Virtual Demand and Stable Mechanisms

Transcription

1 Virtual Demand and Stable Mechanisms Jan Christoph Schlegel Faculty of Business and Economics, University of Lausanne, Switzerland Abstract We study conditions for the existence of stable allocations and of stable, strategyproof mechanisms in a many-to-one matching model with discrete salary space (the discrete Kelso-Crawford model). Firms and workers want to match many-to-one and agree on the terms of their match. Firms demand different sets of workers at different salaries. Workers have preferences over different firm-salary combinations. Workers preferences are monotone in salaries. We show that for this model a descending auction mechanism is the only candidate for a stable mechanism that is strategyproof for workers. Moreover, we identify sufficient and necessary (in a maximal domain sense) conditions on the demand functions of firms such that the mechanism is stable and strategy-proof. For each demand function in our domain, we can construct a related demand function that we call a virtual demand function. Replacing demand functions by virtual demand functions will not change the outcome of our mechanism. Known conditions (gross substitutability and the law of aggregate demand) can be applied to the virtual demand profile to check whether the mechanism is stable and strategy-proof. Our result gives a sense in which gross substitutability and the law of aggregate demand are necessary for a stable and strategy-proof mechanism to exist. JEL-classification: C78, D47 Keywords: Matching with contracts; Matching with salaries; Gross Substitutes; Virtual Demand 1 Introduction Centralized clearing houses based on the deferred-acceptance mechanism are at the heart of many successful real-world matching markets (Roth, 1984; Abdulkadiroglu and Sönmez, 2003; Sönmez and Switzer, 2013; Sönmez, 2013). These kind of matching mechanisms are appealing because they produce stable outcomes meaning that no subgroup I am grateful to my adviser Bettina Klaus for many useful comments that greatly improved the paper. I thank Sangram Kadam for many insightful discussions and Fuhito Kojima and Maciej Kotowski for useful comments on a previous version of this paper. I gratefully acknowledge financial support by the Swiss National Science Foundation (SNSF) under project

2 of agents can find a mutually beneficial deviation and thus would have a reason to contract outside the market. 1 Moreover, it is safe for one side of the market to report their true preferences to the mechanism. Thus, the mechanism successfully aggregates the information in the market and levels the playing field for naive and sophisticated participants. In some applications, the market does not only match agents but also determines the contractual details of the match. In a labor market, firms and workers may have some discretion on how to set the salary. In the cadet-to-branch match (Sönmez and Switzer, 2013), cadets can choose between different lengths of service time in exchange for a higher priority in their branch of choice. These markets can be understood as hybrids between matching markets and auctions and have first been analyzed in a seminal paper of Kelso and Crawford (1982). They propose a generalization of the (firm-proposing) deferredacceptance algorithm that they call the salary adjustment process. They identify a condition on firms demand functions - workers have to be gross substitutes for the firms - that guarantees that the process converges to a stable allocation. In the salary adjustment mechanism, it is in general not optimal for workers to reveal their true preferences or for firms to reveal their true demand functions. Subsequently, Hatfield and Milgrom (2005) have identified conditions on the demand such that the workerproposing (descending) version of the salary adjustment process is stable and strategyproof for the workers. 2 For this to be the case, workers have to be gross substitutes for firms and the law of aggregate demand 3 must hold for each firm. In this paper, we extend the analysis of Kelso and Crawford (1982) and Hatfield and Milgrom (2005) and consider necessary as well as sufficient conditions for the existence of stable and strategy-proof mechanisms. We show that gross substitutability and the law of aggregate demand are essentially necessary for the existence of a stable and strategy-proof mechanism. For this purpose, we introduce the notion of a virtual demand function. For a demand function, the corresponding virtual demand function is a closely related but in general more well-behaved demand function. We show that replacing demand functions by virtual demand functions will not change the outcome of the salary adjustment process. If workers are virtual gross substitutes, i.e. workers are gross substitutes according to the virtual demand, then the salary adjustment process converges to an allocation that is stable both in the original market and the virtual market where we replace demand by virtual demand. We show that the domain of demand functions, under which workers are virtual gross substitutes and the virtual law 1 See Roth (1984, 1991) for evidence that clearing houses that use unstable mechanisms tend to fail in practice. 2 Hatfield and Milgrom actually go beyond the Kelso-Crawford model by allowing for multidimensional contract-terms instead of just salaries in their model. Different contracts can be ranked arbitrarily whereas in the Kelso-Crawford model workers preferences are monotone in the contractdimension, i.e. workers prefer higher salaries to lower salaries. We will discuss this issue in more detail later. See also Echenique (2012) and Schlegel (2015) for a discussion on what this adds in generality to the model. 3 This means that if we enlarge the choice set of a firm by lowering salaries, an equal or smaller number of workers will be chosen. The axiom trivially holds if firms have quasi-linear preferences in salaries and workers are gross substitutes. 2

