Matching with Continuous Bidirectional Investment James C. D. Fisher∗ This Draft: 8.17.2016 (Initial Draft: 3.2014)

Abstract We introduce a one-to-one matching game where workers and firms exert efforts to produce benefits for their partners. We develop natural conditions for the existence of interior stable allocations and we characterize the structure of these allocations, with a focus on the benefits that players produce for and receive from their partners. We show, for instance, that benefit production and receipt are related in a strict rank order fashion and that players with greater incremental returns produce and receive larger benefits. Keywords: bidirectional investment, benefit production and receipt, characteristics, existence, matching with contracts, and stability.

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Ford Motor Company, email: [email protected]. My thanks to Andreas Blume, Federico Echenique, Isa Hafalir, Hans Haller, Asaf Plan, Marek Pycia, Mark Walker, John Wooders, Bumin Yenmez, and two anonymous referees, as well as to seminar participants at Chapman University, Carlos III University, Maastricht University, the University of Arizona, the University of California Los Angeles, the 2014 Southwest Economic Theory Conference, and the 2016 International Conference on Game Theory for many helpful discussions, comments, and suggestions.

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Introduction “Relationships are reciprocal.” — “College Girl” in American Pie (1999).

Partners often exert efforts to produce benefits for each other. A firm, for instance, trains an intern in industry methods and practices, and the intern, in return, completes projects for the firm. Likewise, a mentor provides guidance for her mentee, who provides social prestige in return. Other examples include masters and apprentices (e.g., professors and graduate students), senior managers and organizations (e.g., deans and colleges), buyers and sellers of specialized goods (e.g., lawyers and clients), and, in the vein of Gale and Shapley [16], even men and women. We study this reciprocity from a perspective of rational self-interest and ask two related questions. First, how does the benefit a player produces compare to the benefit she receives from her partner, i.e., the benefit her partner produces? Second, how do a player’s characteristics influence the benefits she produces and receives? This paper develops a general theory of bidirectional investment to answer to these questions. In our game, which we call the “Effort Game,” a finite number of workers and firms simultaneously pair and commit to efforts, which produce benefits for their partners.1 The benefit a player produces is increasing in her effort, which is a continuous choice, and her payoff (i) is determined by the benefit she receives, her effort, and her type and (ii) is increasing in the benefit she receives. We solve the game with an “interior stable allocation,” which specifies each player’s partner and their efforts, and requires that (i) each player earns at least the value of being single and (ii) no worker and firm can do strictly better by pairing and selecting new efforts, i.e., no two players “block” the allocation. After showing that interior stable allocations exist and are Pareto efficient under natural conditions (Proposition 1), we turn to the connection between benefit production and receipt. We establish that there is a strict rank order relationship between the two. Specifically, a worker who produces the l-th highest benefit among workers matches to a firm who produces the l-th highest benefit among firms (Proposition 2); the analogous result holds for firms.2 The strictness of this result is surprising and stems from the fact players “compete” with each other for the best possible partners. To illustrate this competition, suppose that there is an interior stable allocation where workers w and w0 both produce the highest benefit among all workers, but only f 0 , the partner 1

We couch our game, results, and discussion in terms of workers and firms but emphasize that these are only labels for the two sides. 2 It follows that players match based on the efforts they exert when they have the same benefit production technology (Corollary 1). In the context of men and women, this result provides a novel rationalization for Rammstedt and Schupp [33] and Watson et al.’s [36] empirical observation that individuals with higher levels of conscientiousness usually find partners with higher levels of conscientiousness.

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of w0 , produces the highest benefit among all firms, while f , the partner of w, produces a strictly lower benefit. Then w does strictly better if she slightly increases her effort to produce a larger benefit and “wins” f 0 away from w0 . She is able to win f 0 because the firm desires the largest benefit possible. She does strictly better since (i) the small increase in effort only changes her payoff slightly, all else equal, and (ii) she receives a strictly higher benefit from her new partner. Thus, w and f 0 block. It follows that a necessary condition of interior stability is that workers who produce the highest benefit are matched to firms who produce the highest benefit. The proof extends this intuition to all rank order levels of benefit. Subsequently, we investigate the effects of characteristics on benefit production and receipt. We are particularly interested in understanding the conditions under which higher types produce and receive larger benefits. We show that this result obtains when (i) all players have the same benefit production technology and (ii) the (possibly negative) incremental returns to benefit received and to effort are increasing in type.3 Specifically, under these conditions, worker w0 produces a weakly larger benefit than worker w if w0 has a strictly higher type (Proposition 2); the analogous result holds for firms. Thus, w0 receives a weakly larger benefit than w in any interior stable allocation by Proposition 1. The intuition for the former result is that higher types are able to “outcompete” lower types because their greater incremental rewards allow them to profitably offer slightly higher benefits. Our results suggest that higher types match with each other. While this need not happen in every interior stable allocation (due to indifferences), we establish that there is at least one interior stable allocation where it occurs (Proposition 4). We close our results by examining the relationship between payoffs and characteristics. We show that, when benefit production and payoffs are increasing in type,4 worker w0 has a weakly higher payoff than worker w if w0 has a strictly higher type (Proposition 5); the analogous result holds for firms. The intuition is that higher types are able to “imitate” and outcompete lower types. Our two main economic results, Propositions 2 and 3, shed light on the relationships among the benefit produced, the benefit received, and characteristics. Proposition 2 helps us understand, for instance, why some firms attract outstanding workers, while others do not. In doing so, it illuminates why the most productive academics cluster at universities with the best research environments (e.g., Matthews [28]), why the best free agents in the NBA choose to join teams with the most wins and the greatest chance of a championship 3

These conditions hold, for instance, when all players have the same core productive skills and when higher types have lower marginal effort costs; see Section 3 for details. 4 Benefit production and payoffs are increasing in type if, for instance, the players who are more productive have lower costs of effort; see Section 3 for details.

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(e.g., Yang et al. [38]), and why the firms with the most productive workers are often those with the best working environments (e.g., Weakliem and Frenkel [37]). Proposition 3 provides insights into the characteristics that lead certain firms to produce larger benefits and attract their outstanding workers. In doing so, it sheds light on why larger firms with lower incremental costs of providing certain benefits, like health insurance, employ workers who are more productive (e.g., Currie and Madrin [10] and Lazear and Oyer [24]); after all, a lower incremental cost is a greater, though negative, incremental reward. Stepping back, these results offer guidance on how a firm can improve the rank order benefit it provides and attract a higher rank order caliber of worker: by making pre-match investments that increase its incremental returns to benefit and effort.5 Our minor economic results, Proposition 4 and 5, respectively help us understand how players may pair in stable allocations and give us insight into why certain workers and firms do better. The balance of this section discusses the related literature. Section 2 presents the Effort Game, our solution concept, and Proposition 1, as well as two examples. Section 3 presents and discusses Propositions 2 to 5. Section 4 concludes and discusses key aspects of our game, as well as our assumptions and their essentiality. Appendix A contains a novel proof of the existence of stable allocations in one-to-one matching games that leverages a deep connection with the “Deferred Acceptance” algorithm. Appendix B collects the proofs of Propositions 1 to 5. The Online Appendix collects supplemental results. Related Literature Investment can take place (i) before matching (e.g., people go to college before getting a job), (ii) simultaneously with matching (e.g., firms make training commitments to new hires),6 or (iii) after matching (e.g., holdup and moral hazard). While there is a large literature on pre-match investment (e.g., Burdett and Coles [3], Chade and Lindenlaub [4], Cole et al. [7], Dizdar [13], Hatfield et al. [18], Hoppe et al. [20], Mailath et al. [27, 26], Peters [31], Peters and Siow [32], and Zhang [39], among others) and there is research on postmatch investment (e.g., Kaya and Vereshchagina [22]), to the best of our knowledge there is no prior work on simultaneous matching and investment, and this is where we focus.7 The pre-match investment literature considers two-stage games where players first invest in themselves and then enter a matching game, which is usually based on the games of either Gale and Shapley [16] or Demange and Gale [12]. A player’s payoff depends on her and her 5

We formalize this suggestion in the Online Appendix, where we show, for instance, that improvements in a player’s type increase the rank order of the benefits she produces and receives. 6 In the simultaneous case, investment may actually take place long after players have matched (e.g., a firm may begin training after a new hire starts work); what matters is that partners commit to their investments when they match. 7 We examine the relationship between the Effort Game and other “matching with contracts” games (e.g., Hatfield and Milgrom [19]) in Appendix A.

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partner’s identities and investments. One branch of this literature concerns itself with the Pareto efficiency of investment and finds mixed results – see Noldeke and Samuelson [29] for a recent treatment. A second branch examines how market characteristics shape players’ investment and surplus sharing decisions – see, for instance, Chiappori et al. [5]. Our work is complementary since we study environments where investments are co-determined with the matching. Moreover, the questions on which we focus are unexamined in the pre-match investment literature. Post-match investment is characterized by commitment and information problems.8 To model these problems, Kaya and Vereshchagina [22] consider a two-stage “roommates” game were players first match and agree to the division of profits from a joint venture, before privately deciding how much costly and imperfectly observable effort to contribute to the venture. They establish that equilibrium investments are not Pareto efficient and characterize when different monitoring technologies lead higher types to match with each other or match with lower types.9 Our work is complementary because we focus on different questions and we study an environment where players verifiably choose and compete on the basis of their investments.

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The Effort Game, Its Solution, and Examples

In this section, we describe the Effort Game, give our solution concept, which is an interior stable allocation, present an existence result, and develop two examples. Environment The Effort Game is a one-to-one matching game with two finite groups of players: workers W = {1, . . . , W } and firms F = {W + 1, . . . , N }, with N > W > 0. Let N = W ∪ F denote the set of players. We write w for an arbitrary worker, f for an arbitrary firm, and i for an arbitrary player. Each player i is endowed with an observable type θi ∈ Θ, where Θ ⊂ R++ is the finite type set, and may be single or matched to a member of the opposite group. A matching is a function that specifies each player’s match, i.e., is a φ : N → N such that: (i) for each worker w, φ(w) ∈ F ∪ {w}; (ii) for each firm f , φ(f ) ∈ W ∪ {f }; and (iii) for each worker w and each firm f , φ(w) = f ⇐⇒ φ(f ) = w. Let Φ denote the finite set of all matchings. A player is single if she is matched to herself, i.e., if φ(i) = i, and is partnered if she is matched to a member of the opposite group, i.e., if φ(i) 6= i. When a worker w and firm f match, they select an agreement x = (x1 , x2 ) ∈ R2 , where 8

Absent these problems, post-match investment is the same as simultaneous investment. This latter result, as well as our fourth proposition, are related to the large literature on “type based” assortative matching – see, for instance, Legros and Newman [25]. 9

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x1 specifies w’s effort and x2 specifies f ’s effort. For record keeping purposes, we also suppose that each single player i has an agreement x with herself. We write xi for the agreement that ¯ = (x1 , . . . , xW , xW +1 , . . . , xN ) player i has with her partner or with herself, and we write x for the (joint) vector of players’ agreements. Given a φ ∈ Φ, we have that xi = xφ(i) for each ¯ ∈ A(φ) = {(˜ ˜ N ) ∈ RkN | x ˜i = x ˜ φ(i) for all i ∈ N }. The set A(φ) is the player i, so x x1 , . . . , x ¯ ) such collection of possible agreement vectors for the matching φ. An allocation is a (φ, x ¯ ∈ A(φ); it specifies each player’s partner and their agreement. that φ ∈ Φ and x Each player’s payoff depends on her match and their agreement. For each worker w and each firm f , let uw : {F ∪ {w}} × R2 → R and uf : {W ∪ {f }} × R2 → R record w and f ’s respective payoffs. As to the origins of these payoffs, let b : R × Θ → R be continuous and strictly increasing in its first argument and let r : R2 × Θ → R be strictly increasing in its first argument and continuous in its first and second arguments. We refer to b(·) as the benefit function because if a type θ player exerts effort y, then she produces benefit b(y, θ) for her partner. We refer to r(·) as the reward function because if a type θ player exerts effort y and receives benefit b from her partner, then her reward is r(b, y, θ). Thus, when worker w and firm f are matched, their payoffs to agreement (x1 , x2 ) are uw (f, x1 , x2 ) = r(b(x2 , θf ), x1 , θw ) and uf (w, x1 , x2 ) = r(b(x1 , θw ), x2 , θf ). That is, w (f ) exerts effort to produce a benefit for f (w) and, in return, receives a benefit from f (w) that, along with her own effort and type, determines her payoff.10 For simplicity, the payoff of a single player is normalized to zero, i.e., ui (i, x) = 0 for each player i and ¯ ) = (φ, x1 , . . . , xi , . . . , xN ) ∈ Φ × RkN , in a slight abuse of notation every x ∈ R2 . Let (φ, x ¯ ) for the payoff of player i in (φ, x ¯ ), i.e., ui (φ, x ¯ ) ≡ ui (φ(i), xi ). we write ui (φ, x We suppose that, due to time and energy limitations, players can only feasibly exert efforts between 0 and β, where 0 < β < ∞. Thus, X = [0, β]2 is the set of feasible agreements for each matched pair. Before proceeding, we wish to point out that, while it may appear as though players have a common preference (over matched and agreements) because they prefer partners who produce higher benefits, this is not the case.11 Instead, for a fixed vector of efforts, each side agrees on a ranking of the opposite side – e.g., the firms agree on which workers are best, 10

While w and f ’s payoffs are both increasing in the benefit they receive from each other, we place no structure (for now) on (i) how their types effect the benefits they provide and (ii) on how their efforts and types effect their own payoffs. 11 To illustrate, consider two workers w and w0 who have a choice between (a) firm f and agreement 1 ( /4, 9/10) or (b) firm f 0 and agreement (1/2, 1). Let b(y, θ) = y, let r(b, y, θw ) = b + y − y 2 − 21 , and let r(b, y, θw0 ) = b + y − 4y 2 − 14 . A bit of algebra shows that w strictly prefers (b) to (a), while w0 strictly prefers (a) to (b).

