The Complexity of Probabilistic Lobbying

Transcription

1 Gábor Erdélyi 1, Henning Fernau 2, Judy Goldsmith 3 Nicholas Mattei 3, Daniel Raible 2, and Jörg Rothe 1 1 ccc.cs.uni-duesseldorf.de/~{erdelyi,rothe} 2 Univ. Trier, FB 4 Abteilung Informatik 3 Univ. of Kentucky Dept. of Comp. Sci. October 22, 2009

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5 Table of Contents 1 Overview 2 3 Initial Model Bribery Methods Evaluation Criteria 4 Proof Sketches Issue Weighting 5 6

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8 Bribing Politicians??? Shocking!

9 Can We Really Buy Votes? Are Politicians That Easy? How do you target the money?

10 Can We Really Buy Votes? Are Politicians That Easy? How do you target the money?

11 A Few Formalities... Initial Model Bribery Methods Evaluation Criteria Politicians have initial likelihoods of voting for each referendum We assume that there are fixed prices for each probability increase Question: Can we afford to fix all the referenda?

12 P Matrix Construction Initial Model Bribery Methods Evaluation Criteria Q m n [0,1] denotes the set of m n matrices over Q [0,1]. P Q m n [0,1] is a probability matrix (of size m n), Each entry p i,j of P gives the probability that voter v i will vote yes for issue r j. The result of a vote can be either a yes (represented by 1) or a no (represented by 0). The result of any vote on all issues as a 0/1 vector X = (x 1, x 2,..., x n ).

13 Initial Model Bribery Methods Evaluation Criteria The Discrete Price Function C P c i,j maps {0, 1,... k + 1} Z + in a nondecreasing manner, 1 where {0, 1,... k + 1} indicates steps of size k+1 in the probability interval [0, 1], and c i,j (w) is the price of raising the probability of the ith voter voting yes on the jth issue to p i,j + 1 k+1. The domain of c i,j consists of k + 2 elements of Q [0,1] including 0, p i,j, and 1. Note that k = 0 enforces p i,j {0, 1}. The image of c i,j consists of k + 2 nonnegative integers including 0. c i,j (p i,j ) = 0 represents the starting probability of voter v i voting on issue r j. For any two elements a, b in the domain of c i,j, if p i,j a b or p i,j a b, then c i,j (a) c i,j (b). This corresponds to a guarantee of monotonicity on the prices.

14 Example Overview Initial Model Bribery Methods Evaluation Criteria The Probability Matrix (P) r 1 r 2 r 3 v v The Cost Matrix (C P ) c i,j c 1, c 1, c 1, c 2, c 2, c 2,

15 Problem Definition Overview Initial Model Bribery Methods Evaluation Criteria B i -C j -PLP Name: B i -C j Probabilistic Lobbying Problem. Given: A probability matrix P Q m n [0,1] with table C P of price functions, a target vector Z {0, 1} n, and a budget B. Question: Is there a way for The Lobby to influence P (using bribery method B i and evaluation criterion C j, without exceeding budget B) such that the result of the votes on all issues equals Z?

16 Initial Model Bribery Methods Evaluation Criteria

17 Microbribery (B 1 ) Overview Initial Model Bribery Methods Evaluation Criteria Edit individual elements of the P matrix according to the costs in the C P matrix. The Lobby can pick not only the voter but also which issue to influence for that voter.

18 Initial Model Bribery Methods Evaluation Criteria

19 Issue Bribery (B 2 ) Overview Initial Model Bribery Methods Evaluation Criteria Each column of P represents how all the voters think on a particular issue. The Lobby" can pick a column of the matrix and edit it with the donation of some ammount of money. Each voter receives a fraction, d m, of the total dollar ammount and each voter moves accordingly.

20 Initial Model Bribery Methods Evaluation Criteria

21 Voter Bribery (B 3 ) Overview Initial Model Bribery Methods Evaluation Criteria Each row of P represents what an individual voter thinks on all issues. The Lobby" can pick a row of the matrix and edit it with the donation of some ammount of money. Each issue receives a fraction, d n, of the total dollar ammount and each issue moves accordingly.

22 Strict Majority (C 1 ) Initial Model Bribery Methods Evaluation Criteria For each issue, a strict majority of the individual voters must have probability t of voting according to The Lobby s agenda. r 1 r 2 r 3 v v Example Evaluation Threshold: t = 50% Result: X = (0, 0, 0) This is because none of the issues has a strict majority of voters over 50%

23 Average Majority (C 2 ) Initial Model Bribery Methods Evaluation Criteria For each issue, r j, of a given P, we define: p j = ( P m i=1 p i,j)/m An issue, r j is accepted if and only if, p j t. r 1 r 2 r 3 v v Example Evaluation Threshold: t = 50% Result: X = (1, 0, 0) The first issue has a p j 50%

24 B i -C j -PLP Overview Proof Sketches Issue Weighting Bribery Evaluation Criterion Criterion No. Voters Total Prob. Micro P P Issue P P Voter W[2]-complete W[2]-complete Table: Complexity results for B i -C j -PLP

25 Proof Sketches Issue Weighting Microbribery, Number of Voters is in P How much does it cost for each voter/referendum pair? How much does it cost for each referendum?

26 Proof Sketches Issue Weighting Microbribery, Number of Voters is in P How much does it cost for each voter/referendum pair? How much does it cost for each referendum? Can we afford all the referenda?

