Managing congestion in dynamic matching markets

Transcription

1 Managing congestion in dynamic matching markets Nick Arnosti (Stanford), Ramesh Johari (Stanford), Yash Kanoria (Columia) April 20, 2014 Astract We consider a decentralized two-sided matching market in which agents arrive and depart asynchronously. As a result, it is possile that an agent on one side of the market (a uyer ) identifies an agent on the other side of the market (a seller ) who is a suitale match, only to find that the seller is already matched. We find using a mean field approach that lack of knowledge aout availaility can create large welfare losses to oth uyers and sellers. We consider a simple intervention availale to the platform: limiting visiility of sellers. We find that this intervention can significantly improve the welfare of agents on oth sides of the market; sellers pay lower application costs, while uyers are less likely to find that the sellers they screen have already matched. Somewhat counterintuitively, the enefits of showing fewer sellers to each uyer are greatest in markets in which there is a shortage of sellers. Keywords: Matching, market design, mean field 1 Introduction Many matching markets operate asynchronously; in other words, in these markets agents arrive and depart over time, and matches are formed dynamically without a centralized clearing mechanism. Because of this, it is possile that an agent on one side of the market (a uyer ) identifies an agent on the other side of the market (a seller ) who is a suitale match, only to find the seller is unavailale ecause she is already matched. In this paper, we focus on the welfare effects of asynchronous matching, and in particular the resulting incomplete information regarding availaility. A lack of availaility information is a common prolem in a range of online matching markets, e.g., dating markets and laor markets. As one pertinent example, consider odesk, a large online laor market. On odesk clients can post jos and wait for workers to apply, or directly invite workers instead. In practice, the est workers on odesk may prove to e unavailale to the client, a fact that typically has a strong negative effect on a client s satisfaction on the site. 1 A key reason is that uyers invest effort determining whether 1 In particular, their site states that when clients send out invitations and freelancers don t reply, it s a frustrating experience that makes those clients less likely to hire anyone. See What-is-responsiveness-. 1

2 they want to invite someone to work on their jo, and this effort is wasted if the invited worker turns out to e unavailale. In such a situation we are naturally led to ask: what could the platform do to improve the experience of the uyer? One ovious solution is to provide information aout the likelihood that each individual will respond when contacted, and many online marketplaces such as odesk, AirBnB, and OkCupid strive to do just that. However, these inferences are (at est) imperfect. This is ecause online matching platforms y their very nature are served y a community of freelancers, whose availaility at a given time may depend on a variety of factors that are unoservale to the platform and to other market participants. As a consequence, platforms are interested in other ways to mitigate the negative consequences of a lack of availaility information. We consider an intervention that is simple yet powerful: limiting visiility. In particular, we consider whether the market can improve welfare y restricting the numer of uyers to whom each seller ecomes visile. 2 Coffee Meets Bagel 3, eharmony 4 and odesk 5. This approach is popular on a variety of existing wesites/apps like In this paper, we develop a mathematical model of a dynamic matching market and use it to quantitatively study the possile effects of this type of intervention. Informally, our findings show that such limiting visiility can provide enefits to oth uyers and sellers. When the market is demand-limited i.e., when uyers are relatively scarce we show intervention helps mitigate the tragedy of the commons among sellers. Perhaps more surprising is that limiting seller visiility can significantly improve uyer welfare when the market is supply-limited. The reason is sutle: y limiting seller visiility, we ensure that more sellers are availale, hence reducing wasted screening effort y uyers on unavailale sellers. The remainder of the paper is organized as follows. We egin in Section 3 y presenting our asic model. In the game we consider, uyers and sellers arrive over time, and live for (at most) a unit lifetime. Sellers apply to a suset of uyers present in the system upon arrival, and incur a fixed cost per application sent. Upon departure, uyers screen the applications they receive for compatiility, i.e., fitness for a match; they pay a fixed cost per application screened. If a compatile match is found the uyer can make her an offer, and sellers accept the first offer they receive (if any). If a uyer makes an offer to an unavailale seller, the offer fails and the uyer continues to screen for a match. Because there is a delay etween when an application is sent (the application time ) and when the seller is evaluated y the uyer (the screening time ), the state of the seller can change y the time she is screened. In particular, although the seller may have een unmatched at the application time, she may already e matched y the screening time. Unless the uyer knows this, effort will e wasted screening a candidate that is not availale. 2 Behavioral findings, e.g. Iyengar and Lepper [16], suggest that more choice may make agents less likely to act (e.g., match), providing another possile enefit to such restriction. We do not model this effect. 3 Users are shown exactly one profile a day on this dating app. If oth people like each other this results in contact information eing exchanged. 4 Users are shown 5 matches per day on this dating wesite. 5 Users are allowed to send a limited numer of applications. 2