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second best, and so on. However, efforts are not fixed, they are endogenous. Thus, each worker effectively chooses her position in the firms’ ranking by choosing her effort, likewise for firms. (Her choice depends, of course, on the benefits offered by the other workers and firms, as well as her benefit and reward functions; and this choice is one of our objects of study.) Interior Stable Allocations ¯ ) is feaWe need several definitions to develop our solution concept. An allocation (φ, x 2 N ¯ ∈ X . An allocation (φ, x ¯ ) is individually sible if agreements are in X = [0, β] , i.e., if x ¯ ) ≥ 0 for each rational if every player gets at least the value of being single, i.e., ui (φ, x kN ¯ ) ∈ Φ × R if they both obtain strictly player i. A worker w and a firm f block a (φ, x higher payoffs by matching with each other at a feasible agreement than they obtain in ¯ ), i.e., if there exists an x ∈ X such that uw (f, x) > uw (φ, x ¯ ) and uf (w, x) > uf (φ, x ¯ ). (φ, x ¯ ? ) is stable if it is (i) feasible, (ii) individually rational, and (iii) no An allocation (φ? , x ? ? ¯ ? ) = (φ? , x1 , . . . , xN ) is interior stable if it worker and firm block it. An allocation (φ? , x is (i) stable, (ii) if some player is matched to a member of the opposite group, i.e., φ? (i) 6= i for some player i, and (iii) each matched player has an agreement on the interior of X, i.e., ? φ? (i) 6= i implies that xi ∈ (0, β)2 for each player i. Interior stable allocations are our solution concept. When an allocation is interior stable: (i) no player can do strictly better by choosing to be single (per individual rationality) and (ii) no two players can do strictly better by matching with each other and choosing a new ¯ ? ) (per no blocking).12 We focus on interior stable agreement instead of following (φ? , x allocations because (i) they exist, (ii) they are Pareto efficient, and (iii) they are the only stable allocations in which players are matched with one another under natural conditions.13 The next assumption and result clarify these claims. Assumption 1. Sufficient Conditions for the Existence, Efficiency, and Essentiality of Interior Stable Allocations.14 The benefit and reward functions, as well as the players’ endowed types, are such that: 1. There are a worker w and firm f for whom r(b(x2 , θf ), x1 , θw ) ≥ 0 and r(b(x1 , θw ), x2 , θf ) ≥ 0 for some (x1 , x2 ) ∈ (0, β)2 . 12 As in Gale and Shapley [16], we might imagine that a stable allocation is the outcome of a bargaining process where players try to maximize their own payoffs. The rational is that no player can do strictly better by (i) opting out or (ii) by trying to strike a new bargain with some other player j, as j would reject this bargain since it does not make her strictly better off. After bargaining concludes, we imagine that players pair with their agreed upon partners, take their agreed upon actions, and then receive their payoffs. 13 There is also a technical motivation for focusing on interior stable allocations: they allow us to consider the effects of small shifts in the efforts of partnered players. Such shifts lie at the heart of our proofs in the next section and are impossible at boundary allocations. 14 See Section 4 for a weaker, more abstract version of this assumption.

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2. For each θ ∈ Θ, we have that r(b(y, θ0 ), β, θ) < 0 for all y ∈ [0, β] and all θ0 ∈ Θ. 3. Either of the following holds: (a) For each θ ∈ Θ, there is δθ > 0 such that r(b, y, θ) is non-negative and strictly increasing in y on [0, δθ ] for each b ∈ R. (b) For each θ ∈ Θ, we have that r(b(0, θ0 ), y, θ) < 0 for all y ∈ [0, β] and all θ0 ∈ Θ. The first part is a weak requirement that there are at least one worker and one firm who find matching to be at least as beneficial as being single. The second part requires that each player have a negative payoff when she exerts effort β; it may reflect, for instance, that efforts of size β are taxing. The third part requires that either (a) a player can do strictly better by exerting some small effort rather than no effort or (b) a player has a negative payoff when her partner exerts no effort. The former may reflect, for instance, a local complementary between benefit and effort when effort is small (e.g., that a worker benefits more by actively participating, up to a point, in the training her firm provides), while the latter may reflect a large “matching cost.” We illustrate the assumption in Examples 1 and 2, below. ¯ ) is Pareto efficient if there We need a definition before proceeding. An allocation (φ, x ¯ 0 ) such that (i) all players do weakly better in (φ0 , x ¯ 0 ) than is no other feasible allocation (φ0 , x ¯ ), i.e., ui (φ0 , x ¯ 0 ) ≥ ui (φ, x ¯ ) for each player i, and (ii) at least one player does strictly (φ, x ¯ 0 ) than (φ, x ¯ ), i.e., ui (φ0 , x ¯ 0 ) > ui (φ, x ¯ ) for some player i.15 better in (φ0 , x Proposition 1. Existence, Efficiency, and Essentiality of Interior Stable Allocations. ¯ ? ), (ii) all inteLet Assumption 1 hold, then (i) there is a interior stable allocation (φ? , x rior stable allocations are Pareto efficient, and (iii) any stable allocation where a player is partnered is an interior stable allocation. The intuition is three-fold. First, parts (2) and (3) of Assumption 1 ensure that for each matched worker and firm, every agreement in the boundary of X is either payoff dominated for both players by some interior agreement or leaves one player with a negative payoff. Thus, allocations with boundary agreements cannot be stable and we obtain (iii). Second, since a stable allocation exists by Proposition A1 (see Appendix A), (i) follows from (iii) as part (1) of Assumption 1 ensures that a worker and a firm are matched in at least one stable allocation. Third, Pareto efficiency follows from the strict monotonicity of the benefit and reward functions, as well as the interiority of the allocation. The proposition’s proof, along with the proofs of all other results, are given in Appendix B. Examples We give two examples to illustrate the Effort Game and interior stable allocations. Example 1. A Simple Effort Game with Part (3.a) of Assumption 1. 15

We focus on strong Pareto efficiency since all stable allocations are weakly Pareto efficient.

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Suppose there are four players, two workers and two firms, i.e., W = {1, 2} and F = {3, 4}. Let θ1 = θ3 = 1 and θ2 = θ4 = 21 be the players’ types. Let X = [0, 2]2 , let b(y, θ) = y, b+y and let r(b, y, θ) = e 2 − 1θ y 3 . In words, the set of feasible agreements is the two-square, b+y the benefit function is linear in effort, and the reward function is the “take” of e 2 less the “effort cost” of 1θ y 3 . Thus, when worker w and firm f match with agreement (x1 , x2 ), their payoffs are uw (f, x1 , x2 ) = e

x1 +x2 2

−

x1 +x2 1 1 (x1 )3 and uf (w, x1 , x2 ) = e 2 − (x2 )3 . θw θf

Single players have a payoff of zero. This example meets all three parts of Assumption 1. Part (1) holds because there are 1+1 gains to matching: formally, we have that u1 (3, 1, 1) = e 2 − 13 = e − 1 > 0 and that u3 (1, 1, 1) = e − 1 > 0. Part (2) holds since an effort of β is taxing – it costs a player at least eight to exert effort β, while the largest her take can be is e2 ≈ 7.4: formally, y+2 y r(b(y), β, θ) = e 2 − 8θ ≤ e 2 +1 − 8 ≤ e2 − 8 < 0 because θ ∈ { 12 , 1} and y ∈ [0, 2]. Part (3.a) holds because of the complementarity in the take between the benefit received and effort and because of the flatness of the effort cost near zero: formally, we have (i) that b b ∂r = 12 e 2 > 0 for all b ∈ R, which allow us to conclude r(b, 0, θ) = e 2 > 0 and (ii) that ∂y y=0 that (3.a) holds. Since Assumption 1 holds, Proposition 1 implies that there is an interior stable allocation. ? ? ? ? ¯ ? ) = (φ? , x1 , x2 , x3 , x4 ), where It is readily verified that one such allocation is (φ? , x ? ? ? ? φ? (1) = 3 and x1 = x3 ≈ (0.910, 0.910), and φ? (2) = 4 and x2 = x4 ≈ (0.533, 0.533).16 That is, worker 1 matches with firm 3 and they both exert efforts of 0.910, while worker 2 matches with firm 4 and they both exert efforts of 0.533. Thus, worker 1 and firm 3 provide one another a benefit of 0.910 and each has a payoff of 1.731, while worker 2 and firm 4 provide one another a benefit of 0.533 and each has a payoff of 1.401. 4 Example 2. Another Simple Effort Game with Part (3.b) of Assumption 1. Suppose there are four players, two workers and two firms, i.e., W = {1, 2} and F = {3, 4}. Let θ1 = θ4 = 2 and θ2 = θ3 = 1 be the players’ types. Let X = [0, 2]2 , let b(y, θ) = y, and let r(b, y, θ) = b − 1θ y 2 − 18 . The feasible set and benefit function are as in the previous example, while the reward function is the benefit received less a “matching cost” of 81 and an “effort cost” of 1θ y 2 . When worker w and firm f match, their payoffs to agreement (x1 , x2 ) ?

?

We work with the decimal approximations for simplicity. The exact agreements are x1 = x3 = ? ? 1 1 1 1 (−2 W ( 2√ ), −2 W ( 2√ )) and x2 = x4 = (−2 W ( 2√ ), −2 W ( 2√ )), where W (·) denotes the “product3 3 6 6 log” function; see Corless et al. [8] for details on this function. 16

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are uw (f, x1 , x2 ) = x2 −

(x1 )2 1 (x2 )2 1 − and uf (w, x1 , x2 ) = x1 − − . θw 8 θf 8

Recall that single players get zero. This example meets all three parts of Assumption 1. Part (1) holds because u1 (4, 1, 1) = 1 − 12 − 18 = 38 > 0 and u4 (1, 1, 1) = 1 − 12 − 18 = 83 > 0. Part (2) holds since r(b(y), β, θ) = y − θ4w − 18 ≤ y − 2 − 81 ≤ − 18 < 0 because θw ∈ {1, 2} and y ∈ [0, 2]. Part (3.b) holds because there is a large matching cost, which exceeds the benefit provided by one’s partner when she 2 exerts zero effort: formally, r(b(0), y, θ) = 0 − yθ − 18 < − 81 < 0 for all y ∈ [0, 2]. Since Assumption 1 holds, Proposition 1 gives that there is an interior stable allocation. ? ? ? ? ¯ ? ) = (φ? , x1 , x2 , x3 , x4 ), where φ? (1) = 4 It is easily seen that one such allocation is (φ? , x ? ? ? ? and x1 = x4 = (1, 1), and φ? (2) = 3 and x2 = x3 = ( 21 , 12 ). That is, worker 1 matches with firm 4 and they both exert efforts of 1, while worker 2 matches with firm 3 and they both exert efforts of 21 . Thus, worker 1 and firm 4 provide one another a benefit of 1 and each has a payoff of 38 , while worker 2 and firm 3 provide one another a benefit of 12 and each has a payoff of 18 . 4

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The Structure of Interior Stable Allocations

In this section, we examine the structure of interior stable allocations and develop our main results on the relationships among the benefit produced, benefit received, effort, and type, as well as our minor results on partner’s types and on payoffs. Benefit Production and Receipt Our first question asked how the benefit a player produces is related to the benefit she receives from her partner. The next result answers this question. ? ? ¯ ? ) = (φ? , x1 , . . . , xN ) be an interior Before proceeding, we need some notation. Let (φ? , x ? ¯ ? ), i.e., zi? = xi1 if i is a worker and stable allocation. Let zi? denote player i’s effort in (φ? , x ? ? ? ? zi? = xi2 if i is a firm, where (xi1 , xi2 ) = xi . Each partnered player i produces a benefit b?i = b(zi? , θi ). We rank partnered players by the benefits they produce from greatest to least and place them into groups of equivalent benefit.17 For the workers, we label these groups W W W GW 1 , . . ., GJW , where G1 contains the workers who produce the highest benefit, G2 contains the workers who produce the second highest benefit, and so on. For the firms, we label the analogous groups GF1 , . . ., GFJF . (Notice that JW and JF are strictly positive by interior ¯ ? ). stability.) To simplify notation, we suppress the dependence of these objects on (φ? , x Proposition 2. Benefit Production and Receipt. 17

We omit single players because they produce benefits for no one.

