a tutorial on matching theory & contract theory yanru zhang 1

1

A Tutorial on Matching Theory &

Contract TheoryYanru Zhang

2OUTLINE

• INTRODUCTION

• MATCHING THEORY• Stable Matching with Complete Preference Lists• Stable Matching with Relaxed Preference Lists

• CONTRACT THEORY • Optimal Employment Contracts without Uncertainty, Hidden

Information, or Hidden Actions• Optimal Contracts under Uncertainty• Information and Incentives• Optimal Contracting with Multilateral Asymmetric Information

3INTRODUCTION

• Economics• a field that aims to understand the process• by which scarce resources are allocated • to their most efficient uses

• markets • playing a central role in this allocation process

4INTRODUCTION

• Examples• Matching doctors and hospitals• Matching students and high-schools• Matching kidneys and patients• Investment under risk• Insurance• Hiring employees

• Existing Theories of• Matching Theory • Contract Theory

5MATCHING THEORY

• The Nobel Prize in Economic Sciences 2012• to Lloyd Shapley and Alvin Roth

• Lloyd Shapley• Developed the theory in the 1960s

• Alvin Roth• Generated further analytical developments • practical design of market institutions

6MATCHING THEORY

• Definition on Wikipedia• a mathematical framework attempting to describe the formation of

mutually beneficial relationships over time.

• Where is it used• The U.S. National Resident Matching Program (NRMP) for medical

school graduates• Public schools in Boston and NYC• Many medical and other labor markets across

7PART I: Matching Problem with Complete Preference List

• Three types of matching problems• One-to-one (stable marriage)• One-to-many (college admissions)• Many-to-many (complex scenarios, P2P)

8Solution of a Matching Problem

• We seek to find a stable matching such that• There does not exist any pair of players, i and j matches,

respectively to players, a and b but..• j prefers a to b and a prefers j to i

• How can we find a stable matching?• many approaches(minimizing sum/ max of ranks,

minimizing diff of total ranks, Gale and Shapley algorithm)

• Most popular: Deferred acceptance or GS algorithm• Illustrated via an example

9Example 1: Matching partners

Adam

Bob

Carl

David

Fran

Geeta

Irina

Heiki

• women and men be matched• respecting their individual

preferences


• The Gale-Shapley algorithm can be set up in two alternative ways:• men propose to women• women propose to men

• Each man proposing to the woman he likes the best• Each woman looks at the different proposals she has received (if any)• retains what she regards as the most attractive proposal (but defers from accepting it) and

rejects the others

• The men who were rejected in the first round • Propose to their second-best choices • The women again keep their best offer and reject the rest

• Continues until no men want to make any further proposals

• Each of the women then accepts the proposal she holds

• The process comes to an end

spare

tier

11Here Comes the Story…

Adam

Bob

Carl

David

Fran

Geeta

Irina

Heiki

Geeta, Heiki, Irina, Fran

Irina, Fran, Heiki, Geeta

Geeta, Fran, Heiki, Irina

Irina, Heiki, Geeta, Fran

Adam, Bob, Carl, David

Carl, David, Bob, Adam

Carl, Bob, David, Adam

Adam, Carl, David, Bob

12Search for a Matching

Adam Geeta

Bob Irina Carl Fran

David Heiki

Carl likes Geeta better than Fran!

Geeta prefers Carl to Adam!

X

X

Blocking Pair

13Stable Matching

Adam Heiki

Bob Fran GeetaCarl

IrinaDavid

Bob likes Irina better than Fran!

Unfortunately, Irina loves David better!

Stable Matching: a matching without blocking pairs

Bob and Irina are not a blocking pair

14Gale-Shapley Algorithm

Adam

Heiki

Bob

Fran

Geeta

Carl

IrinaDavid

Geeta, Heiki, Irina, Fran

Irina, Fran, Heiki, Geeta

Geeta, Fran, Heiki, Irina

Irina, Heiki, Geeta, Fran

Carl > Adam

David > Bob

This is a stable matching

15

×××

×

×

××

1: a c b d e a: 2 1 3 4 5

2: c a e b d b: 2 1 4 5 3

3: b a e d c c: 1 2 3 5 4

4: c b d e a d: 3 1 4 2 5

5: c d b e a e: 4 3 1 2 5

×

GS Algorithm

no blocking pairs

1, 2, 3, 4 represent mena, b, c, d represent women


• The setup of the algorithm have important distributional consequences

• it matters a great deal whether • the right to propose is given to the women or to the men

