Arrow's impossibility theorem: Two simple single-profile versions

In this short paper we provide two versions of Arrow’s impossibility theorem, in a world with only one preference profile. Both versions are extremely simple and allow a transparent understanding of Arrow’s theorem. The first version assumes a two-agent society; the second version, which is similar to a theorem of Pollak, assumes two or more agents. Both of our theorems rely on diversity of preferences axioms; our first theorem also uses a neutrality-independence assumption, commonly used in the literature; our second theorem uses a neutrality-independence-monotonicity (NIM) assumption, which is stronger and less commonly used. Using the NIM assumption results in substantial gains in terms of simplicity. We provide examples to show the logical independence of the axioms, and to illustrate our points.