Bloch, Genicot, and Ray (2008, JET)

The paper's abstract is available at http://dx.doi.org/10.1016/j.jet.2008.01.008.

Consider a village in a developing country where households engage in bilateral insurance schemes with directly connected households (relatives or friends, for example). If a household deviates from the insurance scheme (ie. fails to transfer money to directly connected households that have negative income shocks), the victim household will tell such behavior to connected households who in turn terminate the insurance scheme with the deviating household as a punishment.

The question is what aspects of the household network characterizes a stable household network, that is, a network in which no household has incentive to deviate from the bilateral insurance schemes.

Denote by q the number of links that the victim's information is passed through. For example, q equals two if the victim tells its directly connected households which in turn tell their own directly connected households, but the information does not spread beyond that.

Then the stable network is characterized by the maximum length of the smallest cycle connecting any three households in the network being q+2.

Why? Suppose household i fails to transfer money to household j. Household j tells its suffering to its connected households with the path length of less than or equal to q. These informed households cut the link to household i. Since the length of the smallest cycle of any three households in the network is at most q+2, the information from household j reaches all the households with (direct or indirect) links both to i and j. As a result, household j will be cut off from the network that includes household i, by deviating from the insurance scheme. A smaller network yields a lower payoff (intuitively, pooling income risk among more households provides better insurance against negative income shocks). Therefore, household i has no incentive to deviate, which means that the household network is stable.