3 Note that our model differs slightly from the odesk scenario descried aove, in that sellers must first apply to ecome visile to uyers. In odesk, a uyer (client) can invite any sellers (freelancers) that she sees in search. 6 Thus, from the perspective of uyers, we should view the sumarket for the est workers on odesk as a limit of our model where application costs approach zero; this is the asymptotic regime in which we conduct the majority of our welfare analysis. Our model defines a fairly complex dynamic game among the agents, which requires higher order eliefs that uyers and sellers must hold aout each other. In practice, of course, in a large market it seems reasonale to expect simplification in the market structure due to averaging, so that relatively simple reasoning y the agents allows them to play an approximate est response. Inspired y such an argument along with a plausile expectation of ounded rationality, we employ a mean field approach to study our game. In particular, in Section 4 we consider a formal stationary mean field model inspired y a regime where the numer of uyers and sellers in the system grows large, and make two key assumptions that simplify the analysis: we assume that from the point of view of a uyer, each applicant is independently availale with the same fixed proaility q; and from the point of view of a seller, each application is independently successful with the same fixed proaility p. Under these two mean field assumptions, solving for the optimal strategies of uyers and sellers ecomes straightforward. On the other hand, oth p and q must satisfy consistency checks that ensure they arise from the optimal decisions of uyers and sellers. Taken together, this pair of conditions optimality and consistency define a notion of equilirium for our mean field model that we call mean field equilirium (MFE); several recent papers have used a similar modeling approach (see, e.g., [17, 13, 3, 4]). We justify our mean field analysis y showing that as the market grows large, the mean field assumptions hold asymptotically (see Section 5). Most importantly, the resulting equiliria are tractale to analyze, and yield rich insight into the welfare of oth uyers and sellers, as well as the potential improvements that can e generated if the platform intervenes. We carry out our welfare analysis for market with and without intervention in Section 6. We study oth markets in a regime where application costs approach zero, and completely characterize how uyer and seller welfare depends on the screening cost and the ratio of uyers to sellers in the market. We conclude with additional discussion in Section 7. 2 Related Work The model presented in this paper ears resemlance to those that appear in the search theory literature; see Diamond [7], Mortensen [24], Pissarides [25, 26], and the early survey of Hosios [15]. This literature concludes that when jo seekers cannot coordinate their search efforts, inefficient levels of jo creation, jo search, and unemployment result. 6 In our model, an invitation from a uyer to a seller is equivalent to an offer, in the sense that the uyer would want to hire the first availale, compatile seller that they find. 3

