Approval Voting for Committees: Threshold Approaches'

1
Approval Voting for Committees Threshold
Approaches. Peter Fishburn Saa Pekec
2
Approval Voting for Committees Threshold
Approaches. Saa Pekec Decision Sciences The
Fuqua School of Business Duke University pekec_at_du
ke.edu http//faculty.fuqua.duke.edu/pekec
3
(No Transcript)
4

• (S)ELECTING A COMMITTEE
• ? choosing a subset S
• from the set of m available alternatives
• ? choosing a feasible (admissible) subset S
• social choice
• voting
• multi-criteria decision-making
• consumer choice (?)

5

• SOCIAL CHOICE
• There are n individuals, each having preferences
  over m alternatives.
• How to aggregate preferences into a consensus
  preference structure?
• Arrows Impossibility Theorem
• - Independence of Irrelevant Alternatives (IIA)
• - Inherent multidimensionality (single-peaked
  prefs.)
• Choosing a subset?

6

• VOTING
• There are n voters, each having preferences over
  m candidates.
• How to aggregate preferences and determine the
  winner?
• Gibbard-Satterhwaite Theorem
• - IIA implying Arrows result
• or
• - manipulability
• Choosing a subset?

7

• MULTI-CRITERIA DECISION-MAKING
• There are n criteria, each defining a preference
  structure over m alternatives
• How to aggregate preferences and determine the
  consensus preference structure over
  alternatives?
• Choosing a subset?

8

• CONSUMER CHOICE?
• Behavioral aspects dominate
• (Normative approach multicriteria
  decision-making)
• BUT
• Automated consumer choice suggestions
• - search engine page rank
• - suggested product (e.g., credit cards,
  computers)
• .
• Choosing a subset?

9

• CONSUMER CHOICE?
• buying a product bundle
• - related products (home theater system
  components)
• - subset of extra options (cars)
• - features of a highly customized commodity-like
  products (PCs, smartphones, software,)
• multiple criteria (functionality, looks, safety,
  price, )

10

• CHOSING A SINGLE ALTERNATIVE
• Information requirement on voters preferences
• SWF ? rankings
• plurality ? top choice
• scoring rules ? constrained cardinal utility
  (IIA???)
• approval voting ? subset choice

11
CHOSING A SINGLE ALTERNATIVE
12
CHOSING A SINGLE ALTERNATIVE
13
CHOSING A SINGLE ALTERNATIVE BORDA
14
CHOSING A SINGLE ALTERNATIVE BORDA
15
CHOSING A SINGLE ALTERNATIVE
16
CHOSING A SINGLE ALTERNATIVE PLURALITY
17
CHOSING A SINGLE ALTERNATIVE PLURALITY
18
CHOSING A SINGLE ALTERNATIVE PLURALITY
19
CHOSING A SINGLE ALTERNATIVE
20
CHOSING A SINGLE ALTERNATIVE APPROVAL
21
CHOSING A SINGLE ALTERNATIVE APPROVAL
22
CHOSING A SINGLE ALTERNATIVE APPROVAL
23
CHOSING A SINGLE ALTERNATIVE APPROVAL
24
APPROVAL VOTE PROFILE
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
25
APPROVAL VOTE PROFILE
V( 37, 1268, 2 , 47 , 45 , 13 , 12 , 48
, 46 )
26

• CHOSING A SUBSET
• Information requirement on voters preferences
• SWF ? rankings on all 2m subsets
• plurality ? top choice among all 2m subsets
• scoring rules ? constrained card. utility on all
  2m subsets
• approval voting ? subset choice on all 2m
  subsets

27

• CHOSING A SUBSET
• using consensus ranking of alternatives
• but
• for all voters 1gt2gt3 or 2gt1gt3
• AND
• 13gt23gt12 or 23gt13gt12
• divide and conquer
• break into several separate singleton choices
• proportional representation
• IGNORING INTERDEPENDANCIES
• (substitutability and complementarity)

28

• CHOSING A SUBSET
• Barbera et al. (ECA91) impossibility
• A manageable scheme that accounts for
  interdependencies?
• Proposal Approval Voting with modified subset
  count.
• Threshold Approach
• - define t(S) for every feasible S
• - ACt(S) of voters i such that Vi ? S ? t(S)

29

• AV THRESHOLD APPROACH
• Define t(S) for every feasible S
• ACt(S) of voters i such that ViS Vi ? S ?
  t(S)
• Threshold functions (TF)
• t(S)1 (favors small committees)
• t(S) S/2 (majority)
• t(S) (S1)/2 (strict majority)
• t(S) S (favors large committees)
• .