3 of aggregate demand holds, is a maximal domain for the salary adjustment mechanism to be stable and strategy-proof and, more generally, for the existence of any stable and strategy-proof mechanism. The class contains demand functions under which workers are gross substitutes and the law of aggregate demand holds. For these demand functions the virtual demand and the original demand are the same. However it also contains many other demand functions for which the virtual demand and the original demand differ. In our analysis, we treat demand functions as primitive and do not necessarily require that demand functions are generated by preference relations. If we restrict attention to demand function that are generated by preferences that are quasi-linear in salaries, then our domain is just the domain of gross substitutes demand functions. For quasi-linear preferences, it follows from our result that gross substitutability is a necessary and sufficient (in the maximal domain sense) condition for the existence of - both - stable allocations and stable and strategy-proof mechanisms. 1.1 Related Literature Stable many-to-one matching mechanisms and their incentive properties have been extensively studied (Hatfield and Kojima, 2010; Chen et al., 2014; Hirata and Kasuya, 2015; Kominers and Sönmez, 2015; Hatfield et al., 2015). Most papers focus on the pure matching model or on the matching with contracts model of Hatfield and Milgrom (2005). The latter model allows for multidimensional contract-terms that can be ranked arbitrarily. A labor contract could, for example, contain a particular job description as well as a salary and it might not be a priori clear how a worker ranks different job and salary combinations. In contrast to this, we consider the more restricted case where there is a natural ordering on the contract set and the workers have monotone preferences with respect to this ordering. If a worker agrees to work for some firm under some salary, then he will also agree to work for the firm under a higher salary. Cadets prefer short service times over long service times etc. Since we are considering a more restricted model, all sufficient conditions for stability and the existence of a stable and strategy-proof mechanism from the literature on matching with contracts also apply to our model. However, conditions that are necessary for the model with contracts are not necessary condition for the model with salaries. For strategy-proofness, this is because certain preference manipulations are ruled out by the model. A worker must report monotone preferences. Thus he cannot rank working for a firm under a low salary above working for the same firm under a high salary to manipulate the outcome of the mechanism in his favor. Similarly, weaker conditions are sufficient to guarantee the existence of stable allocations than those for markets with contracts. 4 Recently, a maximal domain of choice functions for the existence of a stable and strategy-proof mechanism for matching markets with contracts has been identified (Hat- 4 To illustrate this point consider a Kelso-Crawford market with two firms f 1, f 2 and three workers w 1, w 2, w 3. Suppose there are two kinds of contracts: A firm and a worker can match under a low salary 3

4 field et al., 2015). We note that the result of Hatfield et al. (2015) and our result on stable and strategy-proof mechanisms are logically independent. Our domain is larger than the domain of demand functions whose induced choice functions satisfy the conditions of Hatfield et al. (2015). 5 On the other hand, their result applies to a broader class of problems. We think that one advantage of studying the less general model is that a characterization becomes easier to state and the condition on the demand functions is easier to interpret and test. Furthermore, many practically relevant problems fit into the framework with salaries. On the other hand, some problems like the airline seat-upgrade problem of Kominers and Sönmez (2015) where the true preferences are likely to violate monotonicity in the contract-term do not fit well into the model with salaries. In this sense, we think that the results are complementary and both contribute to our understanding of stable and strategy-proof mechanisms in matching markets with endogenous contracting. 2 Model and Known Results 2.1 Model The following model is based on the job matching model of Kelso and Crawford (1982). There are two finite disjoint sets of agents, a set of firms F and a set of workers W. There is a finite set of possible salaries S = {1, 2,..., σ}. Firms can hire workers and pay them salaries. Each firm f has a demand function D f : S W 2 W that for a vector of salaries s = (s w ) w W specifies a set of workers D f (s) W that the firm wants to hire under these salaries. A market is a pair (D, ) consisting of a demand profile D = (D f ) f F and a preference profile = ( w ) w W. We make the following 1 or under a high salary 2. The firms have preferences {(w 1, 1), (w 2, 1), (w 3, 1)} f1 {(w 1, 1)} f1 {(w 1, 2)} f1 {w 1, 1} f2 {w 2, 1} f2. Workers always prefer to work for a firm under the high salary to working for the same firm under the low salary. There is no restriction on how workers rank the different firms under different salaries, i.e. a worker may prefer to work for f 1 under a low salary to working for f 2 under a high salary. Going through all different cases, one can show that for any preferences for the workers satisfying the monotonicity assumption a stable allocation exists. This changes if workers could report non-monotonic preferences as in the model with contracts. Then a stable allocation can fail to exist. Consider, for example, the following preferences: (f 2, 2) w1 (f 1, 1) w1 (f 2, 1) w1 (f 1, 2) w1 (f 2, 2) w2 (f 1, 2) w2 (f 2, 1) w2 (f 1, 1) w2 (f 2, 2) w3 (f 1, 2) w3 (f 2, 1) w3 (f 1, 1) w3 Worker w 1 has non-monotone preferences in salaries. He prefers to work for firm f 1 under a low salary to working for the same firm under a high salary. No stable (in the matching with contracts sense) allocation exists. 5 For example, the preferences of firm f 1 in the example in Footnote 4 induce a demand function that satisfies our condition but no the condition of Hatfield et al. (2015). 4

5 assumptions on demand functions: 1. The maximal salary σ is prohibitively high so that no firm will ever hire a worker under this salary: s w = σ w / D f (s). It will be convenient to extend demand functions to incomplete salary vectors as follows: Suppose we have a salary vector that specifies salaries only for a subset W W of the workers, s = (s w ) w W S W. Define s S W by s w = s w for w W and s w = σ for w / W. Then we let D f (s) := D f ( s). 2. Demand functions satisfy irrelevance of rejected contracts (IRC): 6 Suppose some worker-salary combination was not chosen at some salary vector and we increase the worker s salary further. Then the firm will make the same choice after the salary has increased. Formally, let w W and s, s S W with s w = s w 7 and s w < s w. Then w / D f (s) D f (s) = D f (s ). Each worker w has preferences w over different firm-salary combinations and an outside option which we denote by. We make the following assumption on workers preferences: 1. Preferences are strict, (f, σ) (f, σ ) (f, σ) w (f, σ ) or (f, σ ) w (f, σ). 2. Preferences are increasing in salaries, σ < σ (f, σ) w (f, σ ). We denote the set of all strict and increasing preferences by R. A matching is a function µ : F W F 2 W such that 1. for each f F, we have µ(f) 2 W, 2. for each w W, we have µ(w) F { }, 3. for each f F and w W, we have f = µ(w) if and only if w µ(f). A salary schedule for a matching µ is a salary vector s = (s w ) S µ(f ) that for each matched worker w specifies a salary s w paid by µ(w) to w. For each f F, we let s f := (s w ) w µ(f). For two salary schedules s, s for the same matching µ, we write s f s f if s w s w for each w µ(f) and s s if s f s f for each f F. An allocation 6 The requirement is a form of the weak axiom of revealed preferences and is an adaption of the IRC condition from the matching with contracts literature (Aygün and Sönmez, 2013) to our set-up (see Subsection ). 7 Here and in the following we let s w := (s w ) w W \{w}. 5

