Summary of the Tension

The summary refers to Figure 23 and is organized according to the areas where a network is uniquely efficient (ignoring the frontiers of these areas, where we have a multitude of efficient networks).

Starting at the bottom left (i), in this area the complete network is efficient and uniquely stable. From both, the individual and the collective perspective, a link between any pair of players is worth its costs. Above that, there is the region where the star network is efficient. In the lowest area (iv) of this region the complete network is uniquely stable. Consider a network g : g _{⊇ g}?_{. As discussed in Example C, there are no positive spillovers on closeness from linking}

in this case. However, the addition of a link decreases the betweenness of the players who are directly linked to both (there is at least one such player, the center of the star). In this area (of the parameters), costs c are low enough such that the individual increase in closeness is worth the costs (c < 1−λ

(n−1)(M −1)). Thus, each pair of players uses each opportunity to add a

link. While this might also increase other player’s utility in many situations, at some point, the network becomes a superset of the star network. Then, players continue to add links without considering the negative effects on the utility of other players (some loose betweenness benefits). That is why we observe for λ > 0 such an area where the star is efficient and the complete network is uniquely stable.

Above that, there is an area (iii), where the star network is efficient and also stable but not uniquely so. The emerging networks exhibit welfare close to efficiency. However, for λ _{≈ 0,} there is systematic problem that was discussed in Example B. The addition of links not only increases the utility of the involved agents but might also increase closeness of other agents. If

this increase in closeness is higher than the decrease in betweenness, then not involved agents benefit from the link. Since maintenance of links is costly, there are situations where agents refuse to build the links, although they would increase welfare. Thus, many emerging networks feature distances that would socially be worth bridging but no pair of agents is willing to do so. For λ >> 0 this effect need not be at work, since spillovers on betweenness can be negative. Above that, there is a large area where the star network is still efficient but not stable. Prop. 4.4 (v) states that in this area any non-empty network is inefficient. Specifically, the central player of the star is not willing to keep his links, although they would be socially beneficial, as discussed in Example C. Moreover, we make the conjecture in Example C that for this setting of c, the sequence of link formation will most frequently lead to the empty network, either because the other stable networks (typically we found networks with isolates and large circles) are not very easily reached in the dynamic process, or because for large enough c the empty network is uniquely stable.81

Thus, individual incentives to form some links break down for a level of c, where it can still be socially desirable to keep them.

For high enough c (ii), the empty network is efficient and uniquely stable (Prop. 4.4 (ii)). In that case the individual and collective incentives are aligned, since no link is worth its costs, neither collectively nor individually.

Finally, there is a region where the line network is efficient. Prop. 4.4 (vi) ruled out stability of the line for λ large enough.82

Consider the line network as starting network in the dynamic process. With its distances, the line network offers a maximum of betweenness. However, it is not stable. When two players that are at a large enough distance have the opportunity to form a link, they will do so. For λ large enough, players with zero betweenness, i.e. at a loose end, do not keep their link. This sequence of shortening distances and cutting ties by the loose ends would lead to isolates and a dense component (such as in Figure 8 for (λ, c) = (1, lo)). Similarly, for given λ and c large enough, we expect the empty network to emerge in most of the cases as discussed in Example C.

Thus, in the area where the line network is efficient, for higher c there is the problem that unconnected networks, i.e. the empty network, emerge; for lower c, the problem is that non- critical links are added. Both problems can occur at the same time, e.g. for λ = 1 and c = lo.

81

That is, we argue that the threshold of the empty network to be uniquely stable is lower than the threshold for the empty network to be efficient. See also, Prop. 3.3 (iii) and its discussion in Subsection 3.2.1. 82

Moreover, there is the conjecture that the line network is never stable for λ_{≥ ˆλ and n ≥ 7, see Remark 4.5.1} in proof of Prop. 4.4.

Centrality ₁₀₂

As a consequence, relative efficiency is low, which becomes apparent for λ >> ˆλ. For λ close enough to ˆλ, say λ = 0.5, and c low enough to exclude the first issue, the emerging networks are much denser than efficient, but without exhibiting low relative efficiency. This is because short distances do not severely reduce total betweenness (for λ ≈ ˆλ) and total costs are not dominating total benefits (for c low enough). However, it is still true in this region that networks emerge that are socially improvable by severance of ties.

Individuals in our model myopically increase their utility. What they do not consider are the consequences for other players. This basic problem between individual incentives and collective consequences is one of the main forces driving the discrepancy between the agents’ goals and the social outcome – that is: the tension between stability and efficiency – in various settings of our model. We will analyze this problem in a much more general framework in the next chapter.