Chapter 12

Making Group Decisions

Mechanism design: study solution concepts

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We vote in awarding scholarships, teacher of the year, person to hire.

Rank feasible social outcomes based on agents' individual ranking of those outcomes

A - set of n agents Ω- set of m feasible outcomes Each agent i has a preference relation >i : Ω x Ω, asymmetric

and transitive

Asymmetric: aRb can’t have bRa

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Voting

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Social Welfare and Social Choice Functions Instead of being competitive, we are looking at

a means of making a group decision. Set of outcomes or candidates: Ω = {w1, … wm} Participants rank the outcomes. The

preference over Ω is noted (Ω) Common scenario is voting for a candidate If |Ω| = 2, we have a pairwise election If |Ω| > 2, we have a general election

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Social Welfare Function – gives a complete ranking

Social Choice Function – gives just the winner

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Example voting rules• Each voter gives a vector of ranked choices (best to worst).• Scoring rules are defined by a vector (a1, a2, …, am); being

ranked ith in a vote gives the candidate ai points. Candidate with most points wins.

So, how would you describe the voting indicated by the following scoring rule vectors:

• (1, 0, 0, …, 0)• (1, 1, …, 1, 0) • (m, m-1, …, 1)

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Example voting rules• Scoring rules are defined by a vector (a1, a2, …, am); being

ranked ith in a vote gives the candidate ai 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, m-1, …, 1)

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Social Welfare function

f: (Ω) x (Ω) x (Ω) … x (Ω) (Ω)

a mapping from n different rankings to one which represents the ranking of the group

Input: the agent preference relations (>1, …, >n) Output: elements of O sorted according the input -

gives the social preference relation >* of the agent group

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May not need entire ranking Entire ranking may be expensive to identify.

Examples? Plurality (largest number of votes): selecting a

single candidate rather than needing a complete ranking.

Each person submits their first place candidate

Can select a candidate when another outcome would be preferred by a majority.

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Pairwise electionsEXTRACT pairwise comparison results from complete list

> >

> >

>

> >

two votes prefer Obama to McCain

>

two votes prefer Obama to Nader

>

two votes prefer Nader to McCain

> >

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Pairwise elections -

>

two votes prefer Obama to McCain

>

two votes prefer Obama to Nader

>

two votes prefer Nader to McCain

> >

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2 2

Majority Graph as arcs represent the majority opinion

Edges may be annotated with numberpreferring or not

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Pairwise elimination aka sequential majority

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: (n-1)

competitions

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Sensitivity to agenda setter: order of elimination matters35 agents a > c > b33 agents b > a > c32 agents c > b > aWho is the winner in the following pairings?((a,b) c)((a,c) b)((b,c) a)

One voter ranks c > d > b > aOne voter ranks a > c > d > bOne voter ranks b > a > c > dNotice, just rotates preferences – so no consensus.

Look at sequential majority election:winner (c, (winner (a, winner(b,d)))=awinner (d, (winner (b, winner(c,a)))=d

winner (d, (winner (c, winner(a,b)))=c

winner (b, (winner (d, winner(c,a)))=b!

Majority Graphs Nodes correspond to outcomes. Edge between (ab) if a majority of voters

would prefer a over b. (a defeats b) Properties

Complete graph (edge between each pair) asymmetric (if (a,b) can’t have (b,a) irreflexive ( can’t have (a,a))

Graphs with this property are called a tournament

Figure 12.3 Three voters:

a >b>c b>c>a c>a>b

Can fix an election so any candidate will win

From the majority graph, we can decide the agenda (ordering of pairwise comparison). What should order be if you want b to win?

Terms

Possible winner: there exists some agenda in which the candidate would be the winner

Condorcet (pronounced Condor-say)Winner: the candidate wins no matter the agenda. From majority graph, how can you identify a Condorcet winner?

Condorcet winner may not exist (see previous slide)

a is the condorcet winner

Thought question

Previous discussion on picking agenda assumes we have total information – we know exactly who will win given a pairing.

What if we only knew probabilities of winning? Ongoing research. What would you do to give your candidate

better chances of winning?

Thought question

If want to give wi the best chance of winning, order the voters from most likely to win against wi to least likely to win against wi .

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Goal of a “good” voting mechanism

Condorcet condition: if there is a Condorcet winner, he must be the winner.

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Slater Ranking One way of thinking about the problem of

finding a social choice ranking is to find an ordering (no cycles) which has the fewest disagreements with the majority graph.

How many edges of the majority graph would have to be flipped to agree with the order chosen?

Note here that we are worrying about finding a total order to rank the candidates (rather than just an agenda to cause our choice to win). Both are “orderings” – so don’t get confused.

