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Voting Theory. Toby Walsh NICTA and UNSW. Motivation. Why voting? Consider multiple agents Each declares their preferences (order over outcomes) How do we make some collective decision? Use a voting rule!. Voting rule Social choice: mapping of a profile onto a winner(s)
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Voting Theory Toby Walsh NICTA and UNSW
Motivation • Why voting? • Consider multiple agents • Each declares their preferences (order over outcomes) • How do we make some collective decision? • Use a voting rule!
Voting rule Social choice: mapping of a profile onto a winner(s) Social welfare: mapping of a profile onto a total ordering Agent Usually assume odd number of agents to reduce ties Vote Total order over outcomes Profile Vote for each agent Terminology Extensions include indifference, incomparability, incompleteness
Voting rules: plurality • Otherwise known as “majority” or “first past the post” • Candidate with most votes wins • With just 2 candidates, this is a very good rule to use • (See May’s theorem)
Voting rules: plurality • Some criticisms • Ignores preferences other than favourite • Similar candidates can “split” the vote • Encourages voters to vote tactically • “My candidate cannot win so I’ll vote for my second favourite”
Two rounds Eliminate all but the 2 candidates with most votes Then hold a majority election between these 2 candidates Consider 25 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B 1st round: B knocked out 2nd round: C>A by 70:25 C wins Voting rules: plurality with runoff
Voting rules: plurality with runoff • Some criticisms • Requires voters to list all preferences or to vote twice • Moving a candidate up your ballot may not help them (monotonicity) • It can even pay not to vote! (see next slide)
Consider again 25 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B C wins easily Two voters don’t vote 23 votes: A>B>C 24 votes: B>C>A 46 votes: C>A>B Different result 1st round: A knocked out 2nd round: B>C by 47:46 B wins Voting rules: plurality with runoff
STV If one candidate has >50% vote then they are elected Otherwise candidate with least votes is eliminated Their votes transferred (2nd placed candidate becomes 1st, etc.) Identical to plurality with runoff for 3 candidates Example: 39 votes: A>B>C>D 20 votes: B>A>C>D 20 votes: B>C>A>D 11 votes: C>B>A>D 10 votes: D>A>B>C Result: B wins! Voting rules: single transferable vote
Voting rules: Borda • Given m candidates • ith ranked candidate score m-i • Candidate with greatest sum of scores wins • Example • 42 votes: A>B>C>D • 26 votes: B>C>D>A • 15 votes: C>D>B>A • 17 votes: D>C>B>A • B wins Jean Charles de Borda, 1733-1799
Voting rules: positional rules • Given vector of weights, <s1,..,sm> • Candidate scores si for each vote in ith position • Candidate with greatest score wins • Generalizes number of rules • Borda is <m-1,m-2,..,0> • Plurality is <1,0,..,0>
Voting rules: approval • Each voters approves between 1 and m-1 candidates • Candidate with most votes of approval wins • Some criticisms • Elects lowest common denominator? • Two similar candidates do not divide vote, but can introduce problems when we are electing multiple winners
Voting rules: other • Cup (aka knockout) • Tree of pairwise majority elections • Copeland • Candidate that wins the most pairwise competitions • Bucklin • If one candidate has a majority, they win • Else 1st and 2nd choices are combined, and we repeat
Voting rules: other • Coomb’s method • If one candidate has a majority, they win • Else candidate ranked last by most is eliminated, and we repeat • Range voting • Each voter gives a score in given range to each candidate • Candidate with highest sum of scores wins • Approval is range voting where range is {0,1}
Voting rules: other • Maximin (Simpson) • Score = Number of voters who prefer candidate in worst pairwise election • Candidate with highest score wins • Veto rule • Each agent can veto up to m-1 candidates • Candidate with fewest vetoes wins • Inverse plurality • Each agent casts one vetor • Candidate with fewest vetoes wins
Voting rules: other • Dodgson • Proposed by Lewis Carroll in 1876 • Candidate who with the fewest swaps of adjacent preferences beats all other candidates in pairwise elections • NP-hard to compute winner! • Random • Winner is that of a random ballot • …
Voting rules • So many voting rules to choose from .. • Which is best? • Social choice theory looks at the (desirable and undesirable) properties they possess • For instance, is the rule “monotonic”? • Bottom line: with more than 2 candidates, there is no best voting rule
Axiomatic approach • Define desired properties • E.g. monotonicity: improving votes for a candidate can only help them win • Prove whether voting rule has this property • In some cases, as we shall see, we’ll be able to prove impossibility results (no voting rule has this combination of desirable properties)
May’s theorem • Some desirable properties of voting rule • Anonymous: names of voters irrelevant • Neutral: name of candidates irrelevant
May’s theorem • Another desirable property of a voting rule • Monotonic: if a particular candidate wins, and a voter improves their vote in favour of this candidate, then they still win • Non-monotonicity for plurality with runoff • 27 votes: A>B>C • 42 votes: C>A>B • 24 votes: B>C>A • Suppose 4 voters in 1st group move C up to top • 23 votes: A>B>C • 46 votes: C>A>B • 24 votes: B>C>A
May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • May, Kenneth. 1952. "A set of independent necessary and sufficient conditions for simple majority decisions", Econometrica, Vol. 20, pp. 680–68 • Since these properties are uncontroversial, this about decides what to do with 2 candidates!
May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Plurality rule is clearly anonymous, neutral and monotonic • Other direction is more interesting
May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Anonymous and neutral implies only number of votes matters • Two cases: • N(A>B) = N(B>A)+1 and A wins. • By monotonicity, A wins whenever N(A>B) > N(B>A)
May’s theorem • Thm: With 2 candidates, a voting rule is anonymous, neutral and monotonic iff it is the plurality rule • Proof: Anonymous and neutral implies only number of votes matters • Two cases: • N(A>B) = N(B>A)+1 and A wins. • By monotonicity, A wins whenever N(A>B) > N(B>A) • N(A>B) = N(B>A)+1 and B wins • Swap one vote A>B to B>A. By monotonicity, B still wins. But now N(B>A) = N(A>B)+1. By neutrality, A wins. This is a contradiction.
