Social Choice and Mechanism Design Summer Term 2010 Part ... · Paul Milgrom, Putting auction theory to work, Cambridge University Press [M] Survey articles: Al Roth, The economist

IntroductionThe marriage model

The college admissions problemSummary two-sided matching

The school choice problem

Social Choice and Mechanism DesignSummer Term 2010

Part III: Matching and multi-objectauctions




Introduction

In recent years, theoretically motivated designs have successfullybeen employed to

match university graduates to their first professional position,

assigns students to public schools, and

conduct multi-object auctions (e.g. for radio spectrum).

Problems closely related: Two-sides of a market have to bematched to each other!




Introduction

School Choice:Today school districts often take students’ (or their parents’)preferences into account when assigning students to publicschools⇒ Some popular schools will have to reject studentsHow should students be matched to public schools?

Multi-object auctions:In many auctions multiple objects have to be sold to a set ofheterogenous bidders (who potentially demand more than onegood)Who should get which objects and how much shouldwinning bidders pay?




Introduction

We will...

... learn (some of) the theoretical concepts and ideasunderlying these applications.

... analyze and compare some real-life matching mechanisms.

... learn that there is an interesting interplay between thedesign of real-life market mechanisms and the advancement oftheory.




Literature

Matching:

Al Roth and Marilda Sotomayor, Two-sided matching,Cambridge University Press [RS]

Auctions:

Paul Milgrom, Putting auction theory to work, CambridgeUniversity Press [M]

Survey articles:

Al Roth, The economist as engineer, Econometrica 70(4),1341–1378Paul Milgrom, Package auctions and exchanges, Econometrica75(4), 935–965Tayfun Sonmez and Utku Unver, Matching, allocation, andexchange of discrete resources, forthcoming Handbook ofSocial Economics (available at Unver’s website)




StabilityIncentivesEquilibrium Analysis

The marriage model

The marriage model





The marriage model

We’ll start by considering a (too) simple model of amongamous and conservative matching market proposed byDavid Gale and Lloyd Shapley (1962).

Analysis uses tools from cooperative and non-cooperativegame theory.

Most of the main ideas generalize to more complex (andrealistic) matching models.

Literature: Chapters 2 and 4 in RS





The marriage model

Two types of agents: Men and WomenSet of men is M and set of women is WAssumption: Finite set of agents

Monogamous and conservative marriage market: Each womancan marry at most one man and can never marry anotherwoman (same condition for men).

Man m (woman w) has a strict rational preference rankingRm (Rw ) of W ∪ {m} (M ∪ {w});w acceptable to m under Rm if wPmm.

A marriage problem is given by the three-tuple (M,W ,R)(often we’ll think of it as being given by R)





The marriage model

Outcome is a matching which specifies who marries whomFormally: A matching is a mapping µ : M ∪W → M ∪Wsuch that

(i) µ(m) ∈W ∪ {m} for all m ∈ M,(ii) µ(w) ∈ M ∪ {w} for all w ∈W ,(iii) µ(m) = w if and only if µ(w) = m.

Assumption: Each agent only cares about his/her partner in amatching

Which matchings will result if all a marriage requires ismutual consent of associated agents?





Stability

Definition

A matching µ is (pair-wise) stable, if

(i) no agent is matched to an unacceptable partner, that is,µ(i)Ri i for all i ∈ M ∪W

(ii) there is no pair (m,w) ∈ M ×W such that mPwµ(w) andwPmµ(m).

Remark: Pair-wise stable matchings are in the core, i.e., if µis pair-wise stable there is no A ⊆ M ∪W and matching µ′

such that(i) µ′(i) ∈ A for all i ∈ A,(ii) µ′(i)Riµ(i) for all i ∈ A,(iii) µ′(i) 6= µ(i) for at least one i ∈ A.





Stability

Theorem

(Gale and Shapley 1962) A stable matching always exists in themarriage model.

