1. A Tutorial on Matching Theory & Contract Theory Yanru Zhang 1

2. OUTLINE 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 2

3. INTRODUCTION 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 3

4. INTRODUCTION 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 4

5. MATCHING 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 5

6. MATCHING 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 6

7. PART 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) 7

8. Solution 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 8

9. Example 1: Matching partners Adam Bob Carl David Fran Geeta Irina Heiki 9 women and men be matched respecting their individual preferences

10. Example 1: Matching partners 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 10 spare tier

11. Here 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 11

12. Search 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 12

13. Stable Matching Adam Heiki Bob Fran Geeta Carl Irina David 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 13

14. Gale-Shapley Algorithm Adam Heiki Bob Fran Geeta Carl Irina David 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 14

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 15 1, 2, 3, 4 represent men a, b, c, d represent women

16. Example 1: Matching partners 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 16

17. Different 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: 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: 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: 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 men a, b, c, d represent women Women get best satisfactory Men get better satisfactory 17

18. Extensions 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 18

19. Example 2: Mini Cost Matching A B C D E a b c d e 1 5 324 1 4 352 34251 12435 21354 A B C D E abcde 1 4 235 3 2 154 34521 15432 23154 a b c d e ABCDE 2=1+1 8 6 3 6 19

20. Cost 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 20

21. Stability 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 21

22. To 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 22

23. PART 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 23

24. Stable 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 24

25. Stable 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. 25

26. Matching Games In communication networks Assign base stations to users Resources to device VMs to jobs in a cloud system 26

27. Many-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 27

28. Contract 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 28

29. Major 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 29

30. Contracting 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 30

31. Problems 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 31

32. Example 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, (ľ 1,ť 1 )=(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, (ľ 2,ť 2 )=(1,0) 32

33. PART 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 33

34. Example 1: Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions 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(l 1, t 1 )+μu(l 2, t 2 ) μ, reflect both the individuals' respective reservation utility levels Ū and ū, and their relative bargaining strengths Subject to l 1 +l 2 =ľ 1 +ľ 2 =1 and t 1 +t 2 =ť 1 +ť 2 =1 34

35. Example 1: Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions Take the first-order conditions U l +μu l =0=U t +μu t U l /U t =u l /u t joint surplus maximization is achieved when the marginal rates of substitution between money and leisure for both individuals are equalized 35

36. Example 1: Optimal Employment Contracts without Uncertainty, Hidden Information, or Hidden Actions The highest possible utility that the employee can get: max u(l 2, t 2 ) Subject to U(1-l 2, 1-t 2 )≥Ū The highest payoff the employer can get: max U(l 1, t 1 ) Subject to u(1-l 1, 1-t 1 )≥ū 36

37. PART 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 37

38. Example 2: Pure Insurance In two possible future states of nature, θ L and θ H θ L, an adverse output shock, or a "recession" θ H, 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 (t L,t H ) the state of nature influences only the value of output E(t L,t H ) for the employer e(t L,t H ) for the employee 38

39. Example 2: Pure Insurance the initial endowments for each individual in each state: (ť 1L,ť 1H )=(1,2), for individual 1 (employer) (ť 2L,ť 2H )=(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(t L,t H ), for employer e(t L,t H ), for employee 39

40. Example 2: Pure Insurance the cosinsurance optimization problem: Max E(t L,t H )+μe(t L,t H ) Take the first-order conditions E L +μe L =0=E H +μe H E L /E H =e L /e H joint surplus maximization is achieved when the marginal rates of substitution between nature θ L and θ H for both individuals are equalized pure exchange under certainty can be transposed entirely to the case with uncertainty 40

41. Von 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 P j є (0,1), the probability of occurrence of any particular state of nature θ j the ex ante utility function is the expectation over ex post utility outcomes: E(t 1L,t 1H )= p L U(t 1L )+ p H U(t 1H ) e(t 2L,t 2H )= p L u(t 2L )+ p H u(t 2H ) 41

