Decision TheoryDecision Theory

CHOICE(Social Choice)

Professor : Dr. Liang

Student : Kenwa Chu

22

6.1 The Problem of Social 6.1 The Problem of Social ChoiceChoice

what are the relationships between the demands of rationality and those of justice or fairness

characterized problem of social choiceA group of individuals has two or more A group of individuals has two or more

alternative group actions or policies alternative group actions or policies The members of the group (henceforth, The members of the group (henceforth,

called citizens) have their own called citizens) have their own preferences concerning the group choice preferences concerning the group choice

33

O1-O8 (CH2)O1-O8 (CH2) O1: If xPy, then not yPx. O2: If xPy, then not xIy. O3: If xIy, then not xPy and not yPx. O4: xPy or yPx or xIy, for any relevant

outcomes x and y. O5: If xPy and yPz, then xPz. O6: If xPy and xIz, then zPy. O7: If xPy and yIz, then xPz. O8: If xIy and yIz, then xIz.

44

collectivecollective group preference profile

A set of such individual orderings-one A set of such individual orderings-one for eachfor each

generate an ordinal utility scalegenerate an ordinal utility scaleSatisfies (conditions 01-08 )

collective choice rulea method that operates on preference a method that operates on preference

profiles and yields a social ranking of profiles and yields a social ranking of the alternativesthe alternatives

55

group preference profilegroup preference profile _EX._EX.

same ordering one preference profile

66

social welfare functionsocial welfare function A collective choice rule

a method that operates on preference profila method that operates on preference profiles and yields a social ranking of the alternates and yields a social ranking of the alternatives.ives.

SWFs (social welfare functions) assume that both the number of alternatives and tassume that both the number of alternatives and t

he number of citizens are finite and will concentrathe number of citizens are finite and will concentrate on a a special type of collective choice rulese on a a special type of collective choice rules

operate on all the preference profiles possible for a operate on all the preference profiles possible for a given set of alternatives and citizensgiven set of alternatives and citizens

yield social orderings that satisfy conditions O1-O8.yield social orderings that satisfy conditions O1-O8.

77

voting paradox voting paradox

Majority prefer a to bprefer a to b prefer b to cprefer b to c prefer c to aprefer c to a

cyclical social orderings violating the condition that requires not aPc if violating the condition that requires not aPc if

cPacPa not a social welfare function

88

social orderings social orderings mechanismsmechanisms

condition Uunrestricted domain conditionunrestricted domain conditionrequirementrequirement

it produces a social ordering for every preference profile

condition DSWF not be dictatorialSWF not be dictatorial

majority rulenot satisfy condition Unot satisfy condition UBut it satisfies condition DBut it satisfies condition D

99

PROBLEMSPROBLEMS Does a collective choice rule, which

merely selects one alternative from a set of alternatives and declares it to be the first choice, count as an SWF?

Suppose there are six citizens and three alternatives and the collective choice rule in use is the following: To decide how to rank a pair of alternatives in the social ordering, roll a die and take the ranking of the citizen whose number comes up as the social ranking of that pair.

Will this method necessarily implement an SWF?Will this method necessarily implement an SWF? Could this method yield a dictatorial SWF? Could this method yield a dictatorial SWF?

1010

6-2. Arrow's Theorem6-2. Arrow's Theorem

1111

6-2a. Arrow's Conditions6-2a. Arrow's Conditions narrow the field of SWFs (Only U,D)

do not agree with one and the same citizen on each do not agree with one and the same citizen on each profileprofile

SWFs impose the same social ordering no matter hSWFs impose the same social ordering no matter how the citizens happen to feel about the alternativeow the citizens happen to feel about the alternativess

Other conditions CSCS PAPA PP II

Arrow's theorem negative result: no SWF can satisfy five quite reasonegative result: no SWF can satisfy five quite reaso

nable conditions-two of which happen to be conditnable conditions-two of which happen to be conditions U and D ions U and D