• If the women propose• the outcome is better for them than if the men propose

• Conversely, the men propose • leads to the worst outcome from the women’s perspective

• optimality is defined on each side, difficult to guarantee on both sides

• The matching may not be unique

17Different Stable Matchings

1: a c b d a: 2 4 1 3 2: b d c a b: 3 1 2 4 3: c a d b c: 4 2 3 1 4: d b a c d: 1 3 4 2




1, 2, 3, 4 represent mena, b, c, d represent women

1Maching

2Maching

3Maching

4Maching

Women get best satisfactory

Men get better satisfactory

18Extensions to new markets

• Prices are not part of the process for the previous implementation

• Does the absence of a price mechanism in the basic Gale-Shapley algorithm limit its applicability?

• Not necessarily

• Algorithms including prices work in much the same way • produce stable matches with similar features • Matching with prices is closely related to auctions

• objects are matched with buyers and where prices are decisive

• Matching vs. Auction• Synergistic but more about matching, less about pricing

19Example 2: Mini Cost Matching

A

B

C

D

E

a

b

c

d

e

1 5 3 2 4

1 4 3 5 2

3 4 2 5 1

1 2 4 3 5

2 1 3 5 4

A

B

C

DE

a b c d e

1 4 2 3 5

3 2 1 5 4

3 4 5 2 1

1 5 4 3 2

2 3 1 5 4

a

b

c

de

A B C D E

2=1+1

8

6

3

6

20Cost Minimum but Not Stable

1: a c b a: 1 3 2

2: b a c b: 2 1 3

3: a b c c: 1 2 3

1: a c b a: 1 3 2

2: b a c b: 2 1 3

3: a b c c: 1 2 3

cost=10

cost=8

21Stability of Matching

1: a c b d e a: 2 1 3 4 5

2: c a e b d b: 2 1 4 5 3

3: b a e d c c: 1 2 3 5 4

4: c b d e a d: 3 1 4 2 5

5: c d b e a e: 4 3 1 2 5

Blocking pair

22To Remove Blocking Pairs…

1: a c b d e a: 2 1 3 4 5

2: c a e b d b: 2 1 4 5 3

3: b a e d c c: 1 2 3 5 4

4: c b d e a d: 3 1 4 2 5

5: c d b e a e: 4 3 1 2 5

No blocking pairs any more

23PART II: Relaxed Preference Lists

• National Resident Matching Program • Assigning many medical students to many hospitals

• Complete total order is unrealistic

• Indifferences (ties) in the list

• Incomplete lists

2: c a e b d

2: (c a) (e b d)

2: c a e

24Stable Matching with Ties (SMT)

Theorem: Any SMT instance admits at least one (weakly) stable matching

1: a (c b d) a: 2 4 1 3 2: b d c a b: (3 2 1) 4 3: (c a) d b c: 4 2 (3 1) 4: d b a c d: 1 3 (4 2)

1: a b c d a: 2 4 1 3 2: b d c a b: 1 2 3 4 3: a c d b c: 4 2 3 1 4: d b a c d: 1 3 2 4

25Stable Matching with Incomplete List (SMI)

1: a c b a: 2 1 3 4 5

2: c a b: 2 1

3: b a c: 1 2

4: c b d e d: 3 1 4

5: c d b e: 4 3

Matching may be partial Theorem [Gale, Sotomayor 1985]

– There may be more than one stable matchings, but their size is all the same and one of them can be obtained in poly time.

