Introduction to Matching Theory E. Maskin Jerusalem Summer School in Economic Theory June 2014

much of economics is about markets – exchanges between buyers and sellers commonplace to suppose that sellers are heterogeneous – sell somewhat different goods – so buyers not indifferent between different sellers’ goods 2

distinctive feature of matching markets: in addition to buyers caring about which seller they buy from, sellers care which buyer they sell to e.g., market for education: think of schools as sellers and prospective students as buyers not only do students have preferences over schools but typically, schools have preferences over students ‒ view some students as more desirable than others another (more technical) feature: indivisibilities (student will attend exactly one school - - or no school at all) 3

Matching Theory which buyers are matched with which sellers in equilibrium? what are equilibrium prices? – positive side which matchings between buyers and sellers have desirable properties ? e.g., stability or fairness – normative side 4

how can we find such desirable matchings? – i.e., can we construct algorithms or mechanisms that result in these matchings? – this is market design / implementation side we’ll look at all 3 sides in summer school – particular emphasis on second and third sides – but tomorrow, will look at first side (positive) in order to study wage inequality 5

Model n sellers, each with 1 indivisible good m buyers, each wants to buy at most 1 good – one-to-one matching – in later lectures, will consider many-to-one matching (e.g. each student assigned to one school, but each school assigned many students) buyer i (i= 1,…,m) gets utility from obtaining seller j’s good (being matched with j) – could be positive or negative seller j ( j= 1,…,n) gets utility from selling good to buyer i (being matched with i) each buyer and seller gets 0 utility from remaining unmatched: notationally, (i matched with seller 0) (j matched with buyer 0) two-sided matching (buyers and sellers are different populations) – men matched with women to form marriages – violinists matched with pianists to form duos – will look at one-sided matching (single population) later in summer school roommate problem and house assignment problem 6

For now, assume there exists perfectly transferable good (money) let = price that buyer i pays for seller j’s good (could be negative) buyer i’s payoff = sellers j’s payoff = let matching be matrix such that 7 +

competitive equilibrium is such that claim: competitive equilibrium exists and is (essentially) unique – despite nonconvexity created by indivisibilities 8

to convexify, – let each buyer randomize over seller he buys from – let each seller randomize over buyer he sells to then (random) demand and supply correspondences satisfy standard convex- valuedness and upper hemicontinuity properties so equilibrium exists – with probability 1, no randomization in equilibrium (because equilibrium matching maximizes sum of utilities, and so is generically unique - - see below) – but even if there is randomization, can convert matching into no-randomization equilibrium 9

e.g., can be converted to each buyer and seller indifferent between randomized equilibrium and deterministic equilibrium 10

for any matching, can obtain (by monetary transfers) any payoffs for buyers and for sellers such that (1) hence, from first welfare theorem (equilibrium is Pareto optimal), equilibrium matching solves (2) generically, unique solution to (2) (and no random solutions) so, generically, unique equilibrium matching – can be multiple prices supporting 11

12 (3) (4)

claim: competitive equilibrium in core, where 13

(5) (6) (7) 14

claim: any point 15

Assortative Matching for each i and j, let assume for all i, j think of index i as positively correlated with buyer’s “productivity” (contribution to ) and j as correlated with seller’s productivity – then (11) says that buyer’s marginal productivity is increasing in seller’s productivity and vice versa – e.g., would hold if where f and g increasing 16

claim: given (11), there will be positive assortative matching in competitive equilibrium, i.e., for equilibrium matching 17

Now drop money from model – for some markets (e.g., public schools) buying and selling goods may be problematic can no longer define competitive equilibrium but can still speak of core 18

19

claim: stable matching exists (Gale-Shapley) proof is constructive (algorithmic): in each stage some buyer i, not currently matched, proposes match to favorite seller j (highest ) among those who have not previously rejected him if seller j prefers i to current match rejects algorithm terminates when each unmatched buyer has been rejected by all sellers giving him positive utility called deferred acceptance algorithm, because seller’s “acceptance” of i only provisional 20

and 21

have looked at stable matching when buyers make proposals could do same for proposals by sellers may get different matching – differs from transferable utility case, where stable matching generically unique 22

for example: two buyers, two sellers – if buyers propose, get – if sellers propose, get henceforth, focus on strict preferences 23

claim: order of buyers doesn’t matter when buyers make proposals every buyer (weakly) prefers outcome of buyer-proposal algorithm to any other stable matching 24

be symmetrically, each seller weakly prefers outcome of seller-proposal algorithm to any other stable matching 25

26

27 let in

28

29 then

summary of case II: 30

dominant strategy for buyers to be truthful in buyer-proposal algorithm but sellers may not gain from true revealation of preferences consider earlier example 31

same example shows there is no algorithm guaranteeing stable matchings for which all players always have dominant strategies suppose to contrary there is such a mechanism consider preferences of example if 32