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Lecture 13: Social Choice

Advanced Microeconomics II

Yosuke YASUDA

Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp

January 20, 2015

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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.

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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 compare any two alternatives in X from a social point of view, and we would like those binary comparisons to be consistent.

This is not an easy problem at all. When we insist on transitivity as a criterion for consistency in social choice, certain well-known difficulties can easily arise.

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Paradox of Voting (1)

The following paradox of voting, called Condorcet’s paradox illustrates that the familiar method of majority voting can fail to satisfy the transitivity requirement on social preference R.

Let X = {a, b, c} be a set of alternatives.

Consider a society that consists of three members: 1, 2, and 3. Their rankings of X are

aP1bP1c, bP2cP2a, and cP3aP3b. According to the majority rule,

aP b since a would get two votes while b would get one. bP c since b would get two votes while c would get one. cP asince c would get two votes while a would get one. This clearly conflicts with the transitivity of the social preference P .

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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 in X.

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 in X within a hierarchy from best to worst. The kind of consistency required by transitivity has, therefore, considerable structural implications.

Q How can we go from consistent individual views to a social view that is consistent and respects certain basic values on matters of social choice that are shared by members of the community?

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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 by n).

Ri (or %i) An individual i’s preference relation on X (an binary relation satisfying completeness and transitivity.

→ Let Pi (or ≻i) and Ii (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)

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Arrow’s Requirements of the SWF (1)

Unrestricted Domain (UD) The domain of f must include all possible combinations of individual preference relations on X.

Weak Pareto Principle (WP) For any pair of alternatives x and y in X, if xPiy for all i, then xP y.

Independence of Irrelevant Alternatives (IIA) Let

R= f (R1, . . . , RN), ˜R= f ( ˜R1, . . . , ˜RN), and let x and y be any two alternatives in X. If each individual i ranks x versus y under Ri the same way that he does under ˜Ri, then the social ranking of xversus y is the same under R and ˜R.

Non-dictatorship (ND) There is no individual i such that for all xand y in X, xPiy implies xP y regardless of the preferences Rj of all other individuals j 6= i.

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Arrow’s Requirements of the SWF (2)

UD says that f 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. WP is very straightforward, and one that economists, at least, are quite comfortable with. It says society should prefer x to y 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 and y should depend only on the individual rankings of x and y. Note that the individual preferences Ri and ˜Ri 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.

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Arrow’s Impossibility Theorem

Arrow established that the proposed four conditions that might be considered minimal properties the social welfare function should possess can never be compatible.

Theorem 1 (Arrow’s Impossibility Theorem)

If there are at least three social states in X, then there is no social welfare function f 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 that transitivity, UD, WP, and IIA imply the existence of a dictator. Consequently, if transitivity, UD, WP, and IIA hold, then ND must fail to hold, and so no social welfare function can satisfy all five conditions.

Rm See the detailed discussion on JR, pp.272-274.

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Proof Sketch (1): Extremal Lemma

Lemma 2 (Extremal Lemma)

Let alternative b be chosen arbitrarily. At any profile in which every voter puts alternative b at 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 put aRb and bRc.

2 By IIA, this would continue to hold even if every individual moved c above a, because that could be arranged without disturbing any ab or cb votes (since b occupies an extreme position in each individual’s ranking).

3 By transitivity the social ranking would then continue to put aRc, but by WP it would also put cP a, a contradiction, proving the lemma.

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Proof Sketch (2): Existence of Pivotal Voter

Lemma 3 (Existence of Pivotal Voter)

There is a voter n = n(b) who is extremely pivotal in the sense that by changing his vote at some profile he can move b from the very bottom of the social ranking to the very top.

Proof.

1 Let each voter put b at the very bottom of his (otherwise arbitrary) ranking of alternatives.

2 By WP, society must as well, i.e., xP b for any x(6= b) ∈ X.

3 Let the voter 1, 2, . . . , N successively move b 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 b to change. (Note a change must occur by WP.)

5 By extremal lemma, society must rank b at the very top.

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Proof Sketch (3): Pivotal Voter Becomes Dictator

Pivotal ⇒ Dictator A pivotal voter n must be a dictator. Proof.

1 Let n move a above b, so that aPnbPnc, and let all other agents n 6= n arbitrarily rearrange their relative rankings of a and c while leaving b in its extreme position.

2 Then, by IIA, the society would necessarily put aP b (ab votes are as in the profile where n put b at the bottom), and bP c (bc votes are as in the profile where n put b at the top).

3 By transitivity, society must put aP c.

4 By IIA, the social preference over ac must agree with n whenever aPnc.

5 The above establish that n is a dictator over any pair ac. We can also show that n is a dictator over every pair ab.

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Gibbard-Satterthwaite Theorem

An SWF specifies a preference relation for every profile. A social 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 function C 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, . . . , ˜Ri, . . . , Rn)PiC(R1, . . . , Ri, . . . , Rn) is a dictatorship.

Rm Gibbard-Satterthwaite theorem shows that, in a rich enough setting, it is impossible to design a non-dictatorial system in which social choices are made based on self-reported preferences without introducing the possibility that individuals can gain by lying.

参照

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