Using a Modified Borda Count to Predict the Outcome of a Condorcet Tally on a

Graphical Model

11/19/05

Galen Pickard, MITAdvisor: Dr. Whitman Richards, CSAIL, MIT

Outline

• Background– Information aggregation– Condorcet and Borda methods

• Application to graphical models

• Seeking a sufficiency condition

• Results

• Numerical methods

Information Aggregation

• Set of voters, and candidates C1…Cn

• Each voter supplies a preference order:– C3 > C1 = C2 > …

• Aggregation method is used to determine the social preference order

• Many different types of aggregation methods, none is clearly optimal

Arrow’s Theorem

• Desirable properties of an aggregation method:– Universality– Non-imposition– Non-dictatorship– Pareto efficiency– Independence of irrelevant alternatives

• No method can satisfy all at once!

Borda Method

• Every voter required to provide a complete preference order (no ties allowed)

• Borda vector (b1, …, bm) for m candidates• A voter’s first choice gets b1 points,

second gets b2, etc• For each candidate, sum points over all

voters• Social order is the list of candidates

ranked by total points

Condorcet Method

• Condorcet criterion:– If more voters rank Cx > Cy than Cy > Cx, the

social order should rank Cx > Cy

• Aggregation method follows naturally– For each pair of candidates, count voters who

prefer either– Build social order based on resulting matrix

Condorcet Method

• Non-transitivity:– 30% of voters rank Rock > Scissors > Paper– 34% of voters rank Scissors > Paper > Rock– 36% of voters rank Paper > Rock > Scissors

• Social order is non-transitive– 66% of voters rank Rock > Scissors– 64% of voters rank Scissors > Paper– 70% of voters rank Paper > Rock

• Result: Rock > Scissors > Paper > Rock

Application to Graphical Models

Application to Graphical Models

• Preference order for voters at Q:– Q > P = R > S = T

• Preference order for voters at S:– S > R = T > Q > P

Application to Graphical Models

• Plurality order– S > Q > P > T > R

• Condorcet order– Q > R > P > S > T

Modified Borda Method

• Need to modify Borda to allow for partial preference orders

• Borda vector (b0, …, bm), graph of diameter no more than m

• For each voter, candidates at distance 0 get b0 points, distance 1 get b1 points, etc

• For each candidate, sum points over all voters

Seeking a Sufficiency Condition

• Sufficiency condition for predicting the outcome of the Condorcet tally:– For a graph with some set of properties, for

any pairwise comparison for which counts using Modified Borda vectors B1 … Bn agree, the Condorcet tally will also agree

Known Sufficiency Condition

• For a graph of diameter 2, for any pairwise comparison for which counts using Modified Borda vectors (1, .5, 0) and (1, 1, 0) agree, the Condorcet tally will also agree

Proof Outline

• Define the Borda difference vector D for some Borda vector B as (d1, d2, …) = (b0-b1, b1-b2, …)

• B = (1,.5,0) D = (.5,.5)

• For two candidates X and Y, consider all possible pairs of distances for a voter

• Describe Borda and Condorcet methods as scalar product operations

Proof Outline

P 0 1 2

T

0

-

0

T

4

-

0

1

P

7

-

0

S

10

2

-

0

Q

8

R

0

Proof Outline

• Condorcet method: T + S + -P + -Q

P 0 1 2

T

0

-

0

T

4

-

0

1

P

7

-

0

S

10

2

-

0

Q

8

R

0

P 0 1 2

T

0 0 1 1

1 -1 0 1

2 -1 -1 0

Proof Outline

• Borda method: d1T + d2S + -d1P + -d2Q

P 0 1 2

T

0

-

0

T

4

-

0

1

P

7

-

0

S

10

2

-

0

Q

8

R

0

P 0 1 2

T

0 0 d1

d1 +

d2

1 -d1 0 d2

2

-d1 -

d2-d2 0

Proof Outline

• If the scalar products of the Borda matrix for (d1, d2) = (.5, .5) and (0, 1) are both positive or both negative, the scalar product for the Condorcet matrix will be the same

Sufficiency Implications

• The result for vector D will agree with the result for k*D, for any positive k

• If the results for DA and DB agree, the result for DA+DB will also agree

Sufficiency Implications

• Thus, for any set of difference vectors D1…Dn which all agree, any non-negative linear combination of these vectors will agree.

• For a graph of diameter n, weakest possible sufficiency condition is D1…Dn = (1,0,0,…), (0,1,0,…), …, (0,0,0,…,1)

• This condition implies all other possible sufficiency conditions

Larger Diameter Graphs

• There are graphs for which weakest sufficiency condition is not met!

• Thus, in general, it is impossible to predict the Condorcet social order based solely on social orders of Modified Borda tallies

Larger Diameter Graphs

• A > B for any possible Modified Borda tally

• B > A for the Condorcet tally

Numerical Results

• If we don’t care about the complete preference order, but only the winner, Borda is a good estimator

• Borda vector of (1, .5, 0, 0, …) works very well, for random graphs

Numerical Results

Modified Borda Vector: (1, x, 0, 0, …)

Probability that Borda winner and Condorcet winner match

Conclusion

• For sufficiently small or dense graphs, it is sometimes possible to infer the Condorcet social order from the outcomes of Modified Borda tallies

• In general, however, it is not possible to do so

• But, in many cases, Borda is a good estimator