4 A range of extensions have een added to these models, including the role of wages [29, 23], incomplete information [11], on-the-jo search [28], and adverse selection [12]. Rogerson et al. [27] conduct a survey of the literature on search theory. Much of this literature assumes that workers apply to only one position (and therefore will always accept the jo if offered it), though several papers (see [2, 9, 20]) permit simultaneous search. Our work is rooted in micro-foundations, unlike much of the search literature, which relies on a lack ox matching function. This allows us to not only study whether the market is efficient, ut also to more easily analyze the effect of possile interventions y the market operator. Furthermore, the search literature often relies on calculations that assume that individual agents do not affect aggregate outcomes in large markets, and that strategic considerations can therefore e ignored. Though we make these same assumptions, one of our important technical contriutions is to show that they hold in the limit as the numer of market participants grows (see Section 5). Though similar in spirit to the recent work of Galenianos and Kircher [10], this proof confronts several novel challenges. First, we discuss a dynamic, rather than static, system. Second, in our model each worker applies to multiple firms, which Galenianos and Kircher acknowledge presents many difficulties even in the static setting. There is a separate ody of work which focuses on the importance of availaility in matching markets, and the potentially eneficial effects of reducing the numer of options presented to each agent. For example, Halaurda and Piskorski [14] conclude that agents may e etter off in platform with less choice. However, in their model, welfare loss is primarily driven y the fact that agents who are shown N candidates match with proaility at most 1/N. By contrast, in our model showing a larger numer of candidates (weakly) increases the numer of matches formed, ut also increases the search and screening costs required to find these matches. Coles et al. [5] study the jo market for students graduating with PhDs in economics, and study the effect of a mechanism that allows students to credily signal their preferences. Lee et al. [22], Coles et al. [6], Lee and Schwarz [21] provide additional theoretical and empirical support for similar schemes. One important distinction etween this line of work and our own is that in existing work on availaility, firms are uncertain aout the preferences of workers, and risk having their offers turned down y workers who prefer other firms. In [6], one driver of improved worker welfare is that when signals are used, workers receive etter offers. In our setting, uyers and sellers are ex-ante homogeneous, and this effect is asent. The uncertainty facing uyers is one of timing; they know that their offer will e accepted, so long as the seller is not already employed. Fradkin [8] notes that on AirBnB, uyers might lack information aout seller availaility ecause transactions take time to complete, and sellers may not relialy update the calendars.in his model, inquiries y uyers may e rejected either ecause the seller deems the uyer unsuitale or ecause the seller is no longer availale to transact. It is not clear that a single intervention can simultaneously address these separate concerns. Our work shows that temporal miscoordination can e very costly in dynamic markets. Furthermore, we show that interventions similar to those proposed to address preference-related frictions can also reduce 4

5 timing-related frictions and enhance the welfare of agents on oth sides of the market. 3 The Model In the market we consider, uyers and sellers arrive over time, interact with each other, and eventually depart. Informally, we are trying to model the following ehavior. Buyers arrive to the market, and immediately post an opening. When sellers arrive, they sumit applications to a suset of the uyers currently in the market. Upon exit from the system, a uyer reviews her applicants. Initially, she does not know whether these applicants are still availale or are compatile with the opening. At some cost, she can screen candidates, learning compatiility in the process. Additionally, she can make offers to applicants, who respond immediately y accepting or rejecting. The uyer takes these actions in sequence, and eventually leaves the marketplace. Our dynamic markets are parameterized y n > 0, which descries the market size. We endow uyers and sellers with unique IDs which convey no other information aout the agent. We use B(t) to denote the set of uyers present in the system at time t, and S(t) to denote the set of sellers present in the system at time t. We let B(t) and S(t) denote the magnitudes of these sets, respectively. We assume that B(0) = S(0) =. 3.1 Utility Before descriing market dynamics, we descrie the payoff structure in our matching market. Upon exiting the market, uyers pay a cost of u > 0 for each seller to whom they are matched. Buyers only earn surplus if they match to sellers with whom they are compatile. In particular, they earn a reward of v > u if and only if they are compatile with at least one seller to whom they have matched. Upon exiting the market, sellers earn a payoff of w > 0 if they are matched to a uyer, and 0 otherwise. Sellers cannot match to multiple uyers. Note that our model is asymmetric, in the sense that uyers are sensitive to whether or not the match is compatile, while sellers are not. We assume that that each uyer-seller pair is compatile with proaility β (independently across all such pairs), and that this is common knowledge. Sellers pay a cost c a 0 for each application that they send, and uyers incur a cost c 0 for each applicant that they screen. The net utility to an agent will e the difference etween value otained from any match(es), and costs incurred. We assume that our agents are risk neutral; that is, they seek to maximize their expected utility. For later reference, it will e useful to consider normalized versions of the screening and application costs. We make the following definitions; note that c and c a are oth dimensionless. c = c β(v u) ; (1) c a = c a βw. (2) 5