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W F F ¯ ? ) be an interior stable allocation and let GW Let (φ? , x 1 , . . . , GJW and G1 , . . . , GJF be the associated benefit groups. Then, (i) there are an equal number of worker and firm benefit groups, i.e., JW = JF , and (ii) a worker w is in the l-th benefit group of workers if and only ⇐⇒ φ? (w) ∈ GFl for if her partner φ? (w) is in the l-th benefit group of firms, i.e., w ∈ GW l each l ∈ {1, . . . , JW }. The analogous result holds for firms.

In other words, benefit production and receipt are related in a strict rank order fashion: if a worker produces the l-th highest benefit among all workers, then she receives the l-th highest benefit among all firms from her partner. We can see this outcome in the stable allocations we found in Examples 1 and 2. For instance, in Example 1, the high benefit producing players, worker 1 and firm 3, are matched, as are the low benefit producing players, worker 2 and firm 4. The intuition behind the proposition is that players “compete” for the best possible partners. To illustrate, suppose there is a worker w in the first benefit group of workers who is matched to a firm that is not in the first benefit groups of firms. Let firm f be in the first benefit group of firms. If w increases her effort slightly, then she gives f a strictly higher benefit than f is currently receiving from her partner. Since f desires the highest benefit possible, she will agree to match with w instead of her current partner. Worker w is willing to increase her effort because the increase (i) only changes her payoff slightly, all else equal, and (ii) allows her to obtain a strictly higher benefit f provides; so, she does strictly better. Thus, w and f block. It follows that a necessary condition of stability is that all workers in the first benefit group of workers are matched to firms in the first benefit group of firms, and vice versa. We formalize this argument in the proof and use induction to show that the analogous result holds for the second benefit groups, third benefit groups, and so on. A corollary is that partnered players exert the same rank order efforts, at least when the benefit function is type independent, i.e., where b(y, θ) = b(y, θ0 ) for all θ and θ0 in Θ. When the benefit function has this property (e.g., Examples 1 and 2), its strict monotonicity W implies that workers in GW 1 exert the most effort, workers in G2 exert the second most effort, and, in general, those in GW l exert the l-th most effort among all workers. Analogously, firms F in Gl exert the l-th most effort among all firms. Proposition 2 thus implies the following result. Corollary 1. Partners’ Efforts with Type Independent Benefit. ¯ ? ) be an interior stable allocation. Let the benefit function be type independent and let (φ? , x Then, a worker w exerts the l-th most effort among all workers if and only if her partner φ? (w) exerts the l-th most effort among all firms for all l ∈ {1, . . . , JW }. The analogous result holds for firms.

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We can see this outcome in the stable allocations we found in Examples 1 and 2. For instance, in Example 1, the high effort players, worker 1 and firm 3, are matched, as are the low effort players, worker 2 and firm 4. Benefit and Characteristics Our second question asked how a player’s characteristics, i.e., her benefit and reward functions, influence the benefit she produces and the benefit she receives. While this question is difficult to answer in general (due to the indefinite natures of the benefit and reward functions), the next results provide answers under the following, natural assumption. Assumption 2. Sufficient Conditions for Benefit to Increase with Type. The benefit function b(y, θ) is type independent. The reward function r(b, y, θ) is such that the difference r(b0 , y 0 , θ) − r(b, y, θ) is strictly increasing in θ when b0 > b and y 0 > y. The first part requires that all types are equally capable of producing benefit, while the second part requires that higher types have greater (though possibly negative) incremental/marginal rewards to additional benefit received and to additional effort than lower types. To digress, the second part of the assumption holds whenever the reward function has either (i) weak increasing differences in (b, θ) and strict increasing differences in (y, θ) or (ii) strict increasing differences in (b, θ) and weak increasing differences in (y, θ). The assumption may reflect, for instance, the ideas that all players have the same core skills and that some players have lower marginal effort costs, as is the case in Examples 1 and 2. In both examples, the benefit function is constant in θ, the reward function has weak increasing differences in (b, θ) because type only effects effort cost, and the reward function and has strict increasing differences in (y, θ) since marginal effort cost is strictly decreasing in type (as all types are strictly positive). Proposition 3. Benefit Production and Type. W ¯ ? ) be an interior stable allocation, and let GW Let Assumption 2 hold, let (φ? , x 1 , . . . , GJW and GF1 , . . . , GFJF be the associated benefit groups. Then, workers with strictly higher types produce weakly higher benefits, i.e., if two workers w and w0 are partnered, then θw < θw0 implies 0 W that w ∈ GW l and w ∈ Gj with j ≤ l. The analogous result holds for firms. We can see this outcome in the stable allocations we found in Examples 1 and 2. For instance, in Example 1, the high type worker, worker 1, provides a benefit of 0.910, which exceeds the benefit provided by the low type worker, worker 2, of 0.533. The intuition for the proposition is that higher types “outcompete” lower types because their greater incremental rewards (to benefit received and to effort) allow them to profitably offer slightly higher benefits. To be more precise, suppose there is an interior stable allocation ¯ ? ) where a lower type worker w produces a greater benefit than a strictly higher type (φ? , x 12

worker w0 . Let f and f 0 be the partners of w and w0 respectively. Stability and type independence then imply that w does better with f at her current effort than she does by matching with f 0 at the effort of w0 ; in symbols, r(b?f , zw? , θw ) ≥ r(b?f 0 , zw? 0 , θw ), where b?f and b?f 0 ¯ ? ). Thus, Assumption 2 are the firms’ benefits and zw? and zw? 0 are the workers efforts in (φ? , x implies that r(b?f , zw? , θw0 ) > r(b?f 0 , zw? 0 , θw0 ), i.e., w0 does strictly better by matching with f at ¯ ? ). Hence, w0 also does strictly better by offering agreement (zw? , zf? ) than she does in (φ? , x f a slightly higher benefit than she currently receives from w, i.e., by offering agreement (zw? + , zf? ) for a sufficiently small  > 0. Since f obviously does better at such an agreement ¯ ? ), w0 and f block. It follows that a necessary condition of stability is that than in (φ? , x strictly higher type workers produce higher benefits.18 The proof makes this intuition precise. We have yet to relate a player’s type to the benefit she receives from her partner. Since Proposition 2 gives that players who produce higher benefits receive higher benefits from their partners, Proposition 3 implies that higher types receive higher benefits. The next corollary formalizes this observation. Corollary 2. Benefit Receipt and Type. W ¯ ? ) be an interior stable allocation, and let GW Let Assumption 2 hold, let (φ? , x 1 , . . . , GJW and GF1 , . . . , GFJF be the associated benefit groups. Then, workers with strictly higher types receive weakly higher benefits from their partners, i.e., if two workers w and w0 are partnered, then θw < θw0 implies that φ? (w) ∈ GFl and φ? (w0 ) ∈ GFj with j ≤ l. The analogous result holds for firms. We can see this outcome in the stable allocations we found in Examples 1 and 2. For instance, in Example 1, the high type worker, worker 1, receives a benefit of 0.910 from her partner, while the low type worker, worker 2, receives a smaller benefit of 0.533. Since the benefit function is strictly monotone, Proposition 3 and Corollary 2 give that higher types and their partners exert more effort. The next corollary formalizes this observation. Corollary 3. Effort and Type. ¯ ? ) be an interior stable allocation, and let {zi? }i∈N be the Let Assumption 2 hold, let (φ? , x associated efforts. Then, workers with strictly higher types and their partners exert weakly more effort, i.e., if two workers w and w0 are partnered, then θw < θw0 implies that zw? ≤ zw? 0 and that zφ?? (w) ≤ zφ?? (w0 ) . The analogous result holds for firms. We can see these outcomes in the stable allocations we found in Examples 1 and 2. For instance, in Example 2, the high type worker, worker 1, and her partner both exert efforts 18

If two workers have the same type, then they may be in different benefit groups because neither is able to profitably outcompete the other. A detailed example is available upon request.

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of 1, while the low type worker, worker 2, and her partner both exert lower efforts of 12 . We close this subsection by noting that, under certain conditions on the benefit and reward functions, Proposition 3 can be strengthened to show that strictly higher types produce strictly higher benefits. We develop this result in the Online Appendix, where we leverage it to show that improvements in a player’s type increase the rank order of the benefits she produces and receives (in every interior stable allocation). Partners’ Types Since higher types are in lower indexed benefit groups, it is natural to think that higher type workers are matched to higher type firms. Although this conjecture is not generally true, the next result shows that there is at least one interior stable allocation where it obtains. ¯ ) exhibits assortative Before proceeding, we need a definition. An allocation (φ, x 0 matching in types if, for any two partnered workers w and w with θw < θw0 , we have that w0 matches with a higher type firm than w does, i.e., θφ(w) ≤ θφ(w0 ) . Proposition 4. Assortative Matching in Types. ¯ ? ) that exhibits Let Assumptions 1 and 2 hold, then there is an interior stable allocation (φ? , x assortative matching in types. Since Examples 1 and 2 meet the proposition’s hypothesis, they have interior stable allocations that exhibit assortative matching in types. In fact, the interior stable allocations we found have this property – e.g., in Example 1, the high types, worker 1 and firm 3, are matched to one another, as are the low types, worker 2 and firm 4. The intuition for the proposition is that any interior stable allocation can be transformed (via rematching) into another interior stable allocation that exhibits assortative matching types; so, the proposition follows from Proposition 1. The need for rematching comes from the fact that players may match in arbitrary ways within their benefit groups. For instance, if w and w0 are in the l-th benefit group of workers, with θw < θw0 , and if f and f 0 are in the l-th benefit group of firms, with θf < θf 0 , then w may match with f 0 and w0 may match with f in a stable allocation.19 The reason is that, all else equal, each worker (firm) is indifferent to her partner’s identity because both firms (workers) produce the same benefit. This indifference also allows us to rematch these players so that higher types are paired, without affecting payoffs or stability. Once this rematching is performed for all benefit groups, Proposition 3 implies that matching is assortative in types. The details are given in the proof. Payoffs and Types Corollary 2 is puzzling from a payoff perspective because it shows that while higher types receive greater benefits, they also exert greater efforts; so, it is unclear whether they actually 19

A detailed example where this occurs is available upon request.

14

do better. As the next result shows, the following assumption is sufficient to resolve the confusion and ensure that higher types earn more. Assumption 3. Sufficient Conditions for Payoffs to Increase with Type. The benefit function b(y, θ) is weakly increasing in θ and the reward function r(b, y, θ) is strictly increasing in θ. That is, higher types are better able to produce benefit for their partners and, all else equal, have strictly higher rewards than lower types. These requirements may reflect, for instance, the idea that the players who are more skilled at benefit production have lower effort costs. This is the case in Examples 1 and 2, where the benefit functions are constant in θ and the reward functions are strictly increasing in θ since higher types have lower effort costs. As these examples demonstrate, Assumptions 2 and 3 are mutually consistent. Proposition 5. Payoffs and Types. ¯ ? ) be an interior stable allocation. Then, workers with strictly Let Assumption 3 and let (φ? , x higher types obtain higher payoffs. In particular, for two workers w and w0 , θw ≤ θw0 implies ¯ ? ) ≤ uw0 (φ? , x ¯ ? ). In addition, if w and w0 are partnered, then θw < θw0 implies that uw (φ? , x ¯ ? ) < uw0 (φ? , x ¯ ? ). The analogous result holds for firms. that uw (φ? , x We can see these outcomes in the stable allocations we found in Examples 1 and 2. For instance, in Example 1, the high types, worker 1 and firm 3, each earn 1.731, which exceeds the payoff of the low types, worker 2 and firm 4, who each earn 1.401. The intuition for the proposition is that higher types can “imitate” and outcompete lower types whenever these lower types do strictly better. To elaborate, suppose there is an interior ¯ ? ) where a lower type worker w makes strictly more than a (weakly) stable allocation (φ? , x higher type worker w0 . Since w0 can produce a larger benefit (for less effort) and has a greater reward function, she earns a greater payoff than w when she receives same benefit as w and provides the same benefit as w. Thus, w0 continues to (i) earn more than w and thus (ii) ¯ ? ) when w0 offers f , the partner of w, a slightly higher benefit earn more than w0 did in (φ? , x than f is currently receiving. Since f does strictly better at this higher benefit and she does ¯ ? ), w0 and f block. It follows that a necessary condition of stability is that higher in (φ? , x types make at least as much as lower types. The proof formalizes this intuition.

