How to Rig Elections and
Competitions
Noam Hazon*, Paul E. Dunne+, Sarit Kraus*, Michael Wooldridge+
*Dept. of Computer Science
Bar Ilan University
Israel
+Dept.
of Computer Science
University of Liverpool
United Kingdom
Outline

A common way to aggregate agents preferences - voting

But it can be manipulated!

Current models assume perfect information

We analyzed a model of imperfect information



Theoretically, manipulation is hard: NP-Complete proofs
Practically it is not: Heuristics and experimental evaluation
Conclusion, Future work
Why Voting?

Preference aggregation - to a socially desirable decision

Not only agents…



Human decisions among a collection of alternatives
Sport tournaments
But it can be manipulated if we have perfect information


By a single voter or a coalition of voters
By the election officer
Our Model - Imperfect Information

Agenda rigging by the election officer



Complexity results



Verifying an agenda is in P
Finding a perfect agenda seems to be hard
Empirical results



Linear order
Tree order
Easy-to-implement heuristics performed well
Tested against random and real data
Motivation - better design of voting protocols
Basic Definitions

Set of outcomes/candidates,   {1 , 2 ,..., n }

Imperfect information ballot matrix M,

Voting trees

A
C
B
A
A
C
C
B
C
D
C
A
B
D
Linear Order Tree

Verifying an agenda is O(n2) - with dynamic programming

A candidate can only benefit by going late in an order
 And this is not true for general trees!

Finding an order for a particular candidate to win:
 With a non-zero probability - O(n2)
 With at least a certain probability
Weak Version of Rigged Agenda

A run 


An order agenda together with the intended outcomes
P[r | M] - a probability that the run will result in our candidate
WIIRA Problem 



Set of candidates, Ω
Imperfect information matrix, M
A favored candidate, ω*
A probability p

Is there a run in which the winner is ω*, s.t. P[r | M] ≥ p ?

NP-C, from the k-HCA problem on tournaments
Balanced Tree Order

Verifying an agenda is O(n2)

Finding an order for a particular candidate to win:
 With a non-zero probability - open problem [Lang07]
 With at least a certain probability - NP-C [Vu08]
From Theory to Practice
“Concern: It might be that there are effective heuristics
to manipulate an election, even though
manipulation is NP-complete.
Discussion: True. The existence of effective heuristics
would weaken any practical import of our idea. It
would be very interesting to find such heuristics.”
[Bartholdi89]
Linear Order – Conversion

Probabilistic data  deterministic data


Simple convert
Threshold convert
ω1

ω2
ω1
-
ω2
0.3
ω3
0.4 0.3
ω3
ω1
ω2
ω3
ω1
-
1
0
0.7
ω2
0
-
1
-
ω3
0
0
-
0.7 0.6
-
Not better than random order
General Approach

Heuristics:





Far adversary
Best win
Local search
Random order
1
2
. . . . m-2 m-1 ω*
Two data sets:


Random data- uniform and normal distribution
Real data - 29 teams from the NBA, top 13 tennis players
Linear Order- Normal Distribution
1
0.9
winning probability
0.8
0.7
0.6
0.5
0.4
optimal
far adversary
best win
simple convert
threshold covert
local search
random order
0.3
0.2
0.1
0
2
10
18
26
34
# of candidates
42
50
optimal
far adversary
GO
best win
To
local search
HE
WI
yto
n
Da
vi d
om
as
Lle
ER
,
TT
,
FE
RR
o
om
my
,T
,T
YC
H
HA
AS
nd
o
nd
y
,M
ari
ern
a
AN
CIC
es
an
mm
y
ICK
,A
LE
Z,
F
BE
RD
NZ
A
RO
DD
ED
O,
E,
Ja
m
ICI
C,
Iv
BL
AK
l
lay
afa
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og
er
L,
R
,R
KO
,N
i ko
LJ
UB
DE
N
RO
BR
DA
VY
RE
R
NA
DA
FE
DE
winning probability
Linear Order- 13 Tennis Players
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
random order
Balanced Tree- Normal Distribution
winning probability
0.6
0.5
0.4
0.3
optimal
far adversary
best win
local search
random
0.2
0.1
0
2 6 10 14 18 22 26 30 34 38 42 46 50 54 58 62 66 70
# of candidates
optimal
far adversary
GO
best win
To
local search
HE
WI
yto
n
Da
vi d
om
as
Lle
ER
,
TT
,
FE
RR
o
om
my
,T
,T
YC
H
HA
AS
nd
o
nd
y
,M
ari
ern
a
AN
CIC
es
an
mm
y
ICK
,A
LE
Z,
F
BE
RD
NZ
A
RO
DD
ED
O,
E,
Ja
m
ICI
C,
Iv
BL
AK
l
lay
afa
e
og
er
L,
R
,R
KO
,N
i ko
LJ
UB
DE
N
RO
BR
DA
VY
RE
R
NA
DA
FE
DE
winning probability
Balanced Tree- 13 Tennis Players
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
random order
0
Hawks
Celtics
Bulls
Cavaliers
Mavericks
Nuggets
Pistons
Warriors
Rockets
Pacers
Clippers
Lakers
Grizzlies
Heat
Bucks
Timberwolves
Nets
Hornets
Knickerbockers
Magic
76ers
Suns
Blazers
Kings
Spurs
Supersonics
Raptors
Jazz
Wizards
winning probability
Balanced Tree- 29 NBA Teams
0.3
far adversary
best win
local search
random order
0.2
0.1
Conclusion

Two voting protocols with incomplete information

Agenda verification is in P

Manipulation by the officer seems to be hard

Far adversary, best win and local search can help a lot!

Future: other voting protocols
{hazonn,sarit}@cs.biu.ac.il , {mjw,ped}@csc.liv.ac.uk