6 We make the following assumptions on the model parameters, which remain in force for the remainder of the paper. Assumption 1. βv < u. This assumption ensures that the proaility of compatiility is sufficiently low (or the cost u sufficiently high) that uyers get negative expected utility from hiring an unscreened worker. Assumption 2. c < 1. This assumption ensures that the screening cost is sufficiently low that uyers get positive expected utility from screening a worker who is known to e availale (and hiring them if they are compatile). If Assumption 1 holds and Assumption 2 does not, then regardless of seller ehavior, it will e optimal for uyers to exit the market immediately, and thus no trade will occur in equilirium. 3.2 Arrivals In our market, individual uyers arrive at intervals of 1/n; individual sellers arrive at intervals of 1/(rn). Here r > 0 is a parameter that controls the relative magnitude of the two sides of the market. Buyers remain in the system for a unit lifetime. Sellers depart the system according to a process that we descrie elow. Buyers post a single opening upon arrival to the system; we assume this opening is tagged with their ID. Therefore, B(t) also represents the set of active openings in the market at time t; for this reason we use uyer and opening interchangealy elow. Buyers do not oserve any state or make any decisions upon their arrival. Suppose that seller s arrives at time t. Upon arrival, s selects a value m s efore oserving any market state. At this point, s applies to each uyer in B(t) with proaility m s /n 7. Note that in our model the expected numer of applications sent y any seller arriving after time t 1 is m s. An alternate model of seller applications might e that sellers directly choose the numer of applications they send. We choose a proailistic specification primarily for technical convenience, to ease our later mean field analysis: in a setting where sellers deterministically send a fixed numer of applications, an additional dependency is introduced across the uyers. 3.3 Seller departure Sellers remain in the system for a maximum of one time unit. If they receive and accept an offer from any uyer (as descried elow), they depart from the market at that time. Recall that (excluding application costs) a seller earns a payoff of w as long as she is matched (regardless of compatiility), and zero otherwise. Thus, it is a dominant strategy for sellers to accept the first offer they receive. We assume henceforth that sellers follow this strategy. 7 6

7 3.4 Buyer departure Each uyer stays in the system for one time unit, and then departs. Upon departure, uyers see the set of applicants to their opening. Initially, they do not know which of these applicants are compatile for their jo, nor do they know which would accept the jo if offered it. The uyer takes a sequence of screening and offer actions, instantaneously learning the result of each, until they choose to exit the market. At each stage of this sequential decision process, the uyer may screen an unscreened applicant, make an offer to an applicant to whom they have not made an offer, or exit. Screening an applicant reveals whether that applicant is compatile with the jo, and incurs a cost c. Making an offer is costless, though recall that the uyer incurs a cost u if the offer is accepted. A more formal description of the dynamic optimization prolem solved y uyers appears in the Appendix. 4 The large market: A stationary mean field model In principle, the strategic choices facing an agent in the model descried aove may e quite complicated. Consider the case of a uyer who knows that he has only one competitor. If he finds that his first applicant has already accepted another offer, he learns that every other applicant is still looking for a jo. Even when the exact numer of other uyers is unknown, similar logic indicates that in thin markets, information revealed during screening may induce significant shifts in the uyer s eliefs. This and other dependencies could conceivaly cause optimal agent to e quite complex, oth to descrie and understand. As the market thickens, however, one might expect that the correlations etween agents on the same side of the market ecome weak. In particular, uyers screening a pool of applicants might reasonaly assume that learning that one applicant has already accepted another offer does not inform them aout the availaility of other applicants. Further, if the uyer does not know anything aout individual sellers, each one should appear to e availale with equal proaility. Similarly, sellers who know nothing aout individual uyers may e justified in assuming that each of their applications convert to offers independently and with equal proaility p. In this section we develop a formal stationary mean field model for our dynamic matching market, and introduce a notion of game-theoretic equilirium for this formal model. In our formal model, agents make the following assumptions. Mean Field Assumption 1 (Buyer Mean Field Assumption). Each seller in a uyer s applicant set is availale with proaility q, and the availaility of sellers in the applicant set is independent. Mean Field Assumption 2 (Seller Mean Field Assumption). Each application yields an offer with proaility p, independently across applications to different openings Mean Field Assumption 3 (Large Market Assumption). The numer of applications sent y a seller who chooses m s = m is Poisson distriuted with mean m. If all sellers select m s = m, the numer of applications received y each uyer is Poisson distriuted with mean rm. 7

