social choice theory by shiyan li. history the theory of social choice and voting has had a long...

Social Choice TheorySocial Choice Theory

By Shiyan Li

HistoryHistory

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

Now it is a branch of discrete Now it is a branch of discrete mathematics.mathematics.

PurposePurpose

Social Choice Theory is the study of Social Choice Theory is the study of systems and institution for making systems and institution for making collective choice, choices that affect collective choice, choices that affect a group of people.a group of people.

Be used in multi-agent planning, Be used in multi-agent planning, collective decision, computerized collective decision, computerized election and so on.election and so on.

Voters Alternatives

Simple Majority VotingSimple Majority Voting

Choose one from two possible Choose one from two possible alternatives by a group of voters.alternatives by a group of voters.

Consider a democratic voting Consider a democratic voting situation.situation.

Preferences and OutcomePreferences and Outcome Alternatives: x or yAlternatives: x or y Every voter has a preferences.Every voter has a preferences. Three possible situations of each voter’s Three possible situations of each voter’s

preference: preference: i) x is strictly better than y: +1i) x is strictly better than y: +1ii) y is strictly better than x: -1ii) y is strictly better than x: -1iii) x and y are equivalent: 0iii) x and y are equivalent: 0

After the voting:After the voting:i) x is winner: +1i) x is winner: +1ii) y is winner: -1ii) y is winner: -1iii) x and y tie: 0iii) x and y tie: 0

General ListGeneral List

Use a list to describe a collection of n Use a list to describe a collection of n voters’ preferencesvoters’ preferencese.g. (-1, +1, 0, 0, -1, …, +1, -1)e.g. (-1, +1, 0, 0, -1, …, +1, -1)

General List:General List:D = (dD = (d11, d, d22, d, d33, …, d, …, dn-1n-1, d, dnn))ddii is +1, -1 or 0 depending on whether is +1, -1 or 0 depending on whether individual i strictly prefers x to y, y to x or individual i strictly prefers x to y, y to x or is indifferent between them.is indifferent between them.

n entries

General ListGeneral List

Consider the sum of list D:Consider the sum of list D:When dWhen d11+d+d22+d+d33+…+d+…+dn-1n-1+d+dn n > 0,> 0,x is to be chosen, simple majority x is to be chosen, simple majority voting assigns +1.voting assigns +1. When d When d11+d+d22+d+d33+…+d+…+dn-1n-1+d+dn n < 0,< 0,y is to be chosen, simple majority y is to be chosen, simple majority voting assigns -1.voting assigns -1. When d When d11+d+d22+d+d33+…+d+…+dn-1n-1+d+dn n = 0,= 0,x and y tie, simple majority voting x and y tie, simple majority voting assigns 0.assigns 0.

Formal Definition of Simple Majority VotingFormal Definition of Simple Majority Voting

Use the sign function to formally define Use the sign function to formally define the simple majority voting:the simple majority voting:(d(d11, d, d22, …, d, …, dnn) sgn(d) sgn(d11+d+d22+…+d+…+dnn))

Function NFunction N+1+1 and N and N-1-1::NN+1+1: associates with a list D the number of : associates with a list D the number of ddii‘s that are strictly positive‘s that are strictly positiveNN-1-1: associates with a list D the number of : associates with a list D the number of ddii‘s that are strictly negative‘s that are strictly negative

Formal Definition of Simple Majority VotingFormal Definition of Simple Majority Voting

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

g (d1, d2, d3, …, dn) =

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

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

0 otherwise

if N+1(d1, d2, d3, …, dn) > n/2

if N-1(d1, d2, d3, …, dn) > n/2

Absolute Majority Voting

Rule of Simple Majority VotingRule of Simple Majority Voting

Social Choice Rule:Social Choice Rule:is a function f(dis a function f(d11, d, d2 2 , …, d, …, dn n ), the domain of ), the domain of the function is the set of all list to which f the function is the set of all list to which f assigns some unambiguous outcome: +1, assigns some unambiguous outcome: +1, -1 or 0.-1 or 0.

A social choice rule of simple majority A social choice rule of simple majority voting can be characterized by 4 voting can be characterized by 4 properties (Kenneth O. May, 1952).properties (Kenneth O. May, 1952).

