This document summarizes research on voting procedures with incomplete voter preferences. It introduces the concepts of possible and necessary winners given incomplete information about voter preferences. It shows that for positional scoring procedures and the Condorcet rule, possible and necessary winners can be computed efficiently in polynomial time by looking at minimum and maximum ranks candidates could achieve. The document also discusses manipulation and elicitation in this setting.

1. VOTING PROCEDURES WITH INCOMPLETE
PREFERENCES
BY KATHRIN KONCZAK AND JEROME LANG
Presented by Scott Ames, Steven Frink, and Ellis Mitchell

2. Heroes >Villians
LexLuthor> Joker > Batman > Superman
Everyone > Joker, Batman > Superman
Villians> Heroes, LexLuthor> Joker

3. INCOMPLETE PREFERENCES
 What if the complete preferences of some (or all) of
the voters are unknown (or don’t exist)?
 Reasons it could happen:
 New candidate introduced
 Voter indifference
 Some voters haven’t voted yet
 Can we determine the winners of the election without
getting the rest of the voters’ preferences?
 What is the complexity of manipulation in this setting?
 How much more information to we need to gather in
order to determine the winners of the election?

4. Heroes >Villians
LexLuthor> Joker > Batman > Superman
Everyone > Joker, Batman > Superman
Villians> Heroes, LexLuthor> Joker
Chester A. Arthur > Everyone

5. PREFERENCE RELATIONS
 An order R on a set C is a reflexive, transitive, and
antisymmetric relation on C.
 An order is linear (or complete) iffR(x,y) or R(y,x)
holds for all x, y ∊ C.
 Rʹ extends R iff R ⊆ Rʹ, i.e. R(x,y) implies Rʹ(x,y )
for all x, y ∊ C.
 A linear order T is a complete extension of R iff T
extends R.
 Ext(R) denotes the set of all complete extensions of
R.

6. EXAMPLES
 Let C = {a, b, c}.
 An order on C is R = a ≻Rb, a ≻Rc
 Rʹ = a ≻Rʹ ≻Rʹ is a complete extension of R
b c
 Ext(R) = {a ≻ b ≻ c, a ≻ c ≻ b}

7. VOTING PROCEDURES
 Set of candidates C = {c1, c2, …, cm}
 Set of voters V = {1, …, n}
 A complete preference profile is a vector T
= <T1, T2, …, Tn>
 Each Ti is a linear order on C representing the
preferences of voter i.
 A voting procedure F is a mapping from every T to a
nonempty subset of C.
 F(T) is the set of winners of T w.r.t F
 F(T) must be nonempty
 They assume that the preference orders are all
asymmetric, so voters cannot be irrational

8. EXTENDING VOTING PROCEDURES TO
INCOMPLETE PREFERENCES
 A voting problem under incomplete preferences is
the same as a voting procedure under complete
preferences, except that the preference profile is
composed of orders (as opposed to complete
orders)
 The vector of the voters’ preferences P
= <P1, P2, …, Pn> is called the (collective)
preference profile.
 P is complete iff each Pi is a complete order
 We write Pi(x,y) as x ≻iy.
 We generalize complete extensions of P as
 Ext(P) = Ext(P1) xExt(P2) x … xExt(Pn)

9. EXTENDING VOTING PROCEDURES
 We can interpret incomplete preferences in two
ways.
 Intrinsic incompleteness: The voter refuses to compare
some candidates.
 Epistemic incompleteness: The voter has a complete
preference, but it is only partially known when the voting
procedure is applied.
 These two interpretations lead to different methods
of extending voting procedures.

10. EXTENDING VOTING PROCEDURES
 We can look at all possible results.
 We can look at some subset of the results.
 We can redefine F to work with partial preferences.
 They look at the first approach.

