1. Using a Modified Borda Count to Predict the Outcome of a Condorcet Tally on a Graphical Model 11/19/05 Galen Pickard, MIT Advisor: Dr. Whitman Richards, CSAIL, MIT

2. Outline Background –Information aggregation –Condorcet and Borda methods Application to graphical models Seeking a sufficiency condition Results Numerical methods

3. Information Aggregation Set of voters, and candidates C 1 …C n Each voter supplies a preference order: –C 3 > C 1 = C 2 > … Aggregation method is used to determine the social preference order Many different types of aggregation methods, none is clearly optimal

4. 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!

5. Borda Method Every voter required to provide a complete preference order (no ties allowed) Borda vector (b 1, …, b m ) for m candidates A voter’s first choice gets b 1 points, second gets b 2, etc For each candidate, sum points over all voters Social order is the list of candidates ranked by total points

6. Condorcet Method Condorcet criterion: –If more voters rank C x > C y than C y > C x, the social order should rank C x > C y Aggregation method follows naturally –For each pair of candidates, count voters who prefer either –Build social order based on resulting matrix

7. 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

8. Application to Graphical Models

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

10. Application to Graphical Models Plurality order –S > Q > P > T > R Condorcet order –Q > R > P > S > T

11. Modified Borda Method Need to modify Borda to allow for partial preference orders Borda vector (b 0, …, b m ), graph of diameter no more than m For each voter, candidates at distance 0 get b 0 points, distance 1 get b 1 points, etc For each candidate, sum points over all voters

12. 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 B 1 … B n agree, the Condorcet tally will also agree

13. 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

14. Proof Outline Define the Borda difference vector D for some Borda vector B as (d 1, d 2, …) = (b 0 -b 1, b 1 -b 2, …) 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

15. Proof Outline P012 T 0 -0-0 T4T4 -0-0 1 P7P7 -0-0 S 10 2 -0-0 Q8Q8 R0R0

16. Proof Outline Condorcet method: T + S + -P + -Q P012 T 0 -0-0 T4T4 -0-0 1 P7P7 -0-0 S 10 2 -0-0 Q8Q8 R0R0 P012 T 0011 101 2 0

17. Proof Outline Borda method: d 1 T + d 2 S + -d 1 P + -d 2 Q P012 T 0 -0-0 T4T4 -0-0 1 P7P7 -0-0 S 10 2 -0-0 Q8Q8 R0R0 P012 T 00d1d1 d 1 + d 2 1-d 1 0d2d2 2 -d 1 - d 2 -d 2 0

18. Proof Outline If the scalar products of the Borda matrix for (d 1, d 2 ) = (.5,.5) and (0, 1) are both positive or both negative, the scalar product for the Condorcet matrix will be the same

19. Sufficiency Implications The result for vector D will agree with the result for k*D, for any positive k If the results for D A and D B agree, the result for D A +D B will also agree

20. Sufficiency Implications Thus, for any set of difference vectors D 1 …D n which all agree, any non-negative linear combination of these vectors will agree. For a graph of diameter n, weakest possible sufficiency condition is D 1 …D n = (1,0,0,…), (0,1,0,…), …, (0,0,0,…,1) This condition implies all other possible sufficiency conditions

21. 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

22. Larger Diameter Graphs A > B for any possible Modified Borda tally B > A for the Condorcet tally

23. 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

24. Numerical Results Modified Borda Vector: (1, x, 0, 0, …) Probability that Borda winner and Condorcet winner match

25. 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