Chapter 12
Making Group Decisions
Mechanism design: study
solution concepts
7-1
Voting
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
2
2
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
3


Social Welfare Function – gives a complete
ranking
Social Choice Function – gives just the winner
4
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)
5
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)
6
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)
o Output: elements of O sorted according the input gives the social preference relation >* of the agent
group
7
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.
8
Pairwise elections
EXTRACT pairwise comparison results from complete list
two votes prefer Obama to McCain
>
>
>
two votes prefer Obama to Nader
>
>
>
two votes prefer Nader to McCain
>
>
>
>
>
9
Pairwise elections two votes prefer Obama to McCain
>
2
2
2
two votes prefer Obama to Nader
>
Majority Graph as arcs
represent the majority
opinion
Edges may be annotated with number
preferring or not
two votes prefer Nader to McCain
>
>
>
10
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
11
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)
12
One voter ranks c > d > b > a
One voter ranks a > c > d > b
One voter ranks b > a > c > d
Notice, just rotates preferences – so no consensus.
Look at sequential majority election:
winner (c, (winner (a, winner(b,d)))=a
winner (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 (ab) 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 .
Goal of a “good” voting
mechanism

Condorcet condition: if there is a Condorcet
winner, he must be the winner.
20
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.
21
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
22
Slater Ranking – Example 2

a>b>c>d has
one arc which
is unhappy (d>a)
23
Slater Ranking

ordering a>b>d>c has arcs which are
unhappy






cd
da
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?
24
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?)
25
25
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 < d
a=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 <c
a=15,b=14, c=13
When loser is removed, next loser becomes winner!
26
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.
27
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.
28
c>a>b
c is condorcet winner but does
• b > c >a
Not win in Borda.
• c >a>b
Is tied with a
• a > c> b
• a>c>b
• c > a >b
• a <b <c
a=16,b=12, c=16
•
When loser is removed, next loser becomes winner!
29
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
30
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!!!
31
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 that
o >i o' implies o >* o' for all preferences of the other agents
32
32
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 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
33
Arrow's impossibility theorem
o No social choice rule satisfies all of the six
conditions
o Maybe all aren’t really needed.
o Must relax desired attributes
o May not require >* to always be defined
o We may not require that >* is asymmetic
and transitive
34
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
35
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
36
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
37
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
38
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
39
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

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.
40
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)
41
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, GibbardSatterthwaite says there are no non42
manipulable voting protocols
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>c
How should the last voter state preferences?
43
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
44
SUPPOSE we had 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
• Is this manipulable? Why or why not?
v5
v4
a1
v2
a2
a3
v1
a4
v3
a5
SUPPOSE we had 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?
v5
v4
a1
v2
a2
a3
v1
a4
v3
a5
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).
47
Voting rule based on pairwise elections
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.
a
(once used by NFL as tie-breaker)
65 (35)
68(32)
35 agents c > a> b >d
33 agents b > a > d > c
32 agents c >d > b > a
What is Copeland Score?
What is second order
Copeland Score?
67(33)
b
67(33)
68 (32)
d
67(33)
c
48
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”
49
Other Voting Mechanisms
50
Nanson (Borda variant)

Candidate with the lowest Borda score is
eliminated, then we re-compute Borda counts
and continue.
51
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
52
What is the Smith set? the smallest nonempty set
such that every member of the set pairwise defeats every member
outside the set.
a
35 agents a > c> b >d
33 agents b > a > d > c
32 agents c >d > b > a
68(32)
65 (35)
68(32)
b
67(33)
68 (32)
d
67(33)
c
What is the relationship
between Smith set and
Condorcet?
53
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.
54
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.
55
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
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.
57
Kemeny ranking
What is Kemeny ranking for this majority graph?
35 agents c > a> b >d
33 agents b > a > d > c
32 agents c >d > b > a
a
68(32)
65 (35)
67(33)
b
67(33)
68 (32)
d
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
67(33)
c
58
Kemeny ranking
What is Kemeny ranking for this majority graph? Mark with
margin of victory.
a
68(32)
35 agents c > a> b >d
33 agents b > a > d > c
32 agents c >d > b > a
c>b>d>a has what score?
a>c>b>d has what score?
65 (35)
67(33)
b
67(33)
68(32)
d
67(33)
c
59
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
pairwise election graph
a
15(13)
d
15(13)
b
Kemeny ranking
a
b
d
c
20(8)
20 (8)
28(0)
22(6)
c
(b > d > c > a)
60