4

Discussion and Conclusions

Motivated by many real-world examples where people exert efforts to produce benefits for their partners, we introduced and formally analyzed a matching game where players do the same. We first developed natural conditions for the existence of interior stable allocations 15

and then we characterized the structure of these allocations, with a focus on the benefits that players produce for and receive from their partners. We established that benefit production and receipt are related in a strict rank order fashion and then showed that, under certain conditions, players who have greater incremental rewards (to benefit received and to effort), i.e., higher types, produce and receive larger benefits. Subsequently, we developed conditions under which higher types match with one another and achieve greater payoffs. We close with discussions of two key features of our game – continuity and strict monotonicity – and of Assumptions 1 to 3. The continuity of effort (i.e., the continuum nature of X) and the continuity of the benefit and reward functions are essential for our results. If X is finite or if these functions are discontinuous, then a small change in effort is either infeasible or induces a substantial change in a player’s payoff. In either event, it is clear that the intuitions behind Propositions 2 to 5 cease to be true, so these results may fail to obtain. To illustrate, consider the following two counterexamples to Proposition 2 where the workers produce different benefits and yet are matched to firms who produce the same benefit. In the first counterexample, X is finite, and in the second counterexample, r(b, y, θ) is discontinuous. Example 3. A Counterexample to Proposition 2. Let W = {1, 2}, let F = {3, 4}, let X = {0, 1}, let b(y, θ) = 1+y, and let r(b, y, θ) = b−y. Then it is stable for worker 1 and firm 3 to match with agreement (0, 1) and for worker 2 and firm 4 to match with agreement (1, 1). Workers 1 and 2 produce benefits of 1 and 2 respectively, while the firms both produce a benefit of 1. 4 Example 4. A Second Counterexample to Proposition 2. Let W = {1, 2}, let F = {3, 4}, let X = [0, 2]2 , let b(y, θ) = y + θ, let r(b, y, θ) = b + y if y ≤ 1 and r(b, y, θ) = −100 if y > 1, and let θ1 = θ3 = θ4 = 0 and θ2 = 1. Then it is stable for worker 1 and firm 3 to match with agreement (1, 1) and for worker 2 and firm 4 to match with agreement (1, 1). Workers 1 and 2 produce benefits of 1 and 2 respectively, while the firms both produce a benefit of 1. 4 Counterexamples where b(y, θ) is discontinuous are readily constructed, as are counterexamples to Propositions 3 to 5. The strict monotonicity of the benefit function in effort and the strict monotonicity of the reward function in benefit are also essential for our results. If these functions are not strictly increasing, then (i) a small increase in effort need not make a player’s indented partner strictly better off and (ii) workers (firms) do not agree on which firms (workers) are best, second best, and so on. Thus, the intuitions behind Propositions 2 to 5 are no longer true, so these results may fail to obtain. To illustrate, consider the following counterexample to 16

Proposition 2, where r(b, y, θ) is non-monotone in benefit. Example 5. A Third Counterexample to Proposition 2. Let W = {1, 2}, let F = {3, 4}, let X = [0, 1]2 , let b(y, θ) = y, let r(b, y, θ) = b if θ = 1 and let r(b, y, θ) = 1 + y − y 2 if θ = 2, and let θ1 = θ2 = 1 and θ3 = θ4 = 2. Then it is stable for worker 1 and firm 3 to match with agreement ( 43 , 21 ) and for worker 2 and firm 4 to match with agreement ( 14 , 12 ). Workers 1 and 2 produce benefits of 34 and 14 respectively and yet are matched to firms who both produce a benefit of 12 . 4 Counterexamples where b(y, θ) is nonmonotone are easily constructed, as are counterexamples to Propositions 3 to 5. Assumption 1 is stronger than necessary and can be replaced with the following, weaker assumption. Assumption 10 . Sufficient Conditions for Existence, Efficiency, and Essentiality. The following hold: 1. There are a worker w and firm f for whom uw (f, x) ≥ 0 and uf (w, x) ≥ 0 for some x ∈ (0, β)2 . 2. For each worker w, each firm f , and each x ∈ ∂X, we have that (i) uw (f, x) < 0, (ii) uf (w, x) < 0, or (iii) there is an x0 ∈ int(X) such that uw (f, x0 ) > uw (f, x) and uf (w, x0 ) > uf (w, x).20 The first part is familiar. The second part is new and requires, for each possible worker-firm pair and each boundary agreement, that either (i) the worker or firm has a negative payoff or (ii) there is some interior agreement both the worker and firm prefer to the boundary agreement. It is readily verified that Assumption 1 implies Assumption 10 and that Assumption 10 is sufficient for Proposition 1. The chief advantages of Assumption 1 over Assumption 10 are its simplicity and the fact that it concerns the primitive benefit and reward functions, rather than the induced payoff functions. While Assumption 2’s requirements that the benefit function is type independent and that the reward function has a form of increasing differences are strong, they are often economically reasonable and, more importantly, they are key to Propositions 3 and 4. If these requirements are unmet, then stability does not imply that higher type do strictly better by outcompeting lower types. Thus, the intuitions behind Propositions 3 and 4 no longer hold, so these results may fail to obtain. To illustrate, consider the following counterexample to Proposition 3, where r(b, y, θ) lacks the requisite type of increasing differences. Example 6. A Counterexample to Proposition 3. 20

We write ∂X for the boundary of X = [0, β]2 and int(X) for the interior of X.

17

b+y

Let W = {1, 2}, let F = {3, 4}, let X = [0, 2]2 , let b(y, θ) = y, let r(b, y, θ) = e 8θ − 1θ y 3 , and let θ1 = θ3 = 1 and θ2 = θ4 = 12 . Then it is stable for worker 1 and firm 3 to match with agreement (0.265, 0.265) and for worker 2 and firm 4 to match with agreement (0.312, 0.312). The benefit the high types produce, 0.265, is less than the benefit the low types produce, 0.312. 4 Counterexamples where b(y, θ) is not type independent are readily constructed, as are counterexamples to Proposition 4. Nevertheless, there is an economically important case where Assumption 2 fails to hold and yet Propositions 3 and 4 obtain: the case where each side has its own type independent benefit function and the reward function has the requisite type of increasing differences. Thus, these results extend to environments where one side pays the other for a service – e.g., the firms pay the workers via a linear benefit function, while the workers provide non-linear benefits to the firms. Assumption 3’s requirements that the benefit and reward functions are strictly increasing in type are key to Proposition 5. If these functions are not increasing in θ, then higher types do not do strictly better than lower types when they imitate the lower types, so the intuition for the proposition ceases to hold and the result may not obtain. To illustrate, consider the following counterexample, where r(b, y, θ) is decreasing in θ. Example 7. A Counterexample to Proposition 5. Let W = {1, 2}, let F = {3, 4}, let X = [0, 2]2 , let b(y, θ) = y, let r(b, y, θ) = b − 1θ y 2 + 3 1 ( − θ), and let θ1 = θ3 = 1 and θ2 = θ4 = 12 . Then it is stable for 1 and 3 to match with 10 2 agreement ( 12 , 12 ) and for 2 and 4 to match with agreement ( 41 , 41 ). The payoff of the high 1 types, 10 , is less than the payoff of the low types, 81 . 4 Counterexamples where b(y, θ) is nonmonotone are easily constructed.

A

Appendix: Existence of Stable Allocations

In this appendix, we establish the existence of stable allocations for a broad class of twosided, one-to-one matching games, which includes the Effort Game, Hatfield and Milgrom’s [19] general “Matching with Contracts Game,” and Demange and Gale’s [12] “Assignment Game.” While others have explored this question before (e.g., Kaneko [21] and Alkan and Gale [2]), our methods are novel and illuminate a deep connection between existence in these games and the “Deferred Acceptance” algorithm. Environment and Stability

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We consider general a one-to-one matching game with two finite groups of players: workers W = {1, . . . , W } and firms F = {W + 1, . . . , N }, with N > W > 0. Let N = W ∪ F denote the set of players. We write w for an arbitrary worker, f for an arbitrary firm, and i for an arbitrary player. A matching is a function that specifies each player’s match, i.e., is a φ : N → N such that: (i) for each worker w, φ(w) ∈ F ∪ {w}; (ii) for each firm f , φ(f ) ∈ W ∪ {f }; and (iii) for each worker w and each firm f , φ(w) = f ⇐⇒ φ(f ) = w. Let Φ denote the finite set of all matchings. A player is single if she is matched to herself, i.e., if φ(i) = i, and is partnered if she is matched to a member of the opposite group, i.e., if φ(i) 6= i. When a worker and firm match, they select an agreement x = (x1 , . . . , xk ) ∈ Rk , where k ≥ 1. Their agreement specifies their individual and joint actions x1 , . . ., xk – e.g., x1 may give pay, x2 may give the number of continuing education hours, and so on. For record keeping purposes, we also suppose that each single player i has an agreement x with herself. We write xi for the agreement that player i has with her partner or with herself, and we ¯ = (x1 , . . . , xW , xW +1 , . . . , xN ) for the (joint) vector of players’ agreements. Given a write x ¯ ∈ A(φ) = {(˜ ˜ N ) ∈ RkN | x ˜i = φ ∈ Φ, we have that xi = xφ(i) for each player i, so x x1 , . . . , x ˜ φ(i) for all i ∈ N }. The vector subspace A(φ) is the collection of possible agreement vectors x ¯ ) such that φ ∈ Φ and x ¯ ∈ A(φ); it specifies for the matching φ. An allocation is a (φ, x ¯ ) ∈ Φ × RkN |¯ each player’s partner and their agreement. We write A = {(φ, x x ∈ A(φ)} for the set of allocations. A player’s payoff depends (only) on the identity of her match and their agreement. Formally, each worker w has a payoff function uw : {F ∪{w}}×Rk → R over her possible matches and agreements. Likewise, each firm f has a payoff function uf : {W ∪ {f }} × Rk → R. The value of being single is normalized to zero, i.e., ui (i, x) = 0 for every x ∈ Rk and i ∈ N . Let ¯ ) = (φ, x1 , . . . , xi , . . . , xN ) ∈ Φ × RkN , in a slight abuse of notation we write ui (φ, x ¯) (φ, x i ¯ ), i.e., ui (φ, x ¯ ) ≡ ui (φ(i), x ). for the payoff of player i in (φ, x ¯ ) is Let X ⊂ Rk be the (nonempty) set of feasible agreements. An allocation (φ, x N ¯ ∈ X . An allocation (φ, x ¯ ) is individually rational feasible if agreements are in X, i.e., x ¯ ) ≥ 0 for each player i. A if every player gets at least the value of being single, i.e., ui (φ, x kN ¯ ) ∈ Φ × R if they both obtain strictly higher payoffs worker w and a firm f block a (φ, x ¯ ), i.e., if there by matching with each other at a feasible agreement than they obtain in (φ, x ¯ ) and uf (w, x) > uf (φ, x ¯ ). An allocation exists an x ∈ X such that uw (f, x) > uw (φ, x ¯ ? ) is stable if (i) it is feasible, (ii) individually rational, and (iii) no worker and firm (φ? , x block it. Stable allocations are our solution concept. We call our matching game the “General Game.” It is easily seen that General Game nests the Effort Game. It also nests the Matching with Contracts Game and the Assignment 19

Games, as the next examples illustrate. Example A1. Matching with Contracts Game. In Hatfield and Milgrom [19], workers and firms choose agreements from a finite set of possible agreements when they match and their payoffs depend on their on their partners’ ˜ ⊂ Rk identities and their agreements. Formally, there is a finite set of possible agreements X ˜ → R and u˜f : (W ∪ {f }) × X ˜ → R for each worker w and payoffs are u˜w : (F ∪ {w}) × X ˜ and each firm f , with u˜i (i, x) = 0 for each player i and each x ∈ X. The chief difficulty with embedding lies in the domain of the payoffs: e.g., worker w’s ˜ while payoffs in the General Game are defined payoff u˜w is only defined on (F ∪ {w}) × X, ˜ is discrete, so u˜w is (trivially) continuous and Tietze’s on (F ∪ {w}) × Rk . Fortunately, X Extension Theorem (Ok [30], p.267) guarantees that there is a continuous function uw : ˜ Hence, (F ∪ {w}) × Rk → R such that uw (i, x) = u˜w (i, x) for all (i, x) ∈ (F ∪ {w}) × X. ˜ and we embed the Matching with Contracts Game in the General Game by letting X = X by letting ui be a Tietze extension of u˜i for each player i. 4 Example A2. Assignment Game. In Demange and Gale [12], workers and firms agree to real-valued monetary transfers when they match and their payoffs depend on their partners’ identities and their transfers. Formally, the set of possible transfers is R and payoffs are u˜w : (F × {w}) × R → R and u˜f : (W ×{f })×R → R for each worker w and each firm f , which are continuous and weakly increasing in the transfer x ∈ R; further, u˜i (i, x) = 0 for each player i and each x ∈ R. It is assumed that there is a finite αwf ≥ 0 such that either worker w or firm f has a negative payoff when they are matched and their transfer is not in [−αwf , αwf ]. We embed the Assignment Game by setting ui = u˜i for each player i and by taking X = [−¯ α, α ¯ ], where α ¯ = max(w,f )∈W×F {αwf }. (Although the Assignment game does not restrict players’ transfers to X, this restriction is without loss as in any individually rational allocation all transfers are automatically in X.) 4 The General Game relaxes both the finiteness of agreements in the Matching with Contracts Game and the unidimensionality of agreements, as well as the monotone nature of payoffs, in the Assignment Game. It is also clear that, while the Effort Game is related to the Matching with Contracts Game and the Assignment Game, it is not a specialization of them. Existence of Stable Allocations We establish that stable allocations exist under the following (weak) assumption. Assumption A1. Sufficient Conditions for the Existence of Stable Allocations.

