1. Efficient assignment and the Israeli medical match 25 th Jerusalem School in Economic Theory Assaf Romm Harvard University

2. Outline Israeli medical match: introduction Random Serial Dictatorship Probabilistic Serial Competitive Equilibrium approach Israeli medical match: mechanism Birkhoff-von Neumann decomposition Israeli medical match: couples

3. Medical studies in Israel A long track: 6 years of school, 6 months of exams, one year of internship, 3-6 years of residency. Highly desirable: requires the best psychometric exam scores, some students study abroad (but still need to do internships), most people’s moms are jewish. What do you do during internship: clerkship in various departments, taking blood, paperwork, night shifts, minimum wage.

4. Why do we have internships? Justifies the fact that the state of Israel pays for tuition. Prevents outside immigration of recently trained doctors. Helps the hospitals (bypasses the rural hospital theorem). Similar practices are common throughout the world.

5. Some numbers In Israel there are 23 hospitals participating in the match. In 2013, there were 448 interns that completed their studies in Israel (not abroad). The number of positions is determined by the number of interns.

6. How should we assign doctors? Why do interns care about their assignment? Let the doctors decide where they want to do their internship? – No. We won’t get many interns in the more distant hospitals. Two-sided matching according to preferences reported by interns and hospitals? – No. We will not get uniform quality across hospitals. Random permutation? – No. Really wasteful to ignore doctors’ preferences.

7. Principles in assigning doctors Uniform quality across hospitals Capacities are determined according to size and geography Fairness among doctors (flexible concept!) Subject to the above, make interns as happy as possible.

8. The assignment problem (not to be confused with “the assignment problem”)

9. What do we care about? Two leading requirements: Fairness Efficiency (compare with two-sided matching) Questions we should also ask ourselves when designing: How important is strategy-proofness? (and how difficult is “cheating”?) Do items themselves have preferences? (here we assume they do not have any preferences)

10. Random Serial Dictatorship Agents are randomly sorted, and then each one in her turn picks the object she likes best (among those that are still available). Advantages: super-easy to understand and to implement (why is this important?) fair strategy-proof ex-post efficient easily extended to more general environments.

11. Easy to implement? Given everybody’s preferences, it’s about two lines of code. However…

12. The French « choix des postes d’internes » Choosing medical internship in France Each candidate has to get a position, whose main characteristics are specialization and location. (Non-random) serial dictatorship: interns are sorted according to their result in the national exam. Up until 2011: everybody comes to Paris to state their choice. 700 interns entering the hall every day and pick among remaing positions.

13. The French « choix des postes d’internes » Starting from 2011: the process got computerized! Every student in her turn enters the system and picks among the remaining positions.

14. The French « choix des postes d’internes » Starting from 2011: the process got computerized! Every student in her turn enters the system and picks among the remaining positions.

15. The French « choix des postes d’internes » Question: why won’t they take preferences from students and run the serial dictatorship automatically? Answer: too many positions to rank!

16. Harvard housing Many incoming graduate students, many apartments. The selected method: random serial dictatorship with a twist. The twist: each student is randomly assigned to a time block. Within time blocks, students compete for apartments. Question: why? Answer: too many apartments to rank, and we don’t want it to take forever.

17. The Israeli medical match Only 23 possible hospitals, no specialization at that point. Up until 2011: Everybody gets in the same room, and, to make things as random as possible... we draw names out of a hat.

18. The Israeli medical match And yes, with about 400 interns, that’s one giant hat… Starting from 2011: computerized random serial dictatorship. G-RRRRY-FIN-DDDDO … Oh, sorry, I meant, Soroka

19. Back to theory…

20. Notions of efficiency

21. Probabilistic Serial Based on Bogomolnaia and Moulin, JET, 2001. Simultaneous eating algorithm with uniform eating speeds.

22. Probabilistic Serial – Step 1 Each object is a pizza. Portions of the pizza represent probability to be assigned that object.

23. Probabilistic Serial – Step 2 Each agent eats pizzas at the same rate (one bite/sec) and each second he takes a bite from the pizza he likes best.

24. Probabilistic Serial – Step 3 When all pizzas were consumed, the content of each agent’s stomach represents the probabiliies with which he is assigned each item.

25. Probabilistic Serial – Step 4

27. Properties of Probabilistic Serial

28. PS is not strategy-proof

29. PS is weakly strategy-proof This means, no one can manipulate and get an assignment that first-order stochastically dominates the PS assignment. (and we are not going to prove that)

30. Large markets results Kojima and Manea (2008): PS is approximately strategy-proof in large enough markets. Che and Kojima (2010): PS and RSD allocations converge as the market grows large. Liu and Pycia (2013): if a mechanism is asymptotically efficient, symmetric and asymptotically strategy-proof it converges to RSD.

