Investing before Stable Matching∗ Benjam´ın Tello† February 2016

Abstract

We analyze the investment game induced by matching markets where workers invest and then match to firms in a stable way, and where monetary transfers are not allowed. We assume that workers have common preferences over firms. We show that a profile of investments is a strictly strong Nash equilibrium if and only if the matching it induces is “investment efficient,” and stable in a related market where investments and partnerships are simultaneously determined. We also characterize its pure strategy Nash equilibria by stability and a weaker notion of efficiency called unilateral efficiency. Next, we provide a condition on the domain of preference profiles that generalizes the notion of lexicographic preferences and ensures the existence of stable and investment efficient matchings. Finally, we show that the requirement that each firm has lexicographic preferences cannot be substantially weakened while still guaranteeing the existence of stable and unilaterally efficient matchings. Keywords: pre-matching investment, Nash equilibria, stability, investment efficiency, unilateral efficiency, only-bilateral disagreement, lexicographic preferences. JEL classification: C78, D47, D60, D82.

∗I

am grateful to Flip Klijn for his guidance. I thank David Cantala, Jordi Mass´ o and Isabel Melguizo for

helpful comments. Financial support from the Consejo Nacional de Ciencia y Tecnolog´ıa (CONACyT), Universitat Aut` onoma de Barcelona through PIF grant 412-01-9/2010 and the Spanish Ministry of Economy and Competitiveness through FPI grant BES-2012-055341 (Project ECO2011-29847-C02) is gratefully acknowledged. † Universitat Aut` onoma de Barcelona and Barcelona GSE, email: [email protected].

1

1

Introduction

Consider the market for medical residency positions or the assignment of students to public schools. Participants in these markets often make large human capital investments well before the matching stage. For example, medical doctors study for several years before participating in residency matching, and students engage in extra-curricular activities or prepare for admission tests before applying to schools. Moreover, in these markets, salaries or prices are fixed by law or they are not the main factor determining the allocation. Therefore, they are not useful in solving the matching problem. The goal of this paper is to study the functioning of matching markets where participants can make investments prior to matching. Our main contribution is the characterization of equilibrium investments and equilibrium outcomes. Moreover, our results describe when equilibrium investments are efficient and when the timing of investment does not matter. We assume that, after sinking their investments, workers and firms match in a stable way. That is, the matching between workers and firms is such that no worker and firm prefer to be assigned to each other rather than to their current partners. Stability is a reasonable assumption for decentralized matching markets with no frictions. The reason is that the only robust predictions that can be made about the outcome of these markets are stable matchings. It is also reasonable for centralized matching markets that employ a stable mechanism1 i.e., a mechanism that selects a stable matching with respect to agents’ reported preferences. The reason is that under the restriction we impose on workers’ preferences (see next paragraph) all stable mechanisms coincide (they select the same matching), and for every worker and firm it is a dominant strategy to report their true preferences to the mechanism. We represent markets with pre-matching investment by a model of (one-to-one) matching with contracts where a contract specifies a firm, a worker and the worker’s investment.2 We impose one restriction on the profile of workers’ preferences called unanimous separability. This restriction requires that there is a ranking of firms 1 Examples

of matching markets that employ a stable mechanism are the National Residency Matching Program,

see Roth (1984), and the New York high school assignment system, see Abdulkadiro˘ glu, Pathak, and Roth (2009). 2 In view of the entry-level labor market interpretation, we call the agents on one side workers, and the agents on the other side firms.

2

such that for any two contracts that involve the same investment, the contract that involves the best firm according to the ranking is preferred by all workers to the other contract. This assumption, while restrictive, arises naturally in various settings. For example, it holds when firms and workers produce using a technology that is increasing in firms’ types and the output is split between workers and firms in fixed proportions. Unanimous separability ensures that once workers have sunk their investments, there is a single stable matching between workers and firms. Our main findings are as follows. In general, the investment game may not have a pure strategy Nash equilibrium (Example 1). We show that an investment profile (a list of workers’ investments) is a strictly strong Nash equilibrium of the investment game if and only if the matching it induces is investment efficient,3 and stable at the complete market4 (Corollary 1). Corollary 1 gives conditions under which the investment game has a (strictly) strong (Nash) equilibrium. Moreover, it establishes that the outcome of the strong equilibrium coincides with the outcome of a single-stage centralized market organized by means of the worker-proposing deferred acceptance (DA) mechanism of Fleiner (2003) and Hatfield and Milgrom (2005). Our result suggests that matching markets whose matching stage is organized by means of a stable mechanism can work well in the presence of a pre-matching investment phase. In particular, (i) workers do not need to recur to complex randomizations, (ii) the equilibrium outcomes are efficient for workers, and (iii) no worker (or even group of workers) has incentives to change investments. In addition, Corollary 1 and the lattice structure of the set of stable matchings imply that the strong equilibrium can be efficiently computed, whenever it exists, via the worker-proposing DA algorithm. Our second main result characterizes the (pure strategy Nash) equilibria of the investment game by stability (at the complete market) and a weak notion of efficiency called unilateral efficiency (Corollary 2). A natural question is, under which conditions do stable (at the complete market) and investment efficient matchings exist? To address this question, we provide a restriction on the domain of preference profiles called only-bilateral disagreement that 3A

matching is investment efficient if there is no investment profile that produces a matching that is weakly

preferred by all workers and strictly preferred by some. 4 The complete market is the benchmark situation where investments and partnerships are determined simultaneously. It is captured by the matching with contracts market where every possible contract is available.

3

ensures the existence of stable and investment efficient matchings. This restriction requires that for each worker w and each firm f, the best contract for w and the best contract for f among all contracts that involve w and f are separated in the preferences of f only by contracts involving w. Thus, the disagreement between f and w is “bilateral” in the sense that it cannot involve some other worker. A firm’s preferences are lexicographic if she ranks all contracts involving the same worker consecutively in her preferences. If the preferences of all firms are lexicographic, then only-bilateral disagreement is satisfied regardless of the preferences of workers. Thus, only-bilateral disagreement is a generalization of lexicographic preferences. Pakzad-Hurson (2014) shows that in a model of many-to-one matching with contracts of which our framework is a special case, suitable generalizations of lexicographic preferences and Ergin’s (2002) acyclicity are sufficient for the existence of stable and Pareto efficient matchings. In addition, he shows that these two conditions are necessary in the following sense: if the profile of firms’ preferences is not lexicographic or has an Ergin cycle, then there are preferences for workers such that no stable matching is efficient. Lexicographic preferences and Ergin acyclicity are not necessary for the existence of stable and investment efficient matchings.5 We show that if one firm’s preferences are not lexicographic and satisfy a mild condition, then there are lexicographic preferences for all other firms and unanimously separable preferences for all workers such that no stable matching is unilaterally efficient (Proposition 1). This result shows that it is not possible to weaken the only-bilateral disagreement condition substantially while still guaranteeing the existence of an equilibrium. 1.1

Related literature

Investment in matching markets has been the subject of several studies. Particular attention has been paid to the case where firms can pay continuous salaries and equilibrium is competitive (agents take salaries as given) (Cole, Mailath, and Postlewaite, 2001a,b; N¨oldeke and Samuelson, 2015). In this setting efficient investment, in the sense of maximizing social surplus, is always an equilibrium. However, because of coordination failures, inefficient equilibria may also arise. 5 Example

4 in the Appendix exhibits a profile of firms’ preferences that has Ergin cycles for which a stable and

investment efficient matching exists regardless of workers’ preferences.