1212

Condition CSCondition CS citizen’s sovereignty condition Requires

each pair of distinct alternatives x and y there is each pair of distinct alternatives x and y there is at least one preference profile for which the SWF at least one preference profile for which the SWF yields a social ordering that ranks x above yyields a social ordering that ranks x above y

EX. at least one preference profile for which the SWF at least one preference profile for which the SWF

ranks macaroni above beef ranks macaroni above beef the social ordering could not be

imposed on them in a way entirely independent of their own preferences

1313

Condition PACondition PA positive association between individual and social

values Requires

if an SWF ranks an alternative x above an alternative y for a if an SWF ranks an alternative x above an alternative y for a given profile, it must also rank x above y in any profile that is given profile, it must also rank x above y in any profile that is exactly like ,the original one except that one or more citizens exactly like ,the original one except that one or more citizens have moved x up in their own rankingshave moved x up in their own rankings

any SWF that satisfied PA and socially ordered chicken above beef for profile 1 would do the same for profile 2

1414

Condition PCondition P Pareto condition

also be called the condition of unanimity also be called the condition of unanimity rulerule

Requiresthe SWF must rank x above y for a given the SWF must rank x above y for a given

profile if every citizen ranks x above y in profile if every citizen ranks x above y in that profile. It couldthat profile. It could

Relationship (P-CS-PA)

1515

Condition P-CS-PACondition P-CS-PA The Pareto condition implies the

citizens' sovereignty condition Condition P does not imply

condition PA

1616

Condition P-CS-PA Condition P-CS-PA _EX._EX.

not meet condition PA conditions PA and CS together

imply condition P

1717

Condition ICondition I independence-of-irrelevant-

alternatives Requires

social welfare functions to obtain social social welfare functions to obtain social orderings by comparing alternatives two orderings by comparing alternatives two at a time taken in isolation from the at a time taken in isolation from the other alternativesother alternatives

1818

Condition I Condition I _EX._EX.

1919

condition I condition I _extent_extent Arrow's theorem (and social choice the

ory in general) abstracts from the mechanisms used in deriviabstracts from the mechanisms used in derivi

ng social orderings ng social orderings does not distinguish between SWFs that yield tdoes not distinguish between SWFs that yield t

he same outputs for each possible input.he same outputs for each possible input. Condition I

If each citizen ranks the alternatives x and y in If each citizen ranks the alternatives x and y in the same order in the preference profiles P1 athe same order in the preference profiles P1 and P2, x and y must be in the same order with rnd P2, x and y must be in the same order with respect to each other in the social orderings thespect to each other in the social orderings that the SWF yields for P1 and P2. at the SWF yields for P1 and P2.

2020

condition I condition I _excludes rank-order _excludes rank-order methodsmethods

rank-order methodeach alternative receives six pointseach alternative receives six pointsranked as socially indifferentranked as socially indifferent

2121

condition I condition I _excludes rank-order _excludes rank-order methodsmethods

a and b stand in the same respective order in both profiles

Condition I requires the SWF to rank a and b alike in both social rankingsthe SWF to rank a and b alike in both social rankings

the rank-order method used to generate these two tables (and any other SWF giving rise to them) fails to satisfy condition I

2222

THE PARETO LEMMATHE PARETO LEMMA Any SWF that satisfies conditions PA, CS, and I also

satisfies condition P Proof

Assume Assume the SWF satisfies conditions PA, CS, and I. P1 is a profile in which every citizen ranks alternative x above

another y We will suppose for P1 the SWF does not socially rank x above We will suppose for P1 the SWF does not socially rank x above

y and derive a contradictiony and derive a contradiction there must be some profile P2 different from P1 for which the there must be some profile P2 different from P1 for which the

SWF socially ranks x above y, because CS is in force. SWF socially ranks x above y, because CS is in force. Furthermore, P1 and P2 must differ in their placement of x and Furthermore, P1 and P2 must differ in their placement of x and y. y.

some in P2 do not prefer x to ysome in P2 do not prefer x to y the placement of x and y in P1 can be obtained from that of P2 the placement of x and y in P1 can be obtained from that of P2

by moving x up in the ranking of one or more citizens. Hence by moving x up in the ranking of one or more citizens. Hence by PA, the social ordering for it must rank x above y. by PA, the social ordering for it must rank x above y.