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The set of feasible agreements X is compact and, for each worker w and each firm f , the payoffs uw (f 0 , x) and uf (w0 , x) are upper semi-continuous in x for all firms f 0 and all workers w0 respectively. Proposition A1. Existence of Stable Allocations. ¯ ? ) exists. Let Assumption A1 hold, then a stable allocation (φ? , x Proposition A1 gives, for instance, that there is a stable allocation in the Effort Game, Matching with Contracts Game, and Assignment Games since inspection shows that Assumption A1 holds in each. We argue the proposition via contradiction and show that if there is no stable allocation when X is compact, then there is no stable allocation when agreements are restricted to a finite subset of X. This contradicts a well-known result, see Lemma 1 below, that there is always a stable allocation when the set of feasible agreements is finite.21 The key insight of our proof is that under the contradiction hypothesis and Assumption A1, we can use the Heine-Borel Theorem to ensure the existence of this finite subset of X. We need four preliminary results to make this argument precise. Lemma A1. Finite Existence. ¯ ? ) exists. Let the set of feasible agreements X be finite, then a stable allocation (φ? , x Proof. The proof parallels the proofs of Hatfield and Milgrom’s Theorem 3 [19] and Roth’s Theorem 1 [34]: one constructs a variant of Gale and Shapley’s [16] Deferred Acceptance algorithm that incorporates a “tie-breaking” rule to account for indifferences and then shows that the algorithm converges to a stable allocation in finite time when X is finite.22 The details are omitted.  ¯ ). Lemma A2. Continuity of ui (φ, x ¯ ) is upper semi-continuous in x ¯ for each Let Assumption A1 hold and let φ ∈ Φ, then ui (φ, x player i. Proof. This is an immediate consequence of Assumption A1.  To prove Proposition 1, we need to (i) represent the set of feasible and individually rational allocations as a collection of compact sets and (ii) establish that the set of allocations a worker and firm block with a given agreement is open. To these ends, let φ ∈ Φ and let ¯ ∈ X N ∩ A(φ) and ui (φ, x ¯ ) ≥ 0 for all i ∈ N } be the (possibly empty) F (φ) = {¯ x ∈ RkN | x ¯ ) is a feasible and individually rational set of agreement vectors in RkN such that (φ, x ¯ ∈ F (φ). Let ΦF = {φ ∈ Φ|F (φ) 6= ∅} be the set of matchings such allocation for each x 21

The intuition here is similar to that of Crawford and Knoer [9]; however, our game, assumptions, and formal approach are quite different from theirs. 22 Roth and Sotomayor’s Theorem 2.8 [35] provides an example of the incorporation of a tie-breaking rule.

21

¯ so that (φ, x ¯ ) is feasible and individually rational. that, for each φ ∈ ΦF , there is an x ¯ ) is feasible and individually rational if an only if φ ∈ ΦF and Clearly, an allocation (φ, x ¯ ∈ F (φ). x Lemma A3. Compactness of F (φ). Let Assumption A1 hold and let φ ∈ Φ, then F (φ) is compact. ¯ ) ≥ 0 for all i ∈ N } is Proof. Assumption A1 implies that X N is compact and {¯ x|ui (φ, x closed (by upper semi-continuity). Thus, F (φ) is closed and bounded since A(φ) is closed.  ¯ ) ∈ Φ × RkN with agreement x ∈ X, we When a worker w and a firm f block a (φ, x ¯ ). Let C = W × F × X. For a φ ∈ Φ and c = (w, f, x) ∈ C, let say (w, f, x) blocks (φ, x ¯ ) and uf (f, x) > uf (φ, x ¯ )} be the set of vectors in Dφ (c) = {¯ x ∈ RkN | uw (f, x) > uw (φ, x kN ¯ ) for each x ¯ ∈ Dφ (c). R such that c blocks the pair (φ, x Lemma A4. Openness of Dφ (c). Let Assumption A1 hold, let φ ∈ Φ, and let c ∈ C, then Dφ (c) is open. ¯ ) < uw (f, x)} and Proof. This follows directly from Assumption A1 since {¯ x ∈ RkN |uw (φ, x kN ¯ ) < uf (w, x)} are open by upper semi-continuity.  {¯ x ∈ R |uf (φ, x Proof of Proposition 1.23 Suppose that there is no stable allocation. Since the set of feasible and individually rational allocations is non-empty (because players may always be single), every feasible and individually rational allocation is blocked by some worker and ¯ ∈ F (φ), we have that (φ, x ¯ ) is blocked by a c ∈ C, firm. Let φ ∈ ΦF . Then, for every x which implies that F (φ) ⊂ ∪c∈C Dφ (c). Since Dφ (c) is open, {Dφ (c)}c∈C is an open cover of F (φ). Since F (φ) is compact, the Heine-Borel Theorem gives the existence of a finite lφ ¯ ∈ F (φ), we have that (φ, x ¯ ) is blocked by some sub-cover {Dφ (cφj )}j=1 . Thus, for every x lφ element of {cφj }j=1 . lφ Repeating this argument for all matchings in ΦF gives a set E = ∪φ∈ΦF {cφj }j=1 such that every feasible and individually rational allocation is blocked by an element of E. Since Φ is finite, ΦF and thus E are finite. Let EX = {x ∈ X | (w, f, x) ∈ E for some (w, f ) ∈ W × F} be the finite set of agreements associated with E. To establish the contradiction, suppose that the set of feasible agreements is EX instead ¯ ? ). Since of X. Since EX is finite, Lemma 1 gives that there is a stable allocation (φ? , x ¯ ? ∈ X N . Thus, (φ? , x ¯ ? ) is a feasible allocation when the set of (EX )N ⊂ X N , we have that x feasible agreements is X. Hence, the previous paragraph gives that there is a (w0 , f 0 , x0 ) ∈ E 23

This proof benefited from discussions with Asaf Plan.

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¯ ? ). Since x0 ∈ EX , we have that w0 and f 0 block (φ? , x ¯ ? ) when the set of which blocks (φ? , x feasible agreements is EX , a contradiction.  Discussion and Conclusion It is possible to weaken several of our assumptions and still obtain existence. It is easily seen, for instance, that existence obtains when (i) different workers and firms have different sets of feasible agreements or when (ii) the value of being single differs across players. More importantly, existence also obtains when (iii) agreements live in a general metric space, instead of Rk , provided payoffs are upper semi-continuous and X is compact in the finite open cover sense. The intuition for this is two fold. First, under these assumptions, Dφ (c) is open (by upper semi-continuity) and F (φ) is compact in the finite open cover sense (by upper semi-continuity, the compactness of X N , the closed nature of A(φ), the fact closed subsets of compact metric spaces are compact). Second, the key elements of Proposition A1’s proof are (a) the openness of Dφ (c) and (b) the compactness of F (φ) in the finite open cover sense. So long as these are preserved, the necessary finite sub-covers and EX exist, allowing one to make the requisite contradiction. Thus, existence still obtains. This extension is economically meaningful when the agreements are random variables, as in Chiappori and Reny [6]. Our other assumptions, unfortunately, cannot be easily weakened. In particular, counterexamples to Proposition A1 abound when the set of feasible agreements is not compact or when payoffs are not upper semi-continuous.24 This is not to say that these conditions are necessary – indeed, they are not since it is also possible to construct examples of stable allocations where X is open and the payoffs are not upper semicontinuous. Rather, it is to say that, absent compactness and upper semi-continuity, general results are hard to come by. Limited result, however, are possible. For example, a stable allocation exists when X is compact and payoffs are increasing transformations of upper semi-continuous functions; which is interesting since an increasing transformation does not preserver upper semi-continuity. Additionally, a stable allocation exists when X is open (and bounded) and payoffs are upper semi-continuous, provided part (b) of Assumption 10 holds. We close by discussing three closely related studies – Crawford and Kroner [9], Kaneko [21], and Alkan and Gale [2]. Crawford and Kroner realized that there is a close connection between a version of Gale and Shapley’s [16] Deferred Acceptance and existence in Assignment games with quasi-linear payoffs, and they leveraged this connection via a contradiction argument to establish existence.25 We build on their insights by using novel methods and 24

These examples are available upon request. Deferred Acceptance algorithms are broadly important as they are the principle means by which existence is established in variants of the Matching with Contracts Game – see, for instance, Adachi [1], Echenique 25

23

broadly establish this connection to show the existence of stable allocations. Kaneko [21] and Alkan and Gale [2] study existence in general one-to-one matching games where each worker and firm have a set of achievable payoffs from which they pick some point when they are matched. Kaneko shows that if this set meets certain technical conditions, then the game is balanced and so has a stable allocation by Scarf’s Balancedness Theorem. Alkan and Gale give alternative proof by describing an algorithm that computes a stable allocation in finite time when each pair’s Pareto frontier is described by a continuous and strictly decreasing function that intersects both axes. We differ from these studies in our assumptions, and so can establish existence in examples that meet neither Kaneko nor Alkan and Gale’s technical conditions.26 More importantly, however, our method of proof is novel and illuminates a deep connection that was first hinted at by Crawford and Kroner.

B

Appendix: Proofs

This appendix collects the proofs of our propositions from the main text; lemmas appear as needed. Lemma B1. Payoff Improvements and Stability. ¯ ? ) is a stable allocation and (φ0 , x ¯ 0 ) is a feasible allocation such that ui (φ0 , x ¯ 0) ≥ If (φ? , x ¯ ? ) for each player i, then (φ0 , x ¯ 0 ) is a stable allocation. ui (φ? , x ¯ 0 ) were not stable, then either (a) ui (φ0 , x ¯ 0 ) < 0 for some Proof. Almost obvious. If (φ0 , x ¯ 0 ) and uf (w, x) > uf (φ0 , x ¯ 0 ) for some worker w, firm f , and player i or (b) uw (f, x) > uw (φ0 , x ¯ 0 ) ≥ ui (φ? , x ¯ ? ) for each player i, (a) implies that (φ? , x ¯ ? ) is agreement x ∈ X. Since ui (φ0 , x ¯ ? ) is blocked; in both cases, stability of not individually rational and (b) implies that (φ? , x ¯ ? ) is contradicted.  (φ? , x Proof of Proposition 1. We first establish (iii), then we use it and Proposition A1 (see Appendix A), which is our principle existence result, to deduce (i). Finally, we argue (ii) via contradiction. ? ? ¯ ? ) = (φ? , x1 , . . . , xN ) where some player We first show that any stable allocation (φ? , x is partnered is an interior stable allocation. Suppose not, then there is a worker w and firm ? ? f = φ? (w) for whom (x?1 , x?2 ) = xw = xf is in the boundary of X = [0, β]2 . Thus, (a) x?1 = β and x?2 ∈ [0, β], (b) x?1 ∈ [0, β] and x?2 = β, (c) x?1 = 0 and x?2 ∈ [0, β], or (d) x?1 ∈ [0, β] ? ¯ ? ) = uw (f, xw ) = and x?2 = 0. If (a), then part (2) of Assumption 1 gives that uw (φ? , x and Oviedo [14], Fleiner [15], Hafalir et al. [17], Hatfield and Milgrom [19], Kominers and Sonmez [23], and Roth [34]. 26 A detailed example is available upon request.