4

If hospitals cannot pay continuous salaries, then efficient investments are not an equilibrium in gerenal. However, Hatfield, Kominers, and Kojima (2015) show that if hospitals can pay discrete salaries, then approximately efficient investments are an equilibrium. A remarkable implication of this result is that under the worker proposing DA mechanism, workers have incentives to make approximately efficient investments before the matching stage. The case where transfers are not possible in the matching stage is analyzed by Peters and Siow (2002); Peters (2007) and N¨oldeke and Samuelson (2015). Peters (2007) studies a model similar to ours where transfers are not possible and where agents on both sides engage in costly investment and are then matched assortatively. The author considers the mixed strategy Nash equilibria of the induced investment game and shows that it converges to a degenerate pure strategy Nash equilibrium in which the two sides of the market invest too much. This differs from and complements our study as it focuses on the mixed strategy Nash equilibria of an investment game, whereas we focus on pure strategy Nash equilibria. The model of N¨oldeke and Samuelson (2015) with non-transferabilities subsumes ours.6 However, the equilibrium concept they consider is competitive: agents take other agents’ utilities as given. Since our equilibrium concept is non-competitive, our analysis and results are different from theirs.

2

Model

2.1

A matching with contracts market

Let W and F be two disjoint sets of workers and firms such that |W | = |F | = m. Let N = W ∪F be the set of agents. Let T be the set of investments or investment types. For each w ∈ W, let tw ∈ T be an investment for worker w. Denote by t = (tw )w∈W an investment profile and let T W be the set of all investment profiles. A feasible contract is a triplet (w, f, tw ) = x ∈ X where X ⊆X ≡W ×F ×T is a feasible set of contracts. We only consider feasible sets of contracts that contain at least one contract between each worker and each firm. Moreover, we assume that 6 N¨ oldeke

and Samuelson (2015) consider both the transferable and the non-transferable utility case.

5

no agent (worker or firm) is assigned more than one contract and that there are no outside options. We write w(x), f (x) and t(x) to denote the worker, firm and investment involved in contract x, respectively. For each X ⊆ X , Xw ≡ {x ∈ X : w(x) = w} and Xf ≡ {x ∈ X : f (x) = f } denote the sets of contracts within X involving worker w and firm f , respectively. Each agent i ∈ N has a complete, transitive and strict preference relation Pi over the set Xi . For x, x0 ∈ Xi we write x Pi x0 if agent i prefers x to x0 (x 6= x0 ), and x Ri x0 if i finds x at least as good as x0 , i.e., x Pi x0 or x = x0 . We denote profiles of workers’ and profiles of firms’ preferences by PW = (Pw )w∈W and PF = (Pf )f ∈F , respectively. Let P = (PW , PF ) be a preference profile and P be the set of all preference profiles. We represent agents’ preferences by ordered lists of feasible contracts; for example, Pf : (w, f, tw ), (w0 , f, t0w0 ), (w, f, t0w ), . . . indicates that (w, f, tw ) Pf (w0 , f, t0w0 ) Pf (w, f, t0w ) . . . We impose one restriction on workers’ preference profiles called unanimous separability. It requires that there is a common ranking of firms such that for each worker and for any two contracts that involve the same investment, one contract is preferred over the other if and only if the former contract involves a better firm according to the common ranking. Formally, a profile of workers’ preferences PW satisfies unanimous separability if there is a linear order  over F such that for all w ∈ W, all tw ∈ T and all f, f 0 ∈ F, f  f 0 if and only if (w, f, tw ) Pw (w, f 0 , tw ). Let  denote the weak relation associated with . We fix W , F and T . Therefore, a market is completely described by a feasible set of contracts X ⊆ X and a preference profile P ∈ P. We denote a market by a pair (X, P ). A matching µ for (X, P ) is a mapping from N to X such that m1. for each i ∈ N, µ(i) ∈ Xi m2. for each w ∈ W and each f ∈ F, if x ∈ Xw ∩ Xf , then µ(w) = x if and only if µ(f ) = x. 6

Let M be the set of (all) matchings for (X , P ). Note that any matching µ for (X, P ) with X ⊆ X is an element of M. A matching µ is blocked by w ∈ W, f ∈ F and x ∈ Xw ∩ Xf at (X, P ) if b1. x Pw µ(w) and b2. x Pf µ(f ). If a matching µ is blocked by w, f and x, we may write “µ is blocked by w and f via t(x)”. A matching is stable at market (X, P ) if it is not blocked at (X, P ). Let S(X, P ) be the set of (all) stable matchings at (X, P ). By Theorem 1 in Kelso and Crawford (1982) or Theorem 3 in Hatfield and Milgrom (2005), S(X, P ) is non-empty for any (X, P ).




For each matching µ, t(µ) ≡ t(µ(w))

w∈W

∈ T W denotes the investment profile

associated with µ. 2.2

Markets induced by investment profiles

An investment profile t induces a feasible set of contracts X(t) ≡ {(w, f, tw ) ∈ X : w ∈ W and f ∈ F }. We say that (X(t), P ) is the market induced by investment profile t. In this market all available contracts for a worker involve the same investment. Therefore, this market represents a situation where workers’ investments are fixed in the matching stage. By contrast, the market (X , P ) represents a situation where investments are flexible, or determined together with worker-firm matches in a single stage. We (sometimes) refer to this market as the complete market. Given our assumption on workers’ preferences (unanimous separability) the preferences of each worker w over contracts in Xw (t) are straightforwardly induced by . Therefore, we have the following Lemma. Lemma 1. Let t ∈ T W . For each market (X(t), P ), S(X(t), P ) is a singleton.7 Proof: A market (X(t), P ) corresponds to a one-to-one matching without contracts market where all workers have the same preferences over firms. Therefore, by Eeckhout (2000), S(X(t), P ) is a singleton. 7 From

now on, in view of Lemma 1, we slightly abuse notation by treating S(X(t), P ) as a matching instead of

a singleton.