Proof by describing an algorithm that always produces a stablematching.





Deferred acceptance algorithm

Given a preference profile R, the men-proposing deferredacceptance algorithm (Gale and Shapley (1962)) proceeds asfollows

Round 1: Each man proposes to his most preferred acceptable woman(if any). Each woman temporarily accepts her most preferredacceptable proposer (if any) and rejects all others. If anunmatched man has not yet proposed to all acceptablewomen, proceed to round 2.

Round k : Each man proposes to his most preferred acceptable womanamong those that have not rejected any of his proposals inrounds 1 through k − 1. Each woman temporarily accepts hermost preferred acceptable proposer and rejects all others. Ifan unmatched man has not yet proposed to all acceptablewomen, proceed to round k + 1.






Algorithm ends after a round in which each (temporarily)unmatched man has proposed to all acceptable women.

At this point temporary assignments become permanent.






Example:

Rm1 Rm2 Rm3 Rw1 Rw2 Rw3

w2 w3 w3 m2 m3 m1

w3 w2 w2 m1 m2 m3

w1 w1 w1 m3 m1 m2

.

For this problem we obtain

µ =m1 m2 m3

w3 w1 w2






Theorem

Fix a profile of preferences R and let µM denote the outcome ofthe men-proposing DA.

(i) µM is a stable matching.

(ii) If µ is any other stable matching then

µM(m)Rmµ(m) for all m ∈ M, andµ(w)Rwµ

M(w) for all w ∈W .

Remark:

Referring to (ii), men-proposing DA is often referred to asmen-optimal stable matching mechanism.

Symmetric properties for women-proposing DA.





The set of stable matchings

Theorem

Fix an arbitrary marriage problem R and let µ, µ′ be two arbitrarystable matchings.

(i) Let M(µ′) = {m ∈ M : µ′(m)Pmµ(m)} andW (µ) = {w ∈W : µ(w)Pwµ

′(w)}. Thenµ(M(µ′)) = µ′(M(µ′)) = W (µ).

(ii) If µ(i) = i for some i ∈ M ∪W then µ′(i) = i .

Remark:

First result usually called decomposition lemma.

Second result implies that the set of singles is the same for allstable matchings.





The set of stable matchings

Given two matchings µ and µ′, define common preferences ofmen by µ ≥M µ′ if and only if µ(m)Rmµ

′ for all m ∈ M.

Set µ >M µ′ if and only if µ ≥M µ′ and µ 6= µ′.

Analogous definitions for common preferences of women

Theorem (Opposition of interests)

If µ and µ′ are two stable matchings then µ >M µ′ if and only ifµ′ >W µ.





Incentives

Preferences over potential partners private information!

Do agents have an incentive to behave straightforwardly when theyface a market organized e.g. along the lines of the men-proposingDA?

If not, which matchings can we expect in equilibrium?





Notation and Terminology

Notation:

Ri : Set of possible preference relations of agent i (= all strictrational rankings of potential partners and the option ofremaining unmatched)R = ×i∈M∪WRi : Set of all possible preference profilesM: Set of all possible matchingsS(R): Set of all stable allocations at R ∈ R

A (direct) matching mechanism is a mapping f : R →M andfi (R) denotes i ’s assignment under matching f (R)

A matching mechanism f is stable if f (R) ∈ S(R) for allR ∈ RExample: Men-proposing DA as a direct mechanism f M (themen-optimal stable matching mechanism)





Manipulating stable matching mechanisms

Individual i ∈ M ∪W can manipulate matching mechanism f at Rif there exists R̃i ∈ Ri such that fi (R̃i ,R−i )Pi fi (R).

Theorem

Let R ∈ R be an arbitrary preference profile and f be some stablematching mechanism. The following statements are equivalent:

(i) Agent i ∈ M ∪W can manipulate f at R.

(ii) Agent i ∈ M ∪W does not get his/her best stable partnerunder f (R), that is, there is some µ ∈ S(R) such thatµ(i)Pi fi (R).