6 is a pair (µ, s) consisting of a matching µ and a salary schedule s. We denote the set of allocations by A. For notational convenience, we extend workers preferences over firm-salary pairs to preferences over allocations in the usual way; for each w W we let (µ, s) w (µ, s ) : (µ(w), s w ) w (µ (w), s w). Let (D, ) be a market. Allocation (µ, s) is individually rational in (D, ) if for each f F we have D f (s f ) = µ(f) and for each w W we have (µ, s) w, blocked in (D, ) by firm f F and group of workers W µ(f) if there is a salary vector s S W with s µ(f) = s f such that D f (s ) = W and (f, s w) w (µ, s) for each w W, stable in (D, ) if it is individually rational and not blocked by any firm and group of workers. We denote the set of all stable allocations in (D, ) by S(D, ). The following lemma provides a reformulation of the stability condition that will be useful in some of the proofs. Lemma 1. For (µ, s) A, f F and R W vector s f S W by define the minimal potential blocking s fw := min{σ S : (f, σ) w (µ, s)} { σ}. Let D be a demand profile. Then (µ, s) S(D, ) if and only if (µ, s) is individually rational in (D, ) and for each f F we have D f ( s f ) = µ(f). Proof. The only if direction follows directly from the definition of stability: If (µ, s) is not individually rational, then it is not stable. If there is a f F with D f ( s f ) µ(f), then f and D f ( s f ) block (µ, s) via s f. The if direction follows by IRC: Indeed suppose there is a firm f, group of workers W µ(f) and salary vector s S W with s µ(f) = s f such that D f (s ) = W and (f, s w) w (µ, s) for each w W. If s = s f, then D f (s ) = µ(f) W and we have a contradiction. Otherwise we have s f s with s µ(f) = s µ(f). Thus, by IRC, D f (s ) = D f ( s f ) = µ(f) W and we have a contradiction. A mechanism (for the workers) is a mapping from preference profiles to allocations M : R W A. Mechanism M is strategy-proof if it is a dominant strategy for workers to report their true preferences to the mechanism, i.e. for each w W, w R W \{w} and w, w R we have M( w, w ) w M( w, w ). Let D be a demand profile. Mechanism M is D-stable if for each R W M( ) S(D, ). we have 6

7 2.1.1 Matching with Contracts Our model can be mapped into the matching with contracts model as follows (here we follow Hatfield and Milgrom, 2005): The set of possible contracts is X = F W S. Thus a contract (f, w, σ) is a bilateral agreement between a firm f and a worker w to match under a salary σ. For each f F we define a choice functions C f : 2 X 2 X as follows. For each set of contracts X X define a salary vector s f (X ) = (s w (X )) w W by s fw (X ) := min{σ S : (f, w, σ) X or σ = σ}. Thus the salary s fw (X ) of worker w is the lowest salary with firm f among contracts in X. Firm f chooses from X the contracts with the workers that it demands under the minimal salaries s f (X ): C f (X ) := {(f, w, s fw (X )) : w D f (s(x ))}. The IRC condition on demand functions translates to the IRC condition on choice functions (Aygün and Sönmez, 2013): (f, w, σ) / C f (X ) C f (X \ {(f, w, σ)}) = C f (X ). Worker w s preferences over X { } are given by (f, w, σ) w (f, w, σ ) (f, σ) w (f, σ ) and (f, w, σ) w (f, σ) w with the convention that w (f, w, σ) for w w. It is easy to check that our definition of stability is equivalent to the usual stability condition in the matching with contracts literature. In this sense, our model is just a matching with contracts model with the additional restriction that preferences are monotone in the contract terms. 2.2 Stable Allocations In general, a stable allocation does not need to exist for our model. A sufficient condition for stability is that workers are gross substitutes for firms, i.e. increasing the salary of some worker will not decrease the demand for other workers whose salaries have not changed. Gross Substitutability: For workers w, w W and salary vectors s, s S W s w = s w and s w < s w, with w D f (s) w D f (s ). Not only is gross substitutability sufficient for the existence of a stable allocation but also it imposes a lattice structure on the set of stable allocations. If workers are gross substitutes for firms, then the set of stable allocation forms a lattice with respect to the preferences of workers (Blair, 1988). In particular, there is a unique stable allocation that is most preferred by all workers among all stable allocations. We call this allocation 7