8 Under these assumptions, characterizing optimal agent ehavior simplifies tremendously. In Section 4.1, we first characterize the optimal uyer and seller decisions in the presence of the mean field assumptions. For sellers, we show that there exists a unique optimal choice of m, given p; and for uyers, we show that the optimal decision rule given q involves mixing etween a simple sequential screening strategy, and exiting immediately. Next, in Section 4.2, we derive consistency checks that p and q should satisfy, if they indeed arise from the conjectured optimal uyer and seller decision rules. Finally, we comine these two steps: our first step derives optimal decision rules from p and q, and our second step derives p and q from the decision rules. In Section 4.3, we define (stationary) mean field equilirium as a fixed point of the resulting composite map, and show that it exists and is unique. Informally, a mean field equilirium is a pair of strategies that are est responses to the stationary market dynamics that they induce in the large n limit. Mean field equilirium defined in this way is the oject of study in the remainder of our paper. Theorems 3 and 4 in Section 5 justify our study of mean field equiliria: there we show (in an appropriate sense) that the mean field assumptions hold as n approaches infinity. 4.1 Optimal decision rules We egin y studying the question of how agents should respond when confronted with a world where the mean-field assumptions hold. In the next two sections we study the optimal decision rules of the seller and uyer, respectively Sellers As discussed in Section 3, in our model it is a dominant strategy for sellers to accept the first offer (if any) that they receive, and we assume sellers follow this rule. Therefore the only decision a seller s needs to make on arrival is her choice of m s, the parameter that determines the distriution of the numer of applications sent. If a seller s chooses m s = m, they incur an expected cost of c a m. If the seller applies to Poisson(m) uyers, and each application independently yields an offer with proaility p, then at least one offer is received i.e., the seller matches to a uyer with proaility 1 e mp. Thus, the expected payout of a seller in the mean-field environment who selects m s = m is Π s (m, p) = w(1 e mp ) c a m. (3) Thus, a seller solves the following optimization prolem: maximize m Π s (m, p), suject to m 0. (4) The ojective function is strictly concave and decays to as m, so this prolem possesses a unique 8