7

Remark 1. If a matching µ is stable at (X , P ), then µ = S(X(t(µ)), P ). In words, any matching that is stable at the complete market is also stable at the market induced by its associated investment profile. This observation together with Lemma 1 imply that for any two different matchings µ, µ0 that are stable at (X , P ), t(µ) 6= t(µ0 ). 2.3

The investment game

The investment decisions of workers induce a market with a unique stable outcome that determines how workers and firms match (Lemma 1). Workers, in anticipation, choose investments strategically. Formally, they play a complete information normal form game Γ(P ) = (W, T, P ) where W is the set of players and T is the set of strategies for each player. Given an investment (strategy) profile t the outcome of this game is determined by S(X(t), P ). Each worker w evaluates the outcome according to his true preferences Pw . A coalition is a nonempty subset of workers I ⊆ W . Given an investment profile t, a coalition I has a profitable deviation at t if there exists t0I ∈ T I such that 1. for each w ∈ I, µ0 (w) Rw µ(w) and 2. for some w ∈ I, µ0 (w) Pw µ(w), where µ = S(X(t), P ) and µ0 = S(X(t0I , tW \I ), P ). An investment profile t is a (pure strategy Nash) equilibrium of Γ(P ) if no coalition I with |I| = 1 has a profitable deviation at t. It is a (strictly) strong (Nash) equilibrium of Γ(P ) if no coalition I has a profitable deviation at t. 2.4

Efficiency

We consider three different notions of efficiency. Let µ, µ0 ∈ M. Then, µ Pareto dominates µ0 if p.1. for each w ∈ W, µ(w) Rw µ0 (w) and p.2. for some w ∈ W, µ(w) Pw µ0 (w). A matching µ ∈ M is e.1. efficient if no other matching µ0 ∈ M Pareto dominates µ.

8

e.2. investment efficient if there is no investment profile t ∈ T such that S(X(t), P ) Pareto dominates µ. e.3. unilaterally efficient if there is no tw ∈ T W such that S(X(tw , t(µ)−w ), P ) Pareto dominates µ. Clearly, efficiency implies investment efficiency and investment efficiency implies unilateral efficiency. However, no converse to either of these implications holds. Example 3 exhibits a matching that is unilaterally efficient but not investment efficient and Example 2 exhibits a matching that is investment efficient but not efficient. Remark 2. By Theorems 1 and 4 in Kelso and Crawford (1982) or Theorems 3 and 4 in Hatfield and Milgrom (2005), S(X , P ) forms a non-empty lattice with respect to the Pareto domination relation. By Remark 1, each matching in S(X , P ) is associated with a different investment profile. Thus, the investment profiles associated with matchings in S(X , P ) also form a non-empty lattice with respect to the Pareto domination relation. 2.5

Deferred acceptance

We describe a worker-proposing deferred acceptance algorithm which is a generalization of Gale and Shapley’s (1962) deferred acceptance algorithm to markets with contracts. Fleiner (2003) and Hatfield and Milgrom (2005) show that this algorithm produces a stable matching that Pareto dominates any other stable matching. The description of the algorithm (below) is based on Pakzad-Hurson (2014). Let X ⊆ X be a feasible set of contracts. For each i ∈ N , let Chi (X, Pi ) be i’s most preferred contract in Xi , i.e., Chi (X, Pi ) = argmax{Xi }. Pi

When it is clear from the context we suppress the dependence of Chi from Pi . The worker proposing deferred acceptance (DA) algorithm Input: A market (X, P ). Step 1: An arbitrary worker w1 ∈ W proposes his most preferred contract in Xw1 . This contract involves some firm say f1 ∈ F . Let firm f1 hold contract x1 . Set y2 (f1 ) = x1 and set y2 (f ) = ∅ for each f 6= f1 .

9

Step k: Let Ik be the set of workers involved in a contract which is held by any firm after Step k − 1. An arbitrary worker wk ∈ W \ Ik proposes his most preferred contract xk ∈ Xwk which he has not proposed in a previous step. This contract




involves some firm fk ∈ F . Firm fk holds the contract x ∈ Chf {yk (fk )} ∪ {xk } , and rejects the other (if any). All other f 6= fk continue to hold the contract they held at the end of Step k − 1. Set yk+1 (fk ) = Chf ({yk (fk )} ∪ {xk }) and set yk+1 (f ) = yk (f ) for each f 6= fk . The algorithm terminates at some step K when no worker proposes any new contract. Given that there is an equal number of firms and workers and that there are no outside options, each worker is matched to some firm at the end of the algorithm. The function µ(f ) = yK (f ) gives the final matching and this matching is called the worker optimal stable matching at (X, P ). The firm proposing DA algorithm is defined symmetrically by exchanging the roles of workers and firms in the worker proposing DA algorithm. Hatfield and Milgrom (2005) show that the firm proposing DA algorithm produces a stable matching at (X, P ) that is Pareto dominated by any other stable matching at (X, P ). The next example illustrates the investment game and the possibility that no equilibrium exists. Example 1 (The investment game and the non-existence of equilibrium). Consider a market with W = {w1 , w2 }, F = {f1 , f2 }, T = {t1 , t2 }, and preferences P given by the columns in Table 1. Vertical dots mean that preferences can be arbitrary. Both workers have the same preferences compatible with unanimous separability and f1  f2 . Table 1: Preferences P in Example 1 Pf1

Pf2

Pw 1

Pw2

(w1 , f1 , t1 )

(w2 , f2 , t2 ) .. .

(w1 , f1 , t2 )

(w2 , f1 , t2 )

(w1 , f1 , t1 )

(w2 , f1 , t1 )

(w1 , f1 , t2 )

(w1 , f2 , t2 )

(w2 , f2 , t2 )

(w2 , f1 , t2 )

(w1 , f2 , t1 )

(w2 , f2 , t1 )

(w2 , f1 , t1 )

There is a unique stable matching at (X , P ) given by

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µ:

w1

w2

|

|

t1

t2

|

|

f1

f2

which is the boxed matching in Table 1. This can be verified by running the worker and the firm proposing DA algorithms with input (X , P ) and observing that the outcome of both algorithms is the same. By Remark 1, µ = S(X(t1 , t2 ), P ). Matching µ is not investment efficient. For example, it is Pareto dominated by the matching µ0 = S(X(t2 , t2 ), P ) given by

0

µ :

w1

w2

|

|

t2

t2

|

|

f1

f2

which is the bold face matching in Table 1. Table 2 depicts the firm matched with each worker at S(X(t), P ) for each investment profile t. Table 2: Stable matchings at (X(t), P ) w1 \ w2

t1

t2

t1

(f1 , f2 )

(f1 , f2 )

t2

(f2 , f1 )

(f1 , f2 )

Using Tables 1 and 2 one can verify that no investment profile is an equilibrium of the game Γ(P ). Therefore, this game has no equilibria.

