Voting with Partial Information:
Minimal Sets of Questions to Determine the
Outcome
Fangzhen Lin
(Joint work with Ning Ding)
Guiyang, July 17, 2012
Voting
A voting consists of
a set of candidates,
a set of voters, and
for each voter, a strict ranking of the candidates.
A voting rule is then used to select a winner.
Voting Examples
Three candidates a, b, and c, and three voters A, B, and C:
A
a
b
c
B
a
b
c
C
b
c
a
Voting rules - deciding the outcome of a vote:
Plurality: the candidate who receives the most top votes is
the winner. In the example above: a is the winner.
Borda: the top candidate receives 1 point, the second 2
points, etc. The candidate who receives the least points is
the winner. In our example, a and b tie:
a receives 1+1+3 = 5 points;
b receives 2+2+1 = 5 points;
c receives 3+3+2 = 8 points;
Social Choice Theory - Science of Impossibility
Theorems
What are good voting rules? What properties should a voting
rule supposed to have?
Pareto efficiency: if every one says that a is better than b,
then a is better than b.
All voters are equal, and no dictators.
There should always be one winner.
IIA (Independence of Irrelevant Alternatives), ...
When there are more than 2 candidates, no such voting rules
exist (Arrow’s impossibility theorem, Muller-Satterthwaite’s
impossibility theorem,...)
In practice, many different rules are used: majority, plurality,
Borda, Condorcet, veto, etc.
Voting with Partial Preference
What happen when a voter does not give a full ranking of the
candidates:
A: a > b, a > c.
B: a > c.
C: c > b.
For most voting rules:
Not enough information to decide the winner.
It is now natural to talk about: possible winners, necessary
winners, possible losers (Konczak and Lang 2005).
Voting with Partial Preference
What happen when a voter does not give a full ranking of the
candidates:
A: a > b, a > c.
B: a > c.
C: c > b.
For most voting rules:
Not enough information to decide the winner.
It is now natural to talk about: possible winners, necessary
winners, possible losers (Konczak and Lang 2005).
Vote Elicitation
If you are candidate a, and you want to know whether you’ll win
or lose in the election under plurality, what would you do if all
you know right now is given as below?
A: a > b, a > c.
B: a > c.
C: c > b.
Issues:
What kind of queries can you ask a voter?
Can you afford to ask one question at a time?
If you have just one chance, which set of questions are you
going to ask?
Vote Elicitation
If you are candidate a, and you want to know whether you’ll win
or lose in the election under plurality, what would you do if all
you know right now is given as below?
A: a > b, a > c.
B: a > c.
C: c > b.
Issues:
What kind of queries can you ask a voter?
Can you afford to ask one question at a time?
If you have just one chance, which set of questions are you
going to ask?
Basic Concepts
Given a set O of candidates, and a set N = {1, 2, ..., n} of
voters:
A preference ordering is a total order on O.
A preference profile is a tuple of preference orderings, one
for each voter.
A voting rule f is a function from preference profiles to sets
of candidates.
A candidate a is a winner in p under rule f if a ∈ f (p).
Otherwise, a is a loser.
A partial preference ordering is a partial order on O, and a
partial preference profile is a tuple of partial preference
orderings.
A completion of a partial profile is a profile that extends
each partial preference to a total preference.
A candidate a is a necessary (possible) winner in a partial
profile if a is a winner in every (at least one) completion of
the partial profile.
Queries and Answers
A (comparison) query to voter i is one of the form i:{a, b}
that asks i to rank candidates a and b.
Given such a query, i answers either “a” (prefers a over b)
or “b” (prefers b over a).
An answer to a set Q of queries is a function σ from Q to O
such that for any i:{a, b} ∈ Q, σ(i:{a, b}) ∈ {a, b}.
An answer σ to Q is legal under a partial profile p if for
each i, the transitive closure of the following set
pi (σ, Q) = pi ∪ {(a, b) | i:{a, b} ∈ Q ∧ σ(i:{a, b}) = a}
is a partial order on O.
Given a legal answer σ to Q under p, the resulting partial
preference profile is
p(σ, Q) = (p1 (σ, Q), ..., pn (σ, Q)).
Deciding Sets and Minimal Deciding Sets
A set Q of queries is a deciding set for a candidate a under
a given partial profile p and a voting rule f , if for every
answer σ, a is either a necessary winner or a necessary
loser in the new partial profile σ(p, Q) under f .
Q is a minimal deciding set for a if it is a deciding set and
there is no other deciding set Q 0 such that Q 0 ⊂ Q.
Examples
Consider the voting rule plurality and the following partial vote:
1
2
3
a
b
c
>
>
>
b
c
a
>
c
the minimal deciding set for candidate a is {2:{a, b}, 3:{b, c}}:
it is a deciding set: if σ(2:{a, b}) = a then a is necessary
winner; otherwise if σ(2:{a, b}) = b and σ(3:{b, c}) = c,
then a is also a necessary winner; and otherwise if
σ(2:{a, b}) = b and σ(3:{b, c}) = b, then a is a necessary
loser.
None of its proper subsets are deciding sets.
Some Results
For any voting rule f , and partial preference profile p, and
any candidate a, there is a unique minimal deciding set of
a in p under f .
For the plurality voting rule, there is a polynomial algorithm
to compute a minimal deciding set.
For most others, like Borda rule, the problem is
NP-complete.
Vote Elicitation
Given some incomplete information about voters’ partial
preferences, how does one query voters to determine the
outcome of the vote for candidate a:
Compute the minimal deciding set for a and ask the
queries simultaneously.
Dynamically, pick a question from the minimal deciding set,
ask the question, add the answer to the preference
orderings, compute the new minimal deciding set, pick a
question to ask, and so on.
Other Issues
Possible winning set of queries for a candidate: a set of
questions for which there is a consistent answer under
which the candidate is a necessary winner.
Possible losing set of queries for a candidate: a set of
questions for which there is a consistent answer under
which the candidate is a necessary loser.
Minimality is not unique anymore.
Harder computationally.
Future Work
Application in vote manipulation: which voter to bribe given
limited resource.
Adding probabilities of how likely a question is going to be
answered.
What about communications among voters.
For more technical details, see:
http://www.cs.ust.hk/~flin/papers/mds.pdf