Clubs and Social Networks Chaim Fershtman and Dotan Persitz Eitan Berglas School of Economics in Tel Aviv University CTN, Marseille, June 2010

2 Social Links and Club Membership Social connections are pairwise. However, most acquaintances are formed within social clubs. Social clubs provide the opportunity to create social connections. Thus, club membership is crucial for social network formation. – Clubs: family, university, gym, series of conferences, hobby, etc.

3 Multiple Membership and the Social Network Nowadays, many people have multiple memberships. – Main distinction from “club theory”. Multiple membership has many implications. – Network of clubs - Interlocking directorates. – Simmel (1908) – “The Web of Group Affiliations” Two individuals may know each other if their share a social club. Otherwise, they can communicate through a third party which shares a club with both. – Multiple membership is a necessary condition.

4 Example – Complete Environments

5 Weights Many sets of choices can lead to the same architecture. The architectures differ by the size of the clubs. Congestion - the bigger the club, the weaker are the connections between its members. – Probability of actual meeting. The congestion effect leads to various links’ strength. – The bigger the club the link originates from, the weaker is the link. Links are weighted. – The weights have a component that represents the congestion effect. The architectures of the complete networks (for example) differ by the weights on the links. – Bloch and Dutta (2009).

6 Another Example – Star Environments The star allows for indirect connections. The path’s weight is a function of the weights along the path. Indirect connections result from multiple membership.

7 The Social Environment A finite set of identical individuals: A finite set of clubs: A pairimplies that individual i is affiliated with club s. A social environment G is a list of individuals, a list of clubs and a list of affiliations:

8 “Two Mode” and “One Mode” The social environment can be characterized by a bipartite graph. The social environment is a “two-mode” network. Thus, the social environment induces two “one-mode” networks: – The social network – our main interest. – The clubs’ network.

9 The Social Network The nodes are the set of individuals: Two individuals are linked if they share at least one club. The weight of the link between two connected agents: The congestion factor: The non-congestion factor:

10 Distance The weight of a path: Path p is the shortest path between individuals i and i’ if and only if there is no other path p’ between i and i’ such that: The distance between individuals i and i’ in G is the weight of the shortest path between them, denote by

11 Utilities Let c be the universal membership fees. Letbe the number of clubs with which individual i is affiliated. The utility of individual i from the social environment G is

12 Example – m-complete

13 Example – m-star

14 Clubwise Stability Let be the set of agents affiliated with club s in G. Some notations: where s is vacant in G A social environment G is clubwise stable if the following conditions hold:

15 m-complete

16 All-pairs If a=0, it is identical to the connections model (Jackson & Wolinsky (1996)).

17 Strong Congestion Prop.3 (omitted) shows that in that case the lower bound is much simpler. An implication of Prop. 3 is that if a>0 there exists such that for every, the m-complete environment has a range in which it is clubwise stable. Example: assume a=0.5. For every m there is an interval where the m-complete social environment is clubwise stable. In addition, assume For m>2, these ranges include the interval

18 m-star Identical to the connections model.

19 Congestion vs. connectivity Assume a=0 and. All pairs3-complete 2-star Using Propositions 2,3,4. 2-star is preferred socially over the 3-complete.

20 Issues Clubwise Nash – Demands a definition of the environment formation game. – Ilkilic & Calvo-Armengol (2009) does not hold. – Example: in a clubwise stable 2-star environment a non-central agent may want to leave his club and join another ( ). Admission rules – Currently free admission. – Could consider entrance approval policy with a voting rule for the existing members. Formation within the club. – A two stage game – choose clubs and then play a formation game within the clubs.

21 Conclusion Social connections are pairwise. However, most acquaintances are formed within social clubs. We introduce a model in which individuals: – Choose which clubs to join (costly). – Enjoy benefits from the induced social network. Individuals choices of clubs are derived by – Congestion effect (clubs’ size). – Connectivity effect (indirect connections). – Membership fees.