Efficient assignment and the Israeli medical match 25th Jerusalem School in Economic Theory Assaf Romm Harvard University 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 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. 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. 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. 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. 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. The assignment problem (not to be confused with “the assignment problem”) • We have a set of objects = 1 , … , to be assigned to a set of agents = 1 , … , . • Each agent has preferences over which object she would like to be assigned. • How do we allocate the objects to the agents? • (Important: transfers are not allowed) • More motivation: medical internships, seats in schools, course allocation, thesis advisers, NBA draft, and many more. 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) 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. Easy to implement? • Given everybody’s preferences, it’s about two lines of code. • However… 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. 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. 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. 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! 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. 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. The Israeli medical match • And yes, with about 400 interns, that’s one giant hat… G-RRRRY-FIN-DDDDO… Oh, sorry, I meant, Soroka • Starting from 2011: computerized random serial dictatorship. Back to theory… One small problem with RSD: it is not ex-ante efficient. Example: = 1 , 2 , 3 , 4 and = 1 , 2 , 3 , 4 , and: 1 , 2 : 1 ≻ 2 ≻ 3 ≻ 4 3 , 4 : 2 ≻ 1 ≻ 4 ≻ 3 Resulting distribution: 1 2 3 4 1 5/12 5/12 1/12 1/12 2 1/12 1/12 5/12 5/12 3 5/12 5/12 1/12 1/12 Suggested improvement: 4 1/12 1/12 5/12 5/12 1 2 3 4 1 2 3 4 1/2 0 1/2 0 1/2 0 1/2 0 0 1/2 0 1/2 0 1/2 0 1/2 Notions of efficiency In laymen terms: after the RSD lottery nobody wants to switch, but prior to the lottery people are willing to trade. A mechanism is • Ex-post efficient at ≻ if ≻ can be represented as a probability distribution over deterministic efficient assignments. • Ex-ante efficient at if is Pareto optimal. • Ordinally efficient at ≻ if the random assignment ≻ is not stochastically dominated by another random assignment. • Rank efficient, rank distribution is not dominated. Probabilistic Serial • Based on Bogomolnaia and Moulin, JET, 2001. Simultaneous eating algorithm with uniform eating speeds. Probabilistic Serial – Step 1 Each object is a pizza. Portions of the pizza represent probability to be assigned that object. Object 1 Object 2 Object 3 Object 4 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. Agent 1 Agent 2 Object 1 Agent 3 Object 2 Agent 4 Object 3 Object 4 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. Agent 1 0.51 + 0.53 Agent 2 0.51 + 0.53 0.52 + 0.54 Agent 3 0.52 + 0.54 Agent 4 Probabilistic Serial – Step 4 We conduct a lottery according to the probabilities. In the example: 1 2 3 4 1 1/2 1/2 0 0 2 0 0 1/2 1/2 3 1/2 1/2 0 0 4 0 0 1/2 1/2 We perform a lottery between two deterministic assignments: 1 2 3 4 1 1 0 0 0 2 0 0 1 0 3 0 1 0 0 4 0 0 0 1 and 1 2 3 4 1 0 1 0 0 2 0 0 0 1 3 1 0 0 0 4 0 0 1 0 Probabilistic Serial – Step 4 But what if we got this matrix? 1 2 3 4 1 0.23 0.29 0.46 0.02 2 0.28 0.44 0.12 0.16 3 4 0.39 0.1 0.05 0.22 0.28 0.14 0.28 0.54 Note that we cannot just randomize for each agent independently, we have to randomize between valid deterministic assignments. Theorem (Birkhoff-von Neumann): This is always possible. Properties of Probabilistic Serial • Not very easy to understand, not very easy to implement. • “Fair” (envy-free, but we’re not going to dicuss that) • Ordinally efficient. Proof: Suppose agents and ′ want to trade the probabilities of getting objects and ′ . It means that when ate , ′ was busy eating something better for him, and he only ate ′ later. But then, why did eat when we ′ was available? A similar argument is true for more general trades. PS is not strategy-proof 1 : 1 ≻ 2 ≻ 3 2 : 1 ≻ 3 ≻ 2 3 : 2 ≻ 1 ≻ 3 Resulting PS assignment: 1 2 3 1 2 1/2 1/4 1/2 0 0 3/4 3 1/4 1/2 1/4 If, however, 3 reports 1 ≻ 2 ≻ 3 he gets: Manipulated PS assignment: 1 2 3 1 2 3 1/3 1/2 1/6 1/3 0 2/3 1/3 1/2 1/6 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) 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. Large markets results Here are the ranking distribution for RSD and PS in the Israeli medical match from 2011 (378 doctors, 20 hospitals): Large markets results Here are the ranking distribution for RSD and PS in the Israeli medical match from 2011 (378 doctors, 20 hospitals): 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. 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) CE approach – small problem Requires submitting vNM utility functions! 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. 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. 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). 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. 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) Efficiency gains On the other hand, not being strategy-proof means that we avoid the large market results of Liu and Pycia (2013): 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? BvN decomposition • Except for RSD, the three mechanisms presented require decomposition of matrices containing probability shares into lotteries over deterministic assignments. • The matrices have two very special properties: – All rows sum to 1 (that is, every doctor is guaranteed an internship) – Every hospital’s column sums to that hospital’s capacity. • This reminds us of bi-stochastic matrices, whose every row and every column sum to 1, and all elements are in 0,1 . BvN decomposition Theorem: Any bi-stochastic matrix can be represented as a convex combination of permutation matrices. Proof: Suppose is a × bi-stochastic matrix that has elements equal to zero. Think about a bipartite graph with edge , iff > 0. Note that for every set of rows we have: = = ∈ =1 ≤ ∈ ∈ = ∈ =1 Use Hall’s theorem to find a perfect matching, represented by the permutation matrix . Let ′ = − , where = min Aij > 0 , and normalize ′ so it will be a bi-stochastic matrix with + 1 zero elements. Repeat until = . BvN decomposition - complexity We can decompose a matrix quite quickly: We do at most 2 steps, and find a perfect matching in each step (can be done by Ford-Fulkerson algorithm in 2 ). Run time is 4 . 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) 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). 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) Couples and BvN decomposition (with Noga Alon, Slava Bronfman and Avinatan Hassidim) Example: Assume that couples cannot be split. Both hospitals have two positions. There are two doctors in the market, 1 and 2 , and a couple, 1 . By naively trading probability shares we could possibly get the following matrix: 1 2 1 ℎ1 ℎ2 3/4 1/4 1/4 3/4 1/2 1/2 This matrix cannot be decomposed to determinstic allocations. Couples and BvN decomposition 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. Couples and BvN decomposition Theorem: assuming that: • there are more singles than couples, • all doctors must be assigned, • and couples must be assigned together, we can find a decomposition that approximates a given matrix 1 up to , where = min ℎ . ℎ There are examples showing that this is tight, in the sense that we cannot do better than doctors. 1 or with more couples than 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. THANK YOU! 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).