190 likes | 203 Views
This study examines the evaluation of election outcomes under uncertainty, considering both perfect and imperfect information models. The complexity analysis of imperfect information models and the constant number of candidates are discussed. The evaluation problem and chance-evaluation problem are explored, along with computational aspects and manipulation in voting protocols. The study concludes with future directions and potential solutions.
E N D
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
B! A > B > C Avoid manipulation! Me My “friend”
Why voting? • Preference aggregation - to a socially desirable decision • Computational aspects when moving to agents • Evaluating a voting protocol • Manipulation • Assuming perfect information
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
Social choice domain • Set of voters, V = {V1,…,Vn} • Corresponding weights, w1,…,wn • Set of outcomes/candidates, Ω = {ω1,…, ωm} • Imperfect information, k = 3
Voting systems- a review • Binary Voting systems • Plurality • Approval • Preferential voting systems • Instant-runoff voting (IRV) • Borda • Condorcet systems • Copeland (Tournament) • Minimax
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
Constant number of candidates • Voting scenario Vs. voting result • A dynamic programming approach (for plurality): • Time Complexity: O(n * # of rows * k)
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 • …
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!)
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
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.
# of candidates as a parameter(2) EVALUATION for Borda and Copeland is #P-Hard.
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 ?
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!
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
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