Condorcet’s paradox • Collective preference may be cyclic • Even when individual preferences are not • Consider 3 votes • A>B>C • B>C>A • C>A>B • Majority prefer A to B, and prefer B to C, and prefer C to A! Marie Jean Antoine Nicolas de Caritat, marquis de Condorcet (1743 – 1794)
Condorcet principle • Turn this on its head • Condorcet winner • Candidate that beats every other in pairwise elections • In general, Condorcet winner may not exist • When they exist, must be unique • Condorcet consistent • Voting rule that elects Condorcet winner when they exist (e.g. Copeland rule)
Condorcet principle • Plurality rule is not Condorcet consistent • 35 votes: A>B>C • 34 votes: C>B>A • 31 votes: B>C>A • B is easily the Condorcet winner, but plurality elects A
Condorcet principle • Thm. No positional rule with strict ordering of weights is Condorcet consistent • Proof: Consider • 3 votes: A>B>C • 2 votes: B>C>A • 1 vote: B>A>C • 1 vote: C>A>B • A is Condorcet winner
Condorcet principle • Thm. No positional rule with strict ordering of weights is Condorcet consistent • Proof: Consider • 3 votes: A>B>C • 2 votes: B>C>A • 1 vote: B>A>C • 1 vote: C>A>B • Scoring rule with s1 > s2 > s3 • Score(B) = 3.s1+3.s2+1.s3 • Score(A) = 3.s1+2.s2+2.s3 • Score(C) = 1.s1+2.s2+4.s4 • Hence: Score(B)>Score(A)>Score(C)
Arrow’s theorem • We have to break Condorcet cycles • How we do this, inevitably leads to trouble • A genius observation • Led to the Nobel prize in economics
Arrow’s theorem • Free • Every result is possible • Unanimous • If every votes for one candidate, they win • Independent to irrelevant alternatives • Result between A and B only depends on how agents preferences between A and B • Monotonic
Arrow’s theorem • Non-dictatorial • Dictator is voter whose vote is the result • Not generally considered to be desirable!
Arrow’s theorem • Thm: If there are at least two voters and three or more candidates, then it is impossible for any voting rule to be: • Free • Unanimous • Independent to irrelevant alternatives • Monotonic • Non-dictatorial
Arrow’s theorem • Can give a stronger result • Weaken conditions • Pareto • If everyone prefers A to B then A is preferred to B in the result • If free & monotonic & IIA then Pareto • If free & Pareto & IIA then not necessarily monotonic
Arrow’s theorem • Thm: If there are at least two voters and three or more candidates, then it is impossible for any voting rule to be: • Pareto • Independent to irrelevant alternatives • Non-dictatorial
Arrow’s theorem • With two candidates, majority rule is: • Pareto • Independent to irrelevant alternatives • Non-dictatorial • So, one way “around” Arrow’s theorem is to restrict to two candidates
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • By IIA, this would not change if every voter moved C above A: • B>A>C => B>C>A • B>C>A => B>C>A • A>C>B => C>A>B • C>A>B => C>A>B • Each AB and BC vote the same!
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • By IIA, this would not change if every voter moved C above A • By transitivity A>C in result • But by unanimity C>A • B>A>C => B>C>A • B>C>A => B>C>A • A>C>B => C>A>B • C>A>B => C>A>B
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose not the case and result has A>B>C • A>C and C>A in result • This is a contradiction • B can only be top or bottom in result
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top • B all at bottom. By unanimity, B at bottom in result • B all at top. By unanimity, B at top in result • By monotonicity, B moves to top and stays there when some particular voter moves B up
Proof of Arrow’s theorem • If all voters put B at top or bottom then result can only have B at top or bottom • Suppose voters in turn move B from bottom to top • Exists pivotal voter from whom result changes from B at bottom to B at top • Pivotal voter is dictator
Proof of Arrow’s theorem • Pivotal voter is dictator • Consider profile when pivotal voter has just moved B to top (and B has moved to top of result) • For any AC, let pivotal voter have A>B>C • By IIA, A>B in result as AB votes are identical to profile just before pivotal vote moves B (and result has B at bottom) • By IIA, B>C in result as BC votes are unchanged • Hence, A>C by transitivity
Proof of Arrow’s theorem • Pivotal voter is dictator • Consider profile when pivotal voter has just moved B to top (and B has moved to top of result) • For any AC, let pivotal voter have A>B>C • Then A>C in result • This continues to hold even if any other voters change their preferences for A and C • Hence pivotal voter is dicatator for AC • Similar argument for AB
Arrow’s theorem • How do we get “around” this impossibility • Limit domain • Only two candidates • Limit votes • Single peaked votes • Limit properties • Drop IIA • …
Single peaked votes • In many domains, natural order • Preferences single peaked with respect to this order • Examples • Left-right in politics • Cost (not necessarily cheapest!) • Size • …
Single peaked votes • There are never Condorcet cycles • Arrow’s theorem is “escaped” • There exists a rule that is Pareto • Independent to irrelevant alternatives • Non-dictatorial • Median rule: elect “median” candidate • Candidate for whom 50% of peaks are to left/right
Conclusions • Many voting rules exist • Plurality, STV, approval, Copeland, … • For two candidates • “Best” rule is plurality • For more than two candidates • Arrow’s theorem proves there is no “best” rule • But there are limited ways around this (e.g. single peaked votes)