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### Social Choice Theory

By Shiyan Li

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.

Voters

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.

Simple Majority Voting

- Choose one from two possible alternatives by a group of voters.
- Consider a democratic voting situation.

Preferences and Outcome

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

General List

- Use a list to describe a collection of n voters’ preferencese.g. (-1, +1, 0, 0, -1, …, +1, -1)
- General List:D = (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.

General List

- Consider the sum of list D:When d1+d2+d3+…+dn-1+dn > 0,x is to be chosen, simple majority voting assigns +1. When d1+d2+d3+…+dn-1+dn < 0,y is to be chosen, simple majority voting assigns -1. When d1+d2+d3+…+dn-1+dn = 0,x and y tie, simple majority voting assigns 0.

Formal Definition of Simple Majority Voting

- Use the sign function to formally define the simple majority voting:(d1, d2, …, dn) sgn(d1+d2+…+dn)
- Function N+1 and N-1:N+1: associates with a list D the number of di‘s that are strictly positiveN-1: associates with a list D the number of di‘s that are strictly negative

g (d1, d2, d3, …, dn) =

-1 if N-1(D)>N+1(D)

0 otherwise

Formal Definition of Simple Majority VotingAbsolute Majority Voting

- E.g. for absolute majority voting: for list D = (+1, -1, -1 ,0, +1, +1),∵ n = 6, n/2 = 3, N+1 (+1, -1, -1 ,0, +1, +1) = 3 > n/2 N-1 (+1, -1, -1 ,0, +1, +1) = 2 <n/2∴ g(+1, -1, -1 ,0, +1, +1) = +1

if N+1(d1, d2, d3, …, dn) > n/2

if N-1(d1, d2, d3, …, dn) > n/2

Rule of Simple Majority Voting

- Social Choice Rule:is 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).

Property 1 of Rule f

- Property 1 – Universal Domain:f 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.

S(i)

i

S(i)

S(i)

S(i)

i

i

not one-to-one correspondence

one-to-one correspondence

Property 2 of Rule f- One-to-one Correspondence:is a function s from the set {1, 2, …, n} to itself such that s is defined on every integer from 1 to n and distinct outcomes are assigned to two different integers:s(i) = s(j) implies i = j.

Property 2 of Rule f

- Permutation:Given 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->3, 2->4, 3->7, 4->2, 5->5, 6->1, 7->6.

Property 2 of Rule f

- Property 2 – Anonymity:A 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 Definition:A 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.

Property 3 of Rule f

- Property 3 – Neutrality:A 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 )
- Note:The condition of anonymity is a way of treating individuals equally, the condition of neutrality is a way of treating alternatives x and y equally.

Property 4 of Rule f

- i-Variants:Suppose there are D = (d1, d2 , …, dn )and D’ = (d1’, d2’, …, dn’);D and D’ are i-variants if for all j≠i, dj=dj’. 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 di=di’, di>di’, or di<di’.)
- 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

Property 4 of Rule f

- Purpose:Simple majority voting can not be strictly characterized by property 1~3 yet (unresponsiveness).
- 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.

Property 4 of Rule f

- Property 4 – Positive Responsiveness:f satisfies positive responsiveness if for all i, whenever (d1, d2 , …, dn ) and (d1’, d2’, …, dn’) are i-variants with di’ > di, then f(d1, d2 , …, dn ) ≥ 0implies f(d1’, d2’, …, dn’) = +1.

Property 4 of Rule f

- Positive responsiveness can be inferred by indirect 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 direct i-variants.

Property 4 of Rule f

- “Negative Responsiveness”:Suppose rule f satisfies property 1~4.For all i, whenever D = (d1, d2 , …, dn ) and D’ = (d1‘, d2‘, …, dn‘) are i-variants with di‘ < di (i.e. -di‘ > -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‘ > -di.Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness.So f(D’) = -f(-D’) = -1Summary:If 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‘ < di, such that f(D) ≤ 0 implies f(D’) = -1

May’s Theorem

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

May’s Theorem

- May’s Theorem:If a social choice rule f satisfies all of i) universal domain ii) anonymity iii) neutrality iv) positive responsivenessthen f is simple majority voting.