42. Example 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 (l 1L,t 1L ) and (l 1H,t 1H ) two different state-contingent time/output bundles of the employer (l 2L,t 2L ) and (l 2H,t 2H ) Two different state-continent time/output bundles of the employee (ľ ij, ť ij ) denote initial endowments i=1, 2; j=L, H 42

43. Example 3: Optimal Employment Contracts under Uncertainty the optimal contracting problem for the employer max[p L U(l 1L,t 1L )+p H U(l 1H,t 1H )] Subject to p L u(l 2L,t 2L )+p H u(l 2H,t 2H )≥ū l 1j +l 2j ≤ľ 1j +ľ 2j t 1j +t 2j ≤ť 1j +ť 2j Where ū=p L u(ľ 2L,ť 2L )+p H u(ľ 2H,ť 2H ) 43

44. PART 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 44

45. Adverse 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 45

46. Screening 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? 46

47. Screening 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 47 Second Price Auction

48. Example 4: Screening Problems Extend the previous examples with uncertainty with private information added Suppose that employee time and output enter additively: U[α θ (1-l)-t] for the employer u( θ l+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 48 U(l 1, t 1 ), u(l 2, t 2 )

49. Example 4: Adverse selection The utility function: U[α θ (1-l)-t] for the employer u( θ l+t) for the employee θ, the state of nature The employee knows whether she is skilled with a value of time θ H or unskilled, with a value of time θ L < θ H The employer knows only that the probability of facing a skilled employee is p H 49

50. Example 4: Adverse selection in state θ j ， the employee's endowment ľ j = θ j relevant reservation utility is ū H =u( θ H ), employer faces a skilled employee ū L =u( θ L ), employer faces an unskilled employee incentive compatibility constraints the employee's time is more efficient when sold to the employer α>1 u(t i )≥u( θ j ) 50

51. Example 4: Adverse selection revelation principle offers a key simplification Type θ H must prefer contract ( t H, l H ) over ( t L, l L ) Type θ L contract ( t L, l L ) over ( t H, l H ) the optimal menu of employment contracts under hidden information: Max{p L U[α θ L (1-l L )-t L ]+p H U[α θ H (1-l H )-t H ]} Subject to u(l L θ L +t L )≥ū( θ L ), u(l H θ H +t H )≥ū( θ H ) u(l H θ H +t H )≥u(l L θ H +t L ), u(l L θ L +t L )≥ u(l H θ L +t H ) 51

52. Example 4: Adverse selection 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 52

53. Moral 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 53 Texas Drivers

54. Moral 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 54

55. PART 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? 55

56. Example 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 θ j is realized this action influences the probability of the state of nature 56

57. Example 5: Moral Hazard when the employee chooses action (1-l) output for the employer is simply θ H with a probability function p H [1-l] and θ L with a probability function p L [I-l]=1-p H [1-l] more effort produces higher expected output, at cost 1-l for the employee The employer offers a compensation contract t( θ j ) to the employee under the output-contingent compensation scheme t( θ j ) (1-l) will be chosen by the employee to maximize her own expected payoff 57

58. Example 5: Moral Hazard the effort level chosen by the employee is the outcome of the employee's own optimization problem: (1-l) є argmax{p L [1-l]u[t( θ L )+l]+p H [1-l]u[t( θ H )+l]} the employer solves the following maximization problem: max{p L [1-l]U[ θ L -t( θ L )]+p H [1-l]U[ θ H -t( θ H )]} Subject to p L [1-l]u[t( θ L )-l]+p H [1-l]u[t( θ H )-l]≥ū=u(1) (1-l) є argmax{p L [1-l]u[t( θ L )-l]+p H [1-l]u[t( θ H )-l]} the employer chooses the optimal compensation contract {t( θ j )} to maximize his expected utility 58

59. Example 5: Moral Hazard the effort level chosen by the employee is the outcome of the employee's own optimization problem: (1-l) є argmax{p L [1-l]u[t( θ L )-l]+p H [1-l]u[t( θ H )-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 59

60. PART 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 60

61. Conclusion Matching Theory Contract Theory 61