2,988 research outputs found
A Control Dichotomy for Pure Scoring Rules
Scoring systems are an extremely important class of election systems. A
length- (so-called) scoring vector applies only to -candidate elections.
To handle general elections, one must use a family of vectors, one per length.
The most elegant approach to making sure such families are "family-like" is the
recently introduced notion of (polynomial-time uniform) pure scoring rules
[Betzler and Dorn 2010], where each scoring vector is obtained from its
precursor by adding one new coefficient. We obtain the first dichotomy theorem
for pure scoring rules for a control problem. In particular, for constructive
control by adding voters (CCAV), we show that CCAV is solvable in polynomial
time for -approval with , -veto with , every pure
scoring rule in which only the two top-rated candidates gain nonzero scores,
and a particular rule that is a "hybrid" of 1-approval and 1-veto. For all
other pure scoring rules, CCAV is NP-complete. We also investigate the
descriptive richness of different models for defining pure scoring rules,
proving how more rule-generation time gives more rules, proving that rationals
give more rules than do the natural numbers, and proving that some restrictions
previously thought to be "w.l.o.g." in fact do lose generality.Comment: A shorter version of this paper will appear in the proceedings of the
Twenty-Eighth AAAI Conference on Artificial Intelligence (AAAI 2014
Towards a Dichotomy for the Possible Winner Problem in Elections Based on Scoring Rules
To make a joint decision, agents (or voters) are often required to provide
their preferences as linear orders. To determine a winner, the given linear
orders can be aggregated according to a voting protocol. However, in realistic
settings, the voters may often only provide partial orders. This directly leads
to the Possible Winner problem that asks, given a set of partial votes, whether
a distinguished candidate can still become a winner. In this work, we consider
the computational complexity of Possible Winner for the broad class of voting
protocols defined by scoring rules. A scoring rule provides a score value for
every position which a candidate can have in a linear order. Prominent examples
include plurality, k-approval, and Borda. Generalizing previous NP-hardness
results for some special cases, we settle the computational complexity for all
but one scoring rule. More precisely, for an unbounded number of candidates and
unweighted voters, we show that Possible Winner is NP-complete for all pure
scoring rules except plurality, veto, and the scoring rule defined by the
scoring vector (2,1,...,1,0), while it is solvable in polynomial time for
plurality and veto.Comment: minor changes and updates; accepted for publication in JCSS, online
version available
The Complexity of Online Manipulation of Sequential Elections
Most work on manipulation assumes that all preferences are known to the
manipulators. However, in many settings elections are open and sequential, and
manipulators may know the already cast votes but may not know the future votes.
We introduce a framework, in which manipulators can see the past votes but not
the future ones, to model online coalitional manipulation of sequential
elections, and we show that in this setting manipulation can be extremely
complex even for election systems with simple winner problems. Yet we also show
that for some of the most important election systems such manipulation is
simple in certain settings. This suggests that when using sequential voting,
one should pay great attention to the details of the setting in choosing one's
voting rule. Among the highlights of our classifications are: We show that,
depending on the size of the manipulative coalition, the online manipulation
problem can be complete for each level of the polynomial hierarchy or even for
PSPACE. We obtain the most dramatic contrast to date between the
nonunique-winner and unique-winner models: Online weighted manipulation for
plurality is in P in the nonunique-winner model, yet is coNP-hard (constructive
case) and NP-hard (destructive case) in the unique-winner model. And we obtain
what to the best of our knowledge are the first P^NP[1]-completeness and
P^NP-completeness results in the field of computational social choice, in
particular proving such completeness for, respectively, the complexity of
3-candidate and 4-candidate (and unlimited-candidate) online weighted coalition
manipulation of veto elections.Comment: 24 page
On the Exact Amount of Missing Information that Makes Finding Possible Winners Hard
We consider election scenarios with incomplete information, a situation that arises often in practice. There are several models of incomplete information and accordingly, different notions of outcomes of such elections. In one well-studied model of incompleteness, the votes are given by partial orders over the candidates. In this context we can frame the problem of finding a possible winner, which involves determining whether a given candidate wins in at least one completion of a given set of partial votes for a specific voting rule.
The Possible Winner problem is well-known to be NP-Complete in general, and it is in fact known to be NP-Complete for several voting rules where the number of undetermined pairs in every vote is bounded only by some constant. In this paper, we address the question of determining precisely the smallest number of undetermined pairs for which the Possible Winner problem remains NP-Complete. In particular, we find the exact values of t for which the Possible Winner problem transitions to being NP-Complete from being in P, where t is the maximum number of undetermined pairs in every vote. We demonstrate tight results for a broad subclass of scoring rules which includes all the commonly used scoring rules (such as plurality, veto, Borda, and k-approval), Copeland^alpha for every alpha in [0,1], maximin, and Bucklin voting rules. A somewhat surprising aspect of our results is that for many of these rules, the Possible Winner problem turns out to be hard even if every vote has at most one undetermined pair of candidates
New Candidates Welcome! Possible Winners with respect to the Addition of New Candidates
In voting contexts, some new candidates may show up in the course of the
process. In this case, we may want to determine which of the initial candidates
are possible winners, given that a fixed number of new candidates will be
added. We give a computational study of this problem, focusing on scoring
rules, and we provide a formal comparison with related problems such as control
via adding candidates or cloning.Comment: 34 page
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