Matching with Contracts

Transcription

1 Matching with Contracts Paul Milgrom Lecture June 25, 2014 Thanks to John Ha=ield for sharing his slides, on which these slides rely 1

2 Themes Matching with Contracts nests various matching models and auchon models (including Vickrey), some with and some without money transfers or other contract terms. The Gale- Shapley algorithm is a planning algorithms, related to Walrasian price- adjustments, which can be mimicked in models without prices, but Marshallian quan6ty adjustments, which will be part of the basis of the FCC s incenhve auchon, and Cumula6ve offer algorithms, which apply to both matching models and combinatorial auchons. MathemaHcal connechons are parhcularly Hght among models sahsfying a suitably general subshtutes condihon. 2

3 Literature This lecture is based mainly on Ha=ield and Milgrom (2005), which is part of the large matching theory literature. Kelso and Crawford (1982): Introduced prices into the two- sided matching framework. Ausubel and Milgrom (2002): Introduced cumulahve offer algorithms to address auchon problems without market clearing prices. 3

4 Contract Environment Our matching environment is characterized by three finite sets D is the set of doctors H is the set of hospitals F = D H is the set of market parhcipants T is the set of terms (wages, locahon, hours, etc) A contract is an element of the finite set X D H T. 4

5 Choice Functions These market parhcipants preferences are encapsulated in choice funchons C f (Y). These are restricted so that C d (Y) {x Y x D = d} A doctor chooses among the contracts bearing her name C h (Y) {x Y x H = h} A hospital chooses among the contracts bearing its name Assume that there are also associated preference relahons P f For the market as a whole, the doctors and hospitals choices from their budget sets are denoted by C D (Y) = d D C d (Y) C H (Y) = h H C h (Y) 5

6 Matching with Contracts SUBSTITUTES 6

7 Universal Substitution We want to define the concept of subshtuhon to be universal in two senses. Whether x and y are subshtutes does not depend on the presence of other alternahves. Every contract is a subshtute for every other contract. Defini&ons and Nota&on. 1. Contracts are subs$tutes if for all x,y X and Z X, if x C f (Z {x}) then x C f (Z {x,y}). 2. A funchon F:X Z is isotone if x y F(x) F(y). (Informally, isotone means non- decreasing). 3. The rejec$on func$on is R f (Z) Z C f (Z). Proposi&on. Contracts are subshtutes for f if and only R f is isotone. 7

8 Equivalent De?initions The matching with contracts model embeds tradihonal matching models. Taking T to be a set of wages, D H T is the Kelso- Crawford model. Taking T to be a singleton, we can idenhfy X with D H to get the Gale- Shapley algorithm. Equivalences Contracts are subshtutes in the Gale- Shapley model if and only if preferences are subshtutable in the sense of Roth & Sotomayor. Contracts are subshtutes in the Kelso- Crawford model if and only if they are price- theorehc gross subshtutes. Key step: raising a worker s ( required ) wage means reducing the set of ophons available to the firm. Concept: prices idenhfy the firm s choice set. 8

9 Examples of Substitutes 1. Responsive preferences (addihvely separable values). 2. Separate openings for men and women. 3. Hospital openings with reversion. 4. A hockey team regard players as subshtutes in the case where the team needs a goalie, a defenseman, and a forward the team s payoff is the sum of the values of the players in the posihons to which they are assigned the team can assign any of its players to any posihon 5. 9

10 Dual Characterization, 1 In the classical theory of the firm, let w be a vector of wages and let π be the firm s indirect profit funchon. π(w) = max x S f(x)- w x. By Hotelling s lemma, for almost all wage vectors w (points of differenhability of π), the firm s demand for workers is x*=- dπ/dw. So, in the theory of the firm, workers are subs6tutes if and only π is submodular (roughly: nega6ve mixed par6al deriva6ve). 10

