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The Complexity of Llull’s Thirteenth-Century Election System

The Complexity of Llull’s Thirteenth-Century Election System . Piotr Faliszewski University of Rochester. Edith Hemaspaandra Rochester Institute of Technology. Lane A. Hemaspaandra University of Rochester. J ö rg Rothe Institute f ü r Informatik Heinrich-Heine-Univ. D ü sseldorf.

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The Complexity of Llull’s Thirteenth-Century Election System

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  1. The Complexity of Llull’s Thirteenth-Century Election System Piotr FaliszewskiUniversity of Rochester Edith HemaspaandraRochester Institute of Technology Lane A. HemaspaandraUniversity of Rochester Jörg RotheInstitute für InformatikHeinrich-Heine-Univ. Düsseldorf Henning Schnoor Rochester Institute of Technology The Hebrew University of Jerusalem, June 17, 2009

  2. Outline • Introduction • Computational study of elections • Bribery and control • Llull/Copeland Elections • Model of elections • Representation of votes • Llull/Copeland rule • Results • Bribery and microbribery • Control of elections • Manipulation Hi, I am Ramon Llull. I have come up with the election that these guys now study!

  3. Introduction • Computational study of elections • Applications in AI • Multiagent systems • Multicriterion decision making • Meta search-engines • Planning • Applications in political science • Computational barrier to cheating in elections • Manipulation • Bribery • Control Computational agents can systematically analyze an election to find optimal behavior

  4. Introduction • Many ways to affect the result of election • Evildoer wants to make someone a winner or prevent someone from winning • Evildoer knows everybody else’s votes • Manipulation • Coalition of agents changes their voteto obtain their desired effect • Bribery • External agent, the briber, chooses a group of voters and tells them what votes to cast • The briber is limited by budget • Control • Organizers of the election modify theirstructure to obtain the desired result In my times it was enough that we all promised we would not cheat...

  5. Introduction • Manipulation versus bribery and control • Control attempted by the chair • Bribery  attempted by some outside agent • Manipulation  attempted by the voters themselves • Self-interested voters • Bribed voters might choose to not follow what the briber said (unless the voting is public, or maybe not even then) Okay… I can take the money, but I will vote as I like anyway!

  6. Outline • Introduction • Computational study of elections • Bribery and control • Llull/Copeland Elections • Model of elections • Representation of votes • Llull/Copeland rule • Results • Bribery and microbribery • Control of elections • Manipulation Let me tell you a bit about my system....

  7. Voting and Elections • Candidates and voters • C = {c1, ..., cn} • V = {v1, ..., vm} • Each voter vi is represented via his or her preferences over C. • Assumption: We know all the preferences • Strengthens negative results • Can be justified as well • Voting rule aggregates these preferences and outputs the set of winners. Hi, my name is v7. Hi v7, I hope you are not one of those awful people who support c3! How will they aggregate those votes?!

  8. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Representing Preferences C = { , , } • Example > >

  9. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Not all voters are rational though! People often have cyclical preferences! Irrational voters are represented via preference tables. Representing Preferences C = { , , } • Example > >

  10. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Irrational preferences Representing Preferences C = { , , } • Example > >

  11. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Irrational preferences Representing Preferences C = { , , } • Example > >

  12. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Irrational preferences Representing Preferences C = { , , } • Example > >

  13. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Irrational preferences Representing Preferences C = { , , } • Example > >

  14. Rational voters Preferences are strict total orders No cycles in single voter’s preference list Irrational preferences Representing Preferences C = { , , } • Example > > > > >

  15. Llull/Copeland Rule • The general rule • For every pair of candidates, ci and cj, perform a head-to-head plurality contest. • The winner of the contest gets 1 point • The loser gets zero points. • At the end of the day, candidates with most points are the winners • Difference between various flavors of the Llull/Copeland rule? • What happens if the head-to-head contest ends with a tie? • Llull: Both get 1 pointCopeland0: Both get 0 pointsCopeland0.5: Both get half a point

  16. Outline • Introduction • Computational study of elections • Bribery and control • Llull/Copeland Elections • Model of elections • Representation of votes • Llull/Copeland rule • Results • Bribery and microbribery • Control of elections • Manipulation Mr. Llull. Let us see just how resistant your system is

  17. E-bribery (E – an election system) Given: A set of candidates C, a set of voters V specified via their preference lists, p in C, and budget k Question: Can we make p win via bribing at most k voters? E-$bribery As above, but voters have prices and k is the spending limit. E-weighted-bribery, E-weighted-$bribery As the two above, but the voters have weights. Bribery Hmm... I seem to have trouble with finding the right guys to bribe...

  18. E-bribery (E – an election system) Given: A set of candidates C, a set of voters V specified via their preference lists, p in C, and budget k Question: Can we make p win via bribing at most k voters? E-$bribery As above, but voters have prices and k is the spending limit. E-weighted-bribery, E-weighted-$bribery As the two above, but the voters have weights. Bribery • Result • Llull/Copeland rule is resistant (NP-hard; warning: a worst-case formalism) to all forms of bribery, both for irrational and rational voters Mr. Agent. My system is resistant to bribery!