Proof of the Manipulation Theorem ((i)⇒ (ii))

Proof:

Show that m ∈ M cannot manipulate if he gets best stablepartner at R.Suppose Rm = w1, . . . ,wn,m and fm(R) = f M

m (R) = wk forsome k ∈ {2, . . . , n} (nothing to prove if k = 1; proof virtuallyidentical if fm(R) = m = f M

m (R)).

Step 1: Suppose there was some profitable deviationR̂m ∈ Rm, i.e. fm(R̂m,R−m) = wl for some l < k .Then a simple deviation in which m ranks only wl asacceptable is also profitable, i.e. for R̃m = wl ,m we must havefm(R̃m,R−m) = wl .





Proof of the Manipulation Theorem ((i)⇒ (ii))

Step 2: If there is some profitable simple deviationR̃m = wl ,m for some l ∈ {1, . . . , k − 1}, i.efm(R̃m,R−m) = wl , then a truncation of m’s true preferencesRm after wl is also profitable, i.e. for Rm = w1, . . . ,wl ,m wemust have fm(Rm,R−m)RmwlPmwk .

Step 3: fm(Rm,R−m) is stable at R(This contradicts the assumption that wk = f M

m (R) andcompletes the proof...)





Proof of the Manipulation Theorem ((ii)⇒ (i))

Show that if f does not give m his best stable partner at R, mcan manipulate f at R.

Let w = f Mm (R) and consider the simple manipulation

R̃m = w ,m.

Claim: fm(R̃m,R−m) = w





Remarks on the Manipulation Theorem

Proof of (i)⇒ (ii) relies heavily on the following two facts:

(i) Set of singles is the same for all stable matchings.(ii) For each agent there is an optimal stable partner.

This suggests that this direction of the proof should work aslong as these two properties of the stable set are satisfied...





Manipulating stable matching mechanisms

A matching mechanism f is strategyproof for agenti ∈ M ∪W if there is no R ∈ R, such that f (R ′i ,R−i )Ri f (R)for some R ′i ∈ Ri .

A matching mechanism f is strategyproof, if it isstrategyproof for all agents.

Corollary

(i) A stable matching mechanism is non-manipulable at R ∈ R ifand only if |S(R)| = 1.

(ii) There is no strategyproof stable matching mechanism.

(iii) The men-optimal (women-optimal) stable matchingmechanism is strategyproof for all men (women).





Equilibrium Analysis

Any matching mechanism must be manipulable in thecooperative or non-cooperative dimension.What can we say about the properties of equilibria of stablemechanisms?Complete information Nash-equilibria

There is no stable matching mechanism such that truthtellingis always a Nash-equilibrium (Why?).Any Nash-equilibrium of the men-optimal stable mechanism inwhich men submit preferences truthfully, must be stable w.r.t.true preferences of the agents (and any stable matching is anequilibrium outcome).

Incomplete information (Bayes-)Nash equilibriaFor common prior environments close link between completeinformation Nash-equilibria and incomplete information(Bayes-)Nash equilibria (Ehlers and Masso (2010))




StabilityIncentives

The college admissions problem




StabilityIncentives


Many markets two-sided but at least one side of the marketcan be matched to more than one partner (e.g. collegeadmissions, school choice,...).We’ll think of a two-sided market consisting of students andcolleges (who can potentially admit more than one student)

Colleges have preferences over groups of students

Most of basic results from the marriage model continue tohold, but some important differences

Theoretical basis for analyzing school choice problem

Literature: Chapter 5 in RS (1990), survey by Sonmez andUnver (Chapter 4).




StabilityIncentives


Set of students I , set of colleges C

Many-to-one matching: Each student can attend at most onecollege, colleges can admit multiple students

Student i ∈ I has a strict rational ranking Ri of C ∪ {i}College c ∈ C has a rational ranking Rc of 2I (the set of allgroups of students) and a capacity qc ∈ NNote: We allow c to be indifferent between two distinct setsof students.

