The strong efficiency concept we introduced in section 2.2 of this chaper is a very strong concepts in the way that the set of efficient network structures is relatively small as compared to the set of stable network structures.
Another notion would be if we think of Pareto-efficient networks in which one cannot alter the network structure such that at least one player is better off without making another player worse off. Obviously this notion is a very weak concept and we will compare the results for the connection model with stability.
Definition 3.19. A network L is Pareto-efficient relative to v and Y if there does not exist another network L0 ∈ H such that Yi(L0, v) ≥ Yi(L, v) for all i with strict inequality for at least one i.
It is obvious that Pareto efficiency is a relatively weak notion compared to strong efficiency since the strong efficiency notion allows redistribution of the players’
payoffs whereas under Pareto efficiency we consider a specific allocation rule. It is clear that a strongly efficient network is therefore also Pareto efficient, since even when payoffs can be redistributed no player can improve by altering the structure without reducing the payoff of any other player. And this is exactly the definition of Pareto efficiency given a certain allocation rule. In this notion we fix the allocation rule whereas in the strong efficiency notion payoffs can arbitrarily be allocated and
transferred among the players. The next notion of efficiency lies in-between the other two notions.
Definition 3.20. A network L is called constrained efficient relative to v if there does not exist any network L0 and a component efficient and anonymous allocation rule Y such that Yi(L0, v) ≥ Yi(L, v) for all i with strict inequality for at least one i.
This notion is stronger than Pareto efficiency because here we demand that for all allocation rules that are anonymous and component efficient we cannot improve a single player without reducing the payoff of any other player. In Pareto efficiency we only demand this condition for a specific allocation rule and whereas constrained efficiency considers a class of allocation rules.
We can easily verify that there always exists a Pareto efficient network and a cons-trained efficient network since there always exists a strongly efficient network and every strongly efficient network is Pareto efficient as well as constrained efficient.
To understand the relationship between the three stability notions we will give ex-amples in the following. First we demonstrate the difference between the set of constrained efficient networks and strong efficient networks.
Example 3.7. Consider a network with n = 5 and any component additive and anonymous value function v where the network that consists of a global link between all players has a value of 10, a bilateral link between any two players has a value of 2 and a hyperlink between three players has a value of 9. All the other networks have a value of 0. We can easily verify that a network that consists of two connected components where one is a bilateral link and one is a hyperlink between three players is the unique efficient structure with a total value of 11. The global link between all players cannot be strongly efficient whereas we can show that it is constrained efficient.
Consider any anonymous and component efficient allocation rule that allocates in a global network the payoff 2 to each player. In the network with two connected components each player in the bilateral link would receive a payoff of 1 from each anonymous and component efficient allocation rule such that they would be worse off. Therefore the global network is constrained efficient.
To see that the network that consists of two connected components is constrained
Constrained Efficient and Strongly Efficient.
Constrained Efficient but not Strongly Effi-cient.
Figure 13: Constrained Efficiency versus Strong Efficiency.
efficient consider that from any anonymous and component efficient allocation rule each player in the hyperlink between three players will receive a payoff of 3 whereas in the global network they obtain a payoff of 2. Therefore they will be worse off under the global link and therefore the network that consists of two components is also constrained efficient. This is summarized in Figure 13.
The next example clarifies the relationship between constrained efficient networks and Pareto efficient networks.
Example 3.8. Consider n = 3 and an anonymous allocation rule where a global link between three players has value 9 and a network that consists of two bilateral links has value 8. All the other networks have value 0. The allocation rule is component efficient and anonymous. It allocates the payoff of 3 to each player in the global link and in each network with two bilateral links it allocates the payoff of 4 to the player with two links and the payoff of 2 to both end players.
Since the total value is maximal in the global network, it is Pareto efficient and constrained efficient. We can also verify that a network that consists of two bilateral links is Pareto efficient since any other network will result in a lower payoff for at least one of the players. But we can also verify that it is not constrained efficient since we can find another anonymous and component efficient allocation rule, such that all players will be better off under the global network. This can be seen when we allocate the payoff of 83 to each player in the network and therefore all players will
Pareto Efficient but not Constrained Effi-cient.
Constrained Efficient and Pareto Efficient.
Figure 14: Constrained Efficiency versus Pareto Efficiency.
improve with the global network. This is shown in Figure 14.
In the symmetric connection model of section 2.3 we observed in Proposition 2.1 and 2.2 that the set of efficient networks and stable networks do not always coincide for certain parameter values. When linking costs are very high, c > δ + (n−2)2 · δ2, we obtain that the only stable and efficient network is the empty network. For very low costs each efficient network can be stable. One problematic case occurs when linking cost are at an intermediate level. Here we can observe that there are stable structures that are not efficient. For δ < c < δ +n−22 ·δ2 a star network is efficient but not stable. In this cost range an empty network is stable. In the following we shall investigate whether the star network can be efficient when we relax the efficiency condition and determine the Pareto efficient networks. Furthermore we observed that for low costs of links there are stable networks that are not strongly efficient.
The next result characterizes the set of Pareto efficient networks in the symmetric connection model.
Proposition 3.5. In the symmetric connection model we find that (i) for very high linking costs the empty network is Pareto efficient.
(ii) Whenever δ < c < δ +n−22 ·δ2 the empty network is stable but not necessarily Pareto efficient.
(iii) If δ − c > δ2 > 0 the set of strongly efficient and Pareto efficient networks coincides.
This result shows that for middle costs to link some stable networks are not even Pareto efficient. Consider the network L = {{1, 2, 3}, {2, 3, 4}, {1, 4}} of section 2.3 with n = 4. With K = 2 and δ − 2c > δ2 > 0, L is stable but not efficient. To see that L is not even Pareto efficient player 2 and player 3 can improve without reducing player 1’s and player 4’s payoffs with L0 = {{1, 2, 3}, {2, 4}, {1, 4}, {3, 4}}.
Whenever δ < c < δ + n−22 · δ2 we find that stable network are not necessarily Pareto efficient. In the appendix we provide an example that shows that for certain parameter values a line network Pareto dominates the empty network.