Social Choice and Mechanism Design Summer term 2010€¦ · Introduction: Mechanism Design Theory...

IntroductionPart I: Classical Social Choice Theory

I.1. The collectice preference approach

Social Choice and Mechanism DesignSummer term 2010

Konrad Mierendorff and Alexander Westkamp

University of Bonn

April 2010

Konrad Mierendorff and Alexander Westkamp Social Choice and Mechanism Design Summer term 2010



Introduction

Game Theory/Basic Microeconomics: Predict outcomes ofstrategic behavior in a given game/economic situation.

Social Choice and Mechanism Design: Having a theory of howpeople play games, how can we design games to promote certainsocial goals or maximize revenues?

Important theoretical questions with equally important practicalimplications for e.g. the design of (democratic) voting proceduresor the design of markets (e.g. by governmental regulations).




Introduction: Social Choice Theory

Social choice theory?

Societies often have to make decisions (regarding e.g. tax rates,distribution of public spending, relations to other societies,...) thataffect the welfare of its individual members

Individuals in a society are usually heterogenous and will not agreeon what the best decision is.

How can or should a society aggregate individual opinions into apublic decision?




Introduction: Social Choice Theory

We will ...

... analyze the pros and cons of well known voting rules (suchas majority voting)

... see that democratic preference aggregation is not ingeneral possible (Arrow’s theorem)

... learn how to use the “axiomatic method”

... see that democratic mechanisms cannot in general provide“strong” incentives for individuals to reveal privately knowninformation (the Gibbard-Satterthwaite theorem).




Introduction: Mechanism Design Theory

Mechanism Design Theory?

Economic decisions (regarding e.g. the allocation of public orprivate goods, wages,...) should and do often depend oninformation that is private to individuals.

(How) Can we induce individuals to reveal this information?




Introduction: Mechanism Design Theory

We will...

... see how efficient outcomes in a society of self-interestedindividuals can (sometimes) be achieved when monetarytransfers are possible

... how to maximize a seller’s expected revenue from sellingan object for which demand is uncertain

... that efficient bilateral trade between two privately informedparties is sometimes impossible (the Myerson-Satterthwaitetheorem)




Introduction: Applications

In recent years economic theorists have taken centre stage indesigning mechanisms for important practical applications:

Allocating radio spectrum licenses

Matching university graduates to their first professionalposition

Allocating seats at public schools among students

Exchange of live-donor organ transplants

In the last part of the course, we’ll try to cover (parts of) theunderlying theories.




Introduction: Organizational questions

Slides will be made available online for the first and third partof the course (for the second part we’ll closely follow atextbook).

There will be weekly or bi-weekly problem sets. Students willbe asked to present their own solutions to exercises in class.

There will be a voluntary midterm in May.

Final grade depends solely on performance in one final exam(most likely two hours and closed book).

More questions?




Part I: Classical Social Choice Theory




Part I: Classical Social Choice Theory

A society has to compare and ultimately choose from a (finite) setof “alternatives”, which affect the welfare of its individualmembers.

Examples: Choosing a representative, voting on tax rates,allocation of scarce resources,...

Two approaches (we will see that they are closely connected):

(i) Collective-preferencePossibility that individual preferences directly aggregated intocollective preference relation which is then maximized.

(ii) Game-theoreticIndividuals make indidivual choices of behavior that jointlydetermine collective choice.




Preferences: A quick reminder

For more detail consult Chapter 1 of Mas-Colell, Whinston, Green:Microeconomic Theory.

Let X be some finite set of mutually exclusive alternatives

A preference relation is a binary relation R on X (that is, asubset of X 2)

Write xRy if (x , y) ∈ R, and say that x is weakly preferredover y (according to R).

Write xPy if xRy but not yRx , and say that x is strictlypreferred over y .

Write xIy if xRy and yRx , and say that x is indifferent to y .

If R is a binary relation on X and Y ⊆ X , write R|Y for therestriction of R to Y .





A preference relation R is(i) reflexive if for all x ∈ X , xRx(ii) complete if for all x , y ∈ X either xRy or yRx (or both)(iii) transitive if xRy and yRz imply xRz(iv) rational if complete and transitive(v) quasi-transitive if xPy and yPz imply xPz(vi) acyclic if there is no sequence x1, . . . , xn such that

x1Px2, x2Px3, . . . , xnPx1

(vii) strict if for all x , y ∈ X such that x 6= y either xPy or yPx(equivalently, xRy implies xPy or x = y).

In the following, R denotes set of all rational preference relations(on X ) and B denotes set of all reflexive and complete preferencerelations.