A college admissions problem (CAP) is a four-tuple(I ,C , q,R).

Everything apart from preferences fixed.⇒ CAP = profile of preferences




StabilityIncentives

Matchings

A matching is a mapping µ : I ∪ C → (C ∪ I ) ∪ 2I such that

µ(i) ∈ C ∪ {i} for all i ∈ I ,

µ(c) ∈ 2I and |µ(c)| ≤ qc for all c ∈ C ,

µ(i) = c if and only if i ∈ µ(c).




StabilityIncentives

College preferences

Assumption: For each college c ∈ C , there is a strict rationalranking of individual students R∗c of I ∪ {c} such that:

For any J ⊆ I with |J| < qc and any i , i ′ ∈ I \ J,

(a) J ∪ {i}PcJ ∪ {i ′} if and only if iP∗c i ′, and(b) J ∪ {i}PcJ if and only if iP∗c c

For any J ⊆ I with |J| > qc , ∅PcJ.

In this case we say that Rc is responsive to R∗c (given qc).




StabilityIncentives

College preferences - Example

Suppose R∗c = i1, i2, i3, i4, c , i5, qc = 2, and Rc responsive toR∗c .

Some examples of what we can infer from this assumption(you should check this):

{i1, i3}Pc{i2, i4}{i1}Pc{i1, i5}∅Pc{i1, i2, i3}{i1, i4}Pc∅

Preference between {i1, i4} and {i2, i3} not pinned down byresponsiveness and R∗c .




StabilityIncentives

Stability

Definition

A matching µ is (pair-wise) stable, if

µ(i)Ri i for all i ∈ I ,

iR∗c c for all i ∈ µ(c) and all c ∈ C ,

there is no student-college pair (i , c) such that cPiµ(i), iP∗c c ,and either |µ(c)| < qc , or iP∗c i ′ for some i ′ ∈ µ(c).

Remark: If preferences are responsive any pair-wise stablematching is in the core.




StabilityIncentives

Existence

Theorem

Stable matchings always exist in the college admissions problemwith responsive preferences.

To prove this construct related marriage market:

For each college c , there are qc “women” c1, . . . , cqc and eachof these women has preferences R∗cMen i ’ preferences R̃i derived from Ri by replacing c with thestring c1, . . . , cqcGiven matching µ for CAP, define corresponding matching inrelated marriage market µ by µ(i) = ck (and µ(ck) = s) if andonly if i is the kth most preferred student in µ(c) w.r.t. R∗c .

Claim: Stability in related marriage market ⇔ stability incollege admissions problem.




StabilityIncentives

Set of stable matchings

The following results follow immediately from the correspondingresults for the marriage model

Theorem

The student proposing deferred acceptance algorithm (SDA)produces the student optimal stable matching.

The set of unmatched students and the set of unfilledpositions are the same for all stable matchings.




StabilityIncentives


The following important lemma allows us to transfer most (andeven strengthen some) of our results from the marriage model.

Lemma

Let µ, µ′ be two stable matchings for the associated collegeadmissions problem and µ, µ′ be the corresponding (stable)matchings for the related marriage market.If µ(ck)P∗cµ

′(ck) for some position ck of c, then µ(cl)R∗cµ′(cl) for

all positions cl of c.




StabilityIncentives


Theorem

(a) If there is some stable matching µ such that |µ(c)| < qc thenµ(c) = µ′(c) for all stable matchings µ′.

(b) The college proposing deferred acceptance algorithm producesthe optimal stable matching for all colleges.

(c) Let µ and µ′ be two stable matchings.

(c.1) A college c is indifferent between µ and µ′ if and only ifµ(c) = µ′(c).