26Matching Games In communication networks

• Assign base stations to users

• Resources to device

• VMs to jobs in a cloud system

27Many-to-Many Game: Femto-cells

• Femto-cell access points (FAPs) are • low-power wireless access points that

• operate in macro-cells’ access points (MAPs) licensed spectrum

• using residential DSL or cable broadband connections

• Challenges:• Random spatial placement of FAPs and huge interference• MAPs to FAPs, FAPs to MAPs and FAPs to FAPs interference• No distributed mechanism to handle the final users (FUs)-FAPs

and FAPs-WOs allocation

28Contract Theory

• Definition on Wikipedia• studies how economic actors can and do construct contractual

arrangements, generally in the presence of asymmetric information

• A standard practice in the contract theory to solve the problem

• under certain numerical utility structures• represent the behavior of a decision maker• apply an optimization algorithm to identify optimal decisions

29Major Topics in Contract Theory

• Bilateral contracting • Under no uncertainty• Under hidden information or hidden action

• multilateral contracting • under hidden information or hidden actions• auction theory

• long-term contracts

• incomplete contracts

30Contracting Situation

• Employer and employee• a manager hiring a worker• a farmer hiring a sharecropper• a company owner hiring a manager

• contracting parties are rational individuals • aiming to achieve the highest possible payoff

• a court• between two parties who operate in a market economy with a well-functioning legal

system

• A system• any contract the parties decide to write will be enforced perfectly • The penalties for breaking the contract will be sufficiently severe • no contracting party will ever consider not honoring the contract

31Problems in Contract Theory

• transaction is a exchange of goods or services for money• What is the price per unit the parties shall agree on? • Are there penalty or reward?

• transaction is an insurance contract• determining how the terms vary with the underlying risk

• with the private information the insuree or the insurer have about the nature of the risk

32Example 1: Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions

• a situation involving only two parties • facing no uncertainty and no private information or hidden actions

• U(l,t), employer's utility• l, the quantity of employee time the employer has acquired• t, the quantity of "money" that he has at his disposal (the "output" that

this money can buy)• initial endowment, (ľ1,ť1)=(0,1)

• u(l,t), employee utility• l, the quantity of time the employee has kept for herself• t, the quantity of money that she has at her disposal• initial endowment, (ľ2,ť2)=(1,0)

33PART I: Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions

• Without any trade• The employer gets no employee time

• but has all the money• The employee has all of her time for herself

• but has no money• each achieve an initial utility of Ū=U(0,1) and ū=u(1,0)

• Utility functions U(l,t) and u(l,t)• strictly increasing and concave• both individuals can increase their joint payoff

• exchanging labor services l for money/output


• Question raised• How many hours of work will the employee be willing to offer?• What (hourly) wage will she be paid?

• the joint surplus maximization problem:• Max U(l1,t1)+μu(l2,t2)

• μ, reflect both the individuals' respective reservation utility levels Ū and ū, and their relative bargaining strengths

• Subject to • l1+l2=ľ1+ľ2=1 and t1+t2=ť1+ť2=1


• Take the first-order conditions• Ul+μul=0=Ut+μut

• Ul/Ut=ul/ut

• joint surplus maximization is achieved when

• the marginal rates of substitution between money and leisure for both individuals are equalized


• The highest possible utility that the employee can get:• max u(l2, t2)

• Subject to U(1-l2, 1-t2)≥Ū

• The highest payoff the employer can get:• max U(l1,t1)

• Subject to u(1-l1, 1-t1)≥ū

37PART II: Optimal Contracts under Uncertainty

• There is more uncertainty in reality than Example 1• Insurance• employees are insured against economic downturns

• A question concerning these insurance schemes is • how much risk should be absorbed by employers • how much by employees

• Enrich Example 1 by introducing uncertainty• analyze the question of optimal risk allocation

38Example 2: Pure Insurance

• In two possible future states of nature, θL and θH

• θL, an adverse output shock, or a "recession" • θH, a good output realization, or a "boom"

• a pure insurance problem without production• disregard time endowments in the time/output bundles (l,t) • Adopt the consumption bundles (tL,tH)

• the state of nature influences only the value of output• E(tL,tH) for the employer• e(tL,tH) for the employee


• the initial endowments for each individual in each state:• (ť1L,ť1H)=(1,2), for individual 1 (employer)

• (ť2L,ť2H)=(1,2), for individual 2 (employee)

• The initial utility (before the state of nature is realized) • Ē=E(1,2) for employer• ē=e(1,2) for employee

• the ex ante utility function• E(tL,tH), for employer

• e(tL,tH), for employee


• the cosinsurance optimization problem:• Max E(tL,tH)+μe(tL,tH)