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Slater Ranking – may be easy Our social

welfare ordering might be a>c>b>d

Ideally - Any candidate which appears before another would beat the candidate in a 1-1 election.

Example 1

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Slater Ranking – Example 2

a>b>c>d has one arc which is unhappy (d->a)

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Slater Ranking ordering a>b>d>c has arcs which are

unhappy cd da

Score for a slater ranking is number of unhappy arcs.

Note not just finding a directed path containing all nodes as that would only look at some arcs.

Smallest number of arc conflicts is slater winner

Great idea – NP hard to compute. Why?

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Another Social Welfare Method

Borda protocol (used if binary protocol is too slow) - assigns an alternative |O| points for the highest preference, |O|-1 points for the second, and so on

The counts are summed across the voters and the alternative with the highest count becomes the social choice

Winner turns loser and loser turns winner if the lowest ranked alternative is removed (does this surprise you?)

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Borda Paradox – remove loser, winner changes(notice, c is always ahead of removed item)

a > b > c >d b > c > d >a c > d > a > b a > b > c > d b > c > d> a c >d > a >b a <b <c < da=18, b=19, c=20,

d=13

a > b > c b > c >a c > a > b a > b > c b > c > a c > a >b a <b <ca=15,b=14, c=13

When loser is removed, next loser becomes winner!

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Strategic (insincere) voters

Suppose your choice will likely come in second place. If you rank the first choice of rest of group very low, you may lower that choice enough so yours is first.

True story. Dean’s selection. Each committee member told they had 5 points to award and could spread out any way among the candidates. The recipient of the most points wins. I put all my points on one candidate. Most split their points. I swung the vote! What was my gamble?

Want to get the results as if truthful voting were done.

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Desirable properties of voting procedure Pareto: if every voter ranks a before b, a should precede b in the ranking. - Satisfied in Borda and plurality, but not be sequential majority.

Condorcet winner condition: A condorcet winner should be first. Satisfied only by sequential majority.

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• c > a > b• b > c >a• c > a > b• a > c> b• a > c > b• c > a >b• a <b <ca=16,b=12, c=16

When loser is removed, next loser becomes winner!

c is condorcet winner but doesNot win in Borda.

Is tied with a

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Desirable properties of voting procedure Pareto: if every voter ranks a before b, a should precede b in the ranking. - Satisfied in Borda and plurality, but not be sequential majority.

Condorcet winner condition: A condorcet winner (if it exists) should be first. Satisfied only by sequential majority.

Independence of irrelevant alternatives: if a > b and then you decide to change your rankings of OTHER candidates, a>b shouldn’t change. Satisfied by none of plurality, borda, or sequential

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Bad voting system has good properties

Dictatorship: For some voter i, the social welfare function just uses his preference list. Interesting that this bad voting system, does satisfy both pareto efficiency and independence of irrelevant alternatives.

In fact, the only voting procedures satisfying pareto efficienty and independence of irrelevant alternatives are dictatorships!!!

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Desirable properties of the social choice rule: A social preference ordering >* should exist for all possible inputs

(Note, I am using >* to mean “is preferred to.) >* should be defined for every pair (o, o')O >* should be asymmetric and transitive over O The outcomes should be Pareto efficient: if i A, o >i o' then o >* o‘ (not mis-order if all agree) The scheme should be independent of irrelevant alternatives (if all

agree on relative ranking of two, should retain ranking in social choice):

No agent should be a dictator in the sense thato >i o' implies o >* o' for all preferences of the other agents

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Independence of irrelevant alternatives (a little more general than just saying throwing out the lowest doesn’t change things)• 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 votethen 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

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Arrow's impossibility theorem No social choice rule satisfies all of the six

conditions Maybe all aren’t really needed. Must relax desired attributes

May not require >* to always be defined We may not require that >* is asymmetic

and transitive

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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

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Majority criterion• If a candidate is ranked first by majority of votes that

candidate should win– Relationship to Condorcet criterion?

a > b > c > d > ee > a > b > c > dc > 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?)

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Monotonicity criteria• Informally, monotonicity means that “ranking a candidate

higher should help that candidate,” but there are multiple nonequivalent definitions

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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

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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 winner determination methods satisfy this

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Weak Pareto efficient if there exist a pair of outcomes o1 and o2

such that i o1 >i o2 then C([>]) o2In other words, we cannot select any outcome

that is dominated by another alternative for all agents

Strong Pareto-efficiency:  For all alternatives, for instance x, x must not be selected if there exists another alternative, say y, such that no voters rank x  over y and at least one voter ranks y over x.