3



Results

First we show that if a matching is investment efficient and stable at the complete market, then its associated investment profile is a strong equilibrium of the investment game. In particular, this result establishes conditions under which a strong equilibrium exists and under which the equilibrium outcome of the two stage market coincides with the outcome of a single stage market that is organized by means of the worker proposing DA mechanism and where the agents are truthful.

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Theorem 1. If µ is investment efficient and stable at (X , P ), then t(µ) is a strictly strong Nash equilibrium of Γ(P ). The proof of Theorem 1 is in the Appendix. The next result, Theorem 2, is a partial converse to Theorem 1. It establishes that every equilibrium (and in particular every strong equilibrium) induces a market such that its unique stable matching is stable at the complete market. Theorem 2. If t ∈ T W is a pure strategy Nash equilibrium of Γ(P ), then S(X(t), P ) is stable at (X , P ). The proof of Theorem 2 is in the Appendix. Theorems 1 and 2 and the fact that every strong equilibrium is investment efficient (otherwise the set of all workers would have a profitable deviation) deliver the following characterization of the strong equilibria of the investment game in terms of stability and investment efficiency. Corollary 1. Let t ∈ T W . Then, t is a strictly strong Nash equilibrium of Γ(P ) if and only if S(X(t), P ) is stable at (X , P ) and investment efficient. Proof. The if statement follows from Theorem 1. Let µ = S(X(t), P ). For the only if statement observe that Theorem 2 implies that µ is stable at (X , P ). Assume by contradiction that µ is not investment efficient. Then, the coalition W has a profitable deviation at t, contradicting that t is a strong equilibrium. Corollary 1 and the lattice structure of the set of stable matchings (Remark 2) imply that whenever a strong equilibrium exists it is unique. In fact, it is the investment profile associated with the worker optimal stable matching at (X , P ). Hence, Corollary 1 gives an easy way to check whether a strong equilibrium exists. There may be equilibria that fail to be investment efficient (see Example 3). Theorem 2 allows us to restrict our search of equilibria to stable matchings of the complete market. Next, we characterize equilibria by stability and a weaker notion of efficiency, unilateral efficiency. Corollary 2. Let t ∈ T W . Then, t is a pure strategy Nash equilibrium of Γ(P ) if and only if S(X(t), P ) is stable at (X , P ) and unilaterally efficient. Proof. The proof of the if statement is a slight modification of the proof of Theorem 1 and therefore we omit it. Let µ = S(X(t), P ). For the only if statement observe 12

again that Theorem 2 implies that µ is stable at (X , P ). Assume by contradiction that µ is not unilaterally efficient. Then, there is a worker w with a profitable deviation, which contradicts that t is an equilibrium. We give an example of a market that has a matching that is stable at the complete market, investment efficient, but not efficient. This example also serves as an illustration of Theorem 1. Example 2 (A stable and investment efficient strong equilibrium outcome). Consider a market with preferences P given by the columns in Table 3. Both workers’ preferences are compatible with unanimous separability and f1  f2 . Table 3: Preferences P in Example 2 Pf1

P f2

Pw1

Pw2

(w1 , f1 , t2 )

(w2 , f2 , t2 )

(w1 , f1 , t1 )

(w2 , f1 , t2 )

(w2 , f1 , t1 )

.. .

(w1 , f1 , t2 )

(w2 , f1 , t1 )

(w2 , f1 , t2 )

(w1 , f2 , t1 )

(w2 , f2 , t2 )

(w1 , f1 , t1 )

(w1 , f2 , t2 )

(w2 , f2 , t1 )

Consider the matching given by:

µ:

w1

w2

|

|

t2

t2

|

|

f1

f2

which is the boxed matching in Table 3. Since f1 and f2 obtain their most preferred contracts, µ is stable at (X , P ). Matching µ is not efficient, as it is Pareto dominated by the matching

0

µ :

w1

w2

|

|

t1

t2

|

|

f1

f2

which is the bold face matching in Table 3. However, µ is investment efficient. To see this observe that no investment profile induces a matching that Pareto dominates µ. In particular any profile where w1 makes investment t1 induces a matching under which w1 matches f2 and therefore is worse off. For example, (t1 , t2 ) induces the matching indicated by thick boxes in Table 3. 13

Table 4 depicts the firm matched with each worker at S(X(t), P ) for each investment profile t. Table 4: Stable matchings at (X(t), P ) w1 \ w2

t1

t2

t1

(f2 , f1 )

(f2 , f1 )

t2

(f1 , f2 )

(f1 , f2 )

Using Tables 3 and 4 one can verify that the investment profile t(µ) = (t2 , t2 ) is the unique equilibrium of the game Γ(P ). Therefore, no coalition formed by one agent has profitable deviations. Since µ is investment efficient, the coalition formed by workers w1 and w2 has no profitable deviations either. Thus, t(µ) is also a strong



equilibrium.

In the next example we exhibit a market where no stable matching is investment efficient, but where the investment profile associated with the worker optimal stable matching is an equilibrium. Example 3 (A stable and investment inefficient equilibrium outcome). Consider a market with W = {w1 , w2 , w3 }, F = {f1 , f2 , f3 }, T = {t1 , t2 }, and preferences P given by the columns of Table 5. Workers’ preferences are compatible with unanimous separability and f1  f2  f3 . Table 5: Preferences P in Example 3

Pf1

Pf2

(w1 , f1 , t1 )

(w2 , f2 , t1 )

(w2 , f1 , t2 )

(w1 , f2 , t2 )

(w3 , f1 , t1 ) .. .

(w3 , f2 , t1 ) .. .

Pf3 .. . .. .

Pw 1

Pw2

Pw3

(w1 , f1 , t2 )

(w2 , f1 , t1 )

(w3 , f1 , t1 )

(w1 , f2 , t2 )

(w2 , f1 , t2 )

(w3 , f2 , t1 )

(w1 , f1 , t1 ) .. .

(w2 , f2 , t1 ) .. .

(w3 , f3 , t1 ) .. .

