1. Making Group Decisions Mechanism design: study solution concepts
Chapter 12
Making Group Decisions
Mechanism design: study solution concepts

2. 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

3. 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

4. Social Welfare Function – gives a complete ranking
Social Choice Function – gives just the winner

5. 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)

6. 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)

7. 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

8. 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.

9. 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
>
>
>
>
>

10. Pairwise elections - > >
two votes prefer Obama to McCain
> 2 2
two votes prefer Obama to Nader 2
>
Majority Graph as arcs represent the majority opinion
Edges may be annotated with number
preferring or not
two votes prefer Nader to McCain
>
>
>

11. 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

12. 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)
The last one, in every case.

13. 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 !

14. 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

15. 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?

16. 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)
check to see there is an edge from the winner to every other node.

17. a is the condorcet winner

18. 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?

19. 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 .

20. Goal of a “good” voting mechanism
Condorcet condition: if there is a Condorcet winner, he must be the winner.

21. 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.

22. 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

23. Slater Ranking – Example 2
a>b>c>d has one arc which is unhappy (d->a)

24. 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?

25. 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

26. a=15,b=14, c=13 a > b > c a > b > c >d b > c >a
Borda Paradox – remove loser, winner changes (notice, c is always ahead of removed item)
a > b > c
b > c >a
c > a > b
b > c > a
c > a >b
a <b <c
a=15,b=14, c=13
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
When loser is removed, next loser becomes winner!

27. 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.

28. 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.

29. c > a > b b > c >a c > a > b a > c> b
a <b <c
a=16,b=12, c=16
c is condorcet winner but does
Not win in Borda.
Is tied with a
When loser is removed, next loser becomes winner!

30. 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

31. 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!!!

32. 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

33. 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

34. No social choice rule satisfies all of the six conditions
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

35. 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

36. 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:
c > b > d > e > a
a is the majority winner, but it does not win under Borda (b wins under Borda, right?)

37. Monotonicity criteria
Informally, monotonicity means that “ranking a candidate higher should help that candidate,” but there are multiple nonequivalent definitions

38. 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

39. 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

40. 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.

41. 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)

42. 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

43. Manipulability b > c > a c > a > b a > b > c
Sometimes, a voter is better off revealing her preferences insincerely, aka. manipulating
E.g. plurality voting
b > c > a
c > a > b
a > b > c
How should the last voter state preferences?

44. Manipulability All our rules are (sometimes) manipulable
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

45. 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 v2 v1 v3 a1 a2 a3 a4 a5

46. 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 v2 v1 v3 a1 a2 a3 a4 a5

47. 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).

48. 35 agents c > a> b >d 33 agents b > a > d > c
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. (once used by NFL as tie-breaker) a
65 (35)
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? b
67(33)
68(32)
67(33)
68 (32) d c
67(33)

49. 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”

50. Other Voting Mechanisms

51. Nanson (Borda variant)
Candidate with the lowest Borda score is eliminated, then we re-compute Borda counts and continue.

52. 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

53. 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 >d
33 agents b > a > d > c
32 agents c >d > b > a
What is the relationship between Smith set and Condorcet? a
65 (35) b
68(32)
68(32)
67(33)
68 (32) d c
67(33)
abc is the smith set. If the smith set has one member, it is a condorcet winner.

54. 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.

55. 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.

56. 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

57. 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.

58. 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
Problem is stated as maximizing value of happy edges or minimizing value of unhappy edges.
Score of ranking shown: a
65 (35) b
67(33)
68(32)
67(33)
68 (32) d c
67(33)

59. 35 agents c > a> b >d 33 agents b > a > d > c
Kemeny ranking
What is Kemeny ranking for this majority graph? Mark with margin of victory. a
65 (35)
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? b
67(33)
68(32)
67(33)
68(32) d c
67(33)

60. 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
Kemeny ranking a
15(13) b a b
20(8)
15(13)
20 (8)
28(0) d c d c
22(6)
(b > d > c > a)