Proof of May’s Theory

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

Proof of May’s Theory

- Step 2:Suppose N+1(D) = N-1(D) and f(D) = r.ObviouslyN+1(D) = N-1(D) = N+1(-D) #1N-1(D) = N+1(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 N+1(D) = N-1(D) implies f(D) = 0.

Proof of May’s Theory

- Step 3:Suppose N+1(D) > N-1(D) where there are n elements in D,so that N+1(D) = N-1(D) + m where 0 < m ≤ n - N-1(D).It will be proved that f(D) = +1 by mathematical induction below:D = (d1, d2, …, dn). Basis: m = 1.∴ N+1(D) = N-1(D) + 1∴ There is at least one di = 1. Suppose D’=(d1’, d2’, …, dn’), an i-variant determined by dj’=dj if j≠i, and di’=0. #1 f is defined at D and D’ by universal domain. Obviously N+1(D’) = N-1(D’).∴ f(D’) = 0 by step 2. #2 ∴ f(D) = +1 by #1, #2 and positive responsiveness.Induction: Suppose N+1(D)=N-1(D)+1 implies f(D)=+1. It has to be shown that N+1(D)=N-1(D)+(m+1) implies f(D)=+1. So suppose N+1(D)=N-1(D)+(m+1).∴ There is at least one di = 1. Suppose D’=(d1’, d2’, …, dn’), an i-variant determined by dj’=dj if j≠i, and di’=0. #3 f is defined at D and D’ by universal domain. Obviously N+1(D’) = N-1(D’)+m.∴ f(D’) = 0 by induction hypothesis. #4 ∴ f(D) = +1 by #1, #2 and positive responsiveness.Summary:Follow an analogous derivation, an assertion “when N+1(D) < N-1(D), f(D) = -1” can be proved.So: If N+1(D) > N-1(D), then f(D) = +1If N+1(D) < N-1(D), then f(D) = -1

Proof of May’s Theory

- Summary of Proof:From step 1, 2, and 3:If N+1(D)=N-1(D), then f(D)=0.If N+1(D)>N-1(D), then f(D)=+1.If N+1(D)<N-1(D), then f(D)=-1.These results just satisfy the formal definition of simple majority voting.So May’s theory is proved.

General Social Choice Rules

- X: a nonempty set of alternatives.The elements of X must only be mutually incompatible.
- v: agenda, v ≠ Ø and v ⊆ X, a set of alternatives that are currently available.
- N: a set of individuals.

General Social Choice Rules

- xRiy: i ∈ N; x, y ∈ X; individual i determines alternative x to be at least as good as alternative y; or i weakly prefers x to y.1. Ri is reflexive: xRix for all x ∈ X.2. Ri is complete: xRiy or yRix (or both) for all x, y ∈ X.3. Ri is transitive: For all x, y, z ∈ X, if both xRiy and yRiz then xRiz.

General Social Choice Rules

- xPiy: xRiy and not yRix; i strongly prefers x to y.
- yPix: yRix and not xRiy; i strongly prefers y to x.
- xIiy: xRiy and also yRix; i is indifferent between x and y.

General Social Choice Rules

- Profile: an assignment of one preference relation to each individual.
- C(v): the elements chosen from agenda v by choice function C.(i) C(v) ⊂ v;(ii) C(v) ≠ Ø.

v

profile of preferences,

u

social choice rule,

f

choice function,

C = f(u)

chosen set,

Cu(v)

General Social Choice Rules- Social Choice Rule:A social choice rule assigns to each of a collection of profiles a corresponding choice function.

Standard Domain Constraint

- Standard domain constraint includes: i) there are at least three alternatives in X; ii) there are at least three individuals in N;iii) the social choice rule has as domain all logically possible profiles of preference orderings on X;iv) each choice function that is an output of the rule has in its domain all finite nonempty agendas.

