Social Choice Theory

1
Social Choice Theory
By Shiyan Li
2
History

• The theory of social choice and voting has had a
  long history in the social sciences, dating back
  to early work of Marquis de Condorcet (the 1st
  rigorous mathematical treatment of voting) and
  others in the 18th century.
• Now it is a branch of discrete mathematics.

3
Purpose

• Social Choice Theory is the study of systems and
  institution for making collective choice, choices
  that affect a group of people.
• Be used in multi-agent planning, collective
  decision, computerized election and so on.

4
Simple Majority Voting

• Choose one from two possible alternatives by a
  group of participants.
• Consider a democratic voting situation.

5
Preferences and Outcome

• Alternatives x or y
• Every voter has a preferences.
• Three possible situations of each voters
  preference i) x is strictly better than y
  1ii) y is strictly better than x -1iii) x
  and y are equivalent 0
• After the votingi) x is winner 1ii) y is
  winner -1iii) x and y tie 0

6
General List

• Use a list to describe a collection of n voters
  preferencese.g. (-1, 1, 0, 0, -1, , 1, -1)
• General ListD (d1, d2, d3, , dn-1, dn)di is
  1, -1 or 0 depending on whether individual i
  strictly prefers x to y, y to x or is indifferent
  between them.

7
General List

• Consider the sum of list DWhen
  d1d2d3dn-1dn gt 0,x is to be chosen, simple
  majority voting assigns 1. When
  d1d2d3dn-1dn lt 0,y is to be chosen, simple
  majority voting assigns -1. When
  d1d2d3dn-1dn gt 0,x and y tie, simple
  majority voting assigns 0.

8
Formal Definition of Simple Majority Vote

• Use the sign function to formally define the
  simple majority vote(d1, d2, , dn)
  sgn(d1d2dn)
• Function N1 and N-1N1 associates with a list
  D the number of dis that are strictly
  positiveN-1 associates with a list D the number
  of dis that are strictly positive

9
Formal Definition of Simple Majority Vote

• E.g. for list D (1, -1, -1 ,0, 1, 1),? n
  6, n/2 3, N1 (1, -1, -1 ,0, 1, 1) 3 gt
  n/2 N-1 (1, -1, -1 ,0, 1, 1) 2 ltn/2?
  g(1, -1, -1 ,0, 1, 1) 1

10
Rule of Simple Majority Voting

• Social Choice Ruleis a function f(d1, d2 , ,
  dn ), the domain of the function is the set of
  all list to which f assigns some unambiguous
  outcome 1, -1 or 0.
• A social choice rule of simple majority voting
  can be characterized by 4 properties (Kenneth O.
  May, 1952).

11
Property 1 of Rule f

• Property 1 Universal Domainf satisfies
  universal domain if it has a domain equal to all
  logically possible lists (i.e. any combination of
  the individual voters preferences) of n entries
  of 1, -1 or 0.

12
Property 2 of Rule f

• One-to-one Correspondenceis a function s from
  the set 1, 2, , n to itself such that s is
  defined on every integer from 1 to n and no
  outcome is assigned to two different
  integerss(i) s(j) implies i j.

13
Property 2 of Rule f

• PermutationGiven two lists D
  (d1, d2 , , dn)and D (d1 , d2
  , , dn )say that D and D are permutation
  of one another if there is a one-to-one
  correspondence s on 1, 2, , n such that
  ds(i) di.
• E.g. voter 1 2 3 4 5 6 7
  (1, 1, 1, 0, 0, -1, -1)and
  voter 1 2 3 4 5 6 7
  (-1, 0, 1, 1, 0, -1, 1)are permutation
  of one another via the one-to-one
  correspondence 1-gt3, 2-gt4, 3-gt7,
  4-gt2, 5-gt5, 6-gt1, 7-gt6.

14
Property 2 of Rule f

• Property 2 AnonymityA social choice rule will
  satisfy this property if it does not make any
  difference who votes in which way as long as the
  numbers of each type are the same (i.e. equal
  treatment of each voter).Formal DefinitionA
  social choice rule f satisfies anonymity if
  whenever (d1, d2, , dn) and (d1, d2, , dn)
  in the domain of f are permutations of one
  another then f(d1, d2, , dn)
  f(d1, d2, , dn)E.g.if D
  (1, 1, 1, 0, 0, -1, -1)and
  D (-1, 0, 1, 1, 0, -1, 1)so D and D are
  permutations of each other,and if f(d1,
  d2, , dn) f(d1, d2, , dn) then social
  choice rule f satisfies anonymity.

15
Property 3 of Rule f

• Property 3 NeutralityA social choice rule
  satisifies neutrality if whenever (d1, d2 , , dn
  ) and (-d1, -d2 , , -dn ) are both the domain of
  f thenf(d1, d2 , , dn )-f(-d1, -d2 , , -dn )
• NoteThe condition of anonymity is a way of
  treating individuals equally, the condition of
  neutrality is a way of treating alternatives x
  and y equally.

16
Property 4 of Rule f

• i-VariantsSuppose there are
  D (d1, d2 , , dn )and
  D (d1, d2 , , dn )D and D
  are i-variants if for all j?i, djdj. Thus two
  i-variants differ in at most the ith entry.
  (Note It has not strictly stipulated the
  relationship of di and di, i.e., it is possible
  that didi, digtdi, or diltdi.)
• E.g.Two lists D (1,
  -1, -1, 0, 1, -1, 1)and
  D (1, -1, 0, 0, 1, -1, 1)are 3-variants
  since they differ only at the third place

17
Property 4 of Rule f

• PurposeSimple majority voting can not be
  strictly characterized by property 13 yet
  (unresponsive).
• E.g.Assume a constant rule (function) const0(D)
  that always generates result 0 for any point in
  its domain.i.e.
  const0(D) 0This constant rule satisfies all
  3 properties mentioned above.D contains all
  logically possible lists.
  Property 1For all permutations D, const0(D)
  const0(D) 0. Property 2For all lists in D,
  const0(D) -const0(-D) 0. Property
  3So, we still need a property to constrain rule
  f to simple majority more strictly.

