Voting and social choice

Looking at a problem from the designers point of view

Voting

over alternatives

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voting rule (mechanism) determines winner based on votes

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• Can vote over other things too – Where to go for dinner tonight, other joint plans, …

Voting (rank aggregation)

• Set of m candidates (aka. alternatives , outcomes ) • n voters ; each voter ranks all the candidates – E.g., for set of candidates {a, b, c, d}, one possible vote is b > a > d > c – Submitted ranking is called a vote • A voting rule takes as input a vector of votes (submitted by the voters), and as output produces either: – the winning candidate, or – an aggregate ranking of all candidates • Can vote over just about anything – political representatives, award nominees, where to go for dinner tonight, joint plans, allocations of tasks/resources, … – Also can consider other applications: e.g., aggregating search engine’s rankings into a single ranking

Example voting rules

• Each voter gives a vector of ranked choices.

• Scoring rules ranked i th are defined by a vector (a 1 , a 2 , …, a m ); being in a vote gives the candidate a i points – Plurality is defined by (1, 0, 0, …, 0) (winner is candidate that is ranked first most often, only first choice votes are counted) – Veto (or anti-plurality ) is defined by (1, 1, …, 1, 0) (winner is candidate that is ranked last the least often) – Borda is defined by (m-1, m 2, …, 0)

Nanson (Borda variant)

• Candidate with the lowest Borda score is eliminated, then we re-compute Borda counts and continue.

Runoff voting rules

proceeds in stages • Plurality with (2-candidate) runoff : top two candidates in terms of plurality score proceed to runoff; whichever is ranked higher than the other by more voters, wins How would you describe the idea behind a runoff?

• Single Transferable Vote ( STV , aka. Instant Runoff ): candidate with lowest plurality score drops out; if you voted for that candidate (as your first choice), your vote transfers to the next (live) candidate on your list; repeat until one candidate remains • Similar runoffs can be defined for rules other than plurality

Pairwise elections

select pairwise comparison from complete list two votes prefer Obama to McCain

> > >

two votes prefer Obama to Nader

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two votes prefer Nader to McCain

> > > > >

2

Pairwise elections

select pairwise comparison from complete list two votes prefer Obama to McCain

>

2 two votes prefer Obama to Nader 2

>

May care about numbers of voters or only winner two votes prefer Nader to McCain

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Pairwise elimination

• Candidates given a schedule of pairwise competitions • Loser is eliminated at each stage.

• Winner goes on to compete at next round • Like a single elimination athletic event (but no parallel competitions) • Not every pair is considered

Sensitivity to agenda setter: order of elimination matters 35 agents a > c > b 33 agents b > a > c 32 agents c > b > a Who is the winner in the following pairings?

(a,b) c (a,c) b (b,c) a

Condorcet

•

Condorcet winner

of an election: – A candidate who wins every pairwise election • Condorcet methods are named for the eighteenth-century French mathematician and philosopher Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet (took their title from the town of Condorcet in Dauphine) Wikipedia Oct 2010 • Clearly, there may not be a Condorcet winner • Condorcet condition : if there is a Condorcet winner, he must be the winner.

Condorcet cycles

two votes prefer McCain to Obama

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two votes prefer Obama to Nader

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two votes prefer Nader to McCain

> > > ?

2

Condorcet cycles

two votes prefer McCain to Obama

>

2 2 two votes prefer Obama to Nader

>

two votes prefer Nader to McCain

> ?

Condorcet cycles 500 voters

251 votes prefer McCain to Obama

>

330 400 251 400 votes prefer Obama to Nader

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330 votes prefer Nader to McCain

>

Number of voters who are unhappy with ranking or number of winning arcs who are unhappy

?

• Preference profile : a tuple giving a preference ordering for each agent • #(o>o’) is the number of agents who prefer o to o’ • Smith set: is the smallest set S  O having the property that  o’  S, #(o> o’)  #(o’>o) In other words, every outcome in the Smith set is preferred (by at least half of the agents) to every outcome outside the set.

What is the Smith set?

