Lecture 13: Social Choice
Advanced Microeconomics II
Yosuke YASUDA
Osaka University, Department of Economics
Social Choice and Arrow (1951)
With its publication in 1951,Social Choice and Individual Values initiated the modern theory of social choice, the study of how a society should choose among its various options based on the preferences of the individual members of society.
In a capitalist democracy there are essentially two methods by which social choices can be made: voting, typically used to make “political” decisions, and the market mechanism, typically used to make “economic” decisions.
The methods of voting and the market are methods of amalgamating the tastes of many individuals in the making of social choices.
The problem of achieving a social maximum derived from individual desires is precisely the problem which has been central to the field of welfare economics.
Aggregation of Preferences
A social choice problem arises whenever any group of individuals must make a collective choice from a set of alternatives.
Arrow (1951) has offered a systematic framework for thinking about collective/social choice problem: his abstract formulation of the social choice problem makes it very widely applicable.
He begins with a society and a set of social alternatives X (possible options from which society must choose), which, depending on the context, could be almost anything.
Ideally, we would like to be able to compareany two alternatives in X from a social point of view, and we would like those binary comparisons to beconsistent.
Paradox of Voting (1)
The following paradox of voting, calledCondorcet’s paradox illustrates that the familiar method of majority voting can fail to satisfy the transitivity requirement on social preferenceR.
Let X={a, b, c} be a set of alternatives.
Consider a society that consists of three members: 1,2, and3.
Their rankings of X are
aP1bP1c, bP2cP2a, and cP3aP3b.
According to the majority rule,
aP b since awould get two votes while bwould get one.
bP c sinceb would get two votes whilec would get one. cP asince cwould get two votes while awould get one.
This clearly conflicts with the transitivity of the social preferenceP.
Paradox of Voting (2)
In this example, the mechanism of majority rule is “complete” in that it is capable of giving a best alternative in every possible pairwise comparison of alternatives inX.
The failure of transitivity, however, means that within this set of three alternatives, no single best alternative can be determined by majority rule.
Requiring completeness and transitivity of the social
preference relation implies that it must be capable of placing every element inX within a hierarchy from best to worst.
The kind of consistency required by transitivity has, therefore, considerable structural implications.
Basic Model
A basic model of social choice consists of the following:
X A set of social alternatives that are mutually exclusive.
N A finite set of individuals (denote the number of elements in N byn).
Ri (or %i) An individuali’s preference relation onX (an binary relation satisfying completeness and transitivity. → Let Pi (ori) andIi (or∼i) be the associated relations
of strict individual preference and indifference, respectively. R (or%) A social preference relation on X.
Profile An n-tuple of orderings(R1, . . . , Rn) interpreted as a certain “state of society”.
Social Welfare Function (SWF) A function that assigns a single social preference relation R to every profile.
R=f(R1, . . . , Rn)
Arrow’s Requirements of the SWF (1)
Unrestricted Domain (UD) The domain off must include all possible combinations of individual preference relations onX.
Weak Pareto Principle (WP) For any pair of alternativesx andy in X, if xPiy for all i, thenxP y.
Independence of Irrelevant Alternatives (IIA) Let
R=f(R1, . . . , RN),R˜=f( ˜R1, . . . ,R˜N), and let xand y be any two alternatives inX. If each individual iranksx versus y under Ri the same way that he does underR˜i, then the social ranking of xversus y is the same under R andR.˜
Non-dictatorship (ND) There is no individualisuch that for all
xand y inX,xPiy impliesxP y regardless of the preferencesRj
Arrow’s Requirements of the SWF (2)
UD says thatf is able to generate a social preference ordering regardless of what the individuals’ preference relations happen to be. It formalizes the principle that the ability of a
mechanism to make social choices should not depend on society’s members holding any particular sorts of views. WPis very straightforward, and one that economists, at least, are quite comfortable with. It says society should preferx toy if every single member of society prefers x to y.
IIA is perhaps the trickiest to interpret, so read it over
carefully. In brief, the condition says that the social ranking of x andy should depend only on the individual rankings of x andy. Note that the individual preferencesRi andR˜i are allowed to differ in their rankings over pairs other than x, y. ND is a very mild restriction indeed. It simply says there should be no single individual who “gets his way” on every single social choice, regardless of the views of everyone else in society.
Arrow’s Impossibility Theorem
Arrow established that the proposed four conditions that might be considered minimal properties the social welfare function should possess cannever be compatible.
Theorem 1 (Arrow’s Impossibility Theorem)
If there are at least three social states inX, then there is no social welfare functionf that simultaneously satisfies transitivity,UD, WP,IIA, and ND.
Idea of the simplest proof by Geanakoplos (1996, 2005)
The strategy of the proof is to show thattransitivity,UD,WP, andIIA imply the existence of a dictator. Consequently, if transitivity,UD,WP, andIIA hold, thenND must fail to hold, and so no social welfare function can satisfy all five conditions.
Proof Sketch (1): Extremal Lemma
Lemma 2 (Extremal Lemma)
Let alternativeb be chosen arbitrarily. At any profile in which every voter puts alternativebat the very top or very bottom of his ranking of alternatives, society must as well.
Proof.
1 Suppose to the contrary that for such a profile and for distinct
a,b,c, the social preference putaRb andbRc.
2 By IIA, this would continue to hold even if every individual
moved c abovea, because that could be arranged without disturbing any abor cbvotes (since boccupies an extreme position in each individual’s ranking).
3 By transitivity the social ranking would then continue to put
aRc, but byWPit would also putcP a, a contradiction, proving the lemma.
Proof Sketch (2): Existence of Pivotal Voter
Lemma 3 (Existence of Pivotal Voter)
There is a votern∗ =n(b)who is extremely pivotal in the sense that by changing his vote at some profile he can moveb from the very bottom of the social ranking to the very top.
Proof.
1 Let each voter putb at the very bottom of his (otherwise
arbitrary) ranking of alternatives.
2 By WP, society must as well, i.e.,xP b for anyx(6=b)∈X.
3 Let the voter1,2, . . . , N successively moveb from the bottom
to the very top, leaving the other relative rankings in place.
4 Let n∗ be the first voter whose change causes the social
ranking of bto change. (Note a change must occur by WP.)
Proof Sketch (3): Pivotal Voter Becomes Dictator
Pivotal ⇒ Dictator A pivotal votern∗ must be a dictator.
Proof.
1 Let n∗ move aaboveb, so thataPn∗bPn∗c, and let all other agents n6=n∗ arbitrarily rearrange their relative rankings ofa andc while leaving bin its extreme position.
2 Then, by IIA, the society would necessarily putaP b (ab votes
are as in the profile where n∗ put bat the bottom), andbP c (bcvotes are as in the profile wheren∗ putb at the top).
3 By transitivity, society must put aP c.
4 By IIA, the social preference over acmust agree withn∗
whenever aPn∗c.
5 The above establish thatn∗ is a dictator over any pairac. We
can also show that n∗ is a dictator over every pairab.
Gibbard-Satterthwaite Theorem
An SWF specifies a preference relation for every profile. Asocial choice function (SCF)attaches an alternative to every profile.
The most striking theorem proved in this framework is the Gibbard-Satterthwaite theorem.
Theorem 4
Any social choice functionC satisfying the condition that it is never worthwhile for an individual to misrepresent his preferences (called strategy-proofness), namely, it is never that
C(R1, . . . ,R˜i, . . . , Rn)PiC(R1, . . . , Ri, . . . , Rn)
is a dictatorship.
