Robust Winners and Winner Determination Policies under Cand - PowerPoint Presentation

Slide1

Robust Winners and Winner Determination Policies under Candidate Uncertainty

Joel oren, university of torontoJoint work with Craig boutilier, JÉrôme Lang and HÉctor Palacios.

1Slide2

Motivation – Winner Determination under Candidate Uncertainty

A committee, with preferences over alternatives: Prospective projects.Goals.Costly determination of availabilities:Market research for determining the feasibility of a project: engineering estimates, surveys, focus groups, etc.

“Best” alternative depends on

available

ones.

2

a

b

c

4 voters

3 voters

2 voters

a

b

c

b

c

a

c

a

b

Winner

a

?

?

?

c

 Slide3

Efficient Querying Policies for Winner Determination

Voters submit votes in advance.Query candidates sequentially, until enough is known in order to a determine the winner.Example:

a wins.

a

b

c

4 voters

3 voters

2 voters

a

b

c

b

c

a

c

a

b

Winner

?

?

3

 Slide4

The Formal Model

A set C of candidates.A vector, , of rankings (a preference profile).

Set is partitioned

:

– a priori known

availability

.

– the “unknown” set.Each candidate

is available with probability .

Voting rule:

is the election winner.

a

c

C

b

b

3 voters

2 voters

a

c

b

c

a

c

a

b

Y

(available)

U

(unknown)

4Slide5

Querying & Decision Making

At iteration submit query q(x),

.

Information set

.

Initial

available set

.

Upon querying candidate

:If

available: add to .

If unavailable: remove from .

–

restriction of

pref. profile to the candidate set

.

Stop when

is -sufficient – no additional querying will change

–

the “robust” winner.

a

c

C

b

b

3 voters

2 voters

a

c

b

c

a

c

a

b

0.5

0.7

0.4

?

a

b

?

5Slide6

Computing a Robust Winner

Robust winner: Given

,

is a robust winner if

.

A related question in voting:

[

Destructive

control by candidate addition] Candidate set

, disjoint spoiler set , pref. profile

over

, candidate

, voting rule

.

Question: is there a subset

, s.t.

?

Proposition:

Candidate

is a robust winner there is no destructive control against

, where the spoiler set is

.

Y

Y

x

6Slide7

Computing a Robust Winner

Proposition: Candidate is a robust winner

there is no destructive control

against

, where the spoiler set is

.

Implication: Pluarlity, Bucklin, ranked

pairs – coNP-complete; Copeland, Maximin -

polytime tractable.Additional results: Checking if

is a robust winner for top cycle, uncovered set, and

Borda can be done in polynomial time.Top-cycle & Uncovered set: prove useful criteria for the corresponding majority graph.

7Slide8

The Query Policy

Goal: design a policy for finding correct winner.Can be represented by a decision tree.Example for the vote profile (plurality):abcde, abcde

,

a

dbec, b

caed, bcead

,cdeab, cbade, cdbea

a

b

a

winsc

c

wins

a wins

b

b

wins

c

a

wins

U

a

b

c

d

a

b

b

c

8Slide9

Winner Determination Policies as Trees

r-Sufficient tree:Information set at each leaf is -sufficient.Each leaf is correctly labelled with the winner.

--

cost

of querying candidate/node

.

– expected cost of policy, over dist. of

.

a

b

a

wins

c

c

wins

a wins

b

b wins

c

a

wins

9Slide10

Cost of a tree:

.

For each node

– a training

set: Possible

true underlying sets

A, that agree with

.Example 1:

Example 2:

.

Can solve using a

dynamic-programming

approach.

Running time:

-- computationally heavy.

a

b

a

wins

c

c

wins

a wins

b

b wins

c

a

wins

Recursively Finding

Optimal Decision

Trees

10Slide11

Myopically Constructing Decision Trees

Well-known approach of maximizing information gain at every node until reached pure training sets – leaves (C4.5).Mypoic step: query the candidate for the highest “information gain” (decrease in entropy of the training set)

.

Running time:

11Slide12

Empirical Results

100 votes, availability probability

.

Dispersion

parameter

. (

uniform distribution).Tested for Plurality, Borda, Copeland.

Preference distributions drawn i.i.d. from Mallows -distribution: probabilities decrease exponentially with distance from a “reference” ranking.

=

0.3

0.5

0.9

Method

Plurality,

DP4.13.42.7

Plurality, Myopic4.13.52.8

Borda, DP3.72.7

1.7Borda, Myopic3.72.7

1.7

0.3

0.50.9

Method

Plurality, DP4.13.4

2.7Plurality, Myopic4.13.5

2.8Borda, DP3.7

2.71.7Borda, Myopic

3.72.7

1.7

Average cost (# of queries)12Slide13

Empirical Results

Cost decrease as increases – [ less uncertainty about the available candidates set].Myopic performed very close to the OPT DP alg.Not shown:Cost increases with

the dispersion parameter – “noisier”/more diverse preferences (not shown

).

-Approximation: stop the recursion when training set is

– pure.For plurality, ,

,

.For

,

.

0.3

0.5

0.9

Method

Plurality,

DP

4.1

3.42.7Plurality, Myopic

4.13.52.8Borda

, DP3.72.71.7

Borda, Myopic3.72.71.7

0.3

0.5

0.9Method

Plurality,

DP4.1

3.42.7Plurality, Myopic

4.13.5

2.8Borda, DP

3.72.7

1.7Borda, Myopic

3.72.71.7Average cost (# of queries)

13Slide14

Additional Results

Query complexity: expected number of queries under a worst-case preference profile.Result: For Plurality, Borda, and Copeland, worst-case exp. query complexity is

.

Simplified policies:

Assume

for all

. Then there is a simple iterative query policy that is asymptotically optimal as

.

14Slide15

Conclusions & Future Directions

A framework for querying candidates under a probabilistic availability model.Connections to control of elections. Two algorithms for generating decision trees: DP, Myopic.Future directions:Ways of pruning the decision trees (depend on the voting rules).

Sample-based methods for reducing training set size.

Deeper theoretical study of the query complexity.

15