8 the worker-optimal stable allocation. It can be found by the salary adjustment process 8 that is defined as follows: 1. We start with salaries s fw (0) := σ for each (f, w) F W. 2. In round t, each worker applies to his favorite firm under salaries (s fw (t)) f F,w W or stays alone if he finds no firm acceptable under these salaries. For each firm f F, we let s f (t) S W be the vector of salaries with the best offers that it has received up to and including round t (with (s f (t)) w = σ if w never has made an offer to f). 3. Each firm tentatively accepts the workers D f (s f (t)) and rejects all other workers. If worker w makes an offer to f in round t and gets rejected we let s fw (t + 1) = s fw (t) 1 and repeat Steps 2 and 3 for round t + 1. We use the convention that a worker will not apply to a firm f in round t + 1 if s fw (t + 1) = 0. If all offers are accepted, then we go to Step We match each firm to the workers whose offers it has accepted in the last round under the offered salary. In general, the salary adjustment process does not need to converge to a feasible allocation. It could be the case that a worker w makes an offer in some round to some firm f which tentatively accepts this offer, but also an other firm f now wants to accept an offer made by w to f in an earlier round that f had previously rejected. Thus, in the final allocation multiple firms could be matched to the same worker. The gross substitutes condition rules out this possibility since it guarantees that firms will never want to recall offers made in previous rounds. Moreover, by the definition of the process, if the outcome of the process is feasible then it is stable as well. Later we will see that weaker conditions than gross substitutability guarantee the convergence to a feasible (and stable) allocation. Let D be a demand profile such that the salary adjustment process converges to a feasible outcome for any preference profile. Then the salary adjustment mechanism for D assigns to each R W the outcome of the salary adjustment process in (D, ). Worker-optimality is related to strategy-proofness. Under the following additional condition on the firms demand functions the salary adjustment mechanism is strategyproof (Hatfield and Milgrom, 2005). Law of Aggregate Demand. For salary vectors s, s S W s s D f (s) D f (s ). The condition is automatically satisfies if the demand function is induced by preferences that are quasi-linear in salaries and workers are gross substitutes. 8 For the more general model with arbitrary contracts, this is called the cumulative offer process. As Hatfield and Milgrom (2005), we consider a version of the process were multiple workers per round make new offers. We could also consider a version of the process were offers are made subsequently. If workers are gross substitutes for firms, then this will yield the same outcome (Hirata and Kasuya, 2014). 8

9 The following proposition summarizes known results about side-optimal stable matchings and strategy-proofness. Proposition 1 (Kelso and Crawford, 1982; Blair, 1988; Hatfield and Milgrom, 2005). 1. If workers are gross substitutes for firms, then the salary adjustment process converges to a stable allocation that is most preferred by all workers among all stable allocations. 2. If demand functions satisfy, moreover, the law of aggregate demand, then the salary adjustment mechanism is strategy-proof. 3 Results 3.1 Virtual Demand Functions It is a natural question whether the conditions of Section 2.2 for the salary adjustment mechanism to be stable and strategy-proof are also necessary. Next we provide a counter example showing that the gross substitutes and the law of aggregate demand are not necessary for the salary adjustment mechanism to be stable and strategy-proof. The example will have the following structure: Each but one firm has a demand function such that workers are gross substitutes and the law of aggregate demand holds. One firm has a demand function such that workers are not gross substitutes for that firm. However, the demand function of that firm can be replaced by another demand function such that the outcome of the salary adjustment process is the same under the original demand profile and the profile where the firm s demand function is replaced by the new demand function. Under the second demand function - we will subsequently call it a virtual demand function - workers are gross substitutes and the law of aggregate demand holds. Thus, the salary adjustment process is stable and strategy-proof both for the original market and the market where we have replaced the demand function by the virtual demand function. Example 1. Let W = {w 1, w 2, w 3 } and f F be a firm. Suppose firms F \ {f} have demand functions D f = (D f ) f f under which workers are gross substitutes and the law of aggregate demand holds. For firm f we consider two different demand functions: The demand function D f is induced by preferences {(w 1, 1), (w 2, 1), (w 3, 1)} f {(w 2, 1)} f {(w 2, 2)} f. Note that under D f workers are not gross substitutes as w 3 D(1, 1, 1) = {w 1, w 2, w 3 } but w 3 / D f (2, 1, 1) = {w 2 }. The demand function Df is induced by preferences {(w 2, 1)} f {(w 2, 2)} f. and thus D f (s) = { {w 2 }, if s 2 2, else. 9

10 Workers are gross substitutes under Df. Let R W be an arbitrary preference profile. Note that the salary adjustment process in (D, ) will not end up in an allocation where f is matched to w 1, w 2, w 3 under salaries (1, 1, 1). This is because once the mechanism tentatively matches w 2 to f under salary 2 the firm will not subsequently drop the worker w 2 or accept additional workers. Thus the salary adjustment processes in the market (D, ) and in the market (Df, D f, ) converge to the same allocation. As workers are gross substitutes under Df and the law of aggregate demand holds, the salary adjustment mechanism for (Df, D f ) (and D) is strategy-proof and (Df, D f )-stable. As observed before, if the salary adjustment process converges to a feasible allocation then this allocation is stable. Thus the mechanism is D-stable as well. The argument in the example can be made more generally. Subsequently, we will define for each demand function a corresponding virtual demand function. To define the virtual demand function Df for a demand function D f we consider a market with only one firm f and workers W. Thus we have an auction rather than a matching market. Since there is only one firm, the workers preferences are determined by their reservation salaries, i.e. the smallest salaries under which the workers are willing to work for f. Now suppose we run the salary adjustment process in the market consisting of firm f with demand function D f and workers with reservation salaries s S W. Let s be the terminal offer vector for firm f in the salary adjustment process under reservation salaries s. In the terminal allocation f is matched to D f (s ). We define the virtual demand function for D f to be the demand function Df : SW 2 W where Df (s) := D f (s ). In the following, for each property of a demand function we indicate that the property holds for the virtual demand function by adding the adjective virtual. Thus we say that workers are virtual gross substitutes if they are substitutes according to the virtual demand function. Similarly, we talk about the virtual law of aggregate demand, etc. Similarly, for a demand profile D = (D f ) f F we call D = (Df ) f F the virtual demand profile for D and for a market (D, ) we call the market, ) the virtual market. If workers are virtual gross substitutes, then the virtual demand function is wellbehaved in the sense that it satisfies the properties required for demand functions in Section 2: By definition, a worker is never virtually demanded under the maximal salary σ. Moreover, the virtual demand function satisfies IRC. Remark 1. If workers are virtual gross substitutes, then the virtual demand function satisfies IRC. Proof. Let w W and s, s S W such that s w = s w and s w < s w. We want to show that w / D f (s) w / D f (s ). We show the stronger statement that w D f (s ) D f (s) = D f (s ). Let (s(t)) t=0,...,t and (s (t)) t=0,...,t be the sequences of offer vectors for f in the salary adjustment process for reservation salaries s and s. Suppose D f (s) D f (s ). Then there is a t such that s(t) = s (t), and s w (t) = s w < s w (t + 1) = s w. Moreover, w / D f (s(t)) = D f (s(t)). By virtual gross substitutes, rejected offer are never recalled, 10