• Take the first-order conditions • EL+μeL=0=EH+μeH

• EL/EH=eL/eH

• joint surplus maximization is achieved when• the marginal rates of substitution between

nature θL and θH for both individuals are equalized

• pure exchange under certainty can be transposed entirely to the case with uncertainty

41Von Neumann–Morgenstern Utility Functions

• two important elements are hidden in the optimal insurance contract• The ex post utility once the state of nature has been realized• The probability of each state occurring

• ex post utility functions • U(t) for the employer • u(t) for the employee

• Pj є (0,1), the probability of occurrence of any particular state of nature θj

• the ex ante utility function is the expectation over ex post utility outcomes:

• E(t1L,t1H)= pLU(t1L)+ pHU(t1H)• e(t2L,t2H)= pLu(t2L)+ pHu(t2H)

42Example 3: Optimal Employment Contracts under Uncertainty

• extended to the Optimal Employment Contracts under Uncertainty problem

• the framework of von Neumann and Morgenstern of Example 2• the contracting problem of Example 1 with two goods, leisure l and a

consumption good t

• (l1L,t1L) and (l1H,t1H) • two different state-contingent time/output bundles of the employer

• (l2L,t2L) and (l2H,t2H) • Two different state-continent time/output bundles of the employee

• (ľij, ťij) denote initial endowments i=1, 2; j=L, H

43Example 3: Optimal Employment Contracts under Uncertainty

• the optimal contracting problem for the employer• max[pLU(l1L,t1L)+pHU(l1H,t1H)]

• Subject to• pLu(l2L,t2L)+pHu(l2H,t2H)≥ū

• l1j+l2j≤ľ1j+ľ2j

• t1j+t2j≤ť1j+ť2j

• Where ū=pLu(ľ2L,ť2L)+pHu(ľ2H,ť2H)

44PART III: Information and Incentives

• Two general types of asymmetric imformation problems• hidden-information problem---- adverse selection

• relevant characteristics of the employee are hidden from her employer• distaste for certain tasks, her level of competence

• hidden-action problem---- moral hazard• employee's actions that are hidden from the employer• whether she works or not, how hard she works, how careful she is

45Adverse Selection

• Screening problems• the party making the contract offers is the uninformed party• the uninformed party attempts to screen the different pieces of

information the informed party has

• signaling problems • the informed party makes the contract offers• the party making the offer attempts to signal to the other party

what it knows

46Screening Problems

• In practice, employers try to overcome the informational asymmetry (to improve their pool of applicants) by

• hiring only employees with some training • only high school and college graduates• more-able employees have a lower disutility of education• more willing to educate themselves than less-able employees

• pay greater than market-clearing wages• attract better applicants

• How efficient can contracting under asymmetric information be?

47Screening Problems

• consider an employer who contracts with two possible types of employees• a "skilled" employee and an "unskilled" one• does not know which is which

• employee productivity is private information• All employee types would respond by "pretending to be skilled" to get the higher wage

• revelation principle• employer offers two employment contracts

• one destined to the skilled employee • the other to the unskilled one

• make sure that each contract is incentive compatible• each type of employee wants to pick only the contract that is destined to her

• the problem reduces to a standard contracting problem

Second Price Auction

48Example 4: Screening Problems

• Extend the previous examples with uncertainty • with private information added

• Suppose that employee time and output enter additively: • U[αθ(1-l)-t] for the employer • u(θl+t) for the employee

• (1-l), the employee time sold to the employer• l, the time the employee keeps for herself• t, the monetary/output transfer from the employer to the employee• α, a positive constant• θ, measures the "unit value of time," or the skill level of the employee

U(l1,t1), u(l2,t2)

49Example 4: Adverse selection

• The utility function:• U[αθ(1-l)-t] for the employer • u(θl+t) for the employee

• θ, the state of nature

• The employee knows whether she is skilled• with a value of time θH

• or unskilled, with a value of time θL<θH

• The employer knows only that the probability of facing a skilled employee is pH


• in state θj ， the employee's endowment ľj = θj

• relevant reservation utility is • ūH =u(θH), employer faces a skilled employee• ūL=u(θL), employer faces an unskilled employee

• incentive compatibility constraints

• the employee's time is more efficient when sold to the employer

• α>1

• u(ti)≥u(θj)


• revelation principle offers a key simplification• Type θH must prefer contract (tH,lH) over (tL,lL)• Type θL contract (tL,lL) over (tH,lH)