9 optimal solution identified y first-order conditions. We conclude that if wp < c a, the optimal choice is m = 0. Otherwise, sellers select We define M to e the function that maps p to this unique value of m: Buyers M(p) = m = 1 ( ) wp p log. (5) c a 0, if wp < c a ; ( ) 1 p log wp c a, if wp c a. Next, we consider the optimal decision rule for the uyers, when they make Mean Field Assumption 1. We consider the following simple strategy, which we denote φ 1. A uyer playing φ 1 sequentially screens candidates in her applicant list. When she finds a compatile applicant, she makes an offer to this candidate; otherwise, she considers the next candidate. This process repeats until one applicant accepts or no more applicants remain. As we show in the following proposition, when Assumption 1 holds, either φ 1 or exiting immediately is optimal for uyers. Proposition 1. Let φ 1 e the strategy of sequentially screening applicants, offering them the jo if and only if they are qualified, until either an applicant is hired or no more applicants remain. Suppose that Assumption 1 and Mean Field Assumption 1 hold. Then φ 1 is uniquely optimal if and only if q > c, exiting immediately is uniquely optimal if and only if q < c, and any mixture of these strategies is optimal if q = c. Motivated y this proposition, we define φ α to e the strategy that plays φ 1 with proaility α and exits immediately with proaility 1 α. Further, define the correspondence A(q) y: {0} if q < c A(q) = [0, 1] if q = c {1} if q > c. This correspondence captures the optimal uyer response, as descried in Proposition 1, so that (6) (7) A(q) = {α [0, 1] : φ α is optimal for the uyer}. Consider uyer playing φ α who receives a numer of applicants that is Poisson distriuted with parameter rm. Each of these applicants is (independently) qualified with proaility β and (if Mean Field Assumption 1 holds) availale with proaility q, so the numer of qualified availale applicants to the uyer is Poisson distriuted with mean rmβq. The uyer hires if and only if they choose to screen and they receive a qualified availale applicant, which occurs with proaility α(1 e rmβq ). Thus, uyer s expected match surplus (excluding screening costs) is α(v u)(1 e rmβq ). For each match found, an 9

10 average of 1/(qβ) applicants are screened, so we find that the total expected surplus to a uyer following φ α is Π (α, q, m) = α(v u)(1 e rmβq )(1 c /q). (8) 4.2 Consistency In the previous section, we discussed the est responses availale to uyers and sellers when the mean field assumptions hold; that is, given p and q, we found the decision rules that agents would adopt. However, it is clear that p and q are simultaneously determined y the strategies agents adopt. In other words, there must e consistency conditions that p and q satisfy given specific strategies adopted y the agents. In this section we identify these consistency conditions. Inspired y the results of the preceding section, in this section we focus on the following strategies. We assume that all sellers choose the same m 0. We assume that all uyers play φ α, i.e., they play φ 1 with proaility α and exit immediately otherwise. Throughout the section, we suppose that Mean Field Assumptions 1, 2 and 3 hold. We emphasize at the outset that our analysis here is only to ensure that we formally derive the correct consistency conditions under these assumptions; rigorously justifying our mean field equations requires the approximation theorems proven in Section 5, which show that the mean field assumptions hold asymptotically as n. We start y deriving a consistency condition that q must satisfy, given p and the seller s strategy. Suppose that s applies to k uyers. Select one such uyer, uniformly at random. In our formal mean field model, the proaility that s is availale when screens is P (s availale to s sent k applications) = 1 k 1 (1 p) j = (1 (1 p)k ). (9) k pk From the perspective of a uyer to whom s has applied, the proaility that s sent a total of k applications is k/m e m m k /k!. Thus, averaging (9) over the numer of applications sent y s, the proaility that s is availale when screens is j=0 q = k=1 (1 (1 p) k ) pk k m e m m k k! = 1 e mp mp = g(mp), (10) where we define g(0) = 1, g(x) = 1 e x x for convenience. For future reference, we note the following aout the function g: for x > 0, (11) Lemma 1. The function g : [0, ) (0, 1] defined y (11) is strictly decreasing. Furthermore, g (x) g(x)/x for x > 0. 10