30
APPROVAL TOP 3-SET
S124 gets 10 votes total.
31
CHOOSING 3-SET, t(S) ? 1
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
32
CHOOSING 3-SET, t(S) ? 1
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
S234 is the only 3-set approved by all voters
33
CHOOSING 3-SET, t(S) ? 2
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
34
CHOOSING 3-SET, t(S) ? 2
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
35
CHOOSING 3-SET, t(S) ? 2
37, 1268, 2 , 47 , 45 , 13 , 12 ,
48 , 46
S123 is the only 3-set approved by at least
three voters
36

• COMPLEXITY of AVCT
• If X the set of all feasible subsets, is part
  of the input then
• computing AVCT winner is polynomial in mnX
• Theorem.
• If X is predetermined (not part of the input),
  then computing AVCT winner is NP-complete at
  best.

37

• COMPLEXITY of AVCT
• If X the set of all feasible subsets, is part
  of the input then
• computing AVCT winner is polynomial in mnX
• Theorem.
• If X is predetermined (not part of the input),
  then computing AVCT winner is NP-complete at
  best.
• Proof choosing a k-set, t?1. Suppose Vi2 for
  all i.
• Note alternatives vertices of a graph
• Vi edges of a graph
• k-set approved by all voters vertex cover of
  size k
• Vertex Cover is a fundamental NP-complete
  problem.

38

• COMPLEXITY contd
• not as problematic as it seems.
• Theorem. (Garey-Johnson)
• If X is predetermined (not part of the input),
  then computing
• maxS inX sumi inS score(i)
• is NP-complete.

39
CHOSING A SINGLE ALTERNATIVE BORDA
40
CHOSING A SINGLE ALTERNATIVE PLURALITY
41
CHOSING A SINGLE ALTERNATIVE APPROVAL
42

• LARGER IS NOT BETTER
• Example m8, n12, strict majority TF
  t(S)(S1)/2
• V (123,15,1578,16,278,23,24,34,347,46,567,568)
• 1-set (AC)
• 1,2,3,4,5,6,7 all approved by 4 voters (8 is
  approved by 3 voters)

43

• LARGER IS NOT BETTER
• Example m8, n12, strict majority TF
  t(S)(S1)/2
• V (123,15,1578,16,278,23,24,34,347,46,567,568)
• 1-set (AC)
• 1,2,3,4,5,6,7 all approved by 4 voters (8 is
  approved by 3 voters)
• 2-set
• 15,23,34,56,57,58,78 all approved by 2 voters

44

• LARGER IS NOT BETTER
• Example m8, n12, strict majority TF
  t(S)(S1)/2
• V (123,15,1578,16,278,23,24,34,347,46,567,568)
• 1-set (AC)
• 1,2,3,4,5,6,7 all approved by 4 voters (8 is
  approved by 3 voters)
• 2-set
• 15,23,34,56,57,58,78 all approved by 2 voters
• 3-set 234 approved by 5 voters
• 4-set 5678 approved by 3 voters
• 5-set 15678 approved by 4 voters

45

• LARGER IS NOT BETTER
• Example m8, n12, strict majority TF
  t(S)(S1)/2
• V (123,15,1578,16,278,23,24,34,347,46,567,568)
• 1-set (AC)
• 1,2,3,4,5,6,7 all approved by 4 voters (8 is
  approved by 3 voters)
• 2-set
• 15,23,34,56,57,58,78 all approved by 2 voters
• 3-set 234 approved by 5 voters
• 4-set 5678 approved by 3 voters
• 5-set 15678 approved by 4 voters

46

• TOP INDIVIDUAL NOT IN A TOP TEAM
• Example m5, n6, majority TF t(S)S/2
• V (123,124,135,145,25,34)

47

• TOP INDIVIDUAL NOT IN A TOP TEAM
• Example m5, n6, majority TF t(S)S/2
• V (123,124,135,145,25,34)
• Top individual
• 1 approved by 4 voters (all other alternatives
  approved by 3 voters)

48

• TOP INDIVIDUAL NOT IN A TOP TEAM
• Example m5, n6, majority TF t(S)S/2
• V (123,124,135,145,25,34)
• Top individual
• 1 approved by 4 voters (all other alternatives
  approved by 3 voters)
• Top team
• 2345 is the only team approved by all 5 voters

49

• TOP INDIVIDUAL NOT IN A TOP TEAM
• Example m5, n6, majority TF t(S)S/2
• V (123,124,135,145,25,34)
• Top individual
• 1 approved by 4 voters (all other alternatives
  approved by 3 voters)
• Top team
• 2345 is the only team approved by all 5 voters
• could generalize examples for almost any TF
• could generalize to top k individuals

50

• THRESHOLD SENSITIVITY
• Theorem
• For any Kgt1, there exist n,m and a corresponding
  V such that AVCT winner Sk (where X is the set of
  all K-sets), k1,,K are mutually disjoint.

