Online Manipulation and Control in Sequential Voting Lane A. Hemaspaandra Jörg Rothe Edith Hemaspaandra

Standard Election on Location of COMSOC-14 Vince Edith H. Lane Lena Felix Edith E. Piotr Jeff Ulle Voters simultaneously express their preferences

Standard Election on Location of COMSOC-14 Vince Edith H. Lane Lena Felix Edith E. Piotr Jeff Ulle Manipulator

Sequential Election on Location of COMSOC-14 Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Lane Voters sequentially express their preferences

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Manipulator

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Manipulator

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Manipulator

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Manipulator

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle ManipulatorNonmanipulator It‘s possible, and Lena and Edith E. just don´t know when they vote!

Lane Sequential Election on Location of COMSOC Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle ManipulatorNonmanipulator Lena and Edith E. exerted a destructive online manipulation.

Lane Magnifying Glass Moment Vince Edith H. Lena Felix Edith E. Piotr Jeff Ulle Manipulator Online Manipulation Setting (OMS) is a tuple, where C is a set of candidates, u is a distinguished voter, is an „election snap shot for C and u,“ σ is the preference order of u ´s coalition, and d is a distinguished candidate. What is u ´s „best“ vote to cast now?

Online Manipulation Problems  online-E-Unweighted-Coalitional-Manipulation (online-E-UCM), for a voting system E. Given: An OMS as described above. Question: Does there exist some vote that u can cast (assuming support from the manipulators after u ) such that no matter what votes are cast by the nonmanipulators after u, there exists some E winner c such that ?  In online-E-Weighted-Coalitional-Manipulation (online-E-WCM), each voter comes with a weight.  In online-E-UCM[ k ] and online-E-WCM[ k ], the number of manipulators from u onward is at most k.  online-E-DUCM, online-E-DWCM, online-E-DUCM[ k ], and online-E-DWCM[ k ] are the destructive variants.

Related Work and How It Differs  Xia and Conitzer (AAAI 2010) and, relatedly, Desmedt and Elkind (EC 2010) study the „Stackelberg voting game,“ which is called „roll-call voting game“ by Sloth (GEB 1993): Voters vote in order and Preferences are common knowledge: Everyone knows everyone else´s preferences, everyone knows that everyone knows everyone else´s preferences, and so on out to infinity. Their analysis is fundamentally game-theoretic: with such complete knowledge there is precisely one (subgame perfect Nash) equilibrium, which can be computed from the back end forward. For bounded number of manipulators, manipulation is in P.

Related Work and How It Differs  Tennenholtz (EC 2004) studies „dynamic voting,“ but focuses on axioms and voting rules rather than on coalitions and manipulation.  Parkes and Procaccia (2011) study sequential decision- making for dynamically varying preferences, using Markov Decision Processes.  Also somewhat related to, but quite different from, our work is the study of possible and necessary winners initiated by Konczak and Lang (2005).  By contrast, inspired by „online algorithms“ our work focuses on complexity theory rather than game theory, is in a partial-information model, is not about axioms and is nonprobabilistic, and involves numbers of quantifiers that can grow with the input.

Easy Observations Proposition: 1.For each voting system E such that the (unweighted) winner problem is solvable in polynomial time, E-UCM reduces to online-E-UCM. 2.For each voting system E such that the weighted winner problem is solvable in polynomial time, E-WCM reduces to online-E-WCM. 3.For each voting system E such that the winner problem is solvable in polynomial time, E-DUCM reduces to online-E-DUCM. 4.For each voting system E such that the weighted winner problem is solvable in polynomial time, E-DWCM reduces to online-E-DWCM.

General Results Theorem: 1.For each voting system E whose winner problem can be solved in polynomial time (or even in polynomial space), online-E-UCM is in PSPACE. 2.For each voting system E whose weighted winner problem can be solved in polynomial time (or even in polynomial space), online-E-WCM is in PSPACE. 3.There is a voting system E with a polynomial-time winner problem such that online-E-UCM is PSPACE-complete. 4.There is a voting system E with a polynomial-time weighted winner problem such that online-E-WCM is PSPACE- complete.

General Results ( k Manipulators) Theorem: Fix any 1.For each voting system E whose winner problem can be solved in polynomial time, online-E-UCM[ k ] is in, the 2k -th level of the polynomial hierarchy. 2.For each voting system E whose weighted winner problem can be solved in polynomial time, online-E-WCM is in. 3.There is a voting system E with a polynomial-time winner problem such that online-E-UCM[ k ] is - complete. 4.There is a voting system E with a polynomial-time weighted winner problem such that online-E-WCM[ k ] is -complete.

Voting Systems Positional Scoring Rules (for m candidates)  defined by scoring vector with  each voter gives points to the candidate on position i  winners: all candidates with maximum score Borda:Plurality Voting: k -Approval ( (m-k) -Veto):Veto (Anti-Plurality):

Results for Specific Natural Voting Systems Theorem: 1.online-plurality-WCM (and thus also online-plurality-UCM) is in P. 2.online-plurality-DWCM (and thus also online-plurality-DUCM) is in P. This result is in our standard, the nonunique-winner, model. The next result is on problems in the unique-winner model. Theorem: 1.online-plurality-DWCM-UW is NP-hard, even when restricted to only two (or three, four, etc.) candidates. 2.online-plurality-WCM-UW is coNP-hard, even when restricted to only two (or three, four, etc.) candidates.

Proof: 1. Reduction from Partition: Given a nonempty sequence of positive integers with for some integer does there exist a set such that ?  Let and let be a Partition instance.  Construct an online-plurality-DWCM-UW instance such that V contains voters who vote in that order: Each votes for and has weight Each is a manipulator of weight Results for Specific Natural Voting Systems

 is a yes-instance of Partition if and only if is a yes-instance of online-plurality-DWCM-UW.  If is in Partition, the manipulators can give points to both and and zero points to the other candidates. So and are tied for the most points and there is no unique winner.  Conversely, the only way to avoid having a unique winner is to have a tie for the most points: Only and can tie (all others´ scores are different modulo ), and they tie only with points. So is in Partition. Results for Specific Natural Voting Systems

Theorem: 1.For each scoring rule, online- -WCM is in P if, and is NP-hard otherwise. 2.For each k, online- k -approval-UCM and online- k -veto-UCM are in P. 3.online-3candidate-veto-WCM is P -complete. 4.online-veto-WCM is in P. Results for Specific Natural Voting Systems NP[1] NP

Uncertainty About the Order of Future Voters  Schedule-Robust-online-E-UCM, for a voting system E. Given: An „OMS“ as before, except now we don´t focus on one manipulator u, but at a moment in time, we don´t know the order of the voters still to come. Question: Can our manipulative coalition ensure that d or someone liked more than d w.r.t. σ will win, regardless of what order the remaining voters vote in? Theorem: 1.For each voting system E whose winner problem is in P, Schedule-Robust-online-E-UCM is in. 2.There is a voting system E, whose winner problem is in P, such that Schedule-Robust-online-E-UCM is -complete.

References  E. Hemaspaandra, L. Hemaspaandra, and J. Rothe: The Complexity of Online Manipulation of Sequential Elections. To appear in Proceedings of COMSOC Also related are the papers:  E. Hemaspaandra, L. Hemaspaandra, and J. Rothe: Online Voter Control in Sequential Elections. To appear in Proceedings of ECAI  E. Hemaspaandra, L. Hemaspaandra, and J. Rothe: Controlling Candidate-Sequential Elections. To appear in Proceedings of ECAI-2012.