Minimax: A Multiwinner Election Procedure legrand@cse.wustl.edu
The Minimax Procedure: Problem Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 1
The Minimax Procedure: Problem Problem set of alternatives up for election set of submitted approval ballots procedure should return a set alternatives of winning number of winning alternatives can range from 0 to one motivation (Brams et al.): multilateral treaties 2
, The Minimax Procedure: Problem Approval ballot example 010101 voter approves three out of six alternatives (, ) voter s most preferred outcome: 010101 ( ) voter s least preferred outcome: 101010 ( ) voter prefers outcomes with smaller Hamming distances from 010101 voter is indifferent among outcomes with equal Hamming distances from 010101, e.g. 000000 and 111111 3
The Minimax Procedure: Problem Hamming distance used as measure of disagreement between a ballot and winner set Hamming distance between two sets and : Hamming distance between two bitstrings and : 4
The Minimax Procedure: Two election procedures Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 5
#" $ &%" ' ( The Minimax Procedure: Two election procedures The minisum procedure simulates a majority yes/no vote on each alternative!alternative is elected iff more ballots approve than disapprove equivalent to choosing the winner set with minimal sumscore!sumscore of a set is the total of the Hamming distances between the set and each ballot: ) *+ 6
, 000011-000111. 001011 / 010011 0 111100 43, The Minimax Procedure: Two election procedures Minisum example 65have 40% approval; 000011 1, approval 2 have 80% approval;, has 20% first four voters are quite satisfied with the minisum outcome last voter is completely dissatisfied and effectively ignored 7
$ 78 &%" ' ( The Minimax Procedure: Two election procedures The minimax procedure finds a winner set that minimizes the dissatisfaction of the least satisfied voters equivalent to choosing the winner set with minimal maxscore!maxscore of a set is the largest Hamming distance between the set and any ballot: ) *+ $ 78 8
, 000011 The Minimax Procedure: Two election procedures Minimax example ' ( $ 78 &%" 0 111100 / 010011. 001011-000111 011111 all voters are relatively satisfied with the minimax outcome 9; all other sets have maxscore at least 4 9
The Minimax Procedure: Complexity Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 10
: : The Minimax Procedure: Complexity Complexity finding a minisum set can be done in time!treat it as yes/no elections and report the majority winners finding a minimax set can be done in <; time!brute-force approach calculates maxscore of each of possible winner sets and takes the lowest Can a minimax set be found in polynomial time? ; 11
= The Minimax Procedure: Complexity A minimax variant: Fixed-size minimax like minimax, finds a winner set that minimizes the dissatisfaction of the least satisfied voters... but only considers sets of fixed size likely more useful for real-world committee elections finding FSM winner set is provably NP-complete 12
B C C B A, - C > > The Minimax Procedure: Complexity Fixed-size minimax decision problem FIXED-SIZE MINIMAX (FSM) INSTANCE: Set subsets =and?> @ of with ; collection of ; nonnegative integer nonnegative integer = ;. Is there a subset of QUESTION: such that all Dfor? 13
F I I I B H C L J E B The Minimax Procedure: Complexity Vertex cover decision problem VERTEX COVER (VC) INSTANCE: Graph ; positive integer. QUESTION: Is there a vertex cover of size or less for i.e., a subset FE, such that for each edge Hat least one of KJand belongs to? F K GF?H FFwith 14
B The Minimax Procedure: Complexity Vertex cover problem B A 1 3 E 4 2 5 C D 7 6 F 8 G Is there a vertex cover of size 9? 15
E The Minimax Procedure: Complexity Vertex cover problem B A 1 3 E 4 2 5 C D 7 6 F 8 G?M is a vertex cover of size 3. 16
, 1100000 The Minimax Procedure: Complexity Equivalent FSM problem P P Q 0000011 P O 0010001 N 0001010 0 0011000 / 0001100. 0110000-1001000 0101001 17