27 Proof Sketches Issue Weighting Microbribery, Number of Voters is in P How much does it cost for each voter/referendum pair? How much does it cost for each referendum? Can we afford all the referenda?

28 Proof Sketches Issue Weighting Microbribery, Total Vote Probability is in P Theorem B 1 -C 2 -PLP P. Reduction to Path Scheduling on One Machine Given a vertex-weighted graph a set of disjoint paths and C, q Vertices are jobs, edges are precedence, weights are job costs Can we schedule q jobs at cost C, while respecting precedence?

29 Proof Sketches Issue Weighting

30 Proof Sketches Issue Weighting Microbribery, Total Vote Probability is in P Theorem B 1 -C 2 -PLP P. Reduction to Path Scheduling on One Machine Precedence chains represent increments of probability for a single voter/issue. Can find minimum cost for each issue, add them up.

31 Proof Sketches Issue Weighting Exact-Cost Microbribery is NP-Hard Theorem For j {1, 2}, Exact-B 1 -C j -PLP is NP-complete. Reduction from Subset Sum to one-voter bribery (so evaluation criteria are equivalent, and microbribery is the same as issue bribery)

32 Proof Sketches Issue Weighting

33 Issue Weighting Overview Proof Sketches Issue Weighting B i -C j -PLP-WIW Name: B i -C j -Probabilistic Lobbying Problem with Issue Weighting. Given: A probability matrix P Q m n [0,1] with table C P of price functions and a lobby target vector Z {0, 1} n, a lobby weight vector W Z n, an objective value O Z +, and a budget B. Question: Is there a way for The Lobby to influence P (using bribery method B i and evaluation criterion C j, without exceeding budget B) such that the total weight of all issues for which the result coincides with The Lobby s target vector Z is at least O?

34 B i -C j -PLP-WIW Overview Proof Sketches Issue Weighting Bribery Evaluation Criterion Criterion No. Voters Total Prob. Micro NP-compl., FPT NP-compl., FPT Issue NP-compl., FPT NP-compl., FPT Voter W[2]-complete W[2]-complete Table: Complexity results for B i -C j -PLP-WIW

35 What s The Issue Here? Really, this is an opimization problem: We want to minimize the amount we need to spend.

36 Optimal Bribery is Hard to Approximate Theorem There is a constant c > 0 such that MIN-OL is not approximable within factor c log(n) unless NP DTIME(n log log(n) ), where n denotes the number of issues. Since OL can be viewed as a special case of both B 3 -C i -PLP and B 3 -C i -PLP-WIW for i {1, 2}, we have the following corollary. Corollary For i {1, 2}, there is a constant c i > 0 such that both MIN-B 3 -C i -PLP and MIN-B 3 -C i -PLP-WIW are not approximable within factor c i log(n) unless NP DTIME(n log log(n) ), where n denotes the number of issues.

37 A Useful Definition Definition A cover number c(r j ) is associated with each issue r j, indicating by how many levels voters must raise their acceptance probabilities in order to arrive at average majority for r j.

38 Corollary There is a constant c > 0 such that MIN-B 3 -C 2 -PLP is not approximable within factor c log(n) unless NP DTIME(N log log(n) ), where N is the cover number of the given instance. A fortiori, the same statement holds for MIN-B 3 -C 2 -PLP-WIW. We show the following theorem by analyzing a greedy approximation algorithm. Theorem MIN-B 3 -C 2 -PLP can be approximated within a factor of ln(n) + 1, where N is the cover number of the given instance.

39 Polynomial Time Approximation Schemes Theorem For i, j {1, 2}, MIN-B i -C j -PLP-WIW admits a PTAS.

40 Questions/Future Work Are these the right evaluation criteria? What about donation limits?

41 Questions/Future Work Are these the right evaluation criteria? What about donation limits? What about voter-voter influence?

42 Questions/Future Work Are these the right evaluation criteria? What about donation limits? What about voter-voter influence? What about the other guys and their donations?

43 Questions/Future Work Are these the right evaluation criteria? What about donation limits? What about voter-voter influence? What about the other guys and their donations? Where will the next ADT be?

44 Questions/Future Work Are these the right evaluation criteria? What about donation limits? What about voter-voter influence? What about the other guys and their donations? Where will the next ADT be?

45 Acknowledgements Supported in part by the DFG under grants RO 1202/12-1 (within the European Science Foundation s EUROCORES program LogICCC: Computational Foundations of Social Choice ), RO 1202/11-1, and NSF grant ITR

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47 Proof Ideas For i, j {1, 2}, B i -C j -PLP-WIW is NP-complete. Hardness: Reduction from Knapsack For j {1, 2}, B 3 -C j -PLP (parameterized by the budget) is W[2]-hard. B 3 -C 1 -PLP is W[2]-hard by reduction from OL a B 3 -C 1 -PLP W[2] by reduction to SMNTMC (Short Nondeterministic Turing Machine Computation) a Optimal Lobbying, introduced by Christian, Fellows, Rosamond, and Slinko, 07.

48 Reduction of Voter Bribery to SNTMC Parameter B (budget) reduces to k (number of steps) The reduction does a lot of work, computing: function H i,j (d) which says that d euros brings voter i up to threshold t on referendum j function s(j), the minimum number of voters that are still needed for referendum j (s(j) B + 1) The NTM writes s(j) symbols on tape j (in parallel) using voter names ({v 1,..., v m }) as alphabet, writes B symbols on 0 th tape looks up H i,j (#v i ), maybe moves head on tape j Are all tapes j completed?