(c.2) If µ(c)Pcµ′(c) then iP∗c i ′ for all i ∈ µ(c) and all

i ′ ∈ µ′(c) \ µ(c).(c.3) µ >C µ

′ (i.e. µ(c)Rcµ′(c) for all c ∈ C and µ 6= µ′) if and

only if µ′ >I µ (i.e. µ′(i)Riµ(i) for all i ∈ I and µ 6= µ′).




StabilityIncentives

Incentives

A short reminder:

Matching mechanism is a mapping f : R →M, where R isthe set of all rational preference profiles and M is the set ofall matchings.

Matching mechanism f is strategyproof for agent a ∈ I ∪ C ifthere is no R such that fa(R ′a,R−a)Pafa(R) for some R ′a ∈ Ra

From our analysis of the marriage model we immediately obtain

Corollary

The student optimal stable matching mechanism is strategyprooffor all students.




StabilityIncentives

Incentives

Does a similar result hold for colleges?

Theorem

There is no stable matching mechanism which is strategyproof forcolleges.

Intuition: A college is like a coalition of individual agents...




StabilityIncentives

Further topics

Capacity manipulation by colleges (Sonmez (JET 1997),Ehlers (GEB 2010))

Empirical and experimental evidence of importance of stability(Roth (AER 1991), Roth and Kagel (QJE 2000))

Couples and other variations (Roth and Peranson (AER1999), Kojima et al. (2010))

...




Summary two-sided matching

Stable matchings exist (in CAP if college preferencesresponsive).

There exist optimal stable matchings for each side of themarket.

Opposition of interests on the set of stable matchings.

In marriage model men (women) optimal stable matchingmechanism strategyproof for men (women).

In CAP student optimal stable matching mechanismstrategyproof for students; no corresponding result forcolleges.




The Boston mechanismFurther topics in school choice







Traditionally students assigned to public school closest totheir home.

Today many school choice districts (in particular in the US)take student preferences into account in assigning students topublic schools.

In most applications, schools are not strategic agents butrather objects to be consumed by students (Abdulkadirogluand Sonmez (AER, 2003)).

Admission at overdemanded schools regulated by assigningpriorities to students (based on e.g. distance to school, scoresfrom entry exams,...).

What is a good way of matching students to public schoolsgiven that some students may have to be rejected byoverdemanded schools?






Set of students I , set of schools S

School s can admit at most qs ∈ N students

School s has a strict priority ranking �s of I

Student i has a strict preference ranking Ri of S ∪ {i}A school choice problem is a five-tuple (I , S , q,�,R)

Everything except student preferences assumed fixed (priorityordering determined by applying fixed legal criteria).⇒ School choice problem = profile of student preferences.





Associated college admissions problem

To each school choice problem we can associate a collegeadmissions problem in which

(i) college c(s)’s preferences over individual students given by �s ,and

(ii) c(s)’ preferences over groups are responsive to this ranking.

Important: Welfare evaluations only on basis of studentpreferences.





Non-wasteful and fair matchings

What is a compelling principle to assign students to public schools?

Definition

A matching µ is

non-wasteful if sPiµ(i) for some student i implies|µ(s)| ≥ qs , and

fair, if there is no pair (i , s) such that sPiµ(i) and i �s i ′ forsome i ′ ∈ µ(s).

Remark: Non-wasteful + fair ⇔ stable in associated CAP1

1Note that we assume that all students are acceptable to all schools.





Stability and Efficiency

Fairness may not be compatible with efficiency, i.e. there areschool choice problems R such that for any fair matching µthere is another matching µ′ such that µ′(i)Riµ(i) for allstudents i , with at least one strict preference.

Rationale for stability:

Priorities as society’s preferences regarding access to publicschoolsJustification of rejections.





The student optimal stable mechanism

For any school choice problem R, let f I denote the studentoptimal stable matching mechanism (for the associated CAP).

Theorem

(i) f I Pareto dominates any other fair matching mechanism, i.e.if g is a fair matching mechanism, then for any school choiceproblem R, fi (R)Rigi (R) for all i ∈ I .