51
ANY GOOD PROPERTIES? P1. Nullity. If
every vote is the empty set, any choice is
good. P2. Anonymity. If U is a
permutation of V, the choices for U and V are
identical. P3. Partition Consistency. If S
is chosen in two voter disjoint elections, then S
would be chosen in the joint election.) P4.
Partition Inclusivity. If no S is chosen
by a single voter and in an election of the
remaining n-1 voters, then any choice would also
be chosen in an election w/o one of the voters.
52
SINGLE VOTER PROPERTIES ?(S) minAS S is a
choice for A P5. For every choice S, there
exists votes A and B such that A is a choice for
S but not for B. P6. Let S be a choice for vote
A that does not choose everyone. If BSgtAS then S
is a choice for B P7. For every S, there is an A
such that AS ?(S) -1 P8. Suppose vote B chooses
every committee. For all A1, A2 and for all
choices S, T If A1S ?(S), A2T ?(T), then
BSgtA1S implies BTgtA2T
53
THE LAST THEOREM OF FISHBURN Theorem. If P1-8
hold, then the subset choice function is the AVCT.
54

• AV THRESHOLD APPROACH
• low informational burden
• simplicity
• takes into account subset preferences
• Results
• properties of TFs, axiomatic characterization
• complexity
• robustness properties theorems show what is
  possible and not what is probable
• Need
• Comparison with other methods, data validation
• strategic considerations

55
Approval Voting for Committees Threshold
Approaches. Peter Fishburn Saa Pekec
56

• DIGRESSION
• Subset Choice and Cooperative Games

57

• APPROVAL VOTE
• subset choice
• alternative vote count
• i. For every S find
• u(S) voters whose approval set is S
• ii. AC(j) SS j in S u(S)

58
APPROVAL VOTE PROFILE

• V (3,12,2,4,4,13,12,4,4)
• u(4)4,u(12)2,u(2)u(3)u(13)1 for all other
  S u(S)0
• AC(j) SS j in S u(S)
• e.g. AC(1)u(12)u(13)213

59

• APPROVAL VOTE
• i. For every S find
• u(S) voters whose approval set is S
• ii. AC(j) SS j in S u(S)

60

• APPROVAL VOTE
• For every S find
• u(S) voters whose approval set is S
• ?
• Cooperative Game u
• solution concepts for cooperative games
• - core, nucleolus, Shapley Value ...
• - define how to attribute subset values to
  individual alts.
• - implicitly define rankings on alternatives

61

• APPROVAL VOTE
• For every S find
• u(S) voters whose approval set is S
• ?
• Cooperative Game u
• solution concepts for cooperative games
• - core, nucleolus, Shapley Value ...
• - define how to attribute subset values to
  individual alts.
• - implicitly define rankings on alternatives

62

• APPROVAL VOTE
• i. For every S find
• u(S) voters whose approval set is S
• ii. Use your favorite solution concept
• to define a ranking on alternatives
• Is there a solution concept that generates
  ranking identical to The Approval Count (AC)?

63

• POWER INDICES
• p(j) c SSj in S w(S,j) u(S) u(S\j)
• Shapley-Shubik Index w(S,j) (S-1)!(m-S)!
• Banzhaf-Coleman Index w(S,j)1
• Proposition
• Banzhaf-Coleman Index pBC( ) is the only power
  index such that, for every u, the ranking of
  alternatives induced by pBC( ) is identical to
  the ranking induced by the Approval Vote Count
  AC( ).

