  A traditionally desired goal when designing auction mechanisms is incentive
compatibility, i.e., ensuring that bidders fare best by truthfully reporting
their preferences. A complementary goal, which has, thus far, received
significantly less attention, is to preserve privacy, i.e., to ensure that
bidders reveal no more information than necessary. We further investigate and
generalize the approximate privacy model for two-party communication recently
introduced by Feigenbaum et al.[8]. We explore the privacy properties of a
natural class of communication protocols that we refer to as "dissection
protocols". Dissection protocols include, among others, the bisection auction
in [9,10] and the bisection protocol for the millionaires problem in [8].
Informally, in a dissection protocol the communicating parties are restricted
to answering simple questions of the form "Is your input between the values
\alpha and \beta (under a predefined order over the possible inputs)?".
  We prove that for a large class of functions, called tiling functions, which
include the 2nd-price Vickrey auction, there always exists a dissection
protocol that provides a constant average-case privacy approximation ratio for
uniform or "almost uniform" probability distributions over inputs. To establish
this result we present an interesting connection between the approximate
privacy framework and basic concepts in computational geometry. We show that
such a good privacy approximation ratio for tiling functions does not, in
general, exist in the worst case. We also discuss extensions of the basic setup
to more than two parties and to non-tiling functions, and provide calculations
of privacy approximation ratios for two functions of interest.
