Evaluation of election outcomes
under uncertainty
Noam Hazon*, Yonatan Aumann*, 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

Perfect Vs. imperfect information

Complexity analysis of imperfect information model




Constant number of candidates: P for most of the cases
Number of candidates as a parameter: #P-Hard
EVALUATION Vs. CHANCE-EVALUATION
Conclusion, Future work
Avoid manipulation!
A>B>C
Me
B!
My “friend”
Why voting?

Preference aggregation - to a socially desirable decision

Computational aspects when moving to agents
 Evaluating a voting protocol
 Manipulation

Assuming perfect information
Our model - imperfect information

A probability distribution over a set of preferences

What is the probability of a candidate to win?

Real world motivation


Avoid manipulation
Reduce communication
A>B>C
B!
Our model - results
Candidates number Weights
constant
parameter
Chance-Evaluation Evaluation
equal
P(p,b,c,m,i,…)
P(p,b,c,m,i,…)
bounded
P(p,b,c,m,i,…)
P(p,b,c,m,i,…)
unbounded NP-Hard(b,c,m,i)
NP-Hard(b,c,m,i)
equal
P(p)
#P-Hard(p,b,c)
bounded
NP-Cpmplete(p)
#P-Hard(p,b,c)
unbounded NP-Cpmplete(p)
#P-Hard(p,b,c)
Social choice domain

Set of voters, V = {V1,…,Vn}

Corresponding weights, w1,…,wn

Set of outcomes/candidates, Ω = {ω1,…, ωm}

Imperfect information, k = 3
V1 , W1=1
1/2
V2 , W2=2
(ω1, ω2,
1/4
(ω3, ω1,
3/4
ω3)
1/3
ω2)
1/6
(ω2, ω1,
ω3)
V3 , W3=1
(ω1, ω2,
9/10
(ω1, ω2,
1/10
ω3)
ω3)
(ω1, ω2,
ω3)
(ω1, ω2,
ω3)
Voting systems- a review

Binary Voting systems
 Plurality
 Approval

Preferential voting systems
 Instant-runoff voting (IRV)
 Borda

Condorcet systems
 Copeland (Tournament)
 Minimax
The EVALUATION problem

Given
 social choice domain
 imperfect information model of voters' preferences
 specific candidate ω*

What is the probability that ω* will be chosen?

Computational time complexity depends on n, m and k
Constant number of candidates

Voting scenario Vs. voting result

A dynamic programming approach (for plurality):
0
1
2
V1
V2
0,0,0
1
0
1/2
1/4
ω1
ω1
1,0,0
0
1/2
0,1,0
0
1/3
1/3
3/4
ω2
ω2
0,0,1
0
1/6
2,0,0
0
0
1/2*1/4
1,1,0
0
0
1/3*1/4 + 1/2*3/4
1,0,1
0
0
1/6*1/4
0,2,0
0
0
1/3*3/4
0,1,1
0
0
1/6*3/4
0,0,2
0
0
0
1/6
ω3

(ω1,ω2,ω3)
Time Complexity:
O(n * # of rows * k)
Representing a voting result

A vector of [0,n]m

Plurality
Approval

Borda


A vector of


: O(kn(mn)m)
[0,n]m(m-1)/2


:
2+1
m
O(kn
)
All Condorcet systems
A vector of [0,n]m!

: O(knm+1)
IRV
Borda
…
: O(knm!+1)
m is constant!
Adding weights

Borda, Copeland, Minimax and IRV are NP-Hard
[Conitzer and Sandholm 2002]

Only for Unbounded weights!

Weights in Poly(n) 
O(Poly(n)m),
Candidates number
constant
2
m
O(Poly(n) ),
Weights
Chance-Evaluation
Evaluation
equal
P(p,b,c,m,i,…)
bounded
P(p,b,c,m,i,…)
unbounded
NP-Hard(b,c,m,i)
equal
parameter
O(Poly(n)m!)
bounded
unbounded

# of candidates as a parameter(1)
Even without weights, EVALUATION for Plurality is #P-Hard!
Voters
V1
V2
0
0
Candidates
1/3
v0
C0
1/3
1/3
1/3
v1
1
1
1/3
v2
2
C1
1/3
C2
1/3
2/3
2
R0
Z0
1
R
1
Z
# of candidates as a parameter(2)
EVALUATION for Borda and Copeland is #P-Hard.
1/3
V1
0
V2
0
v0
(c0,g2,c1,c2,c3,z,g1)
1/3
1/3
v1
1/3
1/3
(c1,c0,g2,c2,c3,z,g1)
Pv
1/3
v2
1
2
2/3
1
2
(c2,c0,c1,g2,c3,z,g1)
1/3
v3
1
z0
1
z1
z2
z3
1
1
1
(c3,c0,c1,c2,g2,z,g1)
(z,c3,c2,c1,g1,g2,c0)
(z,c3,c2,g1,c0,g2,c1)
(z,c3,g1,c1,c0,g2,c2)
(z,g1,c2,c1,c0,g2,c3)
Pz
# of candidates as a parameter(2)
EVALUATION for Borda and Copeland is #P-Hard.
Candidates number Weights
constant
parameter
Chance-Evaluation Evaluation
equal
P(p,b,c,m,i,…)
bounded
P(p,b,c,m,i,…)
unbounded
NP-Hard(b,c,m,i)
equal
#P-Hard(p,b,c)
bounded
#P-Hard(p,b,c)
unbounded
#P-Hard(p,b,c)

CHANCE-EVALUATION problem

Given
 social choice domain
 imperfect information model of voters' preferences
 specific candidate ω*

Is the probability that ω* will be chosen greater than 0 ?
Weighted CHANCE-EVALUATION
NP-Complete (in the strong sense) for Plurality!
Vz
B
V0
V1
V2
V3
V4
W0
W1
W2
W3
W4
1/k
B+1
1
…
C0
K
C1
…
Ck
Z
Un-weighted CHANCE-EVALUATION
Polynomial algorithm for Plurality:
V1
V2
V3
V4
V5
V6
V7
V8
1/4
A 1/2 A 1/3 A 1/3 B 1/3 A 1/3 B 1/3 B 1/3 B
1/2
B 1/2 C 1/2 B 2/3 C 2/3 D 2/3 C 2/3 D 2/3 C
1/4
D
1/6
C
V1'
V2
1
1
1
V3
1
1
1
1
V4
1
1
1
s
V2'
1
A
2
B
1
1
V6
V8
1
1
1
2
2
C
t
Conclusion

Recall the previous results table:
Candidates number Weights
constant
parameter

Chance-Evaluation Evaluation
equal
P(p,b,c,m,i,…)
P(p,b,c,m,i,…)
bounded
P(p,b,c,m,i,…)
P(p,b,c,m,i,…)
unbounded NP-Hard(b,c,m,i)
NP-Hard(b,c,m,i)
equal
P(p)
#P-Hard(p,b,c)
bounded
NP-Cpmplete(p)
#P-Hard(p,b,c)
unbounded NP-Cpmplete(p)
#P-Hard(p,b,c)
Future

More voting protocols

Approximation and/or heuristics for the hard problems
{hazonn,aumann,sarit}@cs.biu.ac.il , {mjw}@csc.liv.ac.uk