3.2 Network Formation Games and Hypergraphs
3.2.2 A Model of Network Formation
4 5
Figure 8: Linear Hypergraph
Cliques
A clique is a maximal subset of nodes with the property that every pair of nodes has a link. Formally, a set of nodes N0 = {i1, i2, ...ik} ⊂ N , where k ≥ 3 is a clique if, for every pair i and j ∈ N0, there exists an L ∈ L for that {i, j} ⊆ L and there exists no subset N00⊂ N0 with this property. In the complete network of Figure 7a) there exists one clique that contains all nodes and each pair of nodes has a bilateral link.
3.2.2 A Model of Network Formation
The importance of network structure on economic outcome motivates an examinati-on of the incentives of players to form links and of the strategic stability of different structures when linking decisions depend on players’ payoffs. A simple way to ana-lyze stable network structures is to examine the requirement that individuals do not benefit from altering the structure by single deviations.
In this chapter we adopt the basic concept of pairwise stability but allow players to form and sever multilateral links.
First we investigate the stable and efficient structures for the connection model.
We show that the results of Jackson and Wolinsky (1996) are a special case of ours, whereas in our framework we obtain a larger range of possible stable networks but therefore also a larger range of possible efficient networks. Then we give a simple existence proof of multilaterally stable networks by introducing the concept of an improving path. We also provide examples in which multilaterally stable networks exist and an example in which we obtain that each improving path results in a cycle.
We obtain existence of a multilaterally stable network in the trading example and
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7
8 L1
L2
L3
L4
L5
Figure 9: n = 8, L = {L1, L2, L3, L4, L5} where L1 = {1, 2}, L2 = {5, 6, 7}, L3 = {2, 3, 6}, L4 = {4, 7} and L5 = {3, 4, 8}.
demonstrate with an example that multilaterally stable networks not always exist.
Consider a finite number of identical players N = {1, ..., n} and assume n ≥ 3.
In our setting we concentrate on undirected links as in Jackson and Wolinsky (1996) which means that a link between players needs the consent of all players involved in that link.38
With the aid of hypergraphs we are able to model multilateral agreements between players.
Definition 3.1. Let N = {1, ..., n} be a finite set of nodes. A family of subsets of N , L, where L = {L1, ..., Lm} is a set of links, L ⊆ 2N, is called a hypergraph on N .
In the following the set of nodes will represent the set of players and the term net-work will be used as a synonym for the word hypergraph. Since each player is linked with himself we restrict our attention to hypergraphs L with L ⊆ {L ∈ 2N||L| ≥ 2}.
The set of all possible hypergraphs that satisfy this definition is denoted with H. A hypergraph is shown in Figure 9.
If L ∈ L, we say that all player i ∈ L have a direct link. This could for example mean that a set of countries has a multilateral free trade agreement. In industrial
38In this thesis we will consider non-directed links such that a link between a group of players requires the consent of all players involved. For directed networks see Bala and Goyal (2000).
organization this could mean that a group of firms forms an alliance. In social con-tact networks it means that a group of people shares and exchanges information.
A global hypergraph is denoted by LG and consists of a single link that contains all players in N with LG = {N }.
The complete network LN is the family of subsets of N with LN = {L ∈ 2N||L| = 2}.
The star with center i, which we denote by LSi, has only bilateral links from the central player i to each of the other players with LSi = {L ∈ 2N||L| = 2 and i ∈ L}.
We denote the empty network by Le.
N (L) denotes the set of players i ∈ N that have at least one direct link in L, N (L) = {i ∈ N | ∃L ∈ L : i ∈ L} and, as already mentioned above, Ni(L) denotes the set of players that are directly linked with player i in a multilateral agreement in network L.
Value Function
The value of a hypergraph is represented by a real valued function v : H → R, which specifies for each hypergraph L ∈ H the total value v(L) generated by L. In most applications it will be the aggregate of individual payoffs or productions of a hypergraph, with v(∅) = 0. The set of all possible value functions is denoted by V.
In chapter 2 it can be understood as the aggregate of all countries’ payoffs in a trading system L.39
Allocation Rule
An allocation rule is a function Y : H × V → Rn that describes how the value is distributed among the players and assigns a payoff Yi(L, v) to each player i ∈ N in the network L ∈ H for the value function v. An allocation rule for example assigns to each firm in a collaboration network its total profit. In an international trade net-work an allocation rule represents a country’s welfare in a netnet-work of international trade agreements. In our model of chapter 2 this was represented by equation (5).
39Of course, in chapter 2 and in the following chapters 4 and 5 the value of an empty network can be positive. For simplification, we will maintain the normalization v(∅) = 0 for the rest of the chapter.
Figure 10: Complete graph and the global hypergraph for n = 4
When v is fixed, we will write Yi(L).40
Stability
We introduce the following notations:
• For L 6∈ L, L ∪ {L} is the network we obtain from L when we form the link L.
• For L ∈ L, L\{L} is the network we obtain from L when we sever the link L, if L ∈ L.
Furthermore for any set of links L0 ⊆ L we define
• L\L0 is the network we obtain from L by severing all links in L0.
The formation of a link requires the consent of all players involved, but severance can be done unilaterally.
With ˜L = L ∪ {L} or ˜L = L\{L} the networks ˜L and L are called adjacent.
We introduce the following stability concept:
Definition 3.2. A hypergraph L ∈ H on N with L = {L1, ..., Lm} is called
multila-40In the connection model of section 2.3 we will define a player’s payoff as a function that only depends on the network structure and not on the value function as the value of a network is fixed and defined as the aggregate payoff over all players.
terally stable with respect to Y and v, if
(i) Yi(L, v) ≥ Yi(L\{L}, v) ∀L ∈ L, ∀i ∈ L and (ii) Yi(L ∪ {L}, v) > Yi(L, v) ⇒ ∃j ∈ L,
such that Yj(L ∪ {L}, v) < Yj(L, v) ∀L /∈ L
The above definition describes a situation in which no country has an incentive to sever any of its existing links and no subset of countries wants to form an addi-tional agreement.
Since in the above definition the formation of a new multilateral link needs the con-sent of all players included in the link, this definition differs from the noncooperative Nash equilibrium concept.
A network L that is not multilaterally stable is said to be defeated by either network L = L ∪ {L} if condition (ii) is violated for L 6∈ L, or it is defeated by network˜ L = L\{L} if condition (i) is violated for L ∈ L.˜
Efficiency
In order to study efficient hypergraphs, we consider the aggregate payoff of all players.
Definition 3.3. A hypergraph L∗ ∈ H on N is said to be strongly efficient relative to v, if v(L) =P
i∈NYi(L, v) ≤ v(L∗) = P
i∈NYi(L∗, v), ∀L ∈ H.
The term strong efficiency indicates maximal total value and not Pareto efficien-cy but one can easily verify that the set of efficient networks is a subset of the Pareto efficient structures. In this case, v is fixed and defined as the total aggregate payoff.
In the following we will simply refer to the set of strongly efficient networks as the set of efficient networks.
In the context of chapter 2 one can interpret an efficient network as a trading system that maximizes the aggregate payoff over all countries and can be understood as a trading system that maximizes world welfare.