Property 1 of Rule fProperty 1 of Rule f

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

Property 2 of Rule fProperty 2 of Rule f One-to-one Correspondence:One-to-one Correspondence:

is a function s from the set 1, 2, …, n to itself is a function s from the set 1, 2, …, n to itself such that s is defined on every integer from 1 to n such that s is defined on every integer from 1 to n and distinct outcomes are assigned to two and distinct outcomes are assigned to two different integers:different integers:s(i) = s(j) implies i = j.s(i) = s(j) implies i = j.

one-to-one correspondence

S(i)i

not one-to-one correspondence

S(i)i

i

S(i)

i

S(i)

Property 2 of Rule fProperty 2 of Rule f Permutation:Permutation:

Given two listsGiven two lists

D = (d D = (d11, d, d2 2 , …, d, …, dnn))andand D’ = (d D’ = (d1 1 ’, d’, d2 2 ’’ , …, d, …, dn n ’)’)

say that D and D’ are permutation of one another if there is a one-to-one say that D and D’ are permutation of one another if there is a one-to-one correspondence s on 1, 2, …, n such thatcorrespondence s on 1, 2, …, n such that d ds(i)s(i)’ = d’ = dii..

E.g.:E.g.:

voter: 1 2 3 4 5 6 7 voter: 1 2 3 4 5 6 7 (+1, +1, +1, 0, 0, -1, -1) (+1, +1, +1, 0, 0, -1, -1)andand voter: 1 2 3 4 5 6 7 voter: 1 2 3 4 5 6 7 (-1, 0, +1, +1, 0, -1, +1) (-1, 0, +1, +1, 0, -1, +1)

are permutation of one another via the one-to-one correspondence: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. 1->3, 2->4, 3->7, 4->2, 5->5, 6->1, 7->6.

Property 2 of Rule fProperty 2 of Rule f Property 2 – Anonymity:Property 2 – Anonymity:

A social choice rule will satisfy this property if it does not make 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 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).each type are the same (i.e. equal treatment of each voter).

Formal Definition:Formal Definition:A social choice rule f satisfies anonymity if whenever (dA social choice rule f satisfies anonymity if whenever (d11, d, d22, …, d, …, dnn) ) and (dand (d11’, d’, d22’, …, d’, …, dnn’) in the domain of f are permutations of one ’) in the domain of f are permutations of one another thenanother then f(d f(d11, d, d22, …, d, …, dnn) = f(d) = f(d11’, d’, d22’, …, d’, …, dnn’)’)

E.g.:E.g.:

if D = (+1, +1, +1, 0, 0, -1, -1)if D = (+1, +1, +1, 0, 0, -1, -1)andand D’ = (-1, 0, +1, +1, 0, -1, +1) D’ = (-1, 0, +1, +1, 0, -1, +1)

so D and D’ are permutations of each other,so D and D’ are permutations of each other,and if f(dand if f(d11, d, d22, …, d, …, dnn) = f(d) = f(d11’, d’, d22’, …, d’, …, dnn’) ’) then social choice rule f satisfies anonymity.then social choice rule f satisfies anonymity.


Property 3 – Neutrality:Property 3 – Neutrality:A social choice rule satisifies neutrality if A social choice rule satisifies neutrality if whenever (dwhenever (d11, d, d2 2 , …, d, …, dn n ) and (-d) and (-d11, -d, -d2 2 , …, -, …, -ddn n ) are both the domain of f then) are both the domain of f thenf(df(d11, d, d2 2 , …, d, …, dn n )=-f(-d)=-f(-d11, -d, -d2 2 , …, -d, …, -dn n ))

Note:Note:The condition of anonymity is a way of The condition of anonymity is a way of treating individuals equally, the condition treating individuals equally, the condition of neutrality is a way of treating of neutrality is a way of treating alternatives x and y equally.alternatives x and y equally.