64
Proposition Banzhaf-Coleman Index pBC( ) is the
only power index such that, for every u, the
ranking of alternatives induced by pBC( ) is
identical to the ranking induced by the Approval
Vote Count AC( ). Proof AC(j) SS j in S
u(S) pBC(j) SSj in S u(S) u(S\j)
SSj in S u(S) SSj not in S u(S) Note that
SS u(S) n, so pBC(j) 2 AC(j) n
Converse is a bit tedious, constructing V to
exploit differences in w(S,j) (Recall p(j) c
SSj inS w(S,j)u(S)u(S\j)).
65

• AV AND COOPERATIVE GAMES
• it is all about subset choice
• demonstrated a link between AV and cooperative
  games
• how to use large body of research in cooperative
  games?
• - opens up possibilities for new aggregation
  methods
• - social choice implications for power indices

66

• PLAN OF ACTION
• Motivation/Introduction
• Subset Choice and Cooperative Games
• Approval Voting Threshold Approach (with
  Fishburn)
• Balancing Teams (with Baucells)

67

• BALANCING TEAMS
• MBA student teams
• - N individuals divided into G groups/teams
• Each individual i described by values aij of
  predefined characteristics j

68

• BALANCING TEAMS
• MBA student teams
• - N individuals divided into G groups/teams
• Each individual i described by values aij of
  predefined characteristics j
• Want as perfectly balanced team assignment as
  possible
• For any characteristic j and any value aj, the
  difference across any two teams in the number of
  people with value aj in characteristic j is at
  most one.

69

• BALANCING TEAMS
• MBA student teams
• - N individuals divided into G groups/teams
• Each individual i described by values aij of
  predefined characteristics j
• Want as perfectly balanced team assignment as
  possible
• For any characteristic j and any value aj, the
  difference across any two teams in the number of
  people with value aj in characteristic j is at
  most one.
• other examples consultants
• showroom settings (cars, furniture)

70

• BALANCING TEAMS
• INSEAD, Stern (Weitz and Jelassi, JORS 92)
• Tuck (Baker et al., JORS 02,03)
• Kelley (Cutshall et al., Interfaces 06)
• Rotman (Krass and Ovchinnikov, Interfaces 2006)
• . . .

71

• FEASIBILITY PROBLEM
• Simplify to binary characteristics
• Input N,G and 0-1 matrix A aij. (Let qj Si
  aij /G)
• Feasibility problem

72

• COMPLEXITY
• Theorem. Balancing Teams is NP-complete.

73

• COMPLEXITY
• Theorem. Balancing Teams is NP-complete.
• Proof. Take aij with exactly two ones in each
  column.
• Note individual vertex of a graph,
  characteristic edge
• Balanced team assignment G-equicoloring.

74

• COMPLEXITY
• Theorem. Balancing Teams is NP-complete.
• Proof. Take aij with exactly two ones in each
  column.
• Note individual vertex of a graph,
  characteristic edge
• Balanced team assignment G-equicoloring.
• Claim. k-coloring and k-equicoloring are in the
  same complexity class.(Add (n-k)(k-1)
  independent vertices.)

75

• COMPLEXITY
• Theorem. Balancing Teams is NP-complete.
• Proof. Take aij with exactly two ones in each
  column.
• Note individual vertex of a graph,
  characteristic edge
• Balanced team assignment G-equicoloring.
• Claim. k-coloring and k-equicoloring are in the
  same complexity class.(Add (n-k)(k-1)
  independent vertices.)
• Finally, Graph k-coloring is NP-complete for kgt2.

76

• COMPLEXITY
• Theorem. Balancing Teams is NP-complete.
• Proof. Take aij with exactly two ones in each
  column.
• Note individual vertex of a graph,
  characteristic edge
• Balanced team assignment G-equicoloring.
• Claim. k-coloring and k-equicoloring are in the
  same complexity class. Add (n-k)(k-1)
  independent vertices.)
• Finally, Graph k-coloring is NP-complete for kgt2.
• Theorem. Any reasonable approximate balancing is
  also NP complete. (Reduction to exact cover by
  3sets.)

77

• SIMULATION
• 2500 instances using Solver Premium
• 3000 instances using CPLEX (w/o preprocessing)
• up to 40 binary categories used
• Logistic regression model
• variables N, M, N/G, density, tight
  constraints

78
Q0N/G
79
(No Transcript)
80
(No Transcript)
81

• BALANCING TEAMS IS EASY
• Example N72, G12
• - Easy for Mlt20,
• - M33 probability of success is 0.5
• Sources of hardness
• large K, small N
• many tight constraints
• density
• large G, small N

• Problem instances related to MBA programs are
  easy.

82

• PLAN OF ACTION
• Motivation/Introduction
• Subset Choice and Cooperative Games
• Approval Voting Threshold Approach (with
  Fishburn)
• Balancing Teams (with Baucells)

Its all over but the crying.
83
ASSEMBLING TEAMS Saa Pekec Decision
Sciences The Fuqua School of Business Duke
University pekec_at_duke.edu http//faculty.fuqua.du
ke.edu/pekec thanks to Manel Baucells, Peter
Fishburn