Ties Matter: Complexity of Voting Manipulation Revisited based on joint work with Svetlana Obraztsova (NTU/PDMI) and Noam Hazon (CMU) Edith Elkind (Nanyang Technological University, Singapore)

Synopsis We will talk about voting – and, in particular, voting manipulation We will focus on a frequently neglected aspect of voting: tie-breaking rules We will show that

Setup An election is given by – a set of candidates C, |C| = m – a list of voters V = {1,..., n} – for each voter i in V, a preference order R i each R i is a total order over C – a voting rule F: for each list of voters’ preference orders, F outputs a candidate in C

Examples of Voting Rules (1/2) Scoring rules: – any vector s = (s 1,..., s m ) defines a scoring rule F s : each candidate receives s i points from each voter who ranks him in position i a candidate’s score is his total # of points the candidate with the highest score wins Examples: – Plurality: (1, 0,..., 0) – Borda: (m-1, m-2,..., 2, 1, 0) abcdeabcde

Examples of Voting Rules (2/2) Copeland: – for a, b  C, we say that a beats b in a pairwise election if more than half of the voters rank a above b – the score of a candidate c is # of pairwise elections c wins  # of pairwise elections c loses Maximin: – for a, b  C, let S(a, b) = # of voters who prefer a to b – the score of a candidate c is min a  C\{c} S(c, a) the number of votes c gets against his toughest opponent

Applications Political elections Hiring new faculty Prizes Decision-making in multi-agent systems – voting over joint plans

Manipulation A voting rule is manipulable if there exists a preference profile s.t. some voter has an incentive to lie about their preferences – i prefers F(R 1,..., R’ i,..., R n ) to F(R 1,..., R i,..., R n ) Gibbard’73, Satterthwaite’75: for |C|>2, any non-dictatorial voting rule is manipulable But maybe manipulations are hard to compute? Bartholdi, Tovey, Trick’89: given a profile, one can find a beneficial manipulation in poly-time for most voting rules – Plurality, Borda, Copeland, maximin

A Complication The common voting “rules” are voting correspondences: several candidates may have the top score Ties need to be broken BTT’89 assumes that ties are broken in favor of manipulator The algorithm extends to any lexicographic tie-breaking rule – i.e., one that uses a priority order over C What if the tie-breaking rule is not lexicographic?

This Work We consider two types of tie-breaking rules: – randomized tie-breaking: the winner is selected from the tied candidates uniformly at random the manipulator assigns utilities to candidates, maximizes his E[utility] – arbitrary poly-time tie-breaking tie-breaking rule is given by an oracle Question: do easiness results of BTT’89 still hold under these tie-breaking rules?

Results: Randomized Tie-Breaking, Scoring Rules Theorem: for any scoring rule the manipulation problem is poly-time solvable under randomized tie-breaking What is the best outcome where winners have t points, for each feasible t? manipulator’s vote non-manipulator’ votes t

Results: Randomized Tie-Breaking, Maximin, “Special” Utilities Theorem: for Maximin, if the manipulator’s utility is given by u(p)=1, u(c)=0 for c ≠ p, then the manipulation problem is poly-time solvable under randomized tie-breaking Proof sketch: – let s(c) denote c’s score before we vote – our vote changes each score by at most 1: c’s score goes up iff c appears before each of its toughest opponents – if s(c) > s(p)+1 for some c ≠ p, we lose; suppose this is not the case – rank p first

Maximin Proof, Continued c is good if s(c) < s(p) c is bad if s(c) = s(p) c is ugly if s(c) = s(p)+1 G = directed graph with vertex set C s.t. there is an edge from a to b iff a is b’s toughest opponent Goal: sort G so that each ugly vertex and as many bad vertices as possible have an incoming edge – can be done in poly-time

Results: Randomized Tie-Breaking, Maximin, General Utilities Theorem: for Maximin the manipulation problem is NP-hard under randomized tie-breaking – even if u(d)=0, u(c)=1 for c ≠ d Proof idea: – set up the instance so that d necessarily wins – need to maximize the number of winners i.e., sort G so that as few vertices as possible have an incoming edge reduction from Feedback Vertex Set

Results: Randomized Tie-Breaking, Copeland, General Utilities Theorem: for Copeland the manipulation problem is NP-hard under randomized tie-breaking – even if u(c)  {0, 1} for all c  C Reduction from Independent Set – extends to a hardness of approximation result : assuming P ≠ NP, no poly-time algorithm can find a manipulative vote such that the manipulator’s utility is within a constant factor from optimal

Arbitrary Poly-Time Tie-Breaking Theorem: there exists a poly-time computable tie-breaking rule T s.t. its combinations with Borda, Copeland and maximin are NP-hard to manipulate – T depends on the set of tied candidates only – if we allow tie-breaking rules that depend on the rest of the profile, even Plurality is NP-hard to manipulate Proof idea: – the winning set encodes a Boolean formula  and a truth assignment  for  – the manipulator can affect , but not  – the tie-breaking rule checks if  satisfies 

Conclusions and Future Work Approximation algorithms/inapproximability results for Maximin? Complexity of control/bribery/coalitional manipulation under randomized tie-breaking?