Self-fulfilling Equilibrium in Social Contests: Expectation and Neighborhood Effects

We consider the following social contest model. In our model, individuals share a resource proportionally to their (educational) efforts, and efforts are costly. However, individuals, located in a network, are locally-sighted. That is, each individual only perceives the resource in his neighborhood and responds to the efforts of his neighbors.

Local Sightedness thus reflects a constraint neighborhood imposes on its residents. To capture plausible outcomes of the model, we propose a new equilibrium concept: self-fulfilling equilibrium.

A self-fulfilling equilibrium consists of a share vector and an effort vector: given the total share of his neighborhood, an individual’s effort is a best response to the efforts of his neighbors; and given the effort vector, an individual’s share is proportional to his effort (and consequently, the share of any neighborhood is proportional to the total efforts of the neighborhood). We show that if efforts cost the same across individuals, in any equilibrium, an individual with higher expectation, i.e., exposed to neighbors of more resource, exerts higher effort and obtains higher resource himself. In particular, if the network is strongly connected and the cost is linear, then the (unique) equilibrium effort vector is proportional to the eigenvector centrality of the network. We then look at the case when costs are different, and show in particular when the disadvantage of higher cost may or may not be compensated by different network structures.