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¯ ?) = r(b(x?2 , θf ), β, θw ) < 0, a contradiction of stability. Analogously, if (b), then uf (φ? , x r(b(x?1 , θw ), β, θf ) < 0, a contradiction of stability. If (c), there are two sub-cases depending ¯ ? ). To on whether parts (3.a) or (3.b) hold. When part (3.a) holds, then w and f block (φ? , x see this, consider agreement (x?1 + , x?2 ) with  > 0. Since uf (w, x1 , x2 ) = r(b(x1 , θw ), x2 , θf ) ¯ ? ) for is strictly increasing in x1 , we have that uf (w, x?1 + , x?2 ) > uf (w, x?1 , x?2 ) = uf (φ? , x all  > 0. Since part (3.a) gives that uw (f, x1 , x2 ) = r(b(x2 , θf ), x1 , θw ) is strictly increasing ¯ ? ) for all in x1 when x1 ∈ [0, δθw ], where δθw > 0, we have that uw (f, x?1 + , x?2 ) > uw (φ? , x  sufficiently small. Since (x?1 + , x?2 ) ∈ X when  < β, it follows that w and f block with ¯ ?) agreement (x?1 + , x?2 ) for some sufficiently small  > 0. When part (3.b) holds, then (φ? , x is not individually rational for f : simply, uf (w, x?1 , x?2 ) = r(b(0, θw ), x?2 , θf ) < 0. In both ¯ ? ) is contradicted. If (d), then an analogous argument shows sub-cases, the stability of (φ? , x ¯ ? ) is not stable, a contradiction. that (φ? , x We next show that there is an interior stable allocation. Since X is compact and since the ¯ ? ) exists by Proposition A1. payoffs are continuous in the agreement, a stable allocation (φ? , x There are two cases, either (a) φ? (i) 6= i for some player i or (ii) φ? (i) = i for each player i. If (a), then we have the desired result; so, we proceed under (b). By part (1) of Assumption ˜ ∈ (0, β)2 such that w and f both get at 1, there is a worker w, a firm f , and an agreement x 0 0 ˜ . We construct a new allocation from (φ0 , x ¯ 0 ) = (φ0 , x1 , . . . , xN ) least zero at agreement x ¯ ? ) by taking w and f , matching them, and assigning them agreement x ˜ , while from (φ? , x 0 0 ˜ , set xf = x ˜, leaving everyone else single – i.e., by setting φ0 (w) = f , φ0 (f ) = w, xw = x 0 ? i0 i? 0 0 ¯ ) is interior stable if it and φ (i) = φ (i) and x = x for all i ∈ N \{f, w}. Clearly, (φ , x ¯ ? ) and gets at least zero in (φ0 , x ¯ 0 ), it follows is stable. Since each player gets zero in (φ? , x ¯ 0 ) is stable. from Lemma B1 that (φ0 , x It remains to show that every interior stable allocation is Pareto efficient. We argue by ¯ ? ) is interior stable, but is not Pareto efficient. Then, there contradiction: suppose that (φ? , x 0 0 ¯ 0 ) = (φ0 , x1 , . . . , xN ) with (a) ui (φ0 , x ¯ 0 ) ≥ ui (φ? , x ¯ ? ) for is another feasible allocation (φ0 , x ¯ 0 ) > ui (φ? , x ¯ ? ) for some i ∈ N . Without loss, suppose that worker all i ∈ N and (b) ui (φ0 , x ¯ ? ), w is the player who does strictly better. We will show that w and some firm block (φ? , x a contradiction. But first, we need two preliminary facts: 1. (Fact One.) Worker w is matched to some firm under φ0 , i.e., φ0 (w) 6= w. If not, then ¯ 0 ) = 0, implying that uw (φ? , x ¯ ? ) < 0 by (b), a contradiction of stability. In uw (φ0 , x light of this, let f = φ0 (w). 0 2. (Fact Two.) The agreement xw is on the interior of X. Simply, Lemma B1 gives that 0 ¯ 0 ) is stable, so Fact One and (iii) imply the interiority xw . (φ0 , x This concludes the list of facts. 0 ¯ ? ). Let (x01 , x02 ) = xw and With these facts in hand, we show that w and f block (φ? , x 25

0

¯ 0 ) ≥ uf (φ? , x ¯ ? ) and consider agreement (x01 + , x02 ), with  > 0. Since uf (w, xw ) = uf (φ0 , x since uf (w, x1 , x2 ) = r(b(x1 , θw ), x2 , θf ) is strictly increasing in x1 , we have that uf (w, x01 + 0 ¯ ? ) for all  > 0. Further, since payoffs are continuous and since uw (f, xw ) = , x02 ) > uw (φ? , x ¯ ? ) for all  sufficiently ¯ 0 ) > uw (φ? , x ¯ ? ), we have that uw (f, x01 + , x02 ) > uw (φ? , x uw (φ0 , x small. Additionally, since (x01 , x02 ) is interior, we may take  to be small enough so that (x01 + , x02 ) ∈ X. Thus, there is a small  > 0 such that w and f block with agreement (x01 + , x02 ).  Proof of Proposition 2. Let {zi? }i∈N and {b?i }i∈{j|φ? (j)6=j} be the associated efforts and ¯ ? ). Let J = min{JW , JF } and J = max{JW , JF }. We will show below that, benefits of (φ? , x ⇐⇒ φ? (w) ∈ GFl for all l ∈ {1, . . . , j}. We for each 1 ≤ j ≤ J, we have that w ∈ GW l refer to this as the “induction” result since we prove it by induction on j: we first establish that it holds when j = 1 and next show that if it holds at j − 1, then it also holds at j when j > 1. The first part of the proposition follows the fact J = J: simply if, say J = JW < JF = J, then there are firms in J + 1 who are matched to a worker not in groups 1 to JW , a contradiction as these groups contain all matched workers; an analogous argument applies if J = JF < JW = J. The second part of the proposition also follows from the induction result, take j = JW . Let j = 1. There are two cases: (i) J = 1 and (ii) J > 1. If case (i), then J = 1 and the induction result is trivially true. Thus, we proceed under case (ii). ? F We first establish that w ∈ GW 1 implies φ (w) ∈ G1 via contradiction. Suppose not, 0 0 ? F then there is a w ∈ GW 1 such that f = φ (w) is in Gl0 with l > 1. Let f be a firm in GF1 and let w0 = φ? (f 0 ). Consider agreement (zw? + , zf?0 ), with  > 0. Since b?f 0 > b?f and since the reward function is strictly increasing in benefit and continuous in own effort, ¯ ? ) for all  we have that uw (f 0 , zw? + , zf?0 ) = r(b?f 0 , zw? + , θw ) > r(b?f , zw? , θw ) = uw (φ? , x sufficiently small. Since the benefit function is strictly increasing in effort, we also have b(zw? + , θw ) > b?w ≥ b?w0 for all  > 0 as w0 may be a member of GW 1 . Consequently, ? ? ? ? ? ? ? ? ¯ ) for all  > 0. Further, uf 0 (w, zw +, zf 0 ) = r(b(zw +, θw ), zf 0 , θf ) > r(bw0 , zf 0 , θf ) = uf 0 (φ , x since zw? and zf?0 are interior, we may take  to be small enough that (zw? + , zf?0 ) ∈ X. Thus, ¯ ? ) with agreement (zw? + , zf?0 ), a there is a small  > 0 such that w and f 0 block (φ? , x contradiction. ? The analogous argument gives that f ∈ GF1 implies φ? (f ) ∈ GW 1 . Since φ (f ) = w if and only if φ? (w) = f , we have that w ∈ GW ⇐⇒ φ? (w) ∈ GFl for all l ∈ {1}, i.e., the induction l result is true when j = 1. Let 1 < j ≤ J and assume that w ∈ GW ⇐⇒ φ? (w) ∈ GFl for all l ∈ {1, . . . , j − 1}. l By the induction hypothesis, it suffices to show that w ∈ GW ⇐⇒ φ? (w) ∈ GFj to prove j the induction result. Again, there are two cases: (i) j = J or (ii) j < J. If case (i) 26

then we are done. Simply, when w ∈ GW j , the induction hypothesis gives that she is not matched to a firm in a lower indexed group. Since there is no higher indexed group of firms, we must have φ? (w) ∈ GFj . Analogously, if f ∈ GFj , then φ? (f ) ∈ GW j . It follows that ? F w ∈ GW j ⇐⇒ φ (w) ∈ Gj . Thus, we proceed under case (ii). ? F We establish that w ∈ GW j implies φ (w) ∈ Gj via contradiction. Suppose not, then is a ? F 0 0 w ∈ GW j such that f = φ (w) is in Gl0 with l 6= j. The induction hypothesis gives that l > j: simply, if l0 < j, then w is a member of GW l0 by the induction hypothesis, an impossibility. Let f 0 be a firm in GFj and let w0 = φ? (f 0 ). The induction hypothesis gives that w0 is in GW j 0 F or a higher indexed benefit group: simply, if w0 ∈ GW with p < j, then f is in G by the p p ? ? induction hypothesis, an impossibility. Consider agreement (zw + , zf 0 ), with  > 0. Since ¯ ? ) for all  sufficiently small. We also b?f 0 > b?f , we have that uw (f 0 , zw? + , zf?0 ) > uw (φ? , x have b(zw? + , θw ) > b?w ≥ b?w0 for all  > 0 since w0 is in GW j or a higher indexed benefit ? ? ? ? ¯ ) for all  > 0. Further, since zw? and zf? group. Consequently, uf 0 (w, zw + , zf 0 ) > uf 0 (φ , x are interior, we may take  to be small enough that (zw? + , zf?0 ) ∈ X. Thus, there is a small ¯ ? ), a contradiction.  > 0 such that w and f 0 block (φ? , x The analogous argument gives that f ∈ GFj implies φ? (f ) ∈ GW j . It follows that w ∈ W ? F Gj ⇐⇒ φ (w) ∈ Gj , i.e., the induction result is true at j.  Proof of Proposition 3. Let {zi? }i∈N and {b?i }i∈{j|φ? (j)6=j} be the associated efforts and ¯ ? ). Let w and w0 be two partnered workers with θw < θw0 . Then, w ∈ GW benefits of (φ? , x l 0 W for some l and w ∈ Gj for some j, we need to establish that j ≤ l. If l = JW , then this is trivially true, so we take l < JW and we argue by contradiction. Suppose not, then l < j; so, b?w > b?w0 and zw? > zw? 0 . Let f = φ? (w) and f 0 = φ? (w0 ). Proposition 2 gives that b?f > b?f 0 , ¯ ? ); but, we first need two preliminary so zf? > zf?0 . We will show that w0 and f block (φ? , x facts. 1. (Fact One.) We have that r(b?f , zw? , θw ) ≥ r(b?f 0 , zw? 0 , θw ). To see this, consider a possible match of w and f 0 with agreement (zw? 0 + , zf?0 ), with  > 0. Since the benefit function is type independent, b(zw? 0 + , θw ) > b(zw? 0 , θw0 ) = b?w0 . Thus, uf 0 (w, zw? 0 + , zf?0 ) = ¯ ? ) for all  > 0. If it were the case r(b(zw? 0 + , θw ), zf?0 , θf 0 ) > r(b?w0 , zf?0 , θf 0 ) = uf 0 (φ? , x ¯ ? ) for some  > 0 with zw? 0 +  ∈ [0, β], then w and that uw (f 0 , zw? 0 + , zf?0 ) > uw (φ? , x ¯ ? ). Yet, since (φ? , x ¯ ? ) is stable, this cannot be; so, we must have that f 0 block (φ? , x ¯ ? ) = r(b?f , zw? , θw ) ≥ r(b?f 0 , zw? 0 + , θw ) = uw (f 0 , zw? 0 + , zf?0 ) for all  > 0 with uw (φ? , x zw? 0 +  ∈ [0, β]. The desired result follows from the continuity of the reward function. 2. (Fact Two.) We have that r(b?f , zw? , θw0 ) > r(b?f 0 , zw? 0 , θw0 ). From the previous fact, r(b?f , zw? , θw ) − r(b?f 0 , zw? 0 , θw ) ≥ 0. Since b?f > b?f 0 and zw? > zw? 0 , Assumption 2 implies that r(b?f , zw? , θw ) − r(b?f 0 , zw? 0 , θw ) > 0. This concludes the list of facts. 27

¯ ? ). Consider agreement (zw? + , zf? ), with  > 0. Now we show that w0 and f block (φ? , x Since b(zw? +, θw0 ) > b(zw? , θw ) = b?w , we have that uf (w0 , zw? +, zf? ) = r(b(zw? +, θw0 ), zf? , θf ) > ¯ ? ). The second fact implies that uw0 (f, zw? +, zf? ) = r(b?f , zw? +, θw0 ) > r(b?w , zf? , θf ) = uf (φ? , x ¯ ? ) for  sufficiently small due to continuity. Since (zw? , zf? ) is interior, r(b?f 0 , zw? 0 , θw0 ) = uw0 (φ? , x we have that (zw? + , zf? ) ∈ X for  small enough. It follows that there is an  > 0 such that ¯ ? ) with agreement (zw? + , zf? ), a contradiction.  w0 and f block (φ? , x ¯ 0 ). Proof of Proposition 4. By Proposition 1, there is an interior stable allocation (φ0 , x We proceed in three steps. First, we describe the rematching procedure that constructs a ¯ ? ) from (φ0 , x ¯ 0 ). Second, we establish that (φ? , x ¯ ? ) is an interior stable new allocation (φ? , x ¯ ? ) exhibits assortative matching in types. Throughout, allocation. Third, we show that (φ? , x W F F 0 ¯ 0 ), let zi0 and b0i let GW 1 , . . . , GJW and G1 , . . . , GJF be the benefit groups associated with (φ , x ¯ 0 ), and let zi? and b?i denote player i’s effort and denote player i’s effort and benefit in (φ0 , x ¯ ? ). benefit in (φ? , x ? ? 0 ¯ ? ) = (φ? , x1 , . . . , xN ) from (φ0 , x ¯ 0 ) = (φ0 , x1 , . . . , xN ) as follows: We first construct (φ? , x 1. (Constructing φ? .) For each i ∈ N with φ0 (i) = i, set φ? (i) = i. For each l ∈ {1, . . . , JW }, (a) List the workers in GW l in descending order of their types (breaking ties randomly) and label them w1 , w2 , . . ., w|GWl | , where | · | denotes the cardinality of a set. So w1 is the highest type worker in GW l , w2 is the second highest type worker, and F so on. Likewise, list the firms in Gl in descending order (breaking ties randomly) of their types and label them f1 , . . ., f|GFl | . (b) Set φ? such that φ? (wj ) = fj and φ? (fj ) = wj for j ∈ {1, . . . , |GW l |}. (Proposition W F 2 gives that |Gl | = |Gl |.) ? ? ¯ ? .) If φ? (i) = i, then set xi = (zi0 , 0) when i is a worker and xi = (0, zi0 ) 2. (Constructing x ? when i is a firm. If φ? (i) 6= i, then set xi = (zi0 , zφ0 ? (i) ) when i is a worker and ? xi = (zφ0 ? (i) , zi0 ) when i is a firm. (Thus, zi? = zi0 for all i ∈ N .) This concludes the construction. ¯ ? ) is an interior stable allocation. It is clear from the Second, we establish that (φ? , x ¯ 0 ) that (φ? , x ¯ ? ) is a feasible allocation with construction and the interior stability of (φ0 , x ? ¯ ? ) is xi ∈ (0, β)2 for each player i and with at least one partnered player. Thus, (φ? , x interior stable if it is individually rational and stable. To establish these, observe that ¯ ? ) = ui (φ0 , x ¯ 0 ) for all i ∈ N : this is trivial if i is single, so we take i to be partnered. ui (φ? , x By step (2), i’s effort is unchanged. Since φ? (i) and φ0 (i) are both in the same benefit group ¯ 0 ) (by step (1)) and since we keep their efforts constant (by step (2)), the benefit i in (φ0 , x ¯ ? ) and (φ0 , x ¯ 0 ). It follows that receives is unchanged. Thus, i has the same payoff in (φ? , x ¯ ? ) is individually rational and, by Lemma B1, stable. (φ? , x 28

