Social Choice and Mechanism Design - Part I.2. Part I.2: Social Choice Theory Summer Term 2011

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1 Social Choice and Mechanism Design Part I.2: Social Choice Theory Summer Term 2011 Alexander Westkamp April 2011

2 Introduction Two concerns regarding our previous approach to collective decision making: Why do we need a full social preference relation when we only want to choose some alternative? Individuals preferences are usually private information to individuals that has to be elicited from them. We ll now consider a situation where a society is interested in choosing an alternative and focus on the problem of finding out individual preferences.

3 Literature If you want to read more: MWG, Chapter 23 (only subsection on the Gibbard-Satterthwaite Theorem - will cover the remainder of this chapter in part II of the class) JR, Chapter 6.5 Austen-Smith, Banks: Positive Political Theory II, Ch. 2 (we only cover some parts of sections ) Moulin: Axioms of Cooperative Decision Making, Ch. 9& 10

4 Social choice problems Definition A social choice problem consists of a society of n individuals I, a set of alternatives X, and a profile R = (R i ) i I of individuals rational preferences over X. Aim: Aggregate individual preferences into one social choice.

5 Social Choice Functions Definition A social choice function (SCF) is a mapping f : R n X. A SCF, or voting rule, is a systematic method for transforming individual preferences into a social decision As before, we implicitly assume X and I do not vary, so that social choice problems can be described by a preference profile.

6 Examples 1. Condorcet consistent rules: An alternative x is a Condorcet winner at the profile R = (R 1,..., R n ), if {i I : xp i y} {i I : yp i x} for all y X. A social choice function is Condorcet consistent if it selects a Condorcet winner whenever it exists. Example: Copeland rule (see Moulin, Ch. 9)

7 Examples II 2. Scoring rules: Assume strict preferences and let s = (s t ) m t=1 be a non-increasing sequence of reals with s 1 > s m. Social choice function f is a scoring rule with respect to s if for all strict preference profiles R = (R 1,..., R n ), f (R)F s (R)y for all y X. Remark: There are preference profiles where the Condorcet winner is not selected by any scoring rule. 3. Dictatorial social choice functions: A social choice function f is dictatorial if there is some individual i such that f always chooses one of i s most preferred alternatives.

8 Social choice functions: The axiomatic method Reminder: The axiomatic method Propose a set of normatively appealing properties, or axioms, that any aggregation procedure should satisfy. Characterize the class of aggregation procedures that satisfy these properties (or show that they are incompatible). While our main focus will be the problem of finding out individual preferences, we will start by discussing some other axioms...

9 Properties of SCFs A SCF f is efficient, if there is no preference profile R = (R 1,..., R n ) R n such that for some y X \ {f (R)}, yr i f (R) for all i I, with at least one strict preference. is weakly monotonic, if whenever R and R are such that for all i I, (i) {y X : f (R)R i y} {y X : f (R)R i y}, and (ii) for all y, z X \ {f (R)}, yr i z if and only if yr i z, then f (R) = f (R ). satisfies participation, if for all preference profiles (R 1,..., R n ) R n, f (R 1,..., R n, R n+1 )R n+1 f (R 1,..., R n ) for all R n+1 R.

10 Properties of SCFs II All strict scoring rules are efficient, the Copeland rule is one efficient Condorcet consistent rule (Why?). All strict scoring rules satisfy weak monotonicity, the Copeland rule is one weakly monotonic Condorcet consistent rule (Why?). A popular voting procedure that does not in general satisfy weak monotonicity: Plurality with runoff. All strict scoring rules satisfy participation (Why?). Condorcet consistent rules may violate participation.

11 Properties of SCFs II All strict scoring rules are efficient, the Copeland rule is one efficient Condorcet consistent rule (Why?). All strict scoring rules satisfy weak monotonicity, the Copeland rule is one weakly monotonic Condorcet consistent rule (Why?). A popular voting procedure that does not in general satisfy weak monotonicity: Plurality with runoff. All strict scoring rules satisfy participation (Why?). Condorcet consistent rules may violate participation. So strict scoring rules have a lot of appealing properties...

12 Properties of SCFs III... but individuals might have an incentive to misrepresent their preferences! Why is this a problem? Asymmetries in strategic sophistication of individuals (Fairness motive)! Desirable properties of scoring rules only guaranteed to hold with respect to reported preferences!

13 Properties of SCFs IV Definition A SCF f is strategyproof on R n if for all individuals i I and for all preference profiles R = (R 1,..., R n ) R n, f (R)R i f (R i, R i) for all R i R. If for some preference profile R R n and some individual i, there is a preference relation R i R such that f (R i, R i)p i f (R), say that i can manipulate f at R. A SCF is strategyproof if no individual can ever manipulate it. Unfortunately, of the SCFs discussed so far, only the dictatorial ones can be strategyproof. We ll now see that this is not a coincidence...

14 The Gibbard-Satterthwaite Theorem Theorem (Gibbard (1973), Satterthwaite (1975)) Suppose that X is a finite set with at least three alternatives and f (R n ) = X (i.e. for all x X, there exists R R n such that f (R) = x). If f is strategyproof on R n, then it is dictatorial.