31. Large markets results Here are the ranking distribution for RSD and PS in the Israeli medical match from 2011 (378 doctors, 20 hospitals):

32. Large markets results Here are the ranking distribution for RSD and PS in the Israeli medical match from 2011 (378 doctors, 20 hospitals):

33. Competitive equilibrium approach Based on Hylland and Zeckhauser, JPE, 1979. Algorithm: 1.agents submit their vNM utility functions. 2.Each agent is endowed with equal probabilities to the others for getting any object. 3.Find a price vector that clears the market, and get the equilibrium allocation. 4.A lottery is conducted according to the resulting assignment.

34. Competitive equilibrium approach Not very easy to understand Not very easy to implement Fair Pareto efficient with respect to the reported utilities Proof: by the first welfare theorem (  Ordinally efficient  Ex-post efficient)

35. CE approach – small problem Requires submitting vNM utility functions!

36. CE approach is not strategy-proof Roughly: if an agent can change her report and affect the equilibrium prices, she can manipulate the prices to her advantage.

37. CE approach in large markets If no agent can substantially change the prices, the mechanism is “almost strategy-proof”. By Liu and Pycia (2013), this implies that for any pre-specified utility structure (and only submission of ordinal preferences), the CE approach also converges to RSD in large markets.

38. Israeli medical match – mechanism (with Arnon Afek, Slava Bronfman and Avinatan Hassidim) How did we redesign to do this match? Requirement: do no harm (compared to RSD). Efficency / strategy-proofness tradeoff – Looking to push efficiency even at the cost of strategy-proofness – A good measure for efficiency is rank distribution – Other mechanisms that were designed this way: Teach for America and HBS Field 2 / global immersion (Featherstone and Roth). Described in Featherstone (2014).

39. Israeli medical match – mechanism Sketch of the mechanism: 1.Compute RSD probability shares 2.Solve a linear program to maximize “social welfare” with the constraints that nobody gets less “utility” compared with their RSD probability vector, and that hospitals’ capacities are respected. 3.Run a lottery using BvN decomposition.

40. Properties of the mechanism Not easy to understand or to implement Fair Not strategy-proof: for example, if you like an unpopular hospital (you rank it in the fifth place), you should rank a more popular hospital in front of it so you would get better probability to get to the unpopular hospital you prefer. (Think about “prices”). Moreover, it is not strategy-proof even in large markets! Why do we think it is ok? – If people begin to cheat, we will know – It is not easy to understand how to cheat (classic cryptography style)

41. Efficiency gains On the other hand, not being strategy-proof means that we avoid the large market results of Liu and Pycia (2013):

42. Open questions There is an efficiency / strategy-proofness trade-off. Are there situations in which the trade-off can be shown to be less pronounced? (different domains of preferences) Can we design a mechanism that will require a lot of computing power to find profitable manipulations?

43. BvN decomposition

45. BvN decomposition - complexity

46. Extensions of the theorem Easily extended to our case (hospitals with capacities), just by splitting each hospital to independent positions. Works for more doctors than positions, or more positions than doctors (adding dummy positions or doctors). Budish et al. (2013): multi-unit environments (course allocation) and real-world constraints (group-specific quotas). Also works for certain substitutable preferences. There are also some more mathematical generalizations (e.g. Ellis et al., 2014)

47. Israeli medical match – couples There are medical couples in Israel as well! Even in tiny Israel, couples do not want to be assigned hospitals that are too far apart. This is a form of complementarity, and it’s going to be a problem again. You may expect more couples in Israel compared to the US, but turns out there aren’t that many. In 2014 there were about 15 couples (out of ~500 doctors). Possibly because being a medical couple in Israel is quite a disasterous life decision. Of those, some are actually not romantically-related (but would love to be roommates).

48. Israeli medical match – couples In order to accommodate for couples, it has been decided that they can participate by submitting one rank order list for both of them. When doing RSD (as in the past), it is easy to give the couple one lottery number, and give them their most preferred hospital that still has two vacant internship positions (if there are none, we split the couple). How can we give those couples a similar benefit in our mechanism? (or in any of the ordinally-efficient mechanisms)

49. Couples and BvN decomposition

50. Theorem: Deciding whether a given matrix with couples can be decomposed to a convex combination of valid deterministic assignments is NP-complete. Note : neither the fact that we only deal with couples (and not groups) nor that we restrict their preferences to a certain structure (diagonal elements) help.

51. Couples and BvN decomposition

52. What’s next? The 2014 medical match went well (in the sense that we got no complaints). We are in the process of running surveys to elicit participants’ satisfaction, and to let them compare their probability vectors (RSD vs. our method). MoH people were happy with the procedure. As we speak, Slava and Avinatan sit with them and give them the code so they can use it on interns that come from non-Israeli schools.

53. THANK YOU!

54. Some interesting results from the Israeli psychology match Replacing crazy rounds mechanism 40+ programs in 12 institutions 970 students, 537 matched Program preferences: – Reserved minority slots, gender balance – Scholarships – Professor-specific quotas Still working on analyzing the data (ranking period ended a week ago). But… – Almost 20% failed sanity test (!) – Scholarships seem to have only a minor effect (contrary to departments’ beliefs).