11 i.e. w / D f (s (t)) = D f (s (t)) implies that for t > t we have w / D f (s (t )) = D f (s (t )). In particular, w / D f (s (T ))) = D f (s (T ) = D f (s ). One readily check that in our example the demand function Df satisfies the definition of the virtual demand function. Example 1 (cont.). To see that the virtual demand function is indeed given by { Df (s) = {w 2 }, if s 2 2, else note that for each s = (s 1, s 2, s 3 ) with s 1 2 a descending auction will terminate in an allocation that matches w 2 to f under salary 2 i.e. for s 1 2 we have s = (2, s 2, s 3 ) and D f (s ) = {w 1 }. Otherwise the terminal assignment is empty i.e. for s 1 > 2 we have s = s and D f (s ) =. As illustrated by Example 1, the virtual demand function can be different from the original demand function. However, for two important classes of demand functions the demand function and the virtual demand function coincide - demand functions generated by preferences that are quasi-linear in salaries and demand functions such that workers are gross substitutes and the law of aggregate demand holds. More generally, we show that a demand function and the corresponding virtual demand function coincide under the following regularity condition on the demand function: Demand function D f satisfies the strong law of demand if for s, s D f (s) with s w = s w and s w < s w we have w D f (s ) D f (s) = D f (s ). Remark 2. If the demand function D f satisfies the strong law of demand, then the demand function and the virtual demand function are the same D f = Df. If a demand function is generated by preferences that are quasi-linear in salaries or if the demand function satisfies gross substitutes and the law of aggregate demand, then it satisfies the strong law of demand and thus coincides with the virtual demand function. Proof. Let s S W and s s be the outcome of the salary adjustment process. By definition, for w / D f (s ) we have s w = s w. Thus the strong law of demand implies that D f (s ) = D f (s). To see that the strong law of demand is satisfied for a demand function that it generated by quasi-linear preferences let u : 2 W R be such that D f (s) = argmax W W u(w ) s w. w W Let s, s D f (s) with s w = s w and s w < s w with w D f (s ). By definition, we have u(d f (s )) s w (u(w ) s w) w D f (s ) w W { = u(df (s )) w D f (s ) s w (u(w ) w W s w) if w W < u(d f (s )) w D f (s ) s w (u(w ) w W s w) if w / W 11

12 Thus D f (s ) also maximizes u(w ) w W s w. Next we show that the strong law of demand is satisfied if workers are gross substitutes and the law of aggregate demand holds. Let s, s D f (s) with s w = s w and s w < s w with w D f (s ). By gross substitutes, we have D f (s) D f (s ). By the law of aggregate demand this implies D f (s) = D f (s ) Stability Next we relate stability in the virtual market to stability in the original market. The proof of the following proposition as well as all subsequent proofs are in the appendix. Proposition 2. Let D be a demand profile, D its virtual version and a preference profile. If workers are virtual gross substitutes for each firm, then the outcome of the salary adjustment process in the original market (D, ) and in the virtual market (D, ) is the same and stable in both markets. 3.2 A Maximal Domain Result In this section, we show that the domain of demand functions such that workers are virtual gross substitutes and the virtual law of aggregate demand holds is a maximal (Cartesian) domain for the existence of a stable and strategy-proof mechanism. This means that if we choose a demand profile D = (D f ) f F such that each D f has the two properties then the salary adjustment mechanism is well-defined, stable and strategyproof. On the other hand if either of the conditions fails for the demand function of a firm f we can define a unit demand 9 function D f for a second firm f such that for the profile D = (D f, D f ) there is no D-stable, strategy-proof mechanism. The following lemma will be useful in the proof of the maximal domain result. It states that stable and strategy-proof mechanisms are unique whenever they exist. Similar results are known for the classical matching model (Alcalde and Barberà, 1994) and the model with contracts (Hirata and Kasuya, 2015). Lemma 2. Let D be a demand profile. If there is a D-stable and strategy-proofness mechanism, then it is unique. If workers are moreover virtual gross substitutes, then the stable and strategy-proof mechanism implements the worker-optimal stable allocation in the virtual market. With the lemma we can proof our main result. Theorem 1. The domain of demand functions under which workers are virtual gross substitutes and the virtual law of aggregate demand holds is maximal for the existence of a stable and strategy-proof mechanism. 1. Let D be a demand profile such for each firm workers are virtual gross substitutes and the virtual law of aggregate holds. Then there is a D-stable and strategy-proof mechanism. 9 Here unit demand means that the firm demands at most one worker at each salary vector. This implies in particular gross substitutability. 12

