3.2 Network Formation Games and Hypergraphs
3.2.4 The Compatibility between Efficiency and Stability
We noticed in section 2.3 as we calculated multilaterally stable networks that effi-ciency and stability of networks are not always compatible. Furthermore, we showed
42Here a community is represented by a multilateral link between a group of players.
by means of the connection model that for certain parameter ranges stable networks are inefficient. The next section deals with the question whether it is always possible to find a multilaterally stable network that is efficient when we are free to define the allocation rule.
We first define further characteristics of value functions and allocation rules and introduce a permutation as a bijective function π : N → N of the player set. For each L ∈ H and L ∈ L we can define a network Lπ such that i ∈ L ←→ π(i) ∈ Lπ such that Lπ has the same structure as network L, with all that has changed is the label of the players. We can now define:
Definition 3.4. An allocation rule Y : H × V → Rn is anonymous if, for any permutation π, Yπ(i)(Lπ, vπ) = Yi(L, v).
Thus with anonymity of an allocation rule the payoff allocated to an individual does not depend on the label of an individual but on the network structure L and the corresponding value function v. The allocation only changes according to the relabelling.
Definition 3.5. The value function v is called anonymous if v(L) = v(Lπ) for all permutations π and networks L.
An anonymous value function allocates the same value to networks that have the same architecture independent of the labels.
Definition 3.6. An allocation rule Y : H×V → Rn is efficient ifP
iYi(L, v) = v(L)
∀ v and L.
Under efficiency no value is wasted and the total value generated from the net-work should be allocated among the players.
Definition 3.7. A value function v : H → R is component additive if v(L) = P
C∈C(L)v(C).
For component additive value functions the value of a network is simply the sum of the value of its components such that the value of one component does not depend on the value of the other components. This condition on value functions rules out
externalities across components and is satisfied in the connection model as well as in many other economic situations.
The next definition states that when a value function is component additive, the value that is generated by a component will be allocated among the members of the component.
Definition 3.8. An allocation rule Y : H × V → Rn is called component efficient if for any component additive v, L and all components C ∈ C(L)
X
i∈N (C)
Yi(L, v) = v(C), (8)
where N (C) denotes the set of players in component C.
Thus component efficiency applies when the members of a component have no incentive to allocate value to members outside of the component whenever there are no externalities across components such that the value function is component additive. It is important to note that component efficiency is merely required from value functions that are component additive. Otherwise component efficiency of an allocation rule when the value function does not satisfy component additivity would directly violate efficiency of the allocation rule.
The following result can be shown by means of an extension of the proof in Jackson and Wolinsky (1996).
Theorem 3.1. There does not exist an anonymous and component efficient allocati-on rule Y such that for every v there exists an efficient network that is multilaterally stable.
As in Jackson and Wolinsky (1996) the theorem does not state that there does not exist an allocation rule that satisfies component efficiency and anonymity for which a multilaterally stable network always exists. In section 2.5 we will see that for certain allocation rules that satisfy component additivity and anonymity there always exists a multilaterally stable network.
Theorem 3.1 states that one cannot always design an allocation which is anonymous and component efficient such that at least one efficient network is multilaterally stable.
The next two allocation rules are of particular interest: the egalitarian allocation rule and the component-wise egalitarian allocation rule.
The egalitarian allocation rule Ye is defined by:
Yie(L, v) = v(L)
n , (9)
for all i and L. The egalitarian allocation rule splits the value of the network equally among the players regardless of which role they play in the network. It is obvious that this allocation rule satisfies anonymity and efficiency but not component efficiency, as all player always obtain an equal share of the total value of the network.
The component-wise egalitarian allocation rule for a component additive v satisfies component efficiency and is given by:
Yice(L, v) =
v(C)
|N (C)| if there exists a C ∈ C(L) such that i ∈ N (C), 0 otherwise.
(10)
This allocation rule allocates the total value generated by a component to the mem-ber of the component in the way that each memmem-ber of a component receives the same payoff. As component additivity implies that disconnected players generate no value we have that Yice(L, v) = 0 if there exists no component C ∈ C(L) such that i ∈ N (C).
We start the analysis by introducing certain characteristics for links of a given net-work L. We introduce the term of a critical link. A link L ∈ L is called critical if it is contained in component C of network L and its deletion splits C into components C1, C2, ...,Ck.
Definition 3.9. A link L is called critical to network L if L\{L} has more com-ponents than L or if at least one player in L is only included in link L.
The last condition states that one of the players in L will become disconnected when L is severed.
Example 3.3. Let n = 6 and L = {{1, 2, 3, 4, 5, 6}, {1, 2}, {3, 4}, {5, 6}} as in Figure 12. Now link L = {1, 2, 3, 4, 5, 6} is critical to L and its deletion splits the network into three components C1 = {{1, 2}}, C2 = {{3, 4}} and C3 = {{5, 6}}.
1
2
5
6 3
4
Figure 12. Network with a critical link L = {1, 2, 3, 4, 5, 6}.
Consider the next property on networks and value functions that is needed for the next result.
Definition 3.10. The pair (L, v) satisfies critical link monotonicity if for any cri-tical link in L and its associated components C, C1, C2, ...,Ck we have that v(C) ≥ v(C1) + v(C2) + ... + v(Ck) implies that |N (C)|v(C) ≥ max[|N (Cv(C1)
1)|,|N (Cv(C2)
2)|, ...,|N (Cv(Ck)
k)|].
Note that in bilateral graphs the deletion of a bilateral link can split a component into at most two components. As Figure 12 demonstrates the severance of a critical link in a hypergraph can split a component into more than just two components.
Lemma 3.3. If L is efficient relative to a component additive v, then L is multilater-ally stable for Yce relative to v if and only if (L, v) satisfies critical link monotonicity.
For a given allocation rule Lemma 3.3 describes for which class of value functions the set of multilaterally stable networks and efficient networks coincide.
We shall now consider the characteristics of the egalitarian allocation rule.
Definition 3.11. The allocation rule Y is independent of potential links if Y (L, v) = Y (L, w) for all networks L and value functions v and w such that there exists a link L such that v and w agree on every graph except L ∪ {L}.
Definition 3.12. An allocation rule is pairwise monotonic if L0 defeats L implies that the value v(L0) > v(L).
Pairwise monotonicity implies that each efficient network is multilaterally stable since an efficient network which maximizes total value cannot be defeated under pairwise monotonicity. This property of allocation rules together with the indepen-dence of potential links uniquely characterizes the allocation rule.
The next result is a slight extension of Theorem 3 of Jackson and Wolinsky (1996).
Theorem 3.2. If Y is anonymous, pairwise monotonic, efficient and independent of potential links, then Yi(L, v) = v(L)n , ∀i, L and anonymous v.