¯ ? ) exhibits assortative matching in types. Let w and w0 Third, we establish that (φ? , x be partnered workers such that θw < θw0 ; note that if there is only one partnered worker, ¯ ? ) trivially exhibits assortative matching. Proposition 3 gives that there are two then (φ? , x cases: (i) b?w = b?w0 or (ii) b?w < b?w0 . Let f = φ? (w) and f 0 = φ? (w0 ). If (i), then w0 occupies an earlier position in the step (1) list than w and so is matched to a higher type firm, i.e., θf ≤ θf 0 . If (ii), then Proposition 2 gives that b?f < b?f 0 , so the contraposition of Proposition 3 gives that θf ≤ θf 0 .  Lemma B2. Higher Types are Partnered when Lower Types are Partnered. ¯ ? ) be an interior stable allocation. Consider two workers Let Assumption 3 hold and let (φ? , x w and w0 , with θw < θw0 . If w is partnered, then so too is w0 , i.e., φ? (w) 6= w =⇒ φ? (w0 ) 6= w0 . The analogous result holds for firms. Proof. We argue via contradiction. Suppose that w is partnered, while w0 is not. Let {zi? }i∈N ¯ ? ), and let f = φ? (w). We and {b?i }i∈{j|φ? (j)6=j} be the associated efforts and benefits of (φ? , x ¯ ? ), a contradiction. will show that w0 and f block (φ? , x Consider agreement (zw? + , zf? ), with  > 0. Since b(y, θ) is weakly increasing in θ, we have that b(zw? + , θw0 ) > b(zw? , θw ) = b?w , which implies that uf (w0 , zw? + , zf? ) = r(b(zw? + ¯ ? ) for all  > 0. Since individual rationality implies , θw0 ), zf? , θf ) > r(b?w , zf? , θf ) = uf (φ? , x ¯ ? ) = r(b?f , zw? , θw ) ≥ 0, we have that uw0 (f, zw? + , zf? ) = r(b?f , zw? + , θw0 ) > 0 = that uw (φ? , x ¯ ? ) for all  sufficiently small because the reward function is continuous and strictly uw (φ? , x increasing in type. Since (zw? , zf? ) is interior, we may take  small enough that (zw? +, zf? ) ∈ X. ¯ ? ) with agreement (zw? + , zf? ).  Thus, there is an  > 0 such that w0 and f block (φ? , x Proof of Proposition 5. Let {zi? }i∈N and {b?i }i∈{j|φ? (j)6=j} be the associated efforts and ¯ ? ). Let w and w0 be two workers. We first establish that θw < θw0 implies benefits of (φ? , x ¯ ? ) ≤ uw0 (φ? , x ¯ ? ), with strict inequality if both workers are partnered. Subsequently, uw (φ? , x ¯ ? ) = uw0 (φ? , x ¯ ? ). The proposition follows. we establish that θw = θw0 implies uw (φ? , x Let θw < θw0 . There are four cases to consider: (i) w and w0 are both single, (ii) w is single and w0 is partnered, (iii) w is partnered and w0 is single, and (iv) both w and w0 are partnered. If (i), then both workers earn zero. If (ii), then w earns zero and w0 earns at least zero by individual rationality. Since Lemma B2 gives that (iii) is impossible, only (iv) remains. ¯ ? ) < uw0 (φ? , x ¯ ? ) by contradiction. Suppose Consider case (iv). We argue that uw (φ? , x ¯ ? ) ≥ uw0 (φ? , x ¯ ? ). Let f = φ? (w) and f 0 = φ? (w0 ). We will show that w0 not, then uw (φ? , x and f block, a contradiction. Consider agreement (zw? + , zf? ), with  > 0. Since b(y, θ) is weakly increasing in θ, we have that b(zw? + , θw0 ) > b(zw? , θw ) = b?w . Thus, uf (w0 , zw? + , zf? ) = r(b(zw? + , θw0 ), zf? , θf ) > 29

¯ ? ) for all  > 0. Since the reward function is strictly increasing in r(b?w , zf? , θf ) = uf (φ? , x ¯ ? ) ≥ uw0 (φ? , x ¯ ? ), where the inequality follows type, r(b?f , zw? , θw0 ) > r(b?f , zw? , θw ) = uw (φ? , x ¯ ?) from the contradiction hypothesis. Thus, uw0 (f, zw? + , zf? ) = r(b?f , zw? + , θw0 ) > uw0 (φ? , x for  sufficiently small by continuity. Since (zw? , zf? ) is interior, we have that (zw? + , zf? ) ∈ X ¯ ? ) with for  small enough. It follows that there is an  > 0 such that w0 and f block (φ? , x agreement (zw? + , zf? ). ¯ ? ) = uw0 (φ? , x ¯ ? ). We argue this via It remains to show that θw = θw0 implies uw (φ? , x ¯ ? ) > uw0 (φ? , x ¯ ? ). We will show that w0 contradiction. Suppose, without loss, that uw (φ? , x and f block, a contradiction. Consider agreement, (zw? + , zf? ), with  > 0. Since b(zw? + , θw0 ) > b?w , we have that ¯ ? ) for all  > 0. Since θw = θw0 , we have that uw0 (f, zw? , zf? ) = uf (w0 , zw? + , zf? ) > uf (φ? , x ¯ ? ) > uw0 (φ? , x ¯ ? ). Thus, uw0 (f, zw? + , zf? ) = r(b?f , zw? + r(b?f , zw? , θw0 ) = r(b?f , zw? , θw ) = uw (φ? , x ¯ ? ) for all  sufficiently small. Since (zw? , zf? ) is interior, we have that (zw? + , θw ) > uw0 (φ? , x ¯ ? ).  , zf? ) ∈ X for  small enough. Thus, there is an  > 0 such that w0 and f block (φ? , x

C

Online Appendix: Supplemental Results

This appendix collects several additional results. In the first subsection, we show (via an example) that the set of stable allocations in the Effort Game is not a lattice. In the second subsection, we strengthen the conclusion of Proposition 3 by giving conditions on the benefit and cost functions that are sufficient for strictly higher types to produce strictly higher benefits. In the third subsection, we develop results on the rank order effects of changes in type. Absence of Lattice Structure While the set of stable allocations is a lattice in classical one-to-one matching games (e.g., Roth and Sotomayor [35] and Hatfield and Milgrom [19]), it is not a lattice in the Effort Game; the next example illustrates. Before we give the example, however, some background is helpful. ¯ ) and (φ0 , x ¯ 0 ), Consider a binary relation 4W on A such that, for any two allocations (φ, x ¯ ) 4W (φ0 , x ¯ 0 ) if every worker does weakly better in (φ0 , x ¯ 0 ) than in (φ, x ¯ ). we have (φ, x Let S denote the set of stable allocations. Roth and Sotomayor (Theorem 2.16 [35]) and Hatfield and Milgrom (Theorem 3 [19]) establish that (S, 4W ) is a lattice in their games when preferences (over matches and agreements) are strict. However, (S, 4W ) is generally not a lattice in the Effort Game because players are indifferent and these indifferences prevent 4W from being antisymmetric.

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Example OA1. A Failure of Antisymmetry and Lattice Structure. Let W = {1}, F = {2, 3}, and let X = [0, 1]2 . Let θ1 be arbitrary but set θ2 = θ3 , so the firms have the same benefit and reward functions. In addition, let parts (1), (2), and (3.b) ˜ ∈ arg max(x1 ,x2 )∈X u1 (2, x1 , x2 ) such that u2 (1, x1 , x2 ) ≥ 0; it is of Assumption 1 hold. Let x easily seen that this solution exists and is interior. ¯ ? ) and (φ0 , x ¯ 0 ). In the former, worker 1 Consider two interior stable allocations (φ? , x ˜ and firm 3 is single, while in the latter, 1 and and firm 2 are matched with agreement x ˜ and 2 is single. Since 2 and 3 are clones, we have that 3 are matched with agreement x 0 0 ? ? ¯ ) = u1 (φ , x ¯ ). Thus, (φ? , x ¯ ? ) 4W (φ0 , x ¯ 0 ) and (φ0 , x ¯ 0 ) 4W (φ? , x ¯ ? ). Yet, (φ? , x ¯ ? ) 6= u1 (φ , x ¯ 0 ). Thus, 4W is not antisymmetric on S and (S, 4W ) is not a lattice. 4 (φ0 , x Higher Types and Strictly Higher Benefits Proposition 3 shows that higher types produce weakly higher benefits than lower types. The purpose of this subsection is to give a stronger result: higher types produce strictly higher benefits than lower types under the following assumption. Assumption OA1. Sufficient Conditions for Benefit to Strictly Increase with Type. The benefit function b(y, θ) is continuously differentiable, with by (y, θ) > 0, and the reward function r(b, y, θ) is continuously differentiable, with rb (b, y, θ) > 0 and ry (b, y, θ) < 0. In addition, ry (b, y, θ)/rb (b, y, θ) is strictly increasing in θ.27 That is, the benefit and reward functions are sufficiently smooth, and the reward function is strictly decreasing in effort (i.e., effort is purely costly). In addition, the marginal reward of effort divided by the marginal reward of benefit is strictly increasing in θ. While this last requirement is not implied by Assumption 2, it is closely related: if r(b, y, θ) has either (i) increasing differences in (b, θ) and strict increasing differences in (y, θ) or (ii) strict increasing  differences in (b, θ) and increasing differences in (y, θ), then ry (b, y, θ) rb (b, y, θ) is strictly increasing in θ. It is readily verified that Example 2 meets these three requirements. Proposition OA1. A Stronger Version of Proposition 3. ¯ ? ) be an interior stable allocation, and let Let Assumptions 2 and OA1 hold, let (φ? , x W F F GW 1 , . . . , GJW and G1 , . . . , GJF be the associated benefit groups. Then, workers with strictly higher types produce strictly higher benefits, i.e., if two workers w and w0 are partnered, then 0 W θw < θw0 implies that w ∈ GW l and w ∈ Gj with j < l. The analogous result holds for firms. The proof, which is given at the end of this sub-section, proceeds in two steps. First, we ¯ ? ) imply that a paired worker and firm must have show that the Pareto efficiency of (φ? , x an agreement that sets their marginal rates of substitution (MRS) to be equal. Second, we 27

We write by (b, θ) for ∂b(y, θ)/∂y, we write rb (b, y, θ) for ∂r(b, y, θ)/∂b, and we write ry (b, y, θ) for ∂r(b, y, θ)/∂y.