The worker optimal stable matching at (X , P ) is given by:

µ:

w1

w2

w3

|

|

|

t1

t1

t1

|

|

|

f1

f2

f3

14

which is the boxed matching in Table 5. By Remark 1, µ = S(X(t1 , t1 , t1 ), P ). The investment profile t(µ) = (t1 , t1 , t1 ) is a NE. To see this observe that w3 can never profit by deviating to t2 . Thus, we analyze the incentives of w1 and w2 to deviate to t2 given that w3 chooses t1 . Table 6 gives the firm matched with w1 and w2 for each pair (t1 , t1 ), (t1 , t2 ), (t2 , t1 ), (t2 , t2 ). Table 6: Stable matchings for w1 and w2 at (X(t), P ) given tw3 = t1

w1 \ w2

t1

t2

t1

(f1 , f2 )

(f3 , f2 )

t2

(f1 , f3 )

(f2 , f2 )

If w1 deviates to t2 , he obtains the contract (w1 , f3 , t2 ), but µ(w1 ) Pw1 (w1 , f3 , t2 ). If w2 deviates to t2 he obtains contract (w2 , f3 , t2 ), but µ(w2 ) Pw2 (w2 , f3 , t2 ). Since no worker has incentives to deviate, t(µ) is a NE. By these same arguments one can verify that t(µ) is unilaterally efficient. However, t(µ) is not investment efficient as the matching µ0 = S(X(t2 , t2 , t1 ), P ) given by

0

µ :

w1

w2

w3

|

|

|

t2

t2

t1

|

|

|

f2

f1

f3

Pareto dominates µ (bold face matching in Table 5). 3.1



Restricted domains of preference profiles

Under which conditions on preference profiles do stable and investment efficient matchings exist? We provide a restriction on preference profiles, called only-bilateral disagreement, that ensures the existence of stable and investment efficient matchings. Thus, by Theorem 1 the investment games associated with preference profiles satisfying this restriction have a strong equilibrium. Our restriction requires that each firm and each worker partially agree on what would be the best investment if they were to match. More precisely, for any worker w and any firm f , let x be the best contract for f and let y be the best contract for w, among all contracts that involve f and w. Then, any contract in between x and

15

y in the preferences of f must involve w. In this sense, the disagreement between f and w is bilateral. Formally, a pair (w, f ) ∈ W × F only-bilaterally disagrees at (Pw , Pf ) if there is no z ∈ Xf \ Xw such that Chf (Xw ∩ Xf, Pf ) Pf z Pf Chw (Xw ∩ Xf, Pw ) A preference profile P ∈ P satisfies only-bilateral disagreement if each pair (w, f ) ∈ W × F only-bilaterally disagrees at (Pw , Pf ). Theorem 3. Let P ∈ P. If P satisfies only-bilateral disagreement, then there is an investment efficient and stable matching at (X , P ). Thus, the game Γ(P ) has a strictly strong Nash equilibrium. The proof of Theorem 3 is in the Appendix. Only-bilateral disagreement generalizes the notion of lexicographic preferences. A firm’s preferences are lexicographic if she ranks consecutively all contracts that involve the same worker in her preference.8 Formally, Pf is lexicographic if for any two contracts x, y ∈ Xw ∩ Xf with x Pf y there is no z ∈ Xf \ Xw such that x Pf z Pf y. A profile of firms’ preferences PF is lexicographic if for each f ∈ F, Pf is lexicographic. Remark 3. If PF is lexicographic, then P satisfies only-bilateral disagreement regardless of the preferences of workers. The preferences of a firm f have a brick if in between any two contracts involving worker w (in the preference of f ) there are all contracts involving another worker w0 . The presence of bricks is a form of non lexicographic preferences that does not preclude the existence of stable and unilaterally efficient matchings. In fact, the preferences of firm f1 in Example 2 violate lexicographic preferences by having a brick, but a stable and unilaterally efficient matching exist in that market. Formally, a firm’s preferences Pf have a brick if there are two different workers w, w0 ∈ W and two contracts x, y ∈ Xw ∩ Xf with x Pf y such that for all z ∈ Xw0 ∩ Xf , x Pf z Pf y. Next, we show that if firms’ preferences have no bricks, then it is not possible to weaken lexicographicity and still guarantee the existence of stable and unilaterally efficient matchings. 8 See

Pakzad-Hurson (2014) for a general definition of lexicographic preferences in a many-to-one matching with

contracts framework.

16

Proposition 1. Suppose that there are at least two workers, two firms and two investments. Moreover, assume that one firm’s preferences are not lexicographic and have no bricks. Then, there are lexicographic preferences for all other firms and unanimously separable preferences for all workers such that no stable matching at (X , P ) is unilaterally efficient. Thus, there is no equilibrium of the induced investment game. The proof of Proposition 1 is in the Appendix. To have an idea of the power of Proposition 1, consider a market with two workers w and w0 and T = {t1 , t2 , . . . , tk }. Consider the following preference for a firm f : Pf : (w, f, t1 ), (w0 , f, t1 ), . . . , (w0 , f, tk ), (w, f, t2 ), . . . (w, f, tk ). The preference Pf has a brick. Moreover, it is possible to obtain 2k − 1 non lexicographic preferences from Pf by placing some (possibly empty) subset of contracts involving w0 below the contract (w, f, t2 ). This gives a sense in which Proposition 1 holds for most violations of lexicographic preferences. In particular, if the number of investments is large, then bricks represent a small fraction of non lexicographic preferences.

4

Conclusion

We consider a stylized model of a labor (matching) market where workers have to invest in their human capital and then match to firms. Is there a straightforward choice or advise for workers about what investment to make? Our results tell when the answer is affirmative and in that case what the choice or advice should be. Our main result establishes that a profile of investments is a strictly strong Nash equilibrium of the investment game if and only if the matching it produces is investment efficient and stable in the complete market (Corollary 1). From the prescriptive point of view, this means that when there is no tension between stability and investment efficiency, we can advise an investment for each worker, such that no worker and even no group of workers can do better than following our advice given that the other workers do. Moreover, such advice can be easily found by means of the worker-proposing DA algorithm. Unfortunately, when a market does not admit stable matchings that satisfy a 17

weak notion of efficiency (unilateral efficiency), straightforward advice is not possible (Corollary 2). In this case, coordination failures may lead to inefficient outcomes.

Appendix Proof of Theorem 1 We prove that if µ is investment efficient and stable at (X , P ), then t(µ) is a strong equilibrium of Γ(P ). Suppose by contradiction that some coalition I ⊆ W has a profitable deviation t0I ∈ T I at t(µ). Let t0 = (t0I , tW \I (µ)) and µ0 = S(X(t0 ), P ). We first show the following claim. Claim 1. Suppose that for some worker w ∈ / I, µ(w) Pw µ0 (w). Let f = f (µ(w)) and w0 = w(µ0 (f )). Then, (i) w0 ∈ / I and (ii) µ(w0 ) Pw0 µ0 (w0 ). Proof of Claim 1. Since w ∈ / I, w makes the same investment under t and t0 . Therefore, µ(w) Pw µ0 (w) and unanimous separability imply that f (µ(w))  f (µ0 (w)).

