Coordination under global random interaction and local imitation

Individuals are randomly matched to play a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$2\times 2$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math> </EquationSource> </InlineEquation> coordination game where the Pareto efficient and risk dominant equilibria differ. Players choose strategies by imitating the strategy of the most successful individual they observe. So, while individuals interact globally, their observation and hence imitation, as determined by their social network, may be local. When all individuals observe each other, the most successful individual in the entire population is imitated; here, in the stochastically stable state, the population coordinates on the Pareto-efficient outcome. While this outcome is always stochastically stable, even when observability is incomplete, the state where everyone plays the action of the risk-dominant equilibrium may be stochastically stable as well. Reasonably tight sufficient conditions for unique stochastic stability of the state where all coordinate on the Pareto-efficient equilibrium strategy include each individual observing at least four other individuals or when each individual observes the same number of other individuals. These properties are readily generalisable to larger <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$m\times m$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>×</mo> <mi>m</mi> </mrow> </math> </EquationSource> </InlineEquation> coordination games where the coordination outcomes can be Pareto-ranked. Copyright Springer-Verlag Berlin Heidelberg 2014