11 Dual Characterization, 2 Defini&on: The (indirect uhlity) funchon U f represents the preference/choice relahon P f /C f if U f (Y) > U f (Z) C f (Y) P f C f (Z) Defini&on. U is submodular if for all x X and Y Z X, U(Y {x}) U(Y) U(Z {x}) U(Z). Theorem (Ha6ield & Kominers). Contracts are subshtutes for parhcipant f if and only if the preferences of f can be represented by a submodular indirect uhlity funchon. 11

12 Proof. Suppose that that contracts are NOT subshtutes for f. Then, there exists a set Z X and contracts x,y X such that x C f (Z {x}) and x C f (Z {x,y}). By strict preference, this implies that C f (Z {x,y}) f C f (Z {x}) = C f (Z). Suppose that indirect uhlity funchon U f represents f. Then, U f (Z {x,y}) > U f (Z {x}) = U f (Z). So, U f (Z {x,y}) U f (Z {x}) > 0 = U f (Z {x}) U f (Z), in contradichon to submodularity. If contracts are subshtutes, then the indirect uhlity funchon that assigns uhlity n to the n th most preferred set of contracts is submodular and represents preferences. 12

13 Matching with Contracts STABILITY 13

14 Stability De?ined A set of contracts Y X is stable if it sahsfies two condihons: 1. Individual ra6onality: for all f F, C f (Y)=Y f. 2. Unblocked: There exists no set of contracts Z such that Z- Y and for all f D H, Z f C f (Y Z). Our next task is to characterize stability in terms of agents making ophmal choices from a budget set, and those sets being mutually consistent. Then, we look at an algorithm to find stable allocahons and characterize it as related to both the Gale- Shapley algorithm and to Marshallian dynamics. 14

15 Marshallian Dynamics Supply Price Supply June 26, 2014 Demand Price QuanHty Traded 15

16 Generalized Deferred Acceptance Informally, the state of the algorithm at any moment is a pair (X D,X H ) represenhng the contrachng opportunihes as seen by the doctors and hospitals, respechvely. Formally, we require X D X and X H X. As the algorithm progresses, the doctors budget set consists of all those contracts that the hospitals have not just rejected, and the hospitals consider [symmetrically] F D (X H ) = X - R H (X H ) F H (X D ) = X - R D (X D ), and F(X H,X D ) = (F D (X H ),F H (X D )) 16

17 Substitutes and Stability Theorem. If contracts are subshtutes for all market parhcipants, then A X is a stable allocahon if and only if there exists a fixed point (X D,X H )=F(X D,X H ), such that A=X D X H. Fixed point: X D = X - R H (X H ) = doctors budget sets X H = X - R D (X D ) = hospitals budget sets Note: X D X H = X 17

18 Proof Sketch If A=X D X H, then Any individually unacceptable contract is always rejected and no such contract is included from A. By construchon, contracts not in X D X H would, if proposed, be rejected by one side or the other. By subshtutes, if mulhple addihonal contracts were proposed, they would shll be rejected (because the rejechon funchons are isotone). If A is stable then, we can construct X D and X H as follows: every contract not in A would, if proposed, be rejected by its h or d, so it can be added to X H or X D, respechvely. 18

19 Bounded monotone sequences converge ASIDE ON LATTICE MATHEMATICS 19

20 Lattices and Order Let be a transihve, reflexive relahon on a set X. Defini&on. (X, ) is a la3ce if for all x,y X, inf{x,y} X and sup{x,y} X both exist. Nota&on. x y=inf{x,y} and x y=sup{x,y}. Examples X comprises the subsets of some base set S and x y means x y. x y=x y and x y=x y. X=R N and x y means x i y i for i=1,,n. (x y) i = min(x i,y i ) and (x y) i = max(x i,y i ). R N is an example of a product la3ce. 20