13 2. Let D f be a demand functions such that either workers are not virtual gross substitutes or the virtual law of aggregate demand fails. Then there is a unit demand function D f such that no (D f, D f )-stable and strategy-proof mechanism exists. For the special case of demand functions induced by quasi-linear preferences, we had previously observed that the virtual demand function is the demand function itself. Moreover, for these kind of demand functions gross substitutability implies the law of aggregate demand. Thus for demand functions induced by preferences that are quasilinear in salaries, gross substitutes is sufficient for the salary adjustment process to be stable and strategy-proof. On the other hand, a variant of the argument in the proof of Theorem 1 shows that gross substitutes is now necessary in the maximal domain sense for the existence of a stable allocation. We obtain the following corollary. Corollary 1. Among demand functions induced by preferences that are quasi-linear in salaries the domain of demand functions under which workers are gross substitutes is maximal for the existence of a stable matching and more generally for the existence of a stable and strategy-proof mechanism. 1. Let D be a demand profile induced by a profile of quasi-linear preferences. If workers are gross substitutes for each firm, then there is a D-stable and strategyproof mechanism. 2. Let D f be a demand functions induced by quasi-linear preferences. If workers are not gross substitutes for the firm, then there are quasi-linear preferences inducing a unit demand function D f and a profile of worker-preferences such that no stable allocation in (D f, D f, ) exists. Bibliography Abdulkadiroglu, A. and Sönmez, T. (2003): School Choice: A Mechanism Design Approach. American Economic Review, 93(3): Alcalde, J. and Barberà, S. (1994): Top dominance and the possibility of strategy-proof stable solutions to matching problems. Economic Theory, 4(3): Aygün, O. and Sönmez, T. (2013): Matching with Contracts: Comment. American Economic Review, 103(5): Blair, C. (1988): The Lattice Structure of the Set of Stable Matchings with Multiple Partners. Mathematics of Operations Research, 18(4): Chen, P. C.-W., Egesdal, M., Pycia, M., and Yenmez, M. B. (2014): Manipulability of Stable Mechanisms. Technical report, mimeo. Echenique, F. (2012): Contracts versus Salaries in Matching. Review, 102(1): American Economic 13

14 Hatfield, J. W. and Kojima, F. (2010): Substitutes and Stability for Matching with Contracts. Journal of Economic Theory, 145(5): Hatfield, J. W., Kominers, S. D., and Westkamp, A. (2015): Stability, Strategy- Proofness, and Cumulative Offer Mechanisms. Technical report, mimeo. Hatfield, J. W. and Milgrom, P. R. (2005): Matching with Contracts. Economic Review, 95(4): American Hirata, D. and Kasuya, Y. (2014): Cumulative Offer Process is Order-Independent. Economics Letters, 124(1): Hirata, D. and Kasuya, Y. (2015): On stable and strategy-proof rules in matching markets with contracts. Technical report, mimeo. Kelso, A. and Crawford, V. P. (1982): Job Matching, Coalition Formation, and Gross Substitutes. Econometrica, 50(6): Kominers, S. D. and Sönmez, T. (2015): Matching with Slot-Specific Priorities: Theory. Theoretical Economics, forthcoming. Roth, A. E. (1984): The Evolution of the Labor Market for Medical Interns and Residents: A Case Study in Game Theory. Journal of Political Economy, 92(6): Roth, A. E. (1991): A Natural Experiment in the Organization of Entry-Level Labor Markets: Regional Markets for New Physicians and Surgeons in the United Kingdom. American Economic Review, 81: Schlegel, J. C. (2015): Contracts versus Salaries in Matching: A General Result. Journal of Economic Theory, 159(A): Sönmez, T. (2013): Bidding for Army Career Specialities: Improving the ROTC Branching Mechanism. Journal of Political Economy, 121(1): Sönmez, T. and Switzer, T. B. (2013): Matching With (Branch-of-Choice) Contracts at the United States Military Academy. Econometrica, 81(2):

15 A Proof of Proposition 2 Proof. We prove the result by induction on the length of workers preference lists. For R W let l( ) := w W {(f, σ) F S : (f, σ) w }. Induction Basis: For each R W with l( ) = 0, the only individually rational allocation is the empty matching. Both the salary adjustment process in (D, ) and in (D, ) converge to the empty matching which is stable in both markets. Induction Assumption: For each R W with l( ) n the outcome of the salary adjustment process in the market (D, ) and in the virtual market (D, ) is the same and stable in both markets. Induction Step: Let R W with l( ) = n + 1. Since workers are virtual gross substitutes for each firm, Proposition 1 implies that the salary adjustment process in the virtual market converges to the worker-optimal stable allocation (µ, s) in (D, ). First note that during the salary adjustment process in both markets the worker will apply to all firms under all acceptable salaries. Otherwise we could invoke the induction assumption. For each f F let s f = (s fw ) w W S W be the vector of salaries s fw = min{σ S : (f, σ) w }. We will show that for each f F we have D f (s f ) = Df (s f ) = µ(f) and thus the allocation (µ, s) is also the outcome of the salary adjustment process and stable in (D, ). Suppose for the sake of contradiction that there is a f F with D f (s f ) Df (s f ). Consider the modified salary schedule s where for each w µ(f ) we let s w = s f w and for w / µ(f ) we let s w = s w. Since µ(f ) = Df (s f ) D f (s f ) we have s s. Next we show that (µ, s) S(D, ). This will be a contradiction with (µ, s) being the worker-optimal stable allocation in (D, ). For each f F let s f be the minimal potential blocking vector for (µ, s) under preferences as defined in Lemma 1. For f we have s f = s f and by definition of D f we have D f (s f ) = D f (s f ) = µ(f ). For each f f we have s f s f with s fw = s fw for w µ(f). Thus by the virtual IRC we have Df ( s f ) = Df (s f ). Therefore Lemma 1 (again we implicitly use the virtual IRC) implies (µ, s) S(D, ). We have reached a contradiction. Thus for each f F we have D f (s f ) = Df (s f ) = µ(f) and the allocation (µ, s) is also the outcome of the salary adjustment process and stable in (D, ). B Proof of Lemma 2 Proof. Let M, M be D-stable and strategy-proof mechanisms. We show that for each R W we have M( ) = M ( ) and if workers are virtual gross substitutes under D for each firm then M( ) is the worker-optimal stable allocation in the virtual market (D, ). We prove the result by induction on the length of workers preference lists. For R W let l( ) := w W {(f, σ) F S : (f, σ) w }. Induction Basis: For each R W with l( ) = 0, the empty matching is the only stable allocation in (D, ) and (D, ) and the lemma trivially holds. Induction Assumption: For each R W with l( ) n we have M( ) = M ( ). 15

