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The notion of {em 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 {em simultaneously}, while preserving {em 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 {em 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 {em privacy constraint graph} $G_f$ whose edges correspond to values resulting by applying $f$ to neighboring databases.

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