Property 4 of Rule fProperty 4 of Rule f i-Variants:i-Variants:

Suppose there areSuppose there are

D = (d D = (d11, d, d2 2 , …, d, …, dn n ))andand D’ = (d D’ = (d11’, d’, d22’’ , …, d, …, dnn’’ ););

D and D’ are i-variants if for all j≠i, dD and D’ are i-variants if for all j≠i, d jj=d=djj’. Thus two i-variants ’. Thus two i-variants differ in at most the ith entry. (Note: It has not strictly stipulated differ in at most the ith entry. (Note: It has not strictly stipulated the relationship of dthe relationship of dii and d and dii’, i.e., it is possible that d’, i.e., it is possible that dii=d=dii’, d’, dii>d>dii’, or ’, or ddii<d<dii’.)’.)

E.g.:E.g.:

Two listsTwo lists

D = (+1, -1, -1, 0, +1, -1, +1) D = (+1, -1, -1, 0, +1, -1, +1)andand D’ = (+1, -1, 0, 0, +1, -1, +1) D’ = (+1, -1, 0, 0, +1, -1, +1)

are 3-variants since they differ only at the third placeare 3-variants since they differ only at the third place

Property 4 of Rule fProperty 4 of Rule f Purpose:Purpose:

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

E.g.:E.g.:Assume a constant rule (function) constAssume a constant rule (function) const00(D) that always generates (D) that always generates result 0 for any point in its domain.result 0 for any point in its domain.

i.e. consti.e. const00(D) 0(D) 0

This constant rule satisfies all 3 properties mentioned above.This constant rule satisfies all 3 properties mentioned above.

D contains all logically possible lists. – Property 1D contains all logically possible lists. – Property 1For all permutations D’, constFor all permutations D’, const00(D) = const(D) = const00(D) = 0. – Property 2(D) = 0. – Property 2For all lists in D, constFor all lists in D, const00(D) = -const(D) = -const00(-D) = 0. – Property 3(-D) = 0. – Property 3

So, we still need a property to constrain rule f to simple majority So, we still need a property to constrain rule f to simple majority more strictly.more strictly.


Property 4 – Positive Responsiveness:Property 4 – Positive Responsiveness:

f satisfies positive responsiveness if f satisfies positive responsiveness if for all i, whenever (dfor all i, whenever (d11, d, d2 2 , …, d, …, dn n ) and ) and (d(d11’, d’, d22’’ , …, d, …, dnn’) are i-variants with di’ ’) are i-variants with di’ > di, then> di, then

f(d f(d11, d, d2 2 , …, d, …, dn n ) ≥ 0) ≥ 0impliesimplies f(d f(d11’, d’, d22’’ , …, d, …, dnn’) = +1.’) = +1.


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

E.g.:E.g.:Suppose to apply lists #1 below to f which is a rule satisfies Suppose to apply lists #1 below to f which is a rule satisfies positive responsiveness:positive responsiveness:

f(+1, 0, -1, 0, 0, +1, -1) = 0. 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),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.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),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.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 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 f(+1, 0, 0, +1, 0, +1, -1) = +1, although list #1 and #3 are not direct i-variants.direct i-variants.

Property 4 of Rule fProperty 4 of Rule f ““Negative Responsiveness”:Negative Responsiveness”:

Suppose rule f satisfies property 1~4.Suppose rule f satisfies property 1~4.

For all i, whenever D = (dFor all i, whenever D = (d11, d, d2 2 , …, d, …, dn n ) and D’ = (d) and D’ = (d11‘, d‘, d22‘, …, d‘, …, dnn‘‘ ) are i-) are i-variants with dvariants with dii‘ < d‘ < di i (i.e. -d(i.e. -dii‘ > -d‘ > -di i ).).

If f(D) ≤ 0 then f(-D) = -f(D) ≥ 0 by neutrality.If f(D) ≤ 0 then f(-D) = -f(D) ≥ 0 by neutrality.

So f(-D) ≥ 0.So f(-D) ≥ 0.

There is a list -D’ which together with –D are i-variants with -dThere is a list -D’ which together with –D are i-variants with -d ii‘ > -d‘ > -dii..

Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness.Because f(-D) ≥ 0 so that f(-D’) = +1 by positive responsiveness.

