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Evaluation of election outcomes under uncertainty

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  1. Evaluation of election outcomes under uncertainty

  2. 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

  3. B! A > B > C Avoid manipulation! Me My “friend”

  4. Why voting? • Preference aggregation - to a socially desirable decision • Computational aspects when moving to agents • Evaluating a voting protocol • Manipulation • Assuming perfect information

  5. B! A > B > C 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

  6. Our model - results

  7. Social choice domain • Set of voters, V = {V1,…,Vn} • Corresponding weights, w1,…,wn • Set of outcomes/candidates, Ω = {ω1,…, ωm} • Imperfect information, k = 3

  8. Voting systems- a review • Binary Voting systems • Plurality • Approval • Preferential voting systems • Instant-runoff voting (IRV) • Borda • Condorcet systems • Copeland (Tournament) • Minimax

  9. 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

  10. Constant number of candidates • Voting scenario Vs. voting result • A dynamic programming approach (for plurality): • Time Complexity: O(n * # of rows * k)

  11. m is constant! Representing a voting result • A vector of [0,n]m : O(knm+1) • Plurality • Approval • Borda : O(kn(mn)m) • A vector of [0,n]m(m-1)/2 : O(knm2+1) • All Condorcet systems • A vector of [0,n]m! : O(knm!+1) • IRV • Borda • …

  12. 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), O(Poly(n)m2), O(Poly(n)m!) 

  13. Voters Candidates 1/3 v0 C0 1/3 1/3 1/3 v1 C1 1/3 1/3 v2 C2 1/3 2/3 R0 R 1 Z0 Z 1 # of candidates as a parameter(1) Even without weights, EVALUATION for Plurality is #P-Hard! V1 V2 0 0 1 1 2 2

  14. V1 V2 1/3 (c0,g2,c1,c2,c3,z,g1) v0 0 0 1/3 1 1 1/3 1/3 (c1,c0,g2,c2,c3,z,g1) v1 2 2 1/3 Pv 1/3 (c2,c0,c1,g2,c3,z,g1) v2 1/3 2/3 (c3,c0,c1,c2,g2,z,g1) v3 1 1 (z,c3,c2,c1,g1,g2,c0) z0 (z,c3,c2,g1,c0,g2,c1) 1 z1 Pz 1 (z,c3,g1,c1,c0,g2,c2) z2 1 (z,g1,c2,c1,c0,g2,c3) z3 # of candidates as a parameter(2) EVALUATION for Borda and Copeland is #P-Hard.

  15. # of candidates as a parameter(2) EVALUATION for Borda and Copeland is #P-Hard. 

  16. 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 ?

  17. Vz B+1 V0 V1 V2 V3 V4 W0 W2 W4 W1 W3 1/k 1 B … … C0 C1 Ck Z K Weighted CHANCE-EVALUATION NP-Complete (in the strong sense) for Plurality!

  18. V1 V5 V7 1/4 A 1/3 A 1/3 B 1/2 B 2/3 D 2/3 D 1/4 D V1' V2' 1 V2 A 1 2 1 1 1 V3 1 1 1 2 1 s V4 t B 1 1 1 2 1 V6 1 1 1 V8 C Un-weighted CHANCE-EVALUATION Polynomial algorithm for Plurality: V2 V3 V4 V6 V8 1/2 A 1/3 A 1/3 B 1/3 B 1/3 B 1/2 C 1/2 B 2/3 C 2/3 C 2/3 C 1/6 C

  19. Conclusion • Recall the previous results table: • Future • More voting protocols • Approximation and/or heuristics for the hard problems {hazonn,aumann,sarit}@cs.biu.ac.il , {mjw}@csc.liv.ac.uk