Pareto Condition

- Weak Pareto Condition:Let the social choice rule select choice function Cu at profile u. Suppose at u everyone unanimously strictly prefers one alternative, say x, to another, say y; then if x is available (i.e., x ∈ v), y won’t be chosen (i.e., y ∉ Cu(v))

Pareto Condition

- Strong Pareto Condition:Let the social choice rule select choice function Cu at profile u. Suppose at u everyone unanimously find one alternative, x, to be at least as good as another, y, and at least one individual strictly prefers x to y. Then if x is available (i.e., x ∈ v), ywon’t be chosen (i.e., y ∉ Cu(v))

Pareto Condition

- Example:For agenda: 1: (x y1) y2 2: x y1 y2 3: x (y1 y2)In Weak Pareto Condition: y2∉ Cu(v)In Strong Pareto Condition: y1, y2∉ Cu(v)

Pareto Condition

- X is Pareto-superior to y at profile u = (R1, R2, …, Rn) if:(i) xRiy for all individuals i in N;(ii) xPiy for at least one individual i in N.
- Alternatives for which there are no available Pareto-superior alternatives are called Pareto optimal.

Dictator

- Weak DictatorIndividual i is a weak dictator if for every pair of alternatives, x and y, every profile u = (R1, R2, …, Rn) and every agenda v, if xPiy then y ∈ Cu(v) implies x ∈ Cu(v).

Dictator

- CoalitionA subset S of the set N of all individuals is called a coalition.
- Decisive CoalitionFor a social choice rule that maps u to Cu, A coalition S is called decisive for alternative x against alternative y if: for ∀i: i ∈ S • xRiy; ∃j: j ∈ S • xPjy; then ∀v: v ⊂ X, x ∈ v • y ∉ Cu(v).If ∀x, y: x, y ∈ X • S is decisive for alternative x against alternative y, then we simply say S is decisive.

Dictator

- DictatorIf a decisive coalition S = {i}, then i is a dictator.

Borda Rules

- Borda CountAssume that X is finite. Then associated with any preference ordering Ri there is a ranking function ri that associates an integer with each alternative: ri(x) is the number of alternatives stictly preferred to x. Given a profile u = (R1, R2, …, Rn), there is a ranking function r given by r(x) = ∑iri(x).The value of r(x) is called Borda count of x.

Borda Rules

- Global Borda RuleCu(v) = {x|r(x) ≤ r(y) for all y ∈ v}.This rule has us choose from v those alternatives with minimal Borda count.

Independence of Irrelevant Alternatives

- If two profiles u, u’, restricted to an agenda v are identical, then the choices made from that agenda should be the same: Cu(v) = Cu’(v).

Local Borda Rules

- Local Borda CountGiven a profile u = (R1, R2, …, Rn), there is for each v:rv(x) = ∑iriv(x).
- Local Borda RuleCu(v)= {x|rv (x) ≤ rv (y) for all y ∈ v}.

Transitive Explanation

- Explanation:A choice function C is explainable if there exists a relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}.
- Transitive Explanation:A choice function C has transitive explainable if there is a reflexive, complete and transitive relation Ω such that C(v) = {x ∈ v | xΩy for all y ∈ v}.
- We say a social choice rule has transitive explainable if at every admissible profile u the associated Cu has a transitive explainable.

Arrow’s Impossibility Theorem

- There does not exist any social choice rule satisfying all of:1. the standard domain constraint;2. the strong Pareto condition;3. independence of irrelevant alternatives;4. has transitive explanations;5. absence of a dictator.

Mechanism Design

- Implementing a social choice function f(u1, …, un) using a game.
- Center (auctioneer) does not know the agents’ preferences.
- Agents may lie.
- Goal is to design the rules of the game so that in equilibrium (s1, …, sn), the outcome of the game is f(u1, …, un).

Mechanism Design

- Mechanism designer specifies the strategy sets Si and how outcome is determined as a function of (s1, …, sn) (S1, …, Sn).
- VariantsStrongest: There exists exactly one equilibrium. Its outcome is f(u1, …, un).Medium: In every equilibrium the outcome is f(u1, …, un).Weakest: In at least one equilibrium the outcome is f(u1, …, un).

References

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

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