B, where critical and non-critical links have positive spillovers. Moreover, we discussed in that example that the sequence of link formation might also prevent the dynamic process from reaching the efficient outcome. Finally, Example A shows agents that form links that are socially harmful, not

79

We show this result in the proof of Prop. 4.5 (i). 80

We already discussed this issue in Subsection 3.3.3 where we examined the integration of an isolate to explain the emergence of unconnected networks for λ = 1 and c = med, hi. This was the only setting of the simulation where costs were sufficiently high to observe the phenomenon of unconnected networks frequently.

Centrality ₁₀₀ considering the negative spillovers.

Although the identified effects seem very important in determining the outcome of our model, it must be noted that they are not necessarily the only effects and that they do not occur in isolation. This makes the explanation of the emerging networks a non-trivial task. For example, the most frequently emerging network for λ = 1, c = lo (in the simulation with n = 8), depicted in Figure 8, exhibits the issue of non-connectedness (discussed in Example C) and the issue of addition of non-critical links (discussed in Example A). Still, we try to trace back the emerging networks to those effects when summarizing the tension between stability and efficiency in the next subsection.