So f(D’) = -f(-D’) = -1So f(D’) = -f(-D’) = -1

Summary:Summary:If f satisfies positive responsiveness and neutrality then for all i, whenever If f satisfies positive responsiveness and neutrality then for all i, whenever D = (dD = (d11, d, d2 2 , …, d, …, dn n ) and D’ = (d) and D’ = (d11‘, d‘, d22‘, …, d‘, …, dnn‘‘ ) are i-variants with d) are i-variants with dii‘ < d‘ < dii, , such thatsuch that f(D) ≤ 0 implies f(D’) = -1 f(D) ≤ 0 implies f(D’) = -1

May’s TheoremMay’s Theorem

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

May’s TheoremMay’s Theorem

May’s Theorem:May’s Theorem:

If a social choice rule f satisfies all ofIf a social choice rule f satisfies all of

i) universal domain i) universal domain ii) anonymity ii) anonymity iii) neutrality iii) neutrality iv) positive responsiveness iv) positive responsiveness

then f is simple majority voting.then f is simple majority voting.

Proof of May’s TheoryProof of May’s Theory Step 1:Step 1:

If rule f satisfies conditions i), ii), iii) and iv).If rule f satisfies conditions i), ii), iii) and iv).

So the value of f(D) only depends on the number So the value of f(D) only depends on the number of +1’s, 0’s and -1’s by anonymity.of +1’s, 0’s and -1’s by anonymity.

Suppose there are n elements in D, NSuppose there are n elements in D, N+1+1(D) and N(D) and N--

11(D) is the number of +1’s and -1’s in D (D) is the number of +1’s and -1’s in D correspondingly.correspondingly.

So the number of 0’s is n - NSo the number of 0’s is n - N+1+1(D) - N(D) - N-1-1(D).(D).

Therefore, f(D) is entirely determined by NTherefore, f(D) is entirely determined by N+1+1(D) (D) and Nand N-1-1(D) by anonymity.(D) by anonymity.


Suppose NSuppose N+1+1(D) = N(D) = N-1-1(D) and f(D) = r.(D) and f(D) = r.

ObviouslyObviouslyNN+1+1(D) = N(D) = N-1-1(D) = N(D) = N+1+1(-D) #1(-D) #1NN-1-1(D) = N(D) = N+1+1(D) = N(D) = N-1-1(-D). #2(-D). #2

And because f satisfies universal domain, so f is also And because f satisfies universal domain, so f is also defined at –D.defined at –D.

SinceSincef(-D) = -f(D) = -r by neutrality,f(-D) = -f(D) = -r by neutrality,andandf(-D) = f(D) = r by #1 and #2.f(-D) = f(D) = r by #1 and #2.

Combining above results, –r = r so r = 0.Combining above results, –r = r so r = 0.That is NThat is N+1+1(D) = N(D) = N-1-1(D) implies f(D) = 0.(D) implies f(D) = 0.


Suppose NSuppose N+1+1(D) > N(D) > N-1-1(D) where there are n elements in D,(D) where there are n elements in D,so that Nso that N+1+1(D) = N(D) = N-1-1(D) + m where 0 < m ≤ n - N(D) + m where 0 < m ≤ n - N-1-1(D).(D).

It will be proved that f(D) = +1 by mathematical induction below:It will be proved that f(D) = +1 by mathematical induction below:

D = (dD = (d11, d, d22, …, d, …, dnn). ). Basis: m = 1.Basis: m = 1. ∴ ∴ NN+1+1(D) = N(D) = N-1-1(D) + 1(D) + 1 ∴ There is at least one d∴ There is at least one dii = 1. = 1. Suppose Suppose D’ D’=(d=(d11’’, d, d22’’, …, d, …, dnn’’)), an i-variant determined by, an i-variant determined by d djj’=d’=djj if j≠i, and d if j≠i, and dii’=0. #1’=0. #1 f is defined at D and D’ by universal domain. f is defined at D and D’ by universal domain. Obviously Obviously NN+1+1(D’) = N(D’) = N-1-1(D’).(D’). ∴ f(D’) = 0 by step 2. #2∴ f(D’) = 0 by step 2. #2 ∴ f(D) = +1 by #1, #2 and positive responsiveness. ∴ f(D) = +1 by #1, #2 and positive responsiveness.