21 Tarski s Fixed Point Theorem Defini&ons. 1. A laƒce (X, ) is complete if for every non- empty subset S X, inf(s) X and sup(s) X. Roughly, bounded monotone sequences have their limits in the set. 2. A funchon F:X X is isotone if x y F(x) F(y). Theorem. If (X, ) is a complete laƒce and F:X X is isotone, then the set of fixed points of F is a non- empty laƒce. Moreover, if X is finite, with largest element x max and smallest element x min, then the sequences {F n (x max )} and {F n (x min )} converge in finitely many steps to the largest and smallest fixed points of F, respechvely. 21

22 LATTICE STRUCTURE IN MATCHING 22

23 Existence, Computation Defini&on: (X D,X H ) (Y D,Y H ) if both (1) X D Y D and (2) Y H X H. With this definihon, higher pairs represent expanded opportunihes for hospitals and contracted opportunihes for doctors. the domain of the funchon F described earlier is a finite laice. Meets and joins correspond to unions and intersechons in the first component and to the reverse (intersechons and unions) in the second component. the operator F is isotone Theorem. The set of fixed points of the funchon F is non- empty and has a maximal element and a minimal element. Iterated applicahons of F starhng from the highest (lowest) point in the laƒce converges monotonically down (up) to the highest (lowest) fixed point. 23

24 Relation to Gale & Shapley The highest point in the laƒce is the pair (X D,X H )=(,X), at which doctors have no opportunihes and hospitals have all ophons shll open. Iterated applicahon of F At the first applicahon, hospitals reject all but their most preferred acceptable contracts, so that the next pair of contracts includes some offers to doctors. At the second round, doctors reject all but their most preferred acceptable contracts from among the offered contracts, reducing the set of opportunihes for hospitals. That the extremal fixed points are doctor- best and hospital- best follows from a comparison of the opportunity sets. Similarly, in a compehhve model with subshtutes, there is a seller- best point with the highest prices for all goods, and a buyer- best point with the lowest prices. NoHce as- if cumulahve offers 24

25 INCENTIVES AND RELATED 25

26 Law of Aggregate Demand Defini&on. The preferences of individual f sahsfy the law of aggregate demand if for all Y Z X, C f (Y) C f (Z). In the price- theorehc version of the problem, an expanding opportunity set for the firm corresponds to lower wages for all workers. The definihon requires that when a firm is faced with lower wages, it should hire more workers. 26

27 Rural Hospitals Theorem Theorem. If each parhcipant regards contracts as subshtutes and its preferences sahsfy the law of aggregate demand, then each signs the same number of contracts at every stable allocahon. Proof. At the doctor- ophmal stable allocahon, the doctors allocahon coincide with their choices from the largest opportunity set, so each signs at least as many contracts as in any stable allocahon. So, the total number of contracts signed is at least as large as at any stable allocahon. Similarly, each hospital has its smallest opportunity set, and hence signs the smallest number of contracts, so the total number is at least as small But each contract has one doctor and one hospital, so these numbers are equal to each other, and so equal to the corresponding numbers at any stable allocahon. 27

28 Strategy- Proofness Theorem (Ha6ield & Milgrom). Suppose that each doctor can sign only one contract and that each hospital reports preferences sahsfying the law of aggregate demand and for which contracts are subshtutes. Then, the induced one- sided reporhng game among doctors is strategy- proof. Truth- telling is always ophmal for doctors. 28

29 Proof Sketch The proof proceeds by showing that if any doctor d reports any preference P that leads to outcome x, then each of the following variahons leads her to a weakly be er outcome than the preceding one. 1. Report preference P (leading to a set A, including x for d). 2. Report that only x is acceptable. Fewer allocahons are blocked, so A is shll stable. By LOAD, d is assigned at every stable allocahon, so it must be get contract x. 3. Report that contracts truly worse than x are unacceptable, but otherwise report preferences truthfully. By isotonicity of F, the highest stable set is at least as good for doctors as A, so d must do at least as well. 4. Report preferences truthfully. Outcome unchanged from report #3. 29

