Strategic network formation
In many networks, the relation between nodes is determined by the choice of the participating players involved, not by an arbitrary rule.
A strategic network formation requires that individuals create relations that are beneficial and drop those that are not.
One of the most well-known examples in this context is the marriage network of sixteen families in Florence, which showed how the Medici family gained power and took control of Florence by creating a high number of inter-marriages with the other families.
[3] There are fundamental differences in the way these games are modeled depending on their graph structure.
The benefits that they receive from the network are represented by utility functions.
A network game is a set of linked players and their utility functions.
Some of the modeling methods that separate the utility allocation from the network formation process are extensive form games, simultaneous move games, and pairwise stability.
If a network is modeled according to the extensive form game concept then the players of the network first propose to create links one after the other and afterwards they make decisions to create a link or not.
In simultaneous move game settings, all the players declare at the same time to whom they want to link.
Even though these sorts of games are easy to understand and analyze, their drawback is that they have multiple Nash equilibria.
The concept of Nash equilibrium has a drawback in this case since it does not take into consideration the fact that the players can discuss their decisions.
To model such a situation a stability concept that takes this fact into account is required.
In strategic network formation it is important to look at the overall social benefit and to see if networks that players create manage to be efficient for the society in general.
[3]: 206 The Pareto efficiency notion is more reasonable in settings in which allocation rules are fixed.
[3]: 32 A network can Pareto dominate another network if it has strictly larger benefits for one individual and weakly larger benefits for all individuals.
In the figure "An Example of Efficient, Pareto Efficient, and Pairwise Stable Networks in a Four Person Society" an example with four players is given, where the payoffs of the players are noted by the numbers next to the nodes.
The network in green is Pareto efficient since the payoffs are higher but it is not pairwise-stable because the players that have created only one link would also benefit by adding links to one another.
The only Pairwise Stable network in the figure is the dark blue colored one since none of the players involved would benefit by deleting or creating a link.
Jackson and Wolinsky showed that for homogeneous connection cost, the efficient network can only take one of three forms: a complete graph, a star or an empty graph depending on connection cost and benefits.
Finding general analytical solutions for the efficient networks with heterogeneous costs can be intractable.
[5] The utility the players receive does not just come from the direct links that they form with each other, but also from their indirect relations.
When we consider distance, the utility function takes the form
[3]: 209 The distance-based utility assumes that all players’ utility functions are alike and it takes into account only the benefits from indirect links that depend on minimum path length.
The distance-based utility showed that the payoffs of players do not just depend on the direct links that they form, but also on the links that other players have created in the network.
Players may confront positive or negative externalities in networks.
On the other hand, a model that faces players with negative externalities is the so-called “Co-Author model” presented by Jackson and Wolinsky in the paper of 1996.
Given that working on a research paper requires time and devotion, two researchers can benefit more if are only working with each other at a given period of time and not with many other people.
Therefore, in the “Co-Author model” researchers benefit more if their other colleagues have fewer links.
In this model, if a player’s neighbors have many links, it will bring negative externalities to them.
In different models, positive or negative externalities lead to inefficiency.