16 Moreover, if workers are virtual gross substitutes, then M( ) is the worker-optimal stable allocation in (D, ). Induction Step: Let R W with l( ) = n + 1. Let (µ, s) := M( ) and (µ, s ) := M ( ). Suppose (µ, s) (µ, s ). Since (µ, s), (µ, s ) S(D, ), there is a w W such that either (µ, s ) w (µ, s) w or (µ, s) w (µ, s ) w. If (µ, s ) w (µ, s) w, then truncate w s preferences after (µ (w), s w). If w submits the truncated preferences w and everyone else submits preferences w, then - by strategy-proofness of M - w will receive the same assignment (µ (w), s w) in M ( ) and M ( w, w ). Furthermore, by the induction assumption - M ( w, w ) = M( w, w ). But then, M( w, w ) w M( ) contradicting strategy-proofness. Thus, there is no w W with (µ, s ) w (µ, s) w. A completely analog argument shows that there is no w W with (µ, s) w (µ, s ) w. Next assume that workers are virtual gross substitutes for each firm under D. Let (µ, s) := M( ) and (µ, s ) be the worker-optimal stable allocation in (D, ). Suppose (µ, s) (µ, s ). By Proposition 2, (µ, s ) is stable in (D, ) as well. Since (µ, s), (µ, s ) S(D, ), there is a w W such that either (µ, s ) w (µ, s) w or (µ, s) w (µ, s ) w. If (µ, s ) w (µ, s) w, then truncate w s preferences after (µ (w), s w). Note that (µ, s ) is also stable in the markets (D, w, w ) and (D, w, w ). By the induction assumption, M( w, w ) is the worker-optimal stable allocation in (D, w, w ). By strategy-proofness of M, worker w is unmatched in M( w, w ) since otherwise M( w, ) w M( ). However, then (µ, s ) w M( w, ) contradicting the worker-optimality of M( w, ) in (D, w, w ). If (µ, s) w (µ, s ) w, then truncate w s preferences after (µ(w), s w ). By the induction assumption, M( w, w ) is the worker-optimal stable allocation in (D, w, w ). By strategy-proofness of M, M w ( w, w ) = M( ). Thus M w ( w, w ) is stable in (D, ) as well. But this contradicts the worker-optimality of (µ, s ) in (D ). C Proof of Theorem 1 Proof of 1. By Proposition 2, the salary adjustment mechanism for D and for D is the same and stable with respect to both profiles. By Proposition 1, it is strategy-proof. Proof of 2. Let s 1, s 2 S W be salary vectors with s 1 = (s 2 w, s 2 w + 1) for some w W such that for each s S W with s = s 1 at salary vectors s s s workers are virtual gross substitutes and the virtual law of aggregate demand holds either w / D f (s1 ) but w D f (s2 ) for some w W (a violation of virtual gross substitutes) or D f (s1 ) > D f (s2 ) (a violation of the virtual law of aggregate demand). 16

17 First we show the following fact that will be useful in the subsequent proof: Fact 1. Let w W and s, s S W such that s s s 2, s w = s w and s w < s w. If w Df (s ) then Df (s) = D f (s ). Moreover, D f (s 2 ) = Df (s2 ). To prove the first part, let (s(t)) t=0,...,t and (s (t)) t=0,...,t be the sequences of offer vectors for f in the salary adjustment process for reservation salaries s and s. Suppose Df (s) D f (s ). Then there is a t such that s(t) = s (t), and s w (t) = s w < s w (t+1) = s w. Moreover, w / Df (s(t)) = D f (s(t)). Note that in the market with reservation salaries s virtual gross substitutes holds. Thus rejected offer are never recalled, i.e. w / Df (s (t)) = D f (s (t)) implies that for t > t we have w / D f (s (t )) = Df (s (t )). In particular, w / Df (s (T )) = D f (s (T )) = Df (s ). For the second part, suppose Df (s2 ) D f (s 2 ). Recall that (s 2 ) is the terminal offer vector for f in the salary adjustment process with reservation salaries s 2. By definition of the virtual demand, we have (s 2 ) w = s2 w for w / D f (s2 ) and Df ((s2 ) ) = Df (s2 ). In particular, w Df ((s2 ) ). If (s 2 ) w > s 2 w, then (s 2 ) s 1 and by the first part of Fact 1, Df ((s2 ) ) = Df (s1 ). But this contradicts w / Df (s1 ). If (s 2 ) w = s 2 w, then by the first part of Fact 1, Df ((s2 ) w, (s 2 ) w + 1) = D f (s1 ). Since (s 2 ) s 2, virtual gross substitutes holds in the market with reservation salaries (s 2 ). Thus w Df ((s2 ) ) implies that w Df ((s2 ) w, (s 2 ) w + 1) = D f (s1 ). But this contradicts w / Df (s1 ). Now we are ready to proof the second part of the theorem. We consider two cases: Either for s 1 s 2 we have a violation of virtual gross substitutes or we have a violation of the virtual law of aggregate demand. Case 1. Violation of virtual gross substitutes. We define D f by w if s w < σ, s w < s w + 1 D f (s) = w if s w < σ, s w s w 1 else Let D = (D f, D f ). Suppose there is a D-stable, strategy-proof mechanism M. Consider the profile R W defined by (f, σ) w... w (f, s 2 w) w (f, σ) w (f, σ 1) w (f, σ) w... w (f, s 1 w ) w (f, σ) w (f, σ 1) w (f, s 2 w ) w (f, σ) w... w (f, s 2 w ) w for w w, w. First we show that in the virtual market there is no stable allocation, i.e. S(D, ) =. Indeed suppose (µ, s) S(D, ). Let s f be the minimal potential blocking vector for allocation (µ, s) and firm f under preferences. By Fact 1, we have either D (s f ) f = Df (s1 ) if µ(f ) = {w } or Df (s f ) = Df (s2 ) and s w = s 2 w if µ(f ) {w }. In the first 17