30 Group- Strategy Proofness Theorem (Ha6ield & Kojima). Suppose that each doctor can sign only one contract and that each hospital reports preferences sahsfying the law of aggregate demand and according to which contracts are subshtutes. Then, the induced reporhng game among doctors is weakly group strategy- proof. (No coalihon of doctors can deviate in a way that is strictly beneficial to all of them.) 30

31 A First Lemma Lemma. Suppose that Z is the doctor- ophmal stable allocahon under preference profile P, in which doctor d gets contract z and suppose that y is preferred to z under P d. Let Pʹ be the preference profile in which d s preference only is replaced by the one in which y is moved to the top of d s otherwise unchanged list. Then the doctor best stable match for Pʹ is also Z. Proof. Z is shll stable, since it is shll acceptable and the potenhal blocking pairs are unchanged. Suppose that the resulhng allocahon is Y. Since the mechanism is individually strategy- proof, y Y. But then Y is blocked by whatever set had blocked coalihon Y under P. 31

32 Proof Let S={1,,n} be a minimal coalihon that can, for some preferences, benefit by a joint deviahon, changing its outcome from the result x of truthful reporhng to some preferred outcome y. By the lemma, if n alone deviates, reporhng the preferences y n x n, the outcome is unchanged. But if these were n s true preferences, then the coalihon {1,,n- 1} could make the same deviahon to bring about Y, contrary to the assumed minimality of S. 32

33 Cumulative Offer Processes Consider a variahon on the doctor- offering DA algorithm in which hospitals can go back to accept any collechon of offers from the current round or any past round. When contracts are subshtutes, the rejechon funchon F is isotone, so if an offer is once rejected, it stays rejected. That means that the Gale- Shapley doctor- offering algorithm is equivalent to one in which each hospital can go back to take any offer that she has ever received. If hospitals do not regard contracts as subshtutes, then 1. Fixed points of F may not be stable. 2. The DA algorithm may fail to terminate in a stable match. 3. The contracts that emerge from the cumulahve offer algorithm may be infeasible, assigning doctors to mulhple hospitals. 33

34 Cumulative Offer Auction Suppose that there is just one hospital, which is the auchoneer, so that the final match is always feasible. Imagine that the doctors are bidders who are interested in supplying packages of services to the auchoneer, which may include various services on various terms at various prices. The auchon works like this: Bidders submit rank order lists of their bids ( contracts ). The auchoneer submits a rank- order list of collechons of bids. The cumulahve offer algorithm is run. At each round, Each bidder that is not in the currently winning package makes the next bid on its acceptable list. The auchoneer tentahvely rejects all bids except its most combinahon, which can include at most one bid from each bidder. 34

35 Core Outcome Theorem (Ausubel & Milgrom). The outcome of the cumulahve offer auchon is a core allocahon. Proof. At the end of the auchon, each bidder has made every offer that it prefers to the final outcome, so any blocking coalihon must combine the exishng accumulated offers. the auchoneer has accepted the combinahon of accumulated bids that it most prefers. 35

36 The Case of Substitutes Theorem (Ausubel & Milgrom). If the auchoneer regards contracts as subshtutes, then the outcome of the cumulahve offer auchon is the bidder best core allocahon. This can be proved as a corollary of the matching with contracts result, because in the subshtutes case the two algorithms coincide. 36

37 SUBSTITUTES & QUASI- LINEAR UTILITY 37

38 Aside: Submodular Minimization Defini&on. Given a laƒce (X, ), a funchon f:x R is submodular if for all x,y X, f(x) + f(y ) f(x y) + f(x y). The condihon means that the incremental return to increasing one variable is declining in the other variables. For any smooth funchon f:r N R, f is submodular if and only if its mixed parhal derivahves are non- posihve. 38

39 Indirect Utility and Substitutes Define the indirect uhlity funchon for a bidder by: π(p) = max x v(x) p x. By Hotelling s lemma, the bidder s demand for good l at prices p is: x l (p) = - π l (p) where the second term is the parhal derivahve of π with respect to the price p l. Goods are therefore subshtutes if and only if π l (p) is a non- increasing funchon of each p k for k l, or equivalently π( ) is submodular. 39

