### Random tie breaking & complexity of manipulation

+
Random Tie Breaking
Toby Walsh
NICTA and UNSW
+
Random Tie Breaking
Haris Aziz, Serge Gaspers, Nick Mattei, Nina
Narodytska, Toby Walsh
NICTA and UNSW
+
Ties matter

Manipulators can only change result if election is close!
+
Ties matter

Manipulators can only change result if election is close!

How we deal with ties often matters critically

Typical assumption is ties broken in favour of the manipulators
+
Ties matter

Manipulators can only change result if election is close!

How we deal with ties often matters critically

Typical assumption is ties broken in favour of the manipulators

In real elections, ties broken randomly, by the chair, by age ….
+
Ties matter

Manipulators can only change result if election is close!

How we deal with ties often matters critically

Typical assumption is ties broken in favour of the manipulators

In real elections, ties broken randomly, by the chair, by age ….

Tie breaking can itself be a source of computational complexity

2nd order Copeland,

Copeland with weighted votes: polynomial to manipulate if ties
are scored 1, but NP-hard if ties are scored 0
[Faliszewski, Hemaspaandrea, Schnoor 08]
+
Unique and co-winner problems


Unique winner manipulation problem

Equivalent to tie-breaking against manipulator

Can we construct a strategic vote so given candidate is the unique
winner of the election?
Co-winner manipulation problem

Equivalent to tie-breaking in favour of the manipulator

Can we construct a strategic vote so given candidate is one of the
co-winners of the election?
+
Tie-breaking in practice

Random candidate


Random vote


E.g. UK general elections
E.g. Schulze voting breaks ties according to order of candidates in
a random vote
By the chair
+
Tie-breaking with a random
candidate

See [Obraztsova, Elkind, Hazon AAMAS 2011], [Obratzsova,
Elkind IJCAI 2011]

Agents assign utilities to candidates

Look to maximize expected utility of result

Can get a large way though with a simple model of just asking if
a given candidate can win with probability > p?

Equivalent to u(a)=1, u(b)=0 for all other candidates

Avoids the difficult problem of having to assign utilities!
+
Tie-breaking with a random
candidate

Several common rules have been shown to be (in)tractable

THM: When tie-breaking with a random candidate, all scoring
rules (including Borda) are polynomial to manipulate, as are
plurality with runoff and Bucklin

THM: When tie-breaking with a random candidate, Copeland and
Maximin are NP-hard to manipulate
[Obraztsova & Elkind 2011]
+
Tie-breaking with a random vote

In case of a tie, pick a vote uniformly at random


Order candidates according to this vote
In some forthcoming work, we’ve shown that this has
different computational properties to tie-breaking with a
random candidate

In practice, it seems harder

Indeed, it is often proposed as a barrier to manipulation

Suppose you vote strategic to get a preferred candidate to win,
but then your strategic vote may actually make them loose!
+
Tie-breaking with a random vote

Candidates can have quite different probabilities of winning
than tie-breaking with a random candidate

Suppose we use Borda scoring

Half voters vote a>b>c

Half voters vote c>b>a
+
Tie-breaking with a random vote

Candidates can have quite different probabilities of winning
than tie-breaking with a random candidate

Suppose we use Borda scoring


Half voters vote a>b>c

Half voters vote c>b>a
Tie-breaking with a random vote

a or c win with probability 1/2
+
Tie-breaking with a random vote

Candidates can have quite different probabilities of winning
than tie-breaking with a random candidate

Suppose we use Borda scoring


Half voters vote a>b>c

Half voters vote c>b>a
Tie-breaking with a random vote


a or c win with probability 1/2
Tie-breaking with a random candidate

a, b, or c win with probability 1/3
+
Tie-breaking with a random vote

Formally incomparable to tie-breaking with a random
candidate
+
Tie-breaking with a random vote

Formally incomparable to tie-breaking with a random
candidate

THM: exists voting rule, such that the manipulation problem when
tie-breaking with a random candidate is polynomial but tiebreaking with a random vote is NP-complete, and vice versa
+
Tie-breaking with a random vote

Formally incomparable to tie-breaking with a random
candidate

THM: exists voting rule, such that the manipulation problem when
tie-breaking with a random candidate is polynomial, but tiebreaking with a random vote is NP-complete, and vice versa

Proof: Consider Borda voting, and a single manipulator, then tiebreaking with a random candidate is polynomial [Obraztsova,
Elkind, and Hazon 2011]
+
Tie-breaking with a random vote

