More Than the Sum of Its Parts: Compact Preference Representation Over Combinatorial Domains
Joel Uckelman
Abstract:
In this dissertation we present a framework for compactly
representing cardinal preferences over combinatorial domains and show
the feasibility of using this framework for auctions and voting.
Our framework uses goalbases---sets of weighted propositional
formulas---to represent utility functions. We compute the utility of
an alternative as the aggregated value of the weights of the goals
the alternative satisfies. Goalbase languages are formed by
restricting the formulas and weights which may appear in goalbases.
Due to their parametric nature, these languages are scattered all
across the representational landscape. In order to make practical use
of goalbases, we must first know the lay of the land. In particular,
we explore the landscape of goalbase languages in three directions:
* Expressivity: Given a goalbase language, what utility functions are
representable in it? Many goalbase languages with natural definitions
correspond exactly to classes of utility functions having well-known
properties. Furthermore, we show that some goalbase languages have
precisely one representation for any representable utility function,
and provide methods for finding these representations.
* Succinctness: Given two goalbase languages, are the smallest
representations in one significantly smaller than equivalent smallest
representations in the other? We systematically compare more than two
hundred pairs of languages to determine which languages are more
succinct.
* Complexity: Given a goalbase language, how difficult is it to
answer queries about goalbases contained in it? We consider the
computational complexity of finding optimal states for individuals
and groups, and finding pessimal states for individuals. These
problems tend to be intractible for more expressive languages; for
those which are solvable in polynomial time, we provide algorithms
demonstrating that.
After determining the properties of many goalbase languages, we
consider two possible applications of them:
* Combinatorial Auctions: Combinatorial auctions cannot generally be
conducted without concise bidding languages. Goalbase languages may
be used as bidding languages, and are sometimes more and sometimes
less succinct than bidding languages already in use. We give an
integer programming formulation of the Winner Determination Problem
using goalbases as bids, as well as branch-and-bound heuristics for
solving the WDP directly, and present experimental results which
demonstrate the feasibility of using goalbase languages for auctions
of moderate size.
* Voting: We consider the problem of insufficiently expressive voting
methods, and suggest voting with goalbases as ballots as a possible
remedy. Common single-winner voting methods do not extend well to
multi-winner settings like committee elections due to interactions
between candidates. We suggest an extension to Approval Voting, where
properties of the outcome are approved (or not) rather than
particular outcomes as in standard Approval Voting.
Together, this work provides a clear view of the power of goalbase
languages for preference representation, and points to potential
areas of application.