(1)

Moreover, by (1) and the definition of w0 w 6= w0 .

(2)

Suppose that µ(f ) Pf µ0 (f ). Then, f and w block µ0 at (X(t0 ), P ) contradicting that µ0 is stable at (X(t0 ), P ). Hence, µ0 (f ) Pf µ(f ).

(3)

Suppose by contradiction to (i) that w0 ∈ I. Then, µ0 (w0 ) Pw0 µ(w0 ).

(4)

By (3) and (4), f and w0 block µ at (X , P ) via investment t0w0 , which contradicts that µ is stable at (X , P ). Hence, w0 ∈ / I. This shows (i). Suppose by contradiction to (ii) that µ0 (w0 ) Pw0 µ(w0 ). Then, by (3), f and w0 block µ at (X , P ), contradicting that µ is stable at (X , P ). This shows (ii). 18

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We continue with the proof of Theorem 1. Since µ is investment efficient, there is some w0 ∈ / I such that µ(w0 ) Pw0 µ0 (w0 ).

(5)

0 Consider the sequence (wk )∞ k=1 defined by wk = w(µ (f (µ(wk−1 )))) for k ≥ 1. By

repeatedly applying Claim 1 to w0 , w1 , . . . one can see that the following hold for any wk in the sequence (wk )∞ k=1 (a) wk ∈ / I and (b) µ(wk ) Pwk µ0 (wk ). Conditions (a), (b) and unanimous separability imply that for each k ≥ 1, f (µ(wk ))  f (µ0 (wk )). By definition, f (µ0 (wk )) = f (µ(wk−1 )).9 So, we conclude that for each k ≥ 1 f (µ(wk ))  f (µ(wk−1 )).


∞

That is, the induced sequence f (µ(wk ))

k=1

is strictly increasing in , but this

is impossible as there is only a finite number of firms. Hence, there are no profitable deviations for coalition I. Proof of Theorem 2 We prove that if t ∈ T W is an equilibrium of Γ(P ), then µ = S(X(t), P ) is stable at (X , P ). Suppose by contradiction that µ is not stable at (X , P ). Then, there are w0 ∈ W and f 0 ∈ F that block µ via some t0w0 ∈ T . Let t0 = (t0w0 , t−w0 ) and µ0 = S(X(t0 ), P ).

(6)

We show that f (µ0 (w0 ))  f 0 . Suppose by contradiction that f 0  f (µ0 (w0 )).

(7)

µ0 (f 0 ) Pf 0 (w0 , f 0 , t0w0 ) Pf 0 µ(f 0 ).

(8)

Step 1. We show that

9 By

definition wk = w(µ0 (f (µ(wk−1 )))), applying µ0 to both sides we obtain µ0 (wk ) = µ0 (w(µ0 (f (µ(wk−1 ))))),

the right hand side of this relation is equal to µ0 (f (µ(wk−1 ))). Applying f to both sides we obtain f (µ0 (wk )) = f (µ0 (f (µ(wk−1 )))). The right hand side of this last relation is equal to f (µ(wk−1 )) as desired.

19

By 6, (7) and unanimous separability we have (w0 , f 0 , t0w0 ) Pw0 µ0 (w0 ) .

(9)

f 0 6= f (µ0 (w0 )).

(10)

Moreover, (7) implies

Suppose that the first part of (8) does not hold, i.e., (w0 , f 0 , t0w0 ) Pf 0 µ0 (f 0 ) [strictly by (10)]. Therefore, by (9), w0 and f 0 would block µ0 at (X(t0 ), P ), contradicting that µ0 is stable at (X(t0 ), P ). The second part of (8) follows from the fact that w0 and f 0 block µ at (X , P ) via t0w0 . Step 2. We state and prove a Claim. Let f0 = f 0 and w0 = w(µ0 (f0 )). Consider the sequences of firms (fk )∞ k=0 and workers (wk )∞ k=0 defined by fk+1 = f (µ(wk )), for k ≥ 0 and wk = w(µ0 (fk )), for k ≥ 1.

(11)

fk = f (µ0 (wk )).

(12)

Note that

In words, fk is the firm “matched” with wk under µ0 and fk+1 is the firm “matched” with wk under µ. Claim 2. For each k ≥ 0. (i) wk 6= w0 , (ii) fk+1  fk . We prove Claim 2 using induction on k. Basis. We show that (i) and (ii) hold for k = 0. By (10) and the definition of w0 , (i) holds for w0 . Suppose (ii) does not hold for k = 0. That is, f0  f1 .10 By (i), w0 does not change investment from t to t0 . Then, by (11), observation (12) and unanimous separability, µ0 (w0 ) Pw0 µ(w0 ). This together with (8) implies that w0 and f 0 block µ at (X(t), P ), contradicting that µ is stable at (X(t), P ). Hence (ii) holds for k = 0. Induction step. Assume (i) and (ii) hold for all 0, 1, . . . , k − 1 for some k ≥ 1. We show that (i) and (ii) hold for k. 10 Strict

because f0 = f 0 = f (µ0 (w0 )) 6= f (µ(w0 )) = f1 .

20

To see that (i) holds for k observe that wk and w0 obtain different contracts under µ0 and hence they are different workers. Formally, by induction assumption (ii), fk  f0 = f 0 . Hence, by (7) and (12), wk 6= w0 . So, (i) holds for k. Induction assumptions (i), (ii) and unanimous separability imply that µ(wk−1 ) Pwk−1 µ0 (wk−1 ). If µ(fk ) Pfk µ0 (fk ), then fk and wk−1 block µ0 at (X(t0 ), P ). Hence, µ0 (fk ) Pfk µ(fk ). Since (i) holds for k, wk does not change investment from µ to µ0 . If µ0 (wk ) Pwk µ(wk ) then, wk and fk would block µ at (X(t), P ), contradicting that µ is stable at

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(X(t), P ). Hence (ii) holds for k.

The sequence of firms (fk )∞ k=0 is increasing in , but this is impossible as the number of firms is finite. Therefore, we conclude that f (µ0 (w0 ))  f 0 .