11. POSSIBLE AND NECESSARY WINNERS
 Let F be a voting procedure on C and P, where C is
the set of candidates and P is a (possibly
incomplete) preference profile.
 c ∊ C is a necessary winner for P (w.r.t F) iff for all
complete extensions T of P we have c ∊ F(T).
 NWF(P) is the set of all necessary winners for P w.r.t F
 c ∊ C is a possible winner for P (w.r.t F) iff there exists a
complete extension T of P such that c ∊ F(T).
 PWF(P) is the set of all possible winners for P w.r.t F
 For any voting procedure F
 NWF(P) ⊆ PWF(P)
 For all P, Pʹ where Pʹ extends P, PW(Pʹ)⊆ PWF(P)
F
and NWF(Pʹ) ⊆ NWF(P)

12. Heroes >Villians
LexLuthor> Joker > Batman > Superman
Everyone > Joker, Batman > Superman
Villians> Heroes, LexLuthor> Joker

13. POSSIBLE AND NECESSARY WINNERS
 In general, there are exponentially many extensions
of a partial preference profile, so we don’t
immediately know that computing the possible and
necessary winners can be done in P-time.
 Determining whether c ∊ PWF(R) is in NP
 Determining whether c ∊ NWF(R) is in coNP
 If the voting procedure is polynomial time decidable.
 Are there non-trivial voting procedures where the
possible and necessary winners can be computed
in polynomial time?

14. POSITIONAL SCORING PROCEDURES
 A positional scoring procedure is defined from a
scoring vectors = (s1, s2, …, sm) of integers such
that s1 ≥ s2 ≥ … ≥ sm and s1> sm.
 Let T = <T1, T2, …, Tn> be a complete preference
profile.
 For every candidate c and every voter i, let r(Ti,c)
be one greater than the number of candidates that
voter i prefers to c.
 r(Ti,c) = ||{y | y ≻ic}|| + 1
 S(c,T) = Σni=1sr(Ti,x) is the score of c.
 Fs(T) = { c |S(c,T) is maximal}

15. MINIMAL AND MAXIMAL RANK
 The minimal rank of a candidate c is the lowest
possible rank of c in the complete extensions of P.
 rankminP(c) = minT ∊ Ext(P)r(T,c)
 The maximal rank of a candidate c is the highest
possible rank of c in the complete extensions of P.
 rankmaxP(c) = maxT ∊ Ext(P)r(T,c)
 Note that a lower rank means more preferred.

16. PROPOSITION 1
 Let P be a partial order. Then
 rankminP(c) = ||{ y | y>Pc}|| + 1
 rankmaxP(c) = m - ||{ y | c>Py}||
 Proof sketch:
 For rankminP(c), for each candidate x such that x>Pc is
not in the preference order, have c preferred to x.
 Then the best possible rank for c is equal to the number of
candidates already specified as preferred to c plus one.
 For rankmaxP(c), for each candidate x such that c>Px is
not in the preference order, have x preferred to c.
 Then the worst possible rank for c is equal to the total number
of candidates minus the number of candidates that c is already
specified as preferred to.
 These are the complete extensions of P where c has the
minimum and maximum ranks.

17. PROPOSITION 2
 Let P be a preference profile where each Pi is an
order, Fs be a positional voting procedure, and
 SminP(c) = Σni=1srankmax(R_i,c)
 SmaxP(c) = Σni=1srankmin(R_i,c)
 These are the minimum and maximum possible scores
for c.
 Proposition 2.1: Then c is a necessary winner for P
w.r.t. FsiffSminP(c) ≥ SmaxP(x) for all x ≠ c.

18. PROOF
 c is a necessary winner for P w.r.t. FsiffSminP(c) ≥
SmaxP(x) for all x ≠ c.
 <=:
 Suppose for contradiction that c is not a necessary winner.
 Then there is an extension T of P and an x ≠ c such thatS(x,T)
>S(c,T).
 Therefore SmaxP(x) ≥ S(x,T) >S(c,T) ≥ SminP(c).
 This contradicts the assumption that SminP(c) ≥ SmaxP(x) for all
x≠c
 =>: Let c be a necessary winner.
 Then there is no T ∊ Ext(P) and no x ≠ c such that S(c,T)
<S(x,T).
 Then because SminP(c) ≤ S(c,T) and SmaxP(x) ≥ S(x,T), there is
no x ≠ c such that SminP(x) <SmaxP(c)

19. PART 2 OF PROPOSITION 2
 Recall:
 SminR(x) = Σni=1srankmax(c)
 SmaxR(c) = Σni=1srankmin(c)
 Then c is a possible winner for R w.r.t. FsiffSmaxR(c)
≥ SminR(x) for all x ≠ c.
 This is FALSE!