Induction: SupposeInduction: Suppose N N+1+1(D)=N(D)=N-1-1(D)+1 implies f(D)=+1.(D)+1 implies f(D)=+1. It has to be shown that It has to be shown that N N+1+1(D)=N(D)=N-1-1(D)+(m+1) implies f(D)=+1.(D)+(m+1) implies f(D)=+1. So suppose So suppose N N+1+1(D)=N(D)=N-1-1(D)+(m+1).(D)+(m+1). ∴ There is at least one d∴ There is at least one dii = 1. = 1. Suppose Suppose D’ D’=(d=(d11’’, d, d22’’, …, d, …, dnn’’)), an i-variant determined by, an i-variant determined by d djj’=d’=djj if j≠i, and d if j≠i, and dii’=0. #3’=0. #3 f is defined at D and D’ by universal domain. f is defined at D and D’ by universal domain. Obviously Obviously NN+1+1(D’) = N(D’) = N-1-1(D’)+m.(D’)+m. ∴ f(D’) = 0 by induction hypothesis. #4∴ f(D’) = 0 by induction hypothesis. #4 ∴ f(D) = +1 by #1, #2 and positive responsiveness. ∴ f(D) = +1 by #1, #2 and positive responsiveness.

Summary:Summary:Follow an analogous derivation, an assertion “when Follow an analogous derivation, an assertion “when NN+1+1(D) < N(D) < N-1-1(D), f(D) = -1” can be proved.(D), f(D) = -1” can be proved.So: So: If If NN+1+1(D) > N(D) > N-1-1(D), then f(D) = +1(D), then f(D) = +1 If If NN+1+1(D) < N(D) < N-1-1(D), then f(D) = -1(D), then f(D) = -1

Proof of May’s TheoryProof of May’s Theory

Summary of Proof:Summary of Proof:

From step 1, 2, and 3:From step 1, 2, and 3:If NIf N+1+1(D)=N(D)=N-1-1(D), then f(D)=0.(D), then f(D)=0.If NIf N+1+1(D)>N(D)>N-1-1(D), then f(D)=+1.(D), then f(D)=+1.If NIf N+1+1(D)<N(D)<N-1-1(D), then f(D)=-1.(D), then f(D)=-1.

These results just satisfy the formal These results just satisfy the formal definition of simple majority voting.definition of simple majority voting.So May’s theory is proved.So May’s theory is proved.

General Social Choice RulesGeneral Social Choice Rules

X: a nonempty set of alternatives.X: a nonempty set of alternatives.The elements of X must only be The elements of X must only be mutually incompatible.mutually incompatible.

v: agenda, v v: agenda, v ≠ ≠ Ø and v Ø and v ⊆ X, a set of ⊆ X, a set of alternatives that are currently alternatives that are currently available.available.

N: a set of individuals.N: a set of individuals.


xRxRiiy: i ∈ N; x, y ∈ X; individual i determines y: i ∈ N; x, y ∈ X; individual i determines alternative x to be at least as good as alternative x to be at least as good as alternative y; or i weakly prefers x to y.alternative y; or i weakly prefers x to y.

1. R1. Rii is reflexive: xR is reflexive: xRiix for all x ∈ X.x for all x ∈ X.

2. R2. Rii is complete: xR is complete: xRiiy or yRy or yRiix (or both) for x (or both) for all x, y ∈ X.all x, y ∈ X.

3. R3. Rii is transitive: For all x, y, z ∈ X, if both is transitive: For all x, y, z ∈ X, if both xRxRiiy and yRy and yRiiz then xRz then xRiiz.z.


xPxPiiy: xRy: xRiiy and not yRy and not yRiix; i strongly x; i strongly prefers x to y.prefers x to y.

yPyPiix: yRx: yRiix and not xRx and not xRiiy; i strongly y; i strongly prefers y to x.prefers y to x.

xIxIiiy: xRy: xRiiy and also yRy and also yRiix; i is indifferent x; i is indifferent between x and y.between x and y.


Profile: an assignment of one Profile: an assignment of one preference relation to each preference relation to each individual.individual.