18 case, w / µ(f) = Df (s1 ) and f and w can block with salary σ 1. In the second case, µ(f) = Df (s2 ) and f and w can block with salary σ 1. Thus (µ, s) is not stable, a contradiction. Now let (µ, s) := M( ). We consider two cases: Either there is a w W with (µ, s) w (f, s2 w ) or not. In the first case, truncate w s preferences after (µ( w), s w ). If w submits the truncated preferences w and everyone else submits preferences w, then - by strategy-proofness of M - w will receive the same assignment (µ( w), s w ) in M( ) and M( w, w). By Lemma 2, M( w, w) is the worker-optimal stable allocation in the virtual market (D, w, w). Since w is matched in M( w, w) the allocation is also stable in the untruncated virtual market (D, ). But this contradicts S(D, ) =. In the second case, by Fact 1, µ(f) = D f (s 2 ) = Df (s2 ) and therefore w, w µ(f). Hence µ(f ) =. But then f and w can block (µ, s) with salary σ 1 contradicting the stability of (µ, s). Case 2. Violation of the virtual law of aggregate demand. We may assume that workers are virtual gross substitutes since otherwise we are back to Case 1. Since D (s 1 ) Df (s2 ) the IRC condition for Df implies that w / Df (s1 ) but w Df (s2 ). Thus there are two other workers w 1, w 2 W \ {w } with w 1, w 2 Df (s1 ) but w 1, w 2 / Df (s2 ). We define D f by w 1 if s w1 < σ w 2 if s w1 = σ and s w2 < σ D f (s) = w if s w1 = σ and s w2 = σ and s w < σ else Let D = (D f, D f ). Suppose there is a D-stable, strategy-proof mechanism M. Consider the profile R W defined by (f, σ) w... w (f, s 1 w ) w (f, σ) w (f, σ 1) w (f, s 2 w ) w (f, σ) w1... w1 (f, s 2 w 1 ) w1 (f, σ) w1 (f, σ 1) w1 (f, σ) w2... w2 (f, s 2 w 2 + 1) w2 (f, σ) w2 (f, σ 1) w2 (f, s 2 w 2 ) w2 (f, σ) w... w (f, s 2 w ) w for w w, w 1, w 2 Let (µ, s) := M( ). By Lemma 2, (µ, s) is the worker-optimal stable allocation in (D, ). First consider the case that µ(f ) = {w } with s w = σ 1. Then µ(f) = D f (s1 ). But w 2 / D f (s1 ). Therefore f and w 2 can block with salary σ 1 contradicting stability. Second consider the case that µ(f ) = {w 1 } with s w1 = σ 1. Then µ(f) = D f (s2 ). Now suppose that w 2 changes his preferences to (f, σ) w2... w2 (f, s 2 w 2 ) w2 w2 (f, σ) w

19 Let (µ, s ) := M( w 2, w2 ). By Propositions 2, (µ, s ) is the worker-optimal stable allocation in (D, w 2, w2 ). Thus µ (f ) = {w } with s w = σ 1 and µ (f) = D f (s2 ) with s µ (f) = s 2 µ (f). Hence (µ (w 2 ), s w 2 ) = (f, s w2 ) w2 and w 2 could receive a better assignment by misreporting his preferences contradicting strategy-proofness. Third consider the case that µ(f ) = {w 2 } with s w2 = σ 1. Then µ(f) = D f (s2 w 2, s 2 w 2 +1). Since the virtual demand of f satisfies IRC and w 2 / D f (s2 ) we have D f (s2 w 2, s 2 w 2 + 1) = D f (s2 ). But then w 1 / µ(f). Therefore f and w 1 can block with salary σ 1 contradicting stability. Finally consider the case that µ(f ) =. Then w 1 / µ(f) = D f (s2 ) and f and w 1 can block with salary σ 1. D Proof of Corollary 1 Proof. It is easy to see that if a demand function is generated by quasi-linear preferences in salaries and workers are gross substitutes, then the demand function automatically satisfies the law of aggregate demand. Thus the first part of the corollary follows from Proposition 1. For the second part, let D f be induced by quasilinear preferences and assume that workers are not gross substitutes under D f. In the proof of Theorem 1, we have constructed a unit-demand function D f and worker preferences R W such that for D := (D f, D f ) no stable allocation in the virtual market (D, ) exists. Recall that by Remark 2, the demand and virtual demand are the same for demand functions induced by quasi-linear preferences. Thus it suffices to show that the unit-demand function D f from the proof of Theorem 1 can be generated by preferences that are quasi-linear in salaries. Let u : 2 W R be such that u( ) = 0, u({w}) = σ ɛ, u({w }) = σ 1 + ɛ for a small ɛ > 0 and u(w ) < 0 for W, {w}, {w }. Note that the demand function w if s w < σ, s w < s w + 1 D f (s) = w if s w < σ, s w s w 1 else satisfies D f (s) = argmax W W u(w ) w W s w. 19