• the optimal menu of employment contracts under hidden information:

• Max{pLU[αθL(1-lL)-tL]+pHU[αθH(1-lH)-tH]}

• Subject to• u(lLθL+tL)≥ū(θL), u(lHθH+tH)≥ū(θH)

• u(lHθH+tH)≥u(lLθH+tL), u(lLθL+tL)≥ u(lHθL+tH)


• The solution to this constrained optimization problem• produce the most efficient contracts under hidden information• less efficient allocations than under complete information

• the addition of incentive constraints

• The presence of hidden information may • give rise to allocative inefficiencies such as unemployment

• Soviet Union was notorious for its overmanning problems• It may not have had any official unemployment• but it certainly had huge problems of underemployment

53Moral Hazard

• contracting situations with hidden actions • In contrast to hidden information

• informational asymmetries arising after the signing of a contract• employee is not asked to choose from a menu of contracts

• but from a menu of action-reward pairs

• Phenomenon• When a person gets better protection against a bad outcome

• she will rationally invest fewer resources in trying to avoid it• introduction of laws compelling drivers to wear seat belts

• resulted in higher average driving speeds and a greater incidence of accidents

• How do insurers deal with moral hazard? • By charging proportionally more for greater coverage

Texas Drivers

54Moral Hazard

• if an employee's pay and job tenure are shielded against the risk of bad earnings

• she will work less in trying to avoid these outcomes

• Employers typically respond to moral hazard on the job • rewarding good performance

• through bonus payments, piece rates, efficiency wages, stock options• and/or punishing bad performance

• through layoffs

• there is a basic tradeoff between insurance and incentives in most employment relations

55PART III: Information and Incentives

• the trade-off between incentives and insurance• How far should employee insurance makes way for adequate work

incentives? • How could adequate work incentives be structured while preserving

job security as much as possible?

• If employees will be perfectly insured against business risks• the equilibrium price of such insurance would be too high

• employees need to have adequate incentives to work• if their job security or pay is independent of performance• Why should they put any effort into their work?

56Example 5: Moral Hazard

• introduce hidden actions into the preceding employment problem with uncertainty

• suppose • the amount of time (1-l) worked by the employee is private

information (a hidden action) • the employee chooses the action (1-l) before the state of nature θj is

realized

• this action influences the probability of the state of nature


• when the employee chooses action (1-l)• output for the employer is simply θH with a probability function pH

[1-l]• and θL with a probability function pL[I-l]=1-pH [1-l]

• more effort produces higher expected output, at cost 1-l for the employee

• The employer offers a compensation contract t(θj) to the employee

• under the output-contingent compensation scheme t(θj)

• (1-l) will be chosen by the employee to maximize her own expected payoff


• the effort level chosen by the employee is the outcome of the employee's own optimization problem:

• (1-l) є argmax{pL[1-l]u[t(θL)+l]+pH[1-l]u[t(θH)+l]}

• the employer solves the following maximization problem:• max{pL[1-l]U[θL -t(θL)]+pH[1-l]U[θH -t(θH)]}

• Subject to• pL[1-l]u[t(θL)-l]+pH[1-l]u[t(θH)-l]≥ū=u(1)

• (1-l) є argmax{pL[1-l]u[t(θL)-l]+pH[1-l]u[t(θH)-l]}

• the employer chooses the optimal compensation contract {t(θj)} to maximize his expected utility


• the effort level chosen by the employee is the outcome of the employee's own optimization problem:

• (1-l) є argmax{pL[1-l]u[t(θL)-l]+pH[1-l]u[t(θH)-l]}

• An efficient trade-off between insurance and incentives involves

• rewarding the employee most for output outcomes that are most likely to arise when she puts in the required level of effort

• punishing her the most for outcomes that are most likely to occur when she shirks

60PART IV: Optimal Contracting with Multiiateral Asymmetric Information

• many situations where several contracting parties may possess relevant private information or take hidden actions

• the most important and widely studied problem of contracting with multilateral hidden information

• the design of auctions with multiple bidders• each with his or her own private information about the value of the

objects that are put up for auction

61Conclusion

• Matching Theory

• Contract Theory

a tutorial on matching theory & contract theory yanru zhang 1

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