## Cycling of simple rules in the spatial model |

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at+\Rf^at available alternatives boundary dQf^(x Cf(u Cf{u Cfm(u coalition is decisive convex set core is empty core is nonempty cycle set decisive coalition i.e. declares all alternatives definition denote the set dummy voter Euclidean preferences exists a noncollegial f and profile finite number finite sequence finite set ideal points implies includes all available Let f majority preference McKelvey McKelvey's theorem Nakamura number noncollegial g-rule noncollegial simple rule noncollegial strong simple Northwestern University number of steps preference aggregation rule preference cycles preference relation Rf(u preferences are Euclidean reachable from x resolute than f rule f satisfy diversity set of alternatives set of individuals simple preference aggregation smooth utility profiles Spatial Model strict majority rule strict preference strictly quasi-concave utility strong rule strong simple preference strong simple rule sufficiently high dimensional Tf(u Tf{u Tj(u TJ{u top-cycle set includes transitive closure Vx,y weak top-cycle set xPf^y xPj^y xRf^y