15 Discussion Theorem shows that dictatorial social choice rules are characterized by strategyproofness. Theorem also holds provided that the image of f contains at least three elements. (Why?) A corollary: If m 3, an efficient and strategyproof SCF must be dictatorial. (Why?)

16 Proof We ll only show the following: Let P denote the set of all rational and strict preference relations on X. Suppose X is finite with X 3 and f (P n ) = X. If f is strategyproof on P n, then f is dictatorial. See MWG Chapter 23 C on how to extend this result to the general case where indifferences are allowed.

17 Proof - Part I Step 1: If f is strategyproof, it has to be strongly monotonic. Given a preference profile R, define individual i s lower contour set of x X at R by L i (x, R) := {y X : xr i y}. A preference profile R a monotonic transformation of R at x X, if L i (x, R) L i (x, R ) for all i I. A SCF f is strongly monotonic if f (R) = f (R ) whenever R is a monotonic transformation of R at f (R).

18 Proof - Remarks on Part I Note that the proof that strategyproofness implies strong monotonicity does not depend on the assumption that there are at least three alternatives. In the next exercise session we will show that strong monotonicity implies strategyproofness (thus the two properties are equivalent). Strong monotonicity is also known as Maskin monotonicity. This condition plays a crucial role in the theory of Nash implementation pioneered by Eric Maskin in the 1970s (see Eric Maskin, Nash equilibrium and welfare optimality, Review of Economic Studies 66 (1999), 23 38).

19 Proof - Part II Step 2: Suppose f is strongly monotonic, X 3 and f (P n ) = X. Then f has to be efficient. Note: For general social choice problems, strategyproofness implies efficiency. We will see later on, that this does not always hold when we consider restricted domains.

20 Proof - Part III Step 3: If f is strongly monotonic and efficient, it has to be dictatorial. Given a strict preference profile R = (R 1,..., R n ) and Y X, define the preference profile R Y = (R Y 1,..., RY n ) by setting, for all i I, y if and only if xp i y whenever either {x, y} Y or {x, y} X \ Y, and xp Y i xp Y i y for all x Y and all y X \ Y. Define the social welfare function induced by f by xf f p (R)y if and only if x = f (R {x,y} ). Show that this SWF satisfies all of Arrow s assumptions and must hence be dictatorial.

21 Relation to Arrow s Theorem At first glance, the negative conclusion of the Gibbard-Satterthwaite Theorem might seem more severe than Arrow s Theorem (we used Arrow s result to prove it) but the two results are actually two sides of the same coin! More precisely, for the case of strict preferences there is a one-to-one relationship between SWFs that satisfy Arrow s axioms and SCFs that satisfy the assumptions of the Gibbard-Satterthwaite Theorem.

22 Relation to Arrow s Theorem For the following, fix some finite set X with X = m 3. Let P denote the set of all strict and rational preference relations over X. A strict Arrovian aggregator is a social welfare function F : P n P that satisfies IIA and WP. The social choice function induced by strict Arrovian aggregator F is the mapping C F : P n X which for each R P n chooses the most preferred alternative according to F (R).

23 Relation to Arrow s Theorem Theorem (Satterthwaite (1977)) (i) For any strict Arrovian aggregator F, C F is a strategyproof on P n and C F (P n ) = X. (ii) If F and F are two strict Arrovian aggregators such that C F = C F, then F = F. (iii) For any social choice function f that is strategyproof on P n and satisfies f (P n ) = X, there is a strict Arrovian aggregator F such that f = C F.

24 Relation to Arrow s Theorem Theorem shows that for the case of strict preferences, Arrow s Theorem and the Gibbard-Satterthwaite Theorem are equivalent. In particular, Arrow s conditions ensure that the social choice process cannot be manipulated. Question: Is it true then that whenever a social welfare function violates one of Arrow s conditions, the induced social choice function is not strategyproof? For another illustration of the close relationship between Arrow and Gibbard-Satterthwaite, see Phil Reny,Arrow s Theorem and the Gibbard-Satterthwaite Theorem: A unified approach, Economics Letters 70 (2001),

25 Strategyproof social choice in restricted environments What can we say about strategyproofness in restricted environments? In the exercise sessions you will (hopefully) show that: A plurality voting rule defines a strategyproof SCF if there are only two alternatives. For the single-peaked domain, the Condorcet consistent SCF is strategyproof if the number of individuals is odd.

26 Conclusion to Part I We saw that ideal (in terms of incentive or aggregation properties) aggregation procedures cannot be expected to exist in general environments. But the designer of an aggregation procedure/mechanism often has some knowledge about the environment. For example, the designer may know how individuals value money and use monetary transfers to induce truthtelling.... individuals may not always possess sufficiently detailed information to identify profitable manipulations. In particular, truthful preference revelation may still be an equilibrium for some non-dictatorial SCFs that perform reasonably well.

27 Conclusion to Part I In parts II and III of this course, you will see that such considerations yield a more positive perspective on preference aggregation.