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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
• 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
• 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