Given a subset Y ⊆ X and a preference relation R on X , letC (R,Y ) denote the set of R-most preferred elements of Y , that is,C (Y ,R) = {x ∈ Y : xRy ,∀y ∈ Y }.

Fact: C (R,Y ) nonempty if and only if R|Y is acyclic. (Why?)




Social welfare functionsArrow’s TheoremSocial choice with two alternativesSingle-peaked preferences

I.1 The collective preference approach






If you want to read more:

Mas-Colell, Whinston, Green: Microeconomic Theory, Ch. 21

Austen-Smith, Banks: Positive Political Theory I, Ch. 2-4

Moulin: Axioms of Cooperative Decision Making, Ch. 11






A social choice problem consists of

a (finite) society of individuals I with |I | = n,

a set of alternatives X with |X | = m, and

a profile R = (Ri )i∈I of individuals’ rational preferences overX .

We will usually assume that X and I do not vary, so that a socialchoice problem is given simply by a profile of preferences R.

Aim: Aggregate individual preferences into one social preferencerelation.





Social welfare functions

A social welfare function (or preference aggregation rule) is amapping F : Rn → B that maps individual into social preferences.

Notation: F (R) is the social preference relation chosen when theprofile of individual preferences is R;FP(R) is the strict part of this relation, i.e. xFP(R)y if and only ifxF (R)y but not yF (R)x .

Note: Same aggregation procedure for all social choice problems.





Social welfare functions: Examples

1. Plurality Rule: xF pl(R)y if and only if|{i ∈ I : xPiy}| ≥ |{i ∈ I : yPix}|

Example: Bundestag (Einfache Mehrheit)

2. Majority Rule: xF m(R)y if and only if |{i ∈ I : xPiy}| ≥ n2

Example: Bundestag (Absolute Mehrheit necessary e.g. forappointing the chancellor)

Note: If Ri strict preference relation ∀i ∈ I , xF m(R)y if and onlyif xF pl(R)y .






3. Scoring rules: Assume strict preferences. Let s = (st)mt=1 be a

non-increasing sequence of reals with s1 > sm and set si (z ,R) = st

if and only if st is the tth most preferred element of X accordingRi .

The scoring rule induced by s is defined by xF s(R)y if and only if∑i∈I si (x ,R) ≥

∑i∈I si (y ,R).

Special case: If st = m − t + 1,∀t ∈ {1, . . . ,m}, we get theBorda count (denoted by F b(·)).

Examples (for scoring rules): Eurovision song contest, SportsRankings (e.g. best footbal player of the year)






4. Dictatorship: A SWF F is dictatorial if for some individuali ∈ I , xPiy implies xFP(R)y for all R ∈ Rn and all x , y ∈ X .

Examples:....

Note: An individual is a dictator only if she dictates strictpreferences in 100% of social decision problems.





Social welfare functions: The axiomatic method

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 satisfythese properties (or show that they are incompatible).

Axioms used to characterize a class of aggregation proceduresshould be indenpendent: For each of the axioms, thereshould be a SWF that does not satisfy this particular axiom,but does satisfy the remaining axioms.





Social welfare functions: Properties

A social welfare function F is

1. weakly Paretian (WP), if xFP(R)y whenever xPiy , ∀i ∈ i ,

2. independent of irrelevant alternatives (IIA), if wheneverR,R ′ and x , y ∈ X are such that R|{x , y} = R ′|{x ,y}, thenF (R)|{x ,y} = F (R ′)|{x ,y},

3. non-dictatorial (ND), if it is not dictatorial, and

4. rational (R), if F (R) ∈ R for all R ∈ Rn.





Social welfare functions: Properties

Let’s check whether the SWFs encountered so far satisfy theseproperties:

Plurality and majority rules satisfy WP, IIA and ND but not RWhy not R? Successive pairwise majority voting can generatecycles (Condorcet paradox).

Scoring rules satisfy ND and R for arbitrary scoring vectors,WP if s1 > s2 > . . . > s|X |, but not IIAWhy not IIA? Preference rank important.

If there is some individual i ∈ I such that for all R ∈ Rn,F (R) = Ri (a dictatorship in which the dictator also dictatessocial indifference), get a SWF that satisfies WP, IIA and R.





Arrow’s Theorem

What can we say about the class of social welfare functions thatsatisfy WP, IIA and R?

Theorem (Arrow (1951,1963))

Suppose that m ≥ 3.If a SWF satisfies WP, IIA and R, then it is dictatorial.





Arrow’s Theorem: Interpretation

Arrow’s theorem shows that WP, IIA and R characterizedictatorship.

If we view WP, IIA, R and ND as fundamental axioms thatany SWF should satisfy, Arrow’s theorem is an impossibilityresult: There is no SWF which satisfies all four propertiessimulatenously.