C(v): the elements chosen from C(v): the elements chosen from agenda v by choice function C.agenda v by choice function C.(i) C(v) (i) C(v) ⊂⊂ v; v;(ii) C(v) (ii) C(v) ≠ ≠ Ø.Ø.


Social Choice Rule:Social Choice Rule:

A social choice rule assigns to each A social choice rule assigns to each of a collection of profiles a of a collection of profiles a corresponding choice function.corresponding choice function.

social choice rule,f

profile of preferences,u

choice function,C = f(u)

agenda,v

chosen set,Cu(v)

Standard Domain ConstraintStandard Domain Constraint Standard domain constraint includes:Standard domain constraint includes:

i) there are at least three alternatives in X; i) there are at least three alternatives in X;

ii) there are at least three individuals in N; ii) there are at least three individuals in N;

iii) the social choice rule has as domain all iii) the social choice rule has as domain all logically possible profiles of preference orderings logically possible profiles of preference orderings on X;on X;

iv) each choice function that is an output of the iv) each choice function that is an output of the rule has in its domain all finite nonempty rule has in its domain all finite nonempty agendas.agendas.

Pareto ConditionPareto Condition

Weak Pareto Condition:Weak Pareto Condition:

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


Strong Pareto Condition:Strong Pareto Condition:

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


Example:Example:

For agenda: 1: (x yFor agenda: 1: (x y11) y) y22

2: x y 2: x y11 y y22

3: x (y 3: x (y11 y y22))

In Weak Pareto Condition:In Weak Pareto Condition: y y22 ∉∉ C Cuu(v)(v)In Strong Pareto Condition:In Strong Pareto Condition: y y11, y, y22 ∉∉ C Cuu(v)(v)


X is Pareto-superior to y at profile u = (RX is Pareto-superior to y at profile u = (R11, , RR22, …, R, …, Rnn) if:) if:

(i) xR(i) xRiiy for all individuals i in N;y for all individuals i in N;(ii) xP(ii) xPiiy for at least one individual i in N.y for at least one individual i in N.

Alternatives for which there are no Alternatives for which there are no available Pareto-superior alternatives are available Pareto-superior alternatives are called Pareto optimal.called Pareto optimal.

DictatorDictator

Weak DictatorWeak Dictator

Individual i is a weak dictator if for Individual i is a weak dictator if for every pair of alternatives, x and y, every pair of alternatives, x and y, every profile u = (Revery profile u = (R11, R, R22, …, R, …, Rnn) and ) and every agenda v, if xPevery agenda v, if xPiiy theny then

y y ∈∈ C Cuu(v) implies x (v) implies x ∈∈ C Cuu(v).(v).

DictatorDictator CoalitionCoalition

A subset S of the set N of all individuals is called a coalition. A subset S of the set N of all individuals is called a coalition.

Decisive CoalitionDecisive Coalition

For a social choice rule that maps u to CFor a social choice rule that maps u to Cuu, A coalition S is , A coalition S is called decisive for alternative x against alternative y if:called decisive for alternative x against alternative y if: for for ∀i: i ∈ S • xR∀i: i ∈ S • xRiiy;y; ∃j: j ∈ S • xP ∃j: j ∈ S • xPjjy;y; then ∀v: v ⊂ X, x ∈ v • y ∉ then ∀v: v ⊂ X, x ∈ v • y ∉ CCuu(v).(v).

If If ∀x, y: x, y ∈ X • ∀x, y: x, y ∈ X • S is decisive for alternative x against S is decisive for alternative x against alternative y, then we simply say S is decisive.alternative y, then we simply say S is decisive.

DictatorDictator

DictatorDictator

If a decisive coalition S = i, then i If a decisive coalition S = i, then i is a dictator.is a dictator.

Borda RulesBorda Rules Borda CountBorda Count

Assume that X is finite. Then associated Assume that X is finite. Then associated with any preference ordering Rwith any preference ordering Rii there is a there is a ranking function rranking function rii that associates an that associates an integer with each alternative: rinteger with each alternative: rii(x) is the (x) is the number of alternatives stictly preferred to number of alternatives stictly preferred to x. Given a profile u = (Rx. Given a profile u = (R11, R, R22, …, R, …, Rnn), there is ), there is a ranking function r given bya ranking function r given by

r(x) = r(x) = ∑∑iirrii(x).(x).