40 Substitutes and Prices If there are mulhple copies of each good, market- clearing prices may fail to exist if a small increase in the price of a good can lead to either Bidder reduces demand for the good by two units Bidder reduces demand by one unit but subshtutes two units of another product. Defini&on (Milgrom & Strulovici). Goods are strong subs$tutes for a parhcipant if, when each unit is treated as an individual good, these redefined goods are subshtutes. Kelso- Crawford Existence of clearing prices. 40

41 Vickrey, Substitutes and the Core Theorem. If the goods are strong subshtutes for all bidders and if the auchoneer has a fixed supply to sell, then the Vickrey auchon outcome is the bidder- best core allocahon. Role/InterpretaHon Each bidder gets its highest payoff in the core. Seller gets its lowest revenue in the core. This result connects incenhves in matching theory to those in ascending auchons. 41

42 Proof of Theorem Let v( ) be the coalihonal value funchon constructed from the data, with zero value to coalihons that exclude the auchoneer. Any payoff to a bidder j higher than v(n)- v(n- j) results in an allocahon that is blocked by coalihon N- j, so there are no higher bidder core payoffs. Suppose that, as we will later show, the funchon v( ) is submodular. Consider any coalihon S that includes the seller (player 0). Let it be S = {0,1,,k}. Then, S is not a blocking coalihon, because = v(n) π j j S π j j S j S ( ) = v(n) v(n) v(n j) N j=k+1( ) v(n) v(n { j + 1,..., N }) v(n { j,..., N }) = v(s) 42

43 From Goods to Bidders To finish, we must show that v( ) is submodular. We need to use our hypothesis that indirect uhlity a funchon of prices is submodular to derive the conclusion that coalihonal values a funchon of sets of bidders is submodular. Lemma. Suppose that for every bidder, preferences are quasi- linear and goods are strong subshtutes. Then the funchon v v(s) = max x v j (x j ) subject to x j S j S j is submodular. x 43

44 Aside: Submodular Minimization Theorem (Topkis). If (X Y, ) is a product laƒce and f:x Y R is submodular, then min y f(x,y) is a submodular funchon from X R. Remark. This is similar to the fact that if f(x,y) is convex, then min y f(x,y) is a convex funchon of x. 44

45 A Useful Submodular Function Let F(p,S) = π j (p) + p i x. j S Let us verify that if the indirect uhlity funchons π i (p) are submodular, then F is submodular: S alone: F(p,S {i}) = F(p,S)+F(p,{i}). p alone: submodular π i (p) funchons. S,p: F(p,S {i})- F(p,S )= π i (p) is non- increasing. 45

46 Proof of Lemma We calculate the coalihonal value as follows: v(s) = max x v j (x j ) subject to x j S j S j x p i x j ( ) = min p 0 max x v j (x j ) x j S j S = min p 0 max ( xj v j (x j ) p i x ) j + p i x j S = min p 0 π j (p) + p i x j S = min p 0 F(p,S), where F(p,S) = π j (p) + p i x. j S Because goods are strong subshtutes, the market clearing price vector p exists, which jushfies the second line above. Because goods are subshtutes, each π j ( ) is submodular and non- increasing and these imply that F(p,S) is submodular. Therefore, by Topkis s theorem, v( ) is submodular. 46

47 Summary and Forecast The Matching with Contracts approach exposes strong links between matching theory and both price theory and auchon theory. The submodular dual representahon of subshtutes applies to both. The extended subshtutes condihon plays a key role in all three for Stable matches to exist and the GS algorithm to work Tatonnement stability and the existence of seller- best and buyer- best equilibria (highest and lowest prices). Vickrey outcomes to lie in the core and to result from cumulahve offer auchons. Next lecture exposes more connechons, including a deeper connechon to Marshallian dynamics, as part of the design of the upcoming US incenhve auchon. 47

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