The notion of a universally utility-maximizing privacy mechanism was recently introduced by Ghosh, Roughgarden, and Sundararajan [STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, simultaneously, while preserving Differential Privacy [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh et al. have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan [PODS 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a single count query (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function $f$ by its privacy constraint graph $G_f$ whose edges correspond to values resulting by applying $f$ to neighboring databases.