35 agents a > c> b >d 33 agents b > a > d > c 32 agents c >d > b > a 68(32) a 65 (35) 68(32) d 68 (32) 67(33) c b 67(33) What is the relationship between Smith set and Condorcet?

Cumulative voting : Each voter is given k votes which can be cast arbitrarily (voting for any set of candidates he wants). The candidate with the most votes is selected.

Approval voting : Each voter can cast a single vote for as many of the candidates as he wishes; the candidate with the most votes is selected.

Voting rule based on pairwise elections

Copeland : candidate gets two points for each pairwise election it wins, one point for each pairwise election it ties

Second order Copeland

: sum of Copeland scores of alternatives you defeat. (once used by NFL as tie-breaker) a 65 (35) 68(32) 67(33) 35 agents c > a> b >d 33 agents b > a > d > c 32 agents c >d > b > a d 68 (32) 67(33) c b 67(33) What is Copeland Score?

What is second order Copeland Score?

Another voting rule based on pairwise elections

• Maximin (aka. Simpson ): candidate whose worst pairwise result is the best among candidates – wins. So if there are four candidates and 10 voters and between pairs (me,opponent): (9,1), (10,0), (8,2), and (5,5). If others had a worse pairwise vote than (5,5), I would be the winner.

• Slater : create an overall ranking of the candidates that is inconsistent with as few pairwise elections as possible – NP-hard! Consider all orders and count inconsistencies.

• An instance of the Slater problem can be represented by a “pairwise election” graph whose vertices are the candidates, and which has a directed edge from

a to b if and only if a defeats b in their pairwise

election. The goal, then, is to minimize the number of edges that must be flipped in order to make the graph acyclic.

• Cup/pairwise elimination : pair candidates, losers of pairwise elections drop out, repeat

Slater on pairwise election graphs

• Final ranking = acyclic tournament graph • Slater ranking seeks to minimize the number of inverted edges

pairwise election graph Slater ranking

a b a b d What about b >d>c>a?

c d

(a > b > d > c)

c

Even more voting rules…

• Kemeny : create an overall ranking of the candidates that has as few

disagreements

as possible (where a disagreement is with a vote on a pair of candidates). For each pair of voters (X,Y) count how many times X is preferred to Y. Test all possible order-of-preference sequences, calculate a sequence score for each sequence, and compare the scores. Each sequence score equals the sum of the pairwise counts that are “honored by” the sequence (a is preferred to b and a precedes b in the sequence). The sequence with the highest score is identified as the overall ranking – NP-hard!

– Similar to Slater – but looks at actual numbers of votes not just result of pairing.

• Bucklin : start with k=1 and increase k gradually until some candidate is among the top k candidates in more than half the votes; that candidate wins

Kemeny on pairwise election graphs

• Final ranking = acyclic tournament graph – Edge (a, b) means a ranked above b – Acyclic = no cycles, tournament = edge between every pair • Kemeny ranking seeks to minimize the total weight of the inverted edges

pairwise election graph Kemeny ranking

2 2 2

a

2 10

b

4 2

a b d

4

c d

(b > d > c > a)

c

So…

• SO… how do we choose a rule from all of these rules?

• How do we know that there does not exist another, “perfect” rule?

• Let us look at some criteria that we would like our voting rule to satisfy

Condorcet criterion

• A candidate is the Condorcet winner if it wins all of its pairwise elections • Does not always exist… • … but the Condorcet criterion says that if it does exist, it should win • Many rules do not satisfy this simple criterion • Consider plurality voting: – b > a > c > d – c > a > b > d – d > a > b > c • a is the Condorcet winner, but it does not win under plurality. Explain

Majority criterion

• If a candidate is ranked first by majority of votes that candidate should win – Relationship to Condorcet criterion?

a > b > c > d > e e > a > b > c > d c > b > d > a > e • Some rules do not even satisfy this • E.g. Borda: – a > b > c > d > e – a > b > c > d > e – c > b > d > e > a • a is the majority winner, but it does not win under Borda (b wins under Borda, right?)