  19. Microbribery We pay for each small change we make If we want to make two flips on the preference table of the same voter then we pay 2 instead of 1 Comes in the same flavors as bribery Limitations Could be studied forrational voters... ... But we limit ourselves to the irrational case. Microbribery We do not really need to change each vote completely... Yeah... It’s easier to work with the preference Matrix™ ... Preference table, I mean…

  20. Microbribery We pay for each small change we make If we want to make two flips on the preference table of the same voter then we pay 2 instead of 1 Comes in the same flavors as bribery Limitations Could be studied forrational voters... ... But we limit ourselves to the irrational case. Results Both Llull and Copeland0 are vulnerable to microbribery Microbribery Uh oh... How did they do that?!?!?

  21. Setting C = {p=c0, c1,..., cn} V = {v1, ..., vm} Voters vi are irrational For each two candidates ci, cj: pij– number of flips that switch the head-on-head contest between them Approach If possible, find a bribery that gives p at least B points, ... ... and everyone else at most B points Try all reasonable B’s Validate B via min-cost flow problem Microbribery in Copeland Elections

  22. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p c1 t s c2 cn

  23. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p c1 t s c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success cn sink mesh source

  24. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 c1 s(c1)/0 B/K t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success s(cn)/0 B/K cn sink mesh source

  25. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source

  26. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source

  27. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source

  28. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source

  29. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source

  30. Proof Technique: Flow Networks Notation: s(ci) – ci score before bribery B – the point bound K – large number capacity/cost p B/0 s(p)/0 1/p10 c1 1/p20 s(c1)/0 B/K 1/p21 t s s(c2)/0 B/K c2 source – models pre- bribery scores mesh – models bribery cost sink – models bribery success 1/p2n s(cn)/0 B/K cn sink mesh source Cost = K(n(n+1)/2 - p-score) + cost-of-bribery

  31. Microbribery: Application • Round-robin tournament • Everyone plays with everyone else • Bribery in round-robin tournaments • For every game there we know • Expected (default) result • The price for changing it • We want a minimal price for our guy having most points • Round-robin tournament example • FIFA World Cup, group stage • 3 points for win • 1 point for tie • 0 points for loss • Microbribery?

  32. Microbribery: Application • Round-robin tournament • Everyone plays with everyone else • Bribery in round-robin tournaments • For every game there we know • Expected (default) result • The price for changing it • We want a minimal price for our guy having most points • Round-robin tournament example • FIFA World Cup, group stage • 3 points for win • 1 point for tie • 0 points for loss • Microbribery? • Applies directly!! • Given the table of expected results and prices … • … simply run the Microbribery algorithm • For FIFA: Simply use 1/3 as the tie value.

  33. Outline • Introduction • Computational study of elections • Bribery and control • Llull/Copeland Elections • Model of elections • Representation of votes • Llull/Copeland rule • Results • Bribery and microbribery • Control of elections • Manipulation How will your system deal with my attempts to control, Mr. Llull...?

  34. Control of elections The chair of the election attempts to influence the result via modifying the structure of the election Constructive control (CC) or destructive control (DC) Candidate control Adding candidates (AC) Deleting candidates (DC) Partition of candidates (PC/RPC) with or without the runoff Voter control Adding voters (AV) Deleting voters (DV) Partition of voters (PV) Control My system is resistant to all types of constructive control!! Okay, almost all.

  35. Flavors of control AC – adding candidates DC – deleting candidates (R)PC – (runoff) partition of candidates AV – adding voters DV – deleting voters PV – partition of voters R – NP-complete V– P membership Results: Control CC – constructive control TP – ties promoteDC – destructive control TE – ties eliminate

  36. Outline • Introduction • Computational study of elections • Bribery and control • Llull/Copeland Elections • Model of elections • Representation of votes • Llull/Copeland rule • Results • Bribery and microbribery • Control of elections • Manipulation My coalition of agents must be able to somehow manipulate the system!!

  37. Manipulation • Manipulation problem • Given: • Set of candidates • Set of honest voters (with their preferences) • Set of manipulators • Preferred candidate p • Question: • Can the manipulators set their votes so as to guarantee p’s victory? • Can’t avoid manipulation! • Gibbard-Satterthwaite • Dugan-Schwarz There Llull! By Gibbard-Satterthwaite result I know I can sometimes manipulate your system. Finally math gets useful!

  38. Unweighted manipulation NP-complete for all rational tie-handling values except 0, 0.5, 1 Even for two manipulators! Proof ideas Almost a microbribery instance The two manipulators can only switch head-to-head results from tie to a victory Reduce from X3C,1-in-3-SAT Can’t use microbribery to solve the problem because the two voters might not be able to implement the solution we would get Results: Manipulation Haha Mr. Agent! Gibbard-Satterthwaite say that you may be able to influence my system, but you won’t know when or how!

  39. Unweighted manipulation NP-complete for all rational tie-handling values except 0, 0.5, 1 Even for two manipulators! Aren’t we missing exactly the interesting values? What is magic about the values we miss? 1  no point in causing ties 0.5  switching the result of a head-to-head contest from tie to win affect both candidates in a symmetric way 0  makes “transferring points” difficult Results: Manipulation So… maybe there is hope for me…

  40. Weighted manipulation Completely resolved for three candidates Issues with for and more candidates Unique vs. non-unique winner Tie-value 1 is hard to account for. Strange things happen for four candidates. Weighted, three candidates Results: Manipulation Work in progress

  41. Summary • Copeland/Llull elections • Broad resistance to bribery • Constructive/Destructive • Rational/Irrational • Broad resistance to control • Constructive candidate control • Constructive/Destructive voter control • Rational/Irrational • Vulnerability • Microbribery (irrational case) Arrgh! Llull, my agents are practically helpless against your system!

  42. Thank you! I will happily answer your questions!

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