18
Property 4 of Rule f

• Property 4 Positive Responsivenessf
  satisfies positive responsiveness if for all i,
  whenever (d1, d2 , , dn ) and (d1, d2 , ,
  dn) are i-variants with di gt di, then
  f(d1, d2 , , dn ) 0implies
  f(d1, d2 , , dn) 1.

19
Property 4 of Rule f

• Positive responsiveness can be inferred by multi
  i-variants.E.g.Suppose to apply lists 1
  below to f which is a rule satisfies positive
  responsiveness f(1, 0, -1,
  0, 0, 1, -1) 0.First find a 3-variant list
  2 of 1 (1, 0, 0, 0, 0, 1, -1),so f(1, 0,
  0, 0, 0, 1, -1) 1.Second find a 4-variant
  list 3 of 2 (1, 0, 0, 1, 0, 1, -1),so
  f(1, 0, 0, 1, 0, 1, -1) 1.Then it can be
  concluded that f(1, 0, -1, 0, 0, 1, -1) 0
  implies f(1, 0, 0, 1, 0, 1, -1) 1, although
  list 1 and 3 are not i-variants.

20
Property 4 of Rule f

• Negative ResponsivenessSuppose rule f
  satisfies property 14.For all i, whenever D
  (d1, d2 , , dn ) and D (d1, d2, , dn )
  are i-variants with di lt di (i.e. -di gt -di
  ).If f(D) 0 then f(-D) -f(D) 0 by
  neutrality.So f(-D) 0.There is a list -D
  which together with D are i-variants with -di gt
  -di.Because f(-D) 0 so that f(-D) 1 by
  positive responsiveness.So f(D) -f(-D)
  -1For summaryIf f satisfies positive
  responsiveness and neutrality then for all i,
  whenever D (d1, d2 , , dn ) and D (d1,
  d2, , dn ) are i-variants with di lt di, such
  that f(D) 0
  implies f(D) -1

21
Mays Theorem

• Simple majority voting is the only rule that
  satisfies all four properties (or conditions)
  simultaneously.

22
Mays Theorem

• Mays TheoremIf a social choice rule f
  satisfies all of i) universal domain ii)
  anonymity iii) neutrality iv) positive
  responsivenessthen f is simple majority voting.

23
Proof of Mays Theory

• Step 1If rule f satisfies conditions i), ii),
  iii) and iv).So the value of f(D) only depends
  on the number of 1s, 0s and -1s by
  anonymity.Suppose there are n elements in D,
  N1(D) and N-1(D) is the number of 1s and -1s
  in D correspondingly.So the number of 0s is n
  - N1(D) - N-1(D).Therefore, f(D) is entirely
  determined by N1(D) and N-1(D) by anonymity.

24
Proof of Mays Theory

• Step 2Suppose N1(D) N-1(D) and f(D)
  r.ObviouslyN1(D) N-1(D) N1(-D)
  1N-1(D) N1(D) N-1(-D). 2And
  because f satisfies universal domain, so f is
  also defined at D.Sincef(-D) -f(D) -r by
  neutrality,andf(-D) f(D) r by 1 and
  2.Combining above results, r r so r
  0.That is N1(D) N-1(D) implies f(D) 0.

25
Proof of Mays Theory

• Step 3Suppose N1(D) gt N-1(D) where there are
  n elements in D,so that N1(D) N-1(D) m
  where 0 lt m n - N-1(D).It will be proved that
  f(D) 1 by mathematical induction belowD
  (d1, d2, , dn). Basis m 1. ? N1(D)
  N-1(D) 1 ? There is at least one di
  1. Suppose D(d1,
  d2, , dn), an i-variant determined by
  djdj if j?i, and di0.
  1 f is defined at D and D
  by universal domain. Obviously
  N1(D) N-1(D). ? f(D) 0 by step
  2. 2 ?
  f(D) 1 by 1, 2 and positive
  responsiveness.Induction Suppose
  N1(D)N-1(D)1 implies f(D)1.
  It has to be shown that
  N1(D)N-1(D)(m1) implies f(D)1.
  So suppose
  N1(D)N-1(D)(m1). ? There is
  at least one di 1. Suppose
  D(d1, d2, , dn), an
  i-variant determined by
  djdj if j?i, and di0.
  3 f is defined at D and
  D by universal domain.
  Obviously N1(D) N-1(D)m. ?
  f(D) 0 by induction hypothesis. 4
  ? f(D) 1 by 1, 2 and
  positive responsiveness.SummaryFollow an
  analogous derivation, an assertion when N1(D) lt
  N-1(D), f(D) -1 can be proved.So If N1(D) gt
  N-1(D), then f(D) 1 If N1(D) lt N-1(D),
  then f(D) -1

26
Proof of Mays Theory

• Summary of ProofFrom step 1, 2, and 3If
  N1(D)N-1(D), then f(D)0.If N1(D)gtN-1(D),
  then f(D)1.If N1(D)ltN-1(D), then
  f(D)-1.These results just satisfy the formal
  definition of simple majority voting.So Mays
  theory is proved.

27
Voting Paradox

• To be continued

28
References

• Kelly, Jerry S., 1988, Social Choice Theory An
  Introduction, Springer-Verlag, Berlin Heidelberg.