(13)

By (13) and unanimous separability, µ0 (w0 ) Rw0 (w0 , f 0 , t0w0 ). Moreover, since (w0 , f 0 , t0w0 ) blocks µ at (X , P ), we have (w0 , f 0 , t0w0 ) Pw0 µ(w0 ). Putting these two relations together µ0 (w0 ) Rw0 (w0 , f 0 , t0w0 ) Pw0 µ(w0 ). Therefore, t0w0 is a profitable deviation for w0 at t. This contradicts that t is a NE of Γ(P ). We conclude that µ is stable at (X , P ). Proof of Theorem 3 We show that if P satisfies only-bilateral disagreement, then there is a matching that is stable at (X , P ) and investment efficient. We order firms by  (workers’ common ranking of firms) as f1 , f2 , . . . , fm such that f1  f2  . . .  fm . Consider the matching µ∗ generated by the following modified serial dictatorship algorithm: 21

Input. A market (X , P ) Step 1. Let X1 = X . Let w1 be the worker involved in contract Chf1 (X1 ). Let f1 hold the contract that is most preferred by w1 among contracts involving w1 and f1 . Set µ∗ (f1 ) ≡ Chw1 (Xw1 ∩ Xf1 ). Step k. Let Ik ⊆ W be the set of workers with a contract in {µ∗ (f1 ), . . . , µ∗ (fk−1 )}. Let Xk be the set of contracts that does not involve workers in Ik ,




Xk ≡ W \ Ik × F × T. Let wk be the worker involved in contract Chfk (Xk ). Let fk hold the contract that is most preferred by wk among contracts involving wk and fk . Set µ∗ (fk ) ≡ Chwk (Xwk ∩ Xfk ). The algorithm terminates after m steps, and produces the matching µ∗ . For each k, let tk = t(µ∗ (fk )), . Thus, µ∗ (fk ) = (wk , fk , tk ),

for each k = 1, . . . , m.

We shall show that µ∗ is investment efficient and stable at (X , P ). First, we show µ∗ is stable at (X , P ). Suppose by contradiction that some wi and some fj block µ∗ at (X , P ) via some t ∈ T . Then, (wi , fj , t) Pwi (wi , fi , ti )

and

(wi , fj , t) Pfj (wj , fj , tj ).

(14)

Case 1. j = i. Then we have (wi , fi , t) Pwi (wi , fi , ti ). which violates the definition of ti . Case 2. i < j. Then we have (wi , fi , ti ) Rwi (wi , fi , t) Pwi (wi , fj , t).

(15)

The first part of (15) follows from the fact that under the modified serial dictatorship fi is assigned the contract that is most preferred by wi among all contracts that involve wi and fi . The second part follows from unanimous separability. However, (15) contradicts the first relation in (14). 22

Case 3. j < i. At step j of the modified serial dictatorship algorithm, fj is assigned (wj , fj , tj ). Then, by definition it holds that Chfj (Xj ) Rfj (wj , fj , tj ). Since Chfj (Xj ) involves a contract with wj , only-bilateral disagreement implies (wj , fj , tj ) Pfj (wi , fj , t).

(16)

However, (16) contradicts the second relation in (14). Therefore, µ∗ is stable at

N

(X , P ).

Now we show that µ∗ is investment efficient. In particular, we show the following claim from which investment efficiency follows. Claim 3.

Let t0 ∈ T W and µ0 = S(X(t0 ), P ).

Suppose that for wk ∈ W,

µ0 (wk ) Pwk (wk , fk , tk ). Then, there is wl ∈ W such that (wl , fl , tl ) Pl µ0 (wl ). We show Claim 3 by induction on k. Basis. Claim 3 holds for k = 1. By unanimous separability and only-bilateral disagreement (w1 , f1 , t1 ) is the best contract for w1 . Hence, w1 cannot be made better off. Induction step. Assume that Claim 3 holds for all 1, . . . , k − 1 for some k ≥ 2. We show that Claim 3 holds for k. Suppose µ0 (wk ) Pwk (wk , fk , tk ).

(17)

Let fj = f (µ0 (wk )). Thus, µ0 (wk ) = (wk , fj , t0wk ). Case 1. k = j. Then we have (wk , fk , t0wk ) Pwk (wk , fk , tk ) which violates the definition of tk . Case 2. k < j. Then we have (wk , fk , tk ) Rwk (wk , fk , t0wk ) Pwk (wk , fj , t0wk ).

(18)

The first part of (18) follows from the fact that under the modified serial dictatorship fk is assigned the contract that is most preferred by wk among all contracts that involve wk and fk . The second part follows from unanimous separability. However, (18) contradicts (17). 23

Case 3. j < k. If wj is worse off under µ0 relative to µ∗ , then Claim 3 holds for k. So, suppose wj is better off under µ0 relative to µ∗ (strictly because wj changes contract from µ∗ to µ0 ). Then, by the induction assumption some other worker is worse off under µ0 relative to µ∗ . Therefore, Claim 3 holds for k.

N

We have shown that µ∗ is stable at (X , P ) and investment efficient. Therefore, Theorem 3 holds. Proof of Proposition 1 Label the elements of W and F as {w1 , w2 , . . . , wm } and {f1 , f2 , . . . , fm } respectively. Assume that the preferences of firm f1 are not lexicographic. Without loss of generality (w1 , f1 , t1 ) Pf1 (w2 , f1 , t3 ) Pf1 (w1 , f1 , t2 ),

(19)

where t1 , t2 , t3 ∈ T and t1 6= t2 . Moreover, assume that Pf1 has no bricks. Then, there is t4 ∈ T \ {t3 } such that (w1 , f1 , t1 ) Pf1 (w2 , f1 , t3 ) Pf1 (w1 , f1 , t2 ) Pf1 (w2 , f1 , t4 ).

(20)

We fix the preferences of all other firms. • Let firm f2 have any lexicographic preferences such that (w2 , f2 , t4 ), is her most preferred contract. • Let each firm f3 , f4 , . . . , fm have lexicographic preferences such that each firm fi with i ≥ 3 ranks first all contracts that involve wm consecutively, then all contracts that involve wm−1 consecutively and so on. We fix the preferences of all workers. Let the common ranking of firms be given by : fm , fm−1 , . . . , f3 , f1 , f2 . That is, given investments all workers prefer fm , then fm−1 and so on until f3 , then they prefer f3 to f1 and f1 to f2 . • Let worker w1 have unanimously separable preferences with common ranking  such that (w1 , f1 , t2 ) Pw1 (w1 , f1 , t1 ) Pw1 (w1 , f 0 , t0 ), where f 0 ∈ {f1 , f2 } , t0 ∈ T and (w1 , f 0 , t0 ) 6= (w1 , f1 , t1 ), (w1 , f1 , t2 ).