20. COUNTEREXAMPLE
 Take a plurality election with candidates
Batman, Aquaman, and Spiderman.
 Take the following partial preference orders:
 P1: Batman >Aquaman
 P2: Batman >Aquaman
 P3: Batman >Aquaman
 P4: Aquaman> Spiderman
 SmaxP(Aquaman) = 1
 SminP(Batman) = 0, SminP(Spiderman) = 0
 But Aquaman cannot win the election!
 Aquaman can get a maximum score of 1 because voters
1, 2, and 3 won’t pick Aquaman as their first choice.
 Voters 1, 2, and 3 will either pick Batman or Spiderman, so
either Batman or Spiderman will have a score of at least 2.

21. COROLLARY
 Possible and Necessary winners for positional
scoring procedures can be computed in polynomial
time.
 Clearly the corollary still holds true for necessary
winners.
 Unsure if it is still true for possible winners.

22. CONDORCET WINNERS
 Recall a candidate c is a Condorcet winner for a
complete profile T = <ʹ ,…, ʹ >iff for all y ≠ c, ||{ i |
1 n
c ≻iy}|| > n/2.
 Let P be a possibly incomplete preference profile.
 Then c∈C is a necessary Condorcet winner for P
iffc is a Condorcet winner for all complete
extensions of P.
 c∈C is a possible Condorcet winner for P iff there
exists a complete extension of P, such that c is a
Condorcet winner for that extension.

23. CONDORCET WINNERS
 Define NT(x,y) = ||{ i | x>iy}|| - ||{ i |y>ix}||
 Then
 NminP(x,y) = minT∈Ext(P)NT(x,y)
 NminP(x,y) corresponds to the worst case for x among
extensions of P.
 NmaxP(x,y) = maxT∈Ext(P)NT(x,y)
 NmaxP(x,y) corresponds to the best case for x among
extensions of P.

24. PROPOSITION 3
 Let P be a partial preference profile and x,y two distinct
candidates from C. Define:
 NmaxPi(x,y) = { +1 if not(y ≥ix);
-1 if y>ix }
 NminPi(x,y) = { +1 if x>iy;
-1 if not(x ≥Iy) }
 Proposition 3:
 1. NminP(x,y) = ∑ni=1 NminPi(x,y) and NmaxP(x,y) = ∑ni=1NmaxPi(x,y);
 2. x is a necessary Condorcet winner for P iff for all y ≠
x, NminP(x,y) > 0
 3. x is a possible Condorcet winner for P iff for all y≠
x, NmaxP(x,y) > 0
 We omit the proof. (Part 1 is simple, parts 2 and 3 are
similar to the proof for positional scoring procedures)

25. COROLLARY
 Possible and necessary Condorcet winners can be
computed in polynomial time.
 However, this method does not extend to
Condorcet-consistent voting procedures.

26. MANIPULATION
 Constructive Manipulation:
 Deciding your votes to make a candidate a winner.
 Destructive Manipulation:
 Deciding your votes to prevent a candidate from being a
winner.

27. PROPOSITION 4
 There is a constructive manipulation for ciffc is a
possible winner of the preference profile where the
members of the coalition have completely
unspecified preferences.
 There is a destructive manipulation for ciffc is not a
necessary winner of the same preference profile.

28. ELICITATION
 Determining whether the outcome of the election
can be determined with the information you have
about each voter’s preferences, and if not, what
information you need.
 Proposition 5:
 The elicitation task is over iff the set of possible winners
is equal to the set of necessary winners.

29. CONCLUSION
 Even with incomplete preferences we can
sometimes compute the winners of an election.
 Can be used for voter manipulation.
 If you can determine possible winners, then you can do
constructive manipulation.
 If you can determine necessary winners, then you can
do destructive manipulation.
 Combine the results based on the maximum and
minimum rank that each voter could give to each
candidate.

30. FUTURE WORK
 Extend to more voting protocols.
 Count the number of extensions that make a
candidate a winner.
 If the voting procedure is probabilistic, is it possible
or necessary for a candidate to win based on the
voting system?
 Can possible winners be computed efficiently for
positional scoring procedures?