(ii) f I is strategyproof.

If the prime objective is to achieve a fair matching, f I is acompelling mechanism.





The Boston mechanism

How are actual school choice programs organized?

We’ll now look at a particular algorithm that was/is a popularchoice among policymakers and characterize equilibriaassuming complete information about preferences and strictpriority rankings (strong assumptions!).

Compare equilibrium outcomes to student optimal stablemechanism.

Section follows Ergin and Sonmez (JPubEc 2006).





The Boston mechanism

Round 1: Each student applies to his (reported) top choice.For each school s order applicants according to �s and admitstudents in this ordering until either capacity is exhausted orthere are no more students who ranked s first.Adjust capacities accordingly.

Round k: Each unmatched student applied to kth choice.For each school s order applicants according to �s and admitstudents in this ordering until either capacity remaining afterrounds 1 through k − 1 is exhausted or there are no morestudents who ranked s as their kth choice.Adjust capacities accordingly.

Let f B(R) denote the matching chosen by this algorithm forthe problem R.





The Boston mechanism - Example

Consider the following school choice problem (where weassume qs = 1 for all s ∈ S)

Ri1 Ri2 Ri3 �s1 �s2 �s3

s1 s1 s2 i1 i1 i1s3 s2 s1 i2 i2 i2s2 s3 s3 i3 i3 i3

.

For this problem we obtain

f B(R) =i1 i2 i3s1 s3 s2





The Boston mechanism - Remarks

Outcome is efficient with respect to reported preferences

However, strong incentives to manipulate...

Empirical evidence in Abdulkadiroglu et al. (2006) that

1. students/their parents act upon incentives to manipulate, and2. manipulation particularly harmful to parents who strategize

suboptimally (a strategyproof mechanism would level theplaying field).

Mechanism also plays an important role in (the centralizedpart of) the German university admissions system (Westkamp2010)





The Boston mechanism - Equilibrium characterization

The game induced by the Boston mechanism (completeinformation):

(1) Each student i ∈ I submits a rational preference ranking fromRi ; all students submit their rankings simultaneously

(2) Given profile of reported preferences Q ∈ R = ×i∈SRi ,calculate f B(Q)

Given a (true) preference profile R ∈ R, a profile Q ∈ R is aNash equilibrium of (the game induced by) the Bostonmechanism if for all i ∈ I , f B

i (Q)Ri fBi (Q ′i ,Q−i ) for all

Q ′i ∈ Ri .





The Boston mechanism - Equilibrium characterization

Theorem

Fix a school choice problem R.A profile of reports Q is a Nash equilibrium of the Bostonmechanism at R if and only if f B(Q) is stable with respect to R.

Corollary

Fix a school choice problem R and some Nash equilibrium Q of theBoston mechanism at R.Then f I

i (R)Ri fBi (Q) for all i ∈ I .





Boston mechanism - Remarks

Boston mechanism was replaced by SDA in Boston schoolchoice program in 2006 (Abdulkadiroglu et al. (2006))

Equilibrium characterization can be seen as a theoreticaljustification for this...

Caveat: Result does not hold when

there is uncertainty about students’ preferences (Ergin andSonmez (JPubEc 2006)),some students do not act strategically (Pathak and Sonmez(AER 2008)), orpriorities are not strict (see below).





Further topics in school choice

Ties in the priority structure (Erdil and Ergin (AER 2008),Miralles (2009), Abdulkadiroglu et al. (AER 2010), Kestenand Unver (2010), Ehlers and Westkamp (2010),...)

Efficiency costs of stability (Kesten (QJE 2010))

Cardinal utility (Abdulkadiroglu et al. (2008))

Documents

Social Choice and Mechanism Design Summer Term 2010 Part ... · Paul Milgrom, Putting auction theory to work, Cambridge University Press [M] Survey articles: Al Roth, The economist