Formally incomparable to tie-breaking with a random
candidate

THM: exists voting rule, such that the manipulation problem when
tie-breaking with a random candidate is polynomial, but tiebreaking with a random vote is NP-complete, and vice versa

Proof: Consider Borda voting, and a single manipulator, then tiebreaking with a random candidate is polynomial [Obraztsova,
Elkind, and Hazon 2011].
But when tie-breaking with a random vote, manipulation is NPcomplete [forthcoming 2013]
+
Tie-breaking with a random vote

Tie-breaking with a random vote is incomparable to the
unique and co-winner manipulation problems
+
Tie-breaking with a random vote

Tie-breaking with a random vote is incomparable to the
unique and co-winner manipulation problems

THM: exists voting rule, such that the co-winner and unique
winner manipulation problems are polynomial, but tie-breaking
with a random vote is NP-complete, and vice versa

Contrast this with tie-breaking with a random candidate

If unique winner or co-winner manipulation problems are NPhard then tie-breaking with a random candidate is also
+
Random vote versus Random candidate
Polynomial
NP-complete
Open
Random
vote
Plurality, veto,
(fixed) k-approval,
plurality with runoff
(unbounded) k- Copeland,
approval, Borda, maximin
Bucklin, ranked
pairs, STV
Random
candidate
All scoring rules
(plurality, veto, kapproval, Borda),
plurality with
runoff, Bucklin
Copeland,
maximin,
ranked pairs,
STV
+
Random vote versus Random candidate
Polynomial
NP-complete
Open
Random
vote
Plurality, veto,
(fixed) k-approval,
plurality with runoff
(unbounded) k- Copeland,
approval, Borda, maximin
Bucklin, ranked
pairs, STV
Random
candidate
All scoring rules
(plurality, veto, kapproval, Borda),
plurality with
runoff, Bucklin
Copeland,
maximin,
ranked pairs,
STV
+
Random vote versus Random candidate
Polynomial
NP-complete
Open
Random
vote
Plurality, veto,
(fixed) k-approval,
plurality with runoff
(unbounded) k- Copeland,
approval, Borda, maximin
Bucklin, ranked
pairs, STV
Random
candidate
All scoring rules
(plurality, veto, kapproval, Borda),
plurality with
runoff, Bucklin
Copeland,
maximin,
ranked pairs,
STV
How you break ties impacts on the computational complexity!
+
Control by breaking ties

Somewhat related problem

If I am chair, how do I control the result by breaking ties?

Tie-breaking only once (between co-winners), this is trivial

Pick the person you want to win

Tie-breaking even just twice, control can be NP-hard!
+
Control by breaking ties

Control by tie-breaking with two stage rules

THM: Exists a two stage rule combining veto and plurality for
which control by tie-breaking is NP-hard

Proof: Consider rule that eliminates half the candidates using
veto, then elects the plurality winner
+
Control by breaking ties


Control by tie-breaking with two stage rules

THM: Exists a two stage rule combining veto and plurality for
which control by tie-breaking is NP-hard

Proof: Consider rule that eliminates half the candidates using
veto, then elects the plurality winner
Control by tie-breaking with multi-stage rules

THM: Control by tie-breaking with STV, Baldwin and Coombs is
NP-hard
+
Control by breaking ties


Control by tie-breaking with two stage rules

THM: Exists a two stage rule combining veto and plurality for
which control by tie-breaking is NP-hard

Proof: Consider rule that eliminates half the candidates using
veto, then elects the plurality winner
Control by tie-breaking with multi-stage rules

THM: Control by tie-breaking with STV, Baldwin and Coombs is
NP-hard

THM: Control by tie-breaking with Nanson is polynomial
+
Control by breaking ties
Polynomial
NP-hard
Scoring rules, Cup,
STV, Baldwin,
Nanson, Copeland, maximin,
Ranked pairs,
Bucklin, fallback
Coombs
+
Control by breaking ties

Incomparable to the manipulation problem when breaking
ties with a random candidate, or in a fixed order

THM: Exists voting rule such that control by tie-breaking is
polynomial but manipulation problem breaking ties at random/in
a fixed order is NP-complete, and vice versa

E.g. control by breaking ties for Copeland is polynomial, but
manipulation when breaking ties at random is NP-hard
+
Conclusions

Ties do matter

Breaking ties with a random vote somewhat more
computationally challenging than with a random candidate

For two and multi-stage rules, it can be NP-hard for the chair
to control result by breaking ties

Of course, these are all worst case observations and we need
to consider the difficulty of breaking ties in practice/on
average/…
+
Questions?

PS I’m recruiting PhD students and a PostDoc