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Truthful voters vote for the candidate they think is best.Why would you vote for something you

didn’t want? (run off election – want to pick competition) (more than two candidates, figure your candidate doesn’t have a chance)

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Strategic manipulation If I lie about my ranking, will I prefer the choice

made by the social choice function? Gibbard-Satterthwaite theorem – conditions

under which someone can manipulate the results.

Is any voting system non-manipulable? Yes – dictatorship.

If there are at least three outcomes and we want to satisfy Pareto condition, Gibbard-Satterthwaite says there are no non-manipulable voting protocols

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Manipulability• Sometimes, a voter is better off revealing her preferences

insincerely, aka. manipulating• E.g. plurality voting

• b > c > a• b > c > a• c > a > b• c > a > b• a > b > cHow should the last voter state preferences?

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Manipulability– 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) manipulable

SUPPOSE we had Single-peaked preferences• Suppose candidates are ordered on a line

a1 a2 a3 a4 a5

• Every voter prefers candidates that are closer to her most preferred candidate

• Let every voter report only her most preferred candidate (“peak”)

v1v2 v3v4

v5

• Choose the median voter’s peak as the winner– This will also be the Condorcet winner

• Is this manipulable? Why or why not?

SUPPOSE we had Single-peaked preferences• Suppose candidates are ordered on a line

a1 a2 a3 a4 a5

• Every voter prefers candidates that are closer to her most preferred candidate

• Let every voter report only her most preferred candidate (“peak”)

v1v2 v3v4

v5

• 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?

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So what can we do to control manipulation? There is a voting method which is pareto

efficient and harder to manipulate. It is the second order Copeland. So it is possible, in principle, but NP-complete.

However, NP-complete is a worse case result (so it may not be difficult in some cases).

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Copeland: candidate gets one point for each pairwise election it wins, a half 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)35 agents c > a> b >d33 agents b > a > d > c32 agents c >d > b > a

What is Copeland Score?What is second order

Copeland Score?

a

d

b

c

65 (35)

67(33)

68(32)

67(33)

67(33)

68 (32)

Voting rule based on pairwise elections

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Manipulation Sometimes being able to compute

WHEN/HOW to tell a lie is computationally intensive. (A good result of complexity.) That may help or it may just favor the agents with more processing power.

Sometimes there exists another mechanism with the same good properties (as our original mechanism) such that truthfully reporting preferences is rational. This is called the “revelation principle”

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Other Voting Mechanisms

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Nanson (Borda variant) Candidate with the lowest Borda score is

eliminated, then we re-compute Borda counts and continue.

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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. • If no one new meets the quota, the candidate with the fewest votes is eliminated and

that candidate's votes are transferred.• If you are filling multiple seats, if a candidate has more than the quota needed, the

surplus votes are transferred to the next preferred. (in proportion to the second choices of those voting for winner)

• Similar runoffs can be defined for rules other than plurality

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What is the Smith set? the smallest nonempty set such that every member of the set pairwise defeats every member outside the set.35 agents a > c> b >d33 agents b > a > d > c32 agents c >d > b > a

What is the relationship between Smith set and Condorcet?

a

d

b

c

65 (35)

67(33)

68(32)

67(33)

68(32)

68 (32)

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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. The number of votes per voter can be dependent on share of stock owned, but is often equal.

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.

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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.

Even more voting rules…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

For the system shown here, who is the winner and what is k?

a > b > c >d b > c > d >a c > d > a > b a > b > c > d b > c > d> a c > d > a >b a < b <c < d

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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. Margin of victory. 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.

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What is Kemeny ranking for this majority graph?

35 agents c > a> b >d33 agents b > a > d > c32 agents c >d > b > a

Problem is stated as maximizing value of happy edges or minimizing value of unhappy edges.

Score of ranking shown:67+67+67+32+65+32

a

d

b

c

65 (35)

67(33)

68(32)

67(33)

67(33)

68 (32)

Kemeny ranking

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What is Kemeny ranking for this majority graph? Mark with margin of victory.

35 agents c > a> b >d33 agents b > a > d > c32 agents c >d > b > ac>b>d>a has what score?a>c>b>d has what score?

a

d

b

c

65 (35)

67(33)

68(32)

67(33)

67(33)

68(32)

Kemeny ranking

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Kemeny on pairwise election graphs Final ranking = acyclic tournament graph

Edge (a, b) means a ranked above b Edge (a,b) is weighted by number of voters who prefer a to b minus

number who prefer b to a. Acyclic = no cycles, tournament = edge between every pair

Kemeny ranking seeks to minimize the total weight of the inverted edges

a b

d c

15(13)

15(13)28(0)

22(6)

20 (8)20(8)

pairwise election graph Kemeny ranking

a b

d c(b > d > c > a)