Sharp Thresholds for Monotone Non Boolean Functions and Social Choice Theory

A key fact in the theory of Boolean functions $f:\{0,1{\}}^n\to \{0,1\}$ is that they often undergo sharp thresholds. For example: if the function $f : {0,1}^n \to {0,1}$ismonotoneandsymmetricunderatransitiveactionwith$\E_p[f] = \eps$and$\E_q[f] = 1-\eps$then$q-p \to 0$as$n \to \infty$. Here ${\text{E}}_p$ denotes the product probability measure on $\{0,1{\}}^n$ where each coordinate takes the value $1$ independently with probability $p$. The fact that symmetric functions undergo sharp thresholds is important in the study of random graphs and constraint satisfaction problems as well as in social choice.In this paper we prove sharp thresholds for monotone functions taking values in an arbitrary finite sets. We also provide examples of applications of the results to social choice and to random graph problems. Among the applications is an analog for Condorcet’s jury theorem and an indeterminacy result for a large class of social choice functions.