Limited farsightedness in the housing matching model

Myopic solution concepts like the core, the vNM stable set, and the myopic stable set often allow for multiple stable allocations in the housing matching model of Shapley and Scarf (1974). On the other hand, farsighted concepts like the farsighted core, the farsighted vNM stable set, and the DEM farsighted stable set single out the top trading cycle allocation, even irrespective of whether dominations are required to be strict or weak. Motivated by findings in the experimental literature, we address the question of how much farsightedness is needed to obtain the top trading cycle allocation as the unique prediction by studying the horizon-$K $ versions of these solution concepts. We find that in most cases $ K \geq 2 $ is sufficient, the only exception being the horizon-$K $ farsighted stable set based on strict dominance where we need $ K \geq 3. $ To facilitate the presentation of our results, we introduce the cycle decomposition of an allocation as a convenient notational device.