WP, IIA, R and ND are independent. (Why?)





Arrow’s Theorem: Proof

We will follow a proof by John Geanakoplos (Three brief proofs ofArrow’s Impossibility Theorem, Economic Theory 26 (2005),211–215; downloadable from author’s website).

Given is some SWF F that satisfies WP, IIA and R.

The proof proceeds in three main steps

1. Derive social ranking for certain “extreme” preference profiles

2. Use first step to identify a potential dictator using a series ofreference profiles

3. Show that this individual is indeed a dictator





Arrow’s Theorem: Discussion

Arrow’s Theorem shows that no ideal aggregation methodexists.

Arrow’s Theorem continues to hold when we restrict attentionto strict preferences. (Why?)

Weakening R to quasi-transitivity or acyclicity yields slightlymore permissive results which we won’t cover in class (seeAusten-Smith Banks (PPT I) Ch. 2.3-2.5, Moulin Ch. 11).

Consequences of Arrow’s Theorem?





Social choice with two alternatives: May’s Theorem

One assumption of Arrow’s theorem: At least three alternatives.

Are there appealing voting procedures for m = 2?

In this case, what are the properties of the SWFs we encounteredso far?

In the following, let X = {x , y}.






We now introduce some additional properties of SWFs

1. Let π : I → I be a permutation (of agents) and R be somepreference profile. Define the profile Rπ by xRπ

π(i)y if and onlyif xRiy for all i ∈ I .A SWF F is anonymous, if for all permutations π,F (R) = F (Rπ) for all profiles R.

2. Let σ : X → X be a permutation (of alternatives) and R besome preference profile. Define the profile Rσ by σ(x)Rσ

i σ(y)if and only if xRiy for all i ∈ I .A SWF F is neutral, if for all permutations σ and allpreference profiles R, σ(x)F (Rσ)σ(y) if and only if xF (R)y .






3. A SWF F is positively responsive, if whenever twopreference profiles R,R ′ satisfy

(i) xF (R)y ,(ii) {i ∈ I : xPiy} ⊆ {i ∈ I : xP ′i y},(iii) {i ∈ I : yP ′i x} ⊆ {i ∈ I : yPix}, and(iv) there is at least one i ∈ I such that xP ′i y but yRix ,then xFP(R ′)y .






Theorem (May (1952))

Suppose m = 2.Then a SWF F is neutral, anonymous and positively responsive ifand only if it is the plurality rule.

Remark:

1. Characterization result above actually holds for any number ofalternatives (see Austen-Smith, Banks (PPT I), Ch. 3.5).

2. Three axioms characterizing the plurality rule are independent.(Exercise)

3. For some general classes of aggregation procedures one cancalculate the maximal number of alternatives for which theclass under consideration yields an acyclic social ranking (seeAusten-Smith, Banks (PPT I), Ch. 3).





Single-peaked preferences: The median voter theorem

One assumption of Arrow’s theorem: All rational preferenceprofiles are possible.

Intuitively, restricting preferences should make things easier (lessproblems have to be solved by an aggregation procedure).

We’ll only cover the most prominent restriction here (some otherrestrictions follow in the second and third parts of the class):Single-peaked preferences.






Let X = {x1, . . . , xm} (m arbitrary) and the strict (preference)relation Q be such that xt+1Qxt for t = 1, . . . ,m.

A preference relation Ri ∈ R is single-peaked on X with respectto Q if and only if there exists a ti ≤ m such that

(i) xti Pixti+1Pi . . .Pixm, and

(ii) xti Pixti−1Pi . . .Pix1.

In this case we call x i := xti i ’s peak with respect to Ri .

For the following, fix a preference profile R ∈ Rn such that for alli ∈ I , Ri is single-peaked with respect to Q.






Call alternative x a Condorcet winner at R, if xF pl(R)y for ally ∈ X .

Agent h ∈ I is a median agent for the profile R, if

(i) |{i ∈ I : x iQxh}| ≥ n2 , and

(ii) |{i ∈ I : xhQx i}| ≥ n2 .

Theorem

Suppose R is a preference profile that is single-peaked with respectto Q.Then a Condorcet winner at R exists for any value of n and it isunique if n is odd.






If n even, F pl(R) not in general transitive.

Theorem

Suppose n is odd and, in addition to the above, for all i ∈ I , Ri isa strict preference relation.Then F pl(R) satisfies R.

Continues to hold without assumption of strict preferences, butproof more difficult (see Austen-Smith, Banks (PPT I), Ch. 4.1).


Social Choice and Mechanism Design Summer term 2010€¦ · Introduction: Mechanism Design Theory...

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