24

• Let worker w2 have unanimously separable preferences with common ranking  such that (w2 , f1 , t3 ) Pw2 (w2 , f1 , t4 ) Pw2 (w2 , f2 , t4 ) Pw2 (w2 , f 00 , t00 ), where f 00 ∈ {f1 , f2 }, t00 ∈ T and (w2 , f 00 , t00 ) 6= (w2 , f1 , t3 ), (w2 , f1 , t4 ), (w2 , f2 , t4 ). • Let each worker w3 , w4 , . . . , wm have lexicographic (in particular unanimously separable ) preferences with respect to the common ranking . Also let the preferences of each worker wi , i ≥ 3 be such that wi agrees with fi on what is the best investment if wi and fi were to match. The preferences of w1 , w2 , f1 and f2 are sketched in Table 8. Table 7: Preferences of w1 , w2 , f1 , f2 Pf1 .. . (w1 , f1 , t1 ) .. .

Pf2 (w2 , f2 , t4 ) .. .

(w2 , f1 , t3 ) .. .

Pw 1 .. .

Pw2 .. .

(w1 , f1 , t2 )

(w2 , f1 , t3 )

(w1 , f1 , t1 ) .. .

(w2 , f1 , t4 )

(w1 , f 0 , t0 ) .. .

(w1 , f1 , t2 ) .. .

(w2 , f2 , t4 ) .. . (w2 , f 00 , t00 ) .. .

(w2 , f1 , t4 ) .. .

Let P denote the preference profile we have constructed. We claim that there is a unique stable matching at (X , P ) given by: (i) µ(w1 ) = (w1 , f1 , t1 ), (ii) µ(w2 ) = (w2 , f2 , t4 ), (iii) µ(wi ) = (wi , fi , ti ), for all i ≥ 3 where ti ∈ T denotes the most preferred investment for wi when matched to fi . To show this claim suppose by contradiction to (iii) that wi and fj with i 6= j block µ via some t ∈ T .

The case i = j is not possible because ti is al-

ready the best investment for both. If i > j, then because Pwi is lexicographic, 25

(wi , fi , ti ) Pwi (wi , fj , t). Thus, wi and fj cannot block µ via any t ∈ T . If i < j, then because Pfj is lexicographic, (wj , fj , tj ) Pfj (wi , fj , t). Thus, wi and fj cannot block µ via any t ∈ T . Therefore, (iii) holds. Moreover, by the same arguments any matching under which wi and fj with i 6= j are matched is not stable. Since (iii) holds, we can consider w1 , w2 , f1 and f2 in isolation. The unique stable matching in the isolated market is given by (i) and (ii) (the boxed matching in Table 7). This completes the proof of the claim. Consider the investment profile t∗ = (t2 , t(µ)−w1 ) = (t2 , t4 , (ti )m i=3 ). We claim that the unique stable matching in the market induced by t∗ , µ∗ = S(X(t∗ ), P ) is given by (iv) µ∗ (w1 ) = (w1 , f1 , t2 ), (v) µ∗ (w2 ) = µ(w2 ), (vi) µ∗ (wi ) = µ(wi ), for all i ≥ 3. By the same arguments as in (iii), (vi) holds and any matching under which wi and fj with i 6= j are matched is not stable. So again, we can consider w1 , w2 , f1 and f2 in isolation. The unique stable matching in the isolated market induced by investments t2 for w1 and t4 for w2 is given by (iv) and (v) (the bold face matching in Table 7). This completes the proof of the claim. The matching µ∗ Pareto dominates µ. Moreover, t(µ∗ ) differs from t(µ) only in the investment of worker w1 . Thus, µ is not unilaterally efficient and hence no stable matching at (X , P ) is unilaterally efficient. Therefore, by Corollary 2, there is no equilibrium of the investment game Γ(P ). 4.1

Example 4

The next example exhibits a profile of firms’ preferences that has Ergin cycles for which regardless of workers’ preferences a stable and investment efficient matching exists. Example 4. Consider a market with W = {i, j, k}, F = {f1 , f2 , f3 }, and T = {t1 , . . . tl }, l ≥ 2. Let the preferences of f1 , f2 and f3 be lexicographic. In particular, let f1 rank all contracts with i consecutively, then all contracts with

26

j consecutively and then all contracts with k consecutively. Let f2 rank all contracts with k consecutively, then all contracts with i consecutively, and then all contracts with j consecutively. We illustrate such preference profile in Table 8 with t1 , t2 , t3 , t4 , t5 ∈ T . The firms’ preference profile contains Ergin cycles.11 Since the firms’ preference profile is lexicographic, only-bilateral disagreement is satisfied regardless of workers’ preferences. Thus, by Theorem 3 a stable and investment efficient matching exists regardless of workers’ preferences. However, it is possible to find preferences for workers such that no stable matching is efficient. Table 8: Preferences P in Example 4 Pf1

Pf2

(i, f1 , t1 ) .. .

(k, f2 , t4 ) .. .

(j, f1 , t2 ) .. .

(i, f2 , t5 ) .. .

Pf3 .. . .. .

(k, f1 , t3 )



References Abdulkadiro˘glu, A., P. Pathak, and A. E. Roth (2009). Strategy-proofness versus efficiency in matching with indifferences: Redesigning the NYC high school match. American Economic Review, 99(5), 1954–1978. Cole, H. L., G. J. Mailath, and A. Postlewaite (2001a). Efficient non-contractible investments in finite economies. Advances in Theoretical Economics 1(1), Article 2. Cole, H. L., G. J. Mailath, and A. Postlewaite (2001b). Efficient non-contractible investments in large economies. Journal of Economic Theory 101(2), 333–373. Eeckhout, J. (2000). On the uniqueness of stable marriage matchings. Economics Letters 69(1), 1–8. 11 See

Pakzad-Hurson (2014) for the definition of an Ergin cycle in a matching with contracts model.

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Ergin, H. (2002). Efficient resource allocation on the basis of priorities. Econometrica 70(6), 2489–2497. Fleiner, T. (2003). A fixed-point approach to stable matchings and some applications. Mathematics of Operations Research 28(1), 103–126. Gale, D. and L. S. Shapley (1962). College admissions and the stability of marriage. American Mathematical Monthly 69(1), 9–15. Hatfield, J. W., S. D. Kominers, and F. Kojima (2015). Strategy-proofness, investment efficiency, and marginal returns: An equivalence. Working Paper . Hatfield, J. W. and P. R. Milgrom (2005). Matching with contracts. American Economic Review 95(4), 913–935. Kelso, A. and V. P. Crawford (1982). Job matching, coalition formation, and gross substitutes. Econometrica 50(6), 1483–1504. N¨oldeke, G. and L. Samuelson (2015). Investment and competitive matching. Econometrica 83(3), 835–896. Pakzad-Hurson, B. (2014). Stable and efficient resource allocation with contracts. Working Paper . Peters, M. (2007). The pre-marital investment game. Journal of Economic Theory 137(1), 186–213. Peters, M. and A. Siow (2002). Competing premarital investments. Journal of Political Economy 110(3), 592–608. 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 , 991– 1016.

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