The value of r(x) is called Borda count of x.The value of r(x) is called Borda count of x.

Borda RulesBorda Rules

Global Borda RuleGlobal Borda Rule

CCuu(v) = x|r(x) (v) = x|r(x) ≤≤ r(y) for all y r(y) for all y ∈∈ v. v.

This rule has us choose from v those This rule has us choose from v those alternatives with minimal Borda alternatives with minimal Borda count.count.

Independence of Irrelevant AlternativesIndependence of Irrelevant Alternatives

If two profiles u, u’, restricted to an If two profiles u, u’, restricted to an agenda v are identical, then the agenda v are identical, then the choices made from that agenda choices made from that agenda should be the same:should be the same:

C Cuu(v) = C(v) = Cu’u’(v).(v).

Local Borda RulesLocal Borda Rules

Local Borda CountLocal Borda Count

Given a profile u = (RGiven a profile u = (R11, R, R22, …, R, …, Rnn), there is ), there is for each v:for each v: rrvv(x) = (x) = ∑∑iirrii

vv(x).(x).

Local Borda RuleLocal Borda Rule

CCuu(v)= x|r(v)= x|rvv (x) (x) ≤≤ r rvv (y) for all y (y) for all y ∈∈ v. v.

Transitive ExplanationTransitive Explanation Explanation:Explanation:

A choice function C is explainable if there exists a relation A choice function C is explainable if there exists a relation ΩΩ such thatsuch that

C(v) = C(v) = x x ∈∈ v | x v | xΩy for all y Ωy for all y ∈∈ v v..

Transitive Explanation:Transitive Explanation:A choice function C has transitive explainable if there is a A choice function C has transitive explainable if there is a reflexive, complete and transitive relation reflexive, complete and transitive relation ΩΩ such that such that

C(v) = C(v) = x x ∈∈ v | x v | xΩy for all y Ωy for all y ∈∈ v v..

We say a social choice rule has transitive explainable if at We say a social choice rule has transitive explainable if at every admissible profile u the associated Cevery admissible profile u the associated Cuu has a transitive has a transitive explainable.explainable.

Arrow’s Impossibility TheoremArrow’s Impossibility Theorem There does not exist any social choice rule There does not exist any social choice rule

satisfying all of:satisfying all of:

1. the standard domain constraint;1. the standard domain constraint;

2. the strong Pareto condition;2. the strong Pareto condition;

3. independence of irrelevant alternatives;3. independence of irrelevant alternatives;

4. has transitive explanations;4. has transitive explanations;

5. absence of a dictator.5. absence of a dictator.

Mechanism DesignMechanism Design

Implementing a social choice Implementing a social choice function f(ufunction f(u11, …, u, …, unn) using a game.) using a game.

Center (auctioneer) does not know Center (auctioneer) does not know the agents’ preferences.the agents’ preferences.

Agents may lie.Agents may lie. Goal is to design the rules of the Goal is to design the rules of the

game so that in equilibrium (sgame so that in equilibrium (s11, …, , …, ssnn), the outcome of the game is f(u), the outcome of the game is f(u11, , …, u…, unn).).

Mechanism DesignMechanism Design

Mechanism designer specifies the strategy sets SMechanism designer specifies the strategy sets Sii and how and how outcome is determined as a function of (soutcome is determined as a function of (s11, …, s, …, snn) ) (S (S11, …, , …, SSnn).).

VariantsVariants

Strongest: There exists exactly one equilibrium. Its outcome Strongest: There exists exactly one equilibrium. Its outcome is f(uis f(u11, …, u, …, unn).).

Medium: In every equilibrium the outcome is f(uMedium: In every equilibrium the outcome is f(u11, …, u, …, unn).).

Weakest: In at least one equilibrium the outcome is f(uWeakest: In at least one equilibrium the outcome is f(u11, …, , …, uunn).).

ReferencesReferences

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

social choice theory by shiyan li. history the theory of social choice and voting has had a long...

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