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establish that if a benefit group contains more than one type of player, then the equal MRS condition is violated for some worker and firm; implying that each benefit group is type homogeneous. Proposition OA1 then follows from Proposition 3 since w and w0 must be in different benefit groups. It follows, from Proposition 2, that higher types receive strictly higher benefits from their partners. The next corollary formalizes this observation. Corollary OA1. A Stronger Version of Corollary 2. ¯ ? ) be an interior stable allocation, and let Let Assumptions 2 and OA1 hold, let (φ? , x F W F GW 1 , . . . , GJW and G1 , . . . , GJF be the associated benefit groups. Then, workers with strictly higher types receive strictly higher benefits from their partners, i.e., if two workers w and w0 are partnered, then θw < θw0 implies that φ? (w) ∈ GFl and φ? (w0 ) ∈ GFj with j < l. The analogous result holds for firms. Since the benefit function is strictly increasing, it follows that players with strictly higher types and their partners exert strictly more effort. The next corollary formalizes this observation. Corollary OA2. A Stronger Version of Corollary 3. ¯ ? ) be an interior stable allocation, and let {zi? }i∈N Let Assumptions 2 and OA1 hold, let (φ? , x be the associated efforts. Then, workers with strictly higher types and their partners exert strictly more effort, i.e., if two workers w and w0 are partnered, then θw < θw0 implies that zw? < zw? 0 and that zφ?? (w) < zφ?? (w0 ) . The analogous result holds for firms. We now turn to the proof of Proposition OA1. Lemma OA1. Equal Marginal Rates of Substitution. ¯ ? ) be an interior stable allocation, and let {zi? }i∈N Let Assumptions 2 and OA1 hold, let (φ? , x be the associated efforts. If worker w is matched to firm f , then ? ,z ? ) ∂uw (f,zw f ∂x1 ? ,z ? ) ∂uw (f,zw f ∂x2

ry (b(zf? ), zw? , θw ) rb (b(zw? ), zf? , θf )by (zw? , θw ) = = = rb (b(zf? ), zw? , θw )by (zf? , θf ) ry (b(zw? ), zf? , θf ) ?

?

? ,z ? ) ∂uf (f,zw f ∂x1 ? ,z ? ) . ∂uf (f,zw f ∂x2

¯ ? ) = (φ? , x1 , . . . , xN ) is Pareto efficient – the formal argument is akin Proof. Since (φ? , x to the proof of (ii) in Proposition 1 – we have that (zw? , zf? ) ∈ arg max(x1 ,x2 )∈X uw (f, x1 , x2 ) ¯ ? ). Simply, if (zw? , zf? ) were not a solution to this problem, such that uf (w, x1 , x2 ) ≥ uf (φ? , x ¯ 0 ), then there is a (x01 , x02 ) ∈ X that serves as a Pareto improvement for w and f and so (φ0 , x ? ? ? ? ? ? ¯ 0 = (x1 , . . . , xw−1 , x01 , x02 , xw+1 , . . . , xf −1 , x01 , x02 , xf +1 , . . . , xN ), is a with φ0 = φ? and x Pareto improvement for all players, an impossibility.

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Applying Theorem 1.18 of de la Fuente (p. 293 [11]) to the optimization problem gives that there is a λ > 0 such that ∂uw (f, zw? , zf? )/∂x1 + λ ∂uf (w, zw? , zf? )/∂x1 = 0 and ¯ ? ) is the ∂uw (f, zw? , zf? )/∂x2 + λ ∂uf (w, zw? , zf? )/∂x2 = 0 because (i) uf (w, x1 , x2 ) = uf (φ? , x only binding constraint at (zw? , zf? ) (per interiority and payoff monotonicity) and (ii) payoffs are continuously differentiable with strictly positive derivatives. The lemma follows after some algebra.  Lemma OA2. Type Homogeneous Benefit Groups. ¯ ? ) be an interior stable allocation, and let Let Assumptions 2 and OA1 hold, let (φ? , x F W F GW 1 , . . . , GJW and G1 , . . . , GJF be the associated benefit groups. Then, each benefit group of workers is type homogeneous, i.e., for all l ∈ {1, . . . , JW }, if workers w and w0 are in GW l , then θw = θw0 . The analogous result holds for firms. Proof. We argue by contradiction. Let w and w0 be in GW l and suppose not, i.e., suppose that θw < θw0 ; the argument is analogous if θw > θw0 . Let f = φ? (w) and f 0 = φ? (w0 ). Also, let zw? , zf? , zw? 0 , and zf?0 denote these players’ efforts. There are two cases to consider (i) θf ≤ θf 0 and (ii) θf > θf 0 . Let h(y1 , y2 , θ) = ry (b(y1 ), y2 , θ)/rb (b(y1 ), y2 , θ). Consider case (i). Lemma OA1 gives that h(zf? , zw? , θw )/by (zf? , θf ) = by (zw? , θw )/h(zw? , zf? , θf ) and that h(zf?0 , zw? 0 , θw0 )/by (zf?0 , θf 0 ) = by (zw? 0 , θw0 )/h(zw? 0 , zf?0 , θf 0 ). Hence, h(zf? , zw? , θw )h(zw? , zf? , θf ) = by (zw? , θw )by (zf? , θf ) and h(zf?0 , zw? 0 , θw0 )h(zw? 0 , zf?0 , θf 0 ) = by (zw? 0 , θw0 )by (zf?0 , θf 0 ). Since the benefit function is type independent and since zw? = zw? 0 and zf? = zf?0 (as f and f 0 are in GFl ), we have that h(zf? , zw? , θw )h(zw? , zf? , θf ) = h(zf? , zw? , θw0 )h(zw? , zf? , θf 0 ). Yet, Assumption OA1 gives that h(zf? , zw? , θw ) < h(zf? , zw? , θw0 ) and that h(zf? , zw? , θf ) ≤ h(zf? , zw? , θf 0 ) since θw < θw0 and θf ≤ θf 0 , which implies that h(zf? , zw? , θw )h(zw? , zf? , θf ) 6= h(zf? , zw? , θw0 )h(zw? , zf? , θf 0 ), a contradiction. ¯ 0 ) by (i) pairing w Consider the case where θf > θf 0 . Construct a new allocation (φ0 , x with f 0 at agreement (zw? , zf?0 ), (ii) pairing w0 with f at agreement (zw? 0 , zf? ), and (iii) leaving the matches and agreements of all other players alone – the formal procedure is akin to the one given in the Proof of Proposition 4. Since this rematching does not change payoffs, ¯ 0 ) is stable by Lemma B1. Thus, Lemma OA1 implies that h(zf? , zw? , θw )h(zw? , zf? , θf 0 ) = (φ0 , x h(zf? , zw? , θw0 )h(zw? , zf? , θf ). Yet, h(zf? , zw? , θw ) < h(zf? , zw? , θw0 ) and h(zf? , zw? , θf 0 ) < h(zf? , zw? , θf ) as θw < θw0 and θf 0 < θf , implying that h(zf? , zw? , θw )h(zw? , zf? , θf 0 ) 6= h(zf? , zw? , θw0 )h(zw? , zf? , θf ), a contradiction.  Proof of Proposition OA1. Follows immediately from Proposition 3 and Lemma OA2.  We close this subsection by noting that Assumption OA1 cannot be easily relaxed. Indeed, a detailed example where it does not hold and where workers of different types are in 33

the same benefit group is available upon request. The Rank Order Comparative Statics of Type Proposition 3 suggests that an increase in a player’s type increases the rank order of the benefits she produces and receives. The purpose of this subsection is to show that, under the following assumption, this conjecture is correct and that the player’s rank order payoff increases with her type. Assumption OA2. Sufficient Conditions for Rank Order Comparative Statics. We have that W ≤ N/2 and, for each (θ, θ0 ) ∈ Θ2 , that r(b(x2 , θ0 ), x1 , θ) > 0 and r(b(x1 , θ), x2 , θ0 ) > 0 for some (x1 , x2 ) ∈ [0, β]2 . That is, there are more firms than workers and that every possible pair of players obtains strictly positive payoffs at some interior agreement. The assumption ensures that every worker is partnered and holds an interior agreement in any interior stable allocation.28 It is readily verified that Example 2 meets this assumption. Before proceeding, we need some notation. Let θ = (θ1 , . . . , θi−1 , θi , θi+1 , . . . , θN ) denote the vector of players’ endowed types and let θ −i = (θ1 , . . . , θi−1 , θi+1 , . . . , θN ) denote all players’ types aside from player i. We write (θi , θ −i ) for the type vector (θ1 , . . . , θi−1 , θi , θi+1 , . . . , θN ). Let S(θ) denote the set of interior stable allocations when the type vector is θ. Also, for a finite set S, let |S| denote the cardinality of S. Proposition OA2. Rank Order Comparative Statics of Benefit Production. Let Assumptions 2, OA1, and OA2 hold, and suppose the type of worker w increases from θw to θw0 , while the types of all other players are constant. Then, the rank order benefit w ¯ ? ) ∈ S(θw , θ −w ) produces increases in every interior stable allocation. That is, for each (φ? , x ¯ 0 ) ∈ S(θw0 , θ −w ), with associated benefits {b?i }i∈{j|φ? (j)6=j} and {b0i }i∈{j|φ0 (j)6=j} and each (φ0 , x respectively, we have that |{w0 |b?w0 ≤ b?w }| ≤ |{w0 |b0w0 ≤ b0w }|. If the workers’ benefit groups are singletons,29 it follows that w moves into a weakly lower indexed benefit group as her type increases. (Simply, if the index of w’s benefit group increases, then there are fewer players who produce lower benefits, an impossibility.) This need not happen, however, if the benefit groups are non-singletons as then lower indexed benefit groups can “splinter” as the stable allocation changes and the index of w’s benefit group can increase. Nevertheless, the cumulative number of workers who produce a benefit no greater than (greater than) w increases (decreases) and it is this sense in which we develop our rank order comparative statics. 28

This is easily seen via contradiction. Suppose worker w is single in an interior stable allocation, then there is some single firm f by the first part of Assumption OA2. Since w and f both currently earn zero, they do strictly better by matching per the second part of Assumption OA2 and thus block, a contradiction. 29 This is the case, for instance, if each worker has a unique type – see Lemma OA2.

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Proof. (We take S(θw , θ −w ) and S(θw0 , θ −w ) to be non-empty to avoid trivialities.) Since w’s type increases, there are more workers with lower types; in symbols, |{w0 |θw0 ≤ θw }| ≤ |{w0 |θw0 ≤ θw0 }|. Since Assumption OA2 ensures that all workers are partnered in every interior stable allocation, we have that {w0 |θw0 ≤ θw } ⊂ {w0 |b?w0 ≤ b?w } and that {w0 |θw0 ≤ θw0 } ⊂ {w0 |b0w0 ≤ b0w } by Proposition 3. Thus, we only need to show that {w0 |b?w0 ≤ b?w } ⊂ {w0 |θw0 ≤ θw } and {w0 |b0w0 ≤ b0w } ⊂ {w0 |θw0 ≤ θw0 }. Consider {w0 |b?w0 ≤ b?w } ⊂ {w0 |θw0 ≤ θw }. Suppose not, then there is a w˜ ∈ {w0 |b?w0 ≤ b?w } with θw˜ > θw . But, then Proposition OA1 gives that b?w˜ > b?w , a contradiction. Thus, {w0 |b?w0 ≤ b?w } ⊂ {w0 |θw0 ≤ θw }. An analogous argument gives that {w0 |b0w0 ≤ b0w } ⊂ {w0 |θw0 ≤ θw0 }.  It follows, via Proposition 2, that the rank order benefit worker w receives also increases as her type improves. The next corollary formalizes this observation. Corollary OA3. Rank Order Comparative Statics of Benefit Receipt. Let the antecedents of Proposition OA2 hold, then the rank order benefit worker w receives ¯ ? ) ∈ S((θw , θ −w )) and each (φ0 , x ¯ 0) ∈ increases with her type. That is, for each (φ? , x S((θw0 , θ −w )), with associated benefits {b?i }i∈{j|φ? (j)6=j} and {b0i }i∈{j|φ0 (j)6=j} respectively, we have that |{w0 |b?φ? (w0 ) ≤ b?φ? (w) }| ≤ |{w0 |b0φ0 (w0 ) ≤ b0φ0 (w) }|. If the workers’ benefit groups are singletons, the corollary implies that, as w’s type increases, she matches to firms in lower indexed benefit groups. We close with a related comparative statics result, which is a direct implication of Proposition 5. Corollary OA4. Rank Order Comparative Statics of Payoffs. Let Assumptions 3 and OA2 hold and suppose the type of worker w increases from θw to θw0 , while the types of all other players are constant. Then, w’s rank order payoff increases. ¯ ? ) ∈ S((θw , θ −w )) and each (φ0 , x ¯ 0 ) ∈ S((θw0 , θ −w )), we have that That is, for each (φ? , x ¯ ? ) ≤ uw (φ? , x ¯ ? )}| ≤ |{w0 |uw0 (φ0 , x ¯ 0 ) ≤ uw (φ0 , x ¯ 0 )}|. |{w0 |uw0 (φ? , x Proof. Since all workers are partnered by Assumption OA2, Proposition 5 gives that ¯ ? ) ≤ uw (φ? , x ¯ ? )} and {w0 |θw0 ≤ θw0 } = {w0 |uw0 (φ0 , x ¯ 0) ≤ {w0 |θw0 ≤ θw } = {w0 |uw0 (φ? , x ¯ 0 )}. Thus, the result then from the fact |{w0 |θw0 ≤ θw }| ≤ |{w0 |θw0 ≤ θw0 }|.  uw (φ0 , x

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