B B The Minimax Procedure: Complexity Reduction any vertex cover problem can be reduced to a fixed-size minimax problem the vertices of the VC graph become the FSM alternatives; the edges become the ballots each so-constructed ballot necessarily votes for exactly two alternatives since each edge is a set of exactly two vertices there is a vertex cover of size solution with maxscore SRthere is a FSM minimax is NP-complete also (Frances & Litman) 18
The Minimax Procedure: Heuristic for minimax Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 19
Z[\ ]Z^_ d ` d e T Ya 3 Z[\ ]Z^_ 3 k h d i ` h d i 3bat U Y T V 5 U Y T 3 The Minimax Procedure: Heuristic for minimax Heuristic for minimax 1. Choose ballots W 3 T and U such that W 3. W 3 X 3 is maximized and 2. Start the current solution. 3. Repeat until bdoes not change: (a) Initialize a collection (b) For each alternative If If j e b, add add bb, bgf cof sets to be empty. to c. to c. 3don which and differ: T U (c) Compare the sets in cand, of the ones with the smallest maxscore, call the one with the smallest minisum score If ll. has a smaller maxscore, or equal maxscore and smaller sumscore, than the current make it the new bb,. 4. Take bas the minimax solution. 20
- 101011, 000100 The Minimax Procedure: Heuristic for minimax Minimax heuristic can fail O 111000 N 100001 0 010001 / 100100. 011011 001000 heuristic: 100100 m100000 m100001 21
The Minimax Procedure: Heuristic for minimax Testing the heuristic heuristic only sometimes finds optimal minimax sets testing approach:!generate random ballots using uniform electorate model!find maxscores of optimal minimax set and set found by heuristic!scale heuristic maxscore so that optimal minimax set maxscore is 100% and (worst possible maxscore) is 0% 22
n The Minimax Procedure: Heuristic for minimax Heuristic performance higher percentage means closer to optimal on average 5 25 125 6 99.67% 92.71% 99.08% 12 99.23% 92.07% 94.70% 18 98.69% 93.14% 93.34% heuristic often finds optimal minimax set almost always finds set with near-optimal maxscore 23
The Minimax Procedure: Manipulating minimax Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 24
, 000011 The Minimax Procedure: Manipulating minimax Manipulating minimax sincere votes:. 010011-001011, 000111 6 0 011111 / 010011. 001011-000111 all voters approve 1and 2 and disapprove voter 5 has Hamming distance 2 from each minimax winner set 25
, 000011 The Minimax Procedure: Manipulating minimax Manipulating minimax voter 5 is unscrupulous: 0 111100 / 010011. 001011-000111 011111 by voting insincerely, voter 5 has manipulated the election to give his most preferred outcome decisively 26
The Minimax Procedure: Manipulating minimax Manipulating minimax if there are alternatives on which a voter is in the overwhelming majority, it may be able to vote safely against the majority on those alternatives to force more agreement with its relatively controversial choices it is reasonable to expect many minimax ballots to have been insincerely voted if all voters use the above strategy, minimax elections will become extremely unstable 27
The Minimax Procedure: Conclusions and future work Outline Problem Two election procedures Minisum Minimax Computing minimax sets Computational complexity Heuristic for minimax Manipulating minimax Conclusions and future work 28
The Minimax Procedure: Conclusions and future work Conclusions minimax minimizes maximum voter disagreement with the winner set can be seen as fairer than minisum finding an optimal minimax set is NP-complete good polynomial-time heuristics exist minimax is often easily manipulated 29
The Minimax Procedure: Conclusions and future work Future work Is it possible to find a better approach to a minimax heuristic? Can an approximation ratio be proved? How can the heuristics presented be modified to perform well on the fixed-size minimax problem? Is there a way to change minimax to lessen the effects of manipulability while retaining minimax s coalition-forming properties? 30
The Minimax Procedure: References References Brams, Steven J., D. Mark Kilgour and M. Remzi Sanver. A Minimax Procedure for Negotiating Multilateral Treaties. October 2003. Frances, M., and A. Litman. On Covering Problems of Codes. Theory of Computing Systems, 30:113 119, March 1997. thanks to Ron Cytron, Steven Brams and Morgan Deters Questions? 31