But then by condition I, the social ordering for P1 must also But then by condition I, the social ordering for P1 must also rank x above y. rank x above y.

And that contradicts the assumption that the SWF failed to And that contradicts the assumption that the SWF failed to socially order x above y for P1.socially order x above y for P1.

2323

PROBLEMSPROBLEMS Prove that every dictatorial SWF must satisfy condition P. Explain why we cannot prove the Pareto lemma by

applying condition PA directly to P2 and dispensing, thereby, with the use of condition I.

Suppose that in Heaven God keeps a book that lists for each person and for each possible alternative God's assessment of the value of that alternative to that person-on a scale of -1,000 to +1,000. Now consider an SWF that works as follows: To decide how to rank two alternatives x and y, we first use God's book to find their values for the citizens of the particular society at hand. Then we sum the values for x and y, rank x above y if its sum is greater, y above x if its sum is greater, and rank them as indifferent otherwise. Would this SWF necessarily violate condition I? Condition D? Condition CS?

Prove that the following condition implies condition I: If P1 and P2 are two profiles and S is any subset of the set of alternatives, then if the citizens' relative rankings of the members of S are the same for P1 and Pt, the SWF places the members of S in the same relative positions in both P1 and P2.

Prove that condition I implies the condition of exercise 4.

2424

6-2b. Arrow's Theorem 6-2b. Arrow's Theorem and Its Proofand Its Proof

2525

Proof Arrow's Theorem from PA Proof Arrow's Theorem from PA

THEOREM Where three or more alternatives and two or more Where three or more alternatives and two or more

citizens are involved, there is no SWF that meets all citizens are involved, there is no SWF that meets all five conditions CS, D, I, PA, and U.five conditions CS, D, I, PA, and U.

If no SWF meets conditions D, I, U, and P, neither can any SWF meet conditions CS, D, I, PA, and U.For if it satisfied the latter, it would automatically satisfy the former, given the Pareto lemma.

Consequently, we can prove Arrow's theorem by proving its Pareto version, namely, that no SWF simultaneously meets D, I, U, and P: If an SWF satisfies conditions U, P, and I, it must be dictatorial.

2626

several definitionsseveral definitions A set of citizens is decisive for x over

y just in case x is socially preferred to y whenever each member of the set prefers x to y.

A citizen is a dictator for x over y just in case the set consisting of him alone is decisive for x over y.

A citizen is a dictator if and only if he is decisive for every pair of distinct alternatives.

An SWF is dictatorial just in case some citizen is a dictator under it.

2727

LEMMA 1LEMMA 1 For any set of citizens and any

pair of distinct alternatives there is at least one decisive set.

PROOFThe set of all citizens is decisive for every The set of all citizens is decisive for every

pair of alternatives. For if every citizen pair of alternatives. For if every citizen prefers an alternative x to another one y, prefers an alternative x to another one y, then, by condition P, society prefers x to then, by condition P, society prefers x to y. So there is at least one set that is y. So there is at least one set that is decisive for x over y.decisive for x over y.

2828

several definitions several definitions _con._con.

A set of citizens is almost decisive for x over y just in case the social ordering ranks x above y when (a) all members of the set do and (b) all members outside prefer y to x.

A citizen is almost decisive for x over y if and only if the set consisting of him alone is almost decisive for x over y.

2929

LEMMA 2LEMMA 2 There is a citizen who is almost decisive for

some pair of alternatives PROOF

By lemma 1 there are decisive sets for each pair of By lemma 1 there are decisive sets for each pair of alternatives. alternatives.

there must be sets that are almost decisive for each there must be sets that are almost decisive for each alternative as well. alternative as well.

