Planar Beauty Contests

We introduce a planar beauty contest game where agents must simultaneously guess two, endogenously determined variables, a and b. The system of equations determining the actual values of a and b is a coupled system; while the realization of a depends on the average forecast for a, a', as in a standard beauty contest game, the realization of b depends on both a' and on the average forecast for b, b'. Our aim is to better understand the conditions under which agents learn the steady state of such systems and whether the eigenvalues of the system matter for the convergence or divergence of this learning process. We find that agents are able to learn the steady state of the system when the eigenvalues are both less than 1 in absolute value (the sink property) or when the steady state is saddlepath stable with the one root outside the unit circle being negative. By contrast, when the steady state exhibits the source property (two unstable roots) or is saddlepath stable with the one root outside the unit circle being positive, subjects are unable to learn the steady state of the system. We show that these results can be explained by either an adaptive learning model or a mixed cognitive levels model, while other approaches, e.g., naive or homogeneous level-k learning, do not consistently predict whether subjects converge to or diverge away from the steady state.