Strategic network formation defines how and why networks take particular forms. In many networks, the relation between nodes is determined by the choice of the participating players involved, not by an arbitrary rule. A "strategic2 modeling of network requires defining a network’s costs and benefits and predicts how individual preferences become outcomes.
Introduction
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. “Thus, decisions about profitable relations are not a situation of choice, but a situation of strategic interaction – an aspect that is best covered by Game Theory”. In these kinds of settings, the nodes are usually called players, where N = {\displaystyle N=} {1, 2,… n {\displaystyle n} } is a set of players that have formed links in a network. Social Networks have diverse settings, however the simplest ones can be described by an undirected graph whereas more complicated situations are represented by directed graphs. There are fundamental differences in the way these games are modeled depending on their graph structure. If a link exists between player i {\displaystyle i} and player j {\displaystyle j} it is noted as i j {\displaystyle ij} . In cases of undirected networks, i j {\displaystyle ij} is considered equal to j i {\displaystyle ji} . A network g {\displaystyle g} represents a list of all the links between players. In a more formal setting, a network g {\displaystyle g} is defined as a set of unordered pairs { i , j {\displaystyle i,j} }, with i , j {\displaystyle i,j} element of N {\displaystyle \mathbb {N} } .The set of all possible graphs on the set of players N {\displaystyle N} is denoted with G {\displaystyle G} . The benefits that they receive from the network are represented by utility functions. That is, the payoff to a player i {\displaystyle i} is represented by a function u i {\displaystyle u_{i}} : G ( N ) {\displaystyle G(N)} → {\displaystyle \rightarrow } R {\displaystyle \mathbb {R} } , where u i {\displaystyle u_{i}} ( g {\displaystyle g} ) represents the net benefit that i receives if network g {\displaystyle g} is in place. To model strategic network formation the notion of network games is used. A network game is a set of linked players and their utility functions.
Modeling network formation Network games can be modeled in different ways. 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.
Extensive-form game modeling 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 such settings, a couple of players decide to either form a link or not by being aware of all the previous players’ decisions and by making predictions for the decisions of the following players.
Simultaneous-move game modeling 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.
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