Since the number of citizens and alternatives is finite, there Since the number of citizens and alternatives is finite, there must be at least one nonempty set that is almost decisive must be at least one nonempty set that is almost decisive for some pair of alternatives but that has no nonempty for some pair of alternatives but that has no nonempty subsets that are almost decisive for any alternatives.subsets that are almost decisive for any alternatives.

We can find such a set by starting with society as a whole, We can find such a set by starting with society as a whole, which we already know to be almost decisive for every pair, which we already know to be almost decisive for every pair, and proceed to check all sets obtained from it by deleting and proceed to check all sets obtained from it by deleting one member, and so on, until we find one with the desired one member, and so on, until we find one with the desired property.property.

Let M be such a minimal almost decisive set and let it be Let M be such a minimal almost decisive set and let it be almost decisive for x over y. Since M is nonempty, at least almost decisive for x over y. Since M is nonempty, at least one citizen belongs to it. Let J be such a citizen. We will one citizen belongs to it. Let J be such a citizen. We will prove that only J belongs to M. That will show that J is almost prove that only J belongs to M. That will show that J is almost decisive for x over y.decisive for x over y.

3030

LEMMA 2 LEMMA 2 _con._con.

assume more than one citizen belongs to M and derive a contradictionmore than one citizen belongs to M and derive a contradiction

Alternatives: x, y, z set of citizens: J, M-J only alternative left is that of society preferring z to x, and

that again leads to the contradictory conclusion that J is almost decisive for z over y. Thus we have derived a contradiction from our assumption that M did not consist of J alone.

3131

LEMMA 3LEMMA 3 Any citizen who is almost decisive for

a single pair of alternatives is decisive for every pair of alternatives.

ROOF Assume that J is a citizen who is almost Assume that J is a citizen who is almost

decisive for x over y.decisive for x over y. We will show that he is decisive for all pairs of We will show that he is decisive for all pairs of

distinct alternatives.distinct alternatives. These pairs may be divided into seven cases: x These pairs may be divided into seven cases: x

over y, y over x, x over a, a over x, y over a, a over y, y over x, x over a, a over x, y over a, a over y, and a over by where a and b are over y, and a over by where a and b are alternatives distinct from each other and from alternatives distinct from each other and from x and y.x and y.

3232

Case: x over aCase: x over a Assume : a is an alternative distinct from x and y This is to indicate that no information is given about the

relative ordering of x and a in the rankings of the remaining members of society.

each citizen but J ranks y above both x and a, but each might place the latter two in any order independently of his fellow citizens.

everyone- J included-ranks y over a and condition P holds, society ranks y over a

the ordering condition holds, society must also rank x over a since condition I is in force. Thus whenever J prefers x to a,

society' does

3333

Case: a over yCase: a over y Since every citizen ranks a over x, society does (by

condition P) Society must also rank x over y, because J is almost

decisive for x over y by the ordering condition it follows that society

ranks a over y.

3434

Case: y over aCase: y over a By condition P society prefers y to x. Since we have already established that J is decisive for x

over a, we can conclude that society prefers x to a. But then by the ordering condition, it must prefer y to a. Condition I then lets us conclude that J is decisive for y

over a.

3535

Case: a over xCase: a over x The a-over-y case permits us to infer ( 推論 )that soc

iety prefers a to y; condition P results in its preferring y to x.

The argument then proceeds as in the previous cases to conclude that J is for a over x.

3636

Case: x over yCase: x over y Let a be any alternative distinct

from x and y and consider any profile in which J prefers x to a and a to y. By the x-over-a case, society prefers x to a. By the a-over-y case, society prefers a to y. We then proceed as usual to infer that J is decisive for x over y.

3737

Problems (Case Study)Problems (Case Study) Case: y over x Case: a over b