Jackson and Wolinsky (1996) introduced the connection model in which links bet-ween players can be interpreted as social relationships that provide benefits to players in the social network. Forming a link with another individual requires to make some costly effort but in turn it allows access to the benefits available to the latter via his own links. In the following model we can think of a network of social relations and social contacts in which players benefit from indirect connections as from the ”friends of a friend”.
This idea of players forming costly links but benefit by indirect links is known in the literature under the name symmetric connection model. There exist two versions of the connection model. The first one was introduced in Bala and Goyal (2000) where individuals can form links unilaterally34. The other version was first mentio-ned in Jackson and Wolinsky (1996) and considers non-directed formation of links, where players form links on the basis of bilateral agreements. Both versions have been extensively studied in the literature.35 As the present thesis concentrates on the formation of trade agreements which require the consent of all parties involved
33Goyal and Joshi (2006a) as well as Furusawa and Konishi (2007) model the formation of trade agreements as a network formation game. The models consider exclusively the formation of bilateral links.
34In section 1.4 of this chapter we will briefly mention the literature on unilaterally link formation where networks of social contacts are modelled by means of directed graphs. It is based on the pioneering paper by Bala and Goyal (2000).
35See for example Johnson and Gilles (2000) and Jackson and Rogers (2005).
and are therefore described by means of non-directed networks, we will in the fol-lowing report the main findings of the symmetric connection model of Jackson and Wolinsky (1996).
In the symmetric connection model each player pays cost c for the formation of a link and obtains from each direct and indirect link a payoff, where the payoff from a link decreases with the distance two (indirectly) linked players have. Formally, the payoff player i obtains from a direct link to another player is given by δ, where δ ∈ (0, 1), and this payoff decreases with the distance that two players have in a network. δ denotes the decay factor from indirect links such that the payoff from an indirect link decreases with the distance between the two players in the social network.
Consider for example the network with 4 players where player 1 is connected to player 2 and player 3 but not directly connected with player 4. Player 4 however is directly connected with player 3. Player 1 gets the benefit δ from his direct links with player 2 and 3 and the benefit of δ2 from the indirect link with player 4. As it is assumed that δ ∈ (0, 1) this network leads to a smaller payoff for player 1 from the connection with player 4 than from the connection with player 2 and 3.
Jackson and Wolinsky (1996) defined efficiency of networks in the way that a network structure is efficient whenever it maximizes the total payoff of all players, where the payoff of a network is calculated as the sum of the payoffs of all players. In the sym-metric connection model we find that when costs are very low, the complete network is the unique efficient network structure as the formation of a direct link is always more profitable than the gains from an indirect link. For very high linking costs the empty network is efficient, whereas for moderate costs a star network, which descri-bes a situation in which all players are directly linked to one center player and there are no other links such that each player is indirectly linked to each other player, will be efficient. In Figure 5 we illustrate the star network and the complete network for the case of 4 players. Jackson and Wolinsky (1996) adopt the notion of pairwise stability as a necessary requirement for networks to be stable which implies that the formation of a new link requires both players to be formed whereas deletion can be done unilaterally.
a) Star Network b) Complete Network
Figure 5
The results on stability can be summarized as follows: When the linking costs are very high, the unique stable network is empty since no player will ever have an incentive to form a link. When linking costs are very low, the complete network will be formed in which each player is linked bilaterally to any other player since direct links are more beneficial than indirect links. When linking costs are of a moderate level, players will form a star network, in which one player has bilateral links to all the other players and there are no additional links.
One of the central questions Jackson and Wolinsky (1996) address is whether ef-ficient networks will form when self-interested players can choose to form and severe links. They point out that the set of efficient networks and equilibrium networks does not always coincide. In the symmetric connection model, in spite of its simpli-city of the payoff structure, there exists a trade-off between individual and social incentives. Consider the case of moderate costs when the costs of a direct link ex-ceed the benefits from a direct link with δ < c. Then relationships are only beneficial whenever a player is involved in some direct and some indirect links. It is clear that a star network cannot be an equilibrium as the center player will always severe its direct links. It can however be shown that in this cost range a star network might be efficient for some values of c > δ, in the case when gains from indirect connec-tions (δ2) are high enough such that the total value of a star network exceeds the
total value of an empty network (which is zero). This model of social connections between players depicts the conflict that can occur between social and individual incentives.36
It is clear that this stability concept is not very strong as it only considers the de-viation of single links. In our discussion in section 2.6 of chapter 3 we introduce the notion of a stronger stability concept in which players are allowed to delete more than just one link at a time.
The connection model concentrates on bilateral link formation, where networks are modelled by means of non-directed bilateral graphs. In section 2.3 of this chapter we will report the main findings of Jackson and Wolinsky (1996) on stability and efficiency when we extend the framework of Jackson and Wolinsky (1996) toward multilateral link formation, where networks are modelled by means of non-directed hypergraphs. In a hypergraph links can include more than just two players. We defi-ne efficiency and stability notions for hypergraph defi-networks and compare the results of the connection model with the ones in Jackson and Wolinsky (1996). Afterwards we will discuss the conflict between the two concepts.
The following example illustrates that there are cases in which there are no pairwise stable networks and only cycles exist.
3.1.3 Trading Example (Non-existence of Pairwise Stable Networks)