Monotonicity criteria

• Informally, monotonicity means that “ranking a candidate higher should help that candidate,” but there are multiple nonequivalent definitions

Monotonicity criteria

• A weak monotonicity requirement: if – candidate w wins given the current votes, – we then improve the position of w in some of the votes and leave everything else the same, then w should still win.

• E.g., Single Transferable Voting does not satisfy this: – 7 votes b > c > a – 7 votes a > b > c – 6 votes c > a > b • c drops out first (lowest plurality), its votes transfer to a (next candidate), a wins • **But if 2 votes b > c > a change to a > b > c (we improve a’s ranking), b drops out first, its 5 votes transfer to c, and c wins – 5 votes b > c > a – 9 votes a > b > c – 6 votes c > a > b

Monotonicity criteria…

• A strong monotonicity requirement: if – candidate w wins for the current votes, – we then change the votes in such a way that for each vote, if candidate c was ranked below w originally, c is still ranked below w in the new vote then w should still win.

• Note the other candidates can jump around in the vote, as long as they don’t jump ahead of w • None of our rules satisfy this

Independence of irrelevant alternatives

• Independence of irrelevant alternatives criterion: if – the rule ranks a above b for the current votes, – we then change the votes but do not change which is ahead between a and b in each vote then a should still be ranked ahead of b . (The other votes are irrelevant to the relationship between a and b .) • None of our rules satisfy this

Arrow’s impossibility theorem [1951]

• Suppose there are at least 3 candidates • Then there exists no rule that is simultaneously: – Pareto efficient the rule ranks a (if all votes rank a above b ), above b , then Explain use of term – nondictatorial (there does not exist a voter such that the rule simply always copies that voter’s ranking), and – independent of irrelevant alternatives

Weak Pareto efficient

• if there exist a pair of outcomes o1 and o2 such that  i o1 > i o2 then C([>])  o2 In other words, we cannot select any outcome that is dominated by another alternative for all agents Why is it called weak?

Muller-Satterthwaite impossibility theorem

[1977] • Suppose there are at least 3 candidates • Then there exists no rule that simultaneously: – satisfies unanimity should win), (if all votes rank a first, then a – is nondictatorial (there does not exist a voter such that the rule simply always selects that voter’s first candidate as the winner), and – is monotone (in the strong sense).

• The Shoham Leyton-Brown text has proofs of the impossibility theorems. • We won’t go over them – but interested readers should take a look.

Manipulability

• Sometimes, a voter is better off revealing her preferences insincerely, aka. manipulating • E.g. plurality – Suppose a voter prefers a > b > c – Also suppose she knows that the other votes are • 2 voters b > c > a • 2 voters c > a > b – Voting truthfully will lead to a tie between b and c – She would be better off voting e.g. b > a > c, guaranteeing b wins • All our rules are (sometimes) manipulatable

Single-peaked

preferences

• Suppose candidates are ordered on a line • Every voter prefers candidates that are closer to her most preferred candidate • Let every voter report only her most preferred candidate (“ peak ”) • Choose the median voter’s peak as the winner – This will also be the Condorcet winner • Nonmanipulable!

Impossibility results do not necessarily hold when the space of preferences is restricted.

Why would you guess this is true?

v 5 v 4 a 1 v 2 a 2 a 3 v 1 a 4 v 3 a 5

Some computational issues in social choice

• Sometimes computing the winner/aggregate ranking is hard – E.g. for Kemeny and Slater rules this is NP-hard • For some rules (e.g., STV), computing a successful manipulation is NP-hard – Manipulation being hard is a

good

stronger than NP-hardness thing … But would like something – Also: work on the complexity of controlling the outcome of an election by influencing the list of candidates/schedule of the Cup rule/etc.

• Preference elicitation: – We may not want to force each voter to rank

all

candidates; – Rather, we can selectively query voters for parts of their ranking, according to some algorithm, to obtain a good aggregate outcome • Combinatorial alternative spaces: – Suppose there are multiple interrelated issues that each need a decision – Exponentially sized alternative spaces • Different models such as ranking webpages each other by linking) (pages “vote” on