11 Next, we derive a consistency condition that p must satisfy, given q and the uyer s strategy φ α. We consider a uyer s screening process from the vantage point of a particular seller s. Suppose that s is one of l + 1 availale applicants and that uyer follows φ α. Then screens s if and only if they decide to screen at all and all availale sellers screened efore s are unqualified for the position. If screens sellers in a random order, then the proaility that screens s is: α l + 1 l j=0 (1 β) j = α(1 (1 β)l+1 ) (l + 1)β. (12) If each seller that arrives while the uyer is in the market applies to independently with proaility m/n, and is availale when screens with proaility q, then from the perspective of seller s, the numer of competing applicants l is Poisson with parameter rmq in large n limit, y Mean Field Assumption 3. Since hires s if and only if screens s and they are compatile, the proaility that hires s is given y e rmq (rmq) l p = αβ l! l=0 1 (1 β)l+1 (l + 1)β = α(1 e rmβq ) rmq = αβg(rmβq). (13) The equations (10) and (13) are a system for p and q, given the values of m and α (as well as the parameters r and β). Straightforward algeraic manipulation reveals that any solution to this system also satisfies α(1 e rβmq ) = r(1 e mp ). (14) Because the numer of qualified availale applicants to each uyer is assumed to e Poisson with parameter rβmq and the numer of offers received y a seller is assumed to e Poisson with parameter mp, (14) can e interpreted as saying that the proportion of uyers who match must equal r times the proportion of sellers who match. In the next proposition we prove that the system defined y consistency equations (10) and (13) always has a unique solution. Theorem 1. For fixed m, α, r, and β, there exists a unique solution (p, q) to (10) and (13). We defer a proof of Theorem 1 to the Appendix. We refer to the unique pair (p, q) that solve (10) and (13) as a mean field steady state (MFSS). This pair provides a prediction of how a large market should ehave, given specific strategic choices of the agents. For later reference, given m and α, let P(m, α) and Q(m, α) denote the unique values of p and q guaranteed y Theorem 1, respectively. 4.3 Mean field equilirium In this section we define mean field equilirium (MFE), a notion of game theoretic equilirium for our stationary mean field model. 11

12 Informally, we would like a MFE to ensure that (1) agents are playing optimally given their eliefs aout the marketplace, i.e., the values of p and q in the mean field assumptions; and (2) their eliefs aout the marketplace are consistent with what the agents are playing, i.e., that p and q constitute a MFSS given the agents decision rules. Section 4.1 addressed the first point; and section 4.2 addressed the second. To define a mean field equilirium, we compose the maps defined in those sections. Formally, we have the following definition Definition 1. A mean field equilirium (MFE) is a pair (m, α ) such that m = M(P(m, α )) and α A(Q(m, α )). In other words, in an MFE, m and α are optimal responses (under the mean field assumptions) to the mean field steady-state (p, q) that they induce. We also define p = P(m, α ) and q = Q(m, α ). Our main theorem in this section is the following: Theorem 2. Fix r, β, c a, c, w, v, u, and suppose Assumptions 1 and 2 hold. Then there exists a unique mean field equilirium (m, α ). Furthermore, m > 0 if and only if c a < 1. For future reference, we define Π s, Π to e the expected seller and uyer surplus in mean-field equilirium, respectively. In other words, Π s = Π s (m, p ), Π = Π (α, q, m ), (15) where Π s and Π are defined y (3) and (8). 4.4 A regulated market: Application limits As noted in the introduction, we are interested in comparing a market that operates without any intervention, to one where the platform operator intervenes to try to improve the welfare of uyers and/or sellers. We consider a particular type of intervention: where the market operator restricts the numer of applications that can e sent y any individual sender.we refer to the unregulated market as the market without any application limit, and a regulated market as any market with an application limit. (In Section 7, we also riefly discuss the consequences of an alternative intervention: raising the application cost c a.) To formally study a model with application limits, consider a model in which uyer choices and payoffs are identical to our previous model, ut sellers are restricted to selecting m [0, l]. In the corresponding mean field model, this restriction leaves the analysis of the uyer optimal response unchanged, ut alters the seller s optimal response. In particular, in the mean field model, sellers solve the following optimization prolem given p: max Π s(m, p). m [0,l] 12

13 As discussed in Section 4.1.1, the ojective function is strictly concave, so this prolem has a unique solution identified y first-order conditions. It is given y M l (p) = min(l, M(p)). (16) We can then define the consistency conditions analogous to Section 4.2. Analogous to the unregulated market, we define a mean field equilirium of the market with application limit l as a pair (m l, α l ) solving the following pair of equations: m l = M l(p(m l, α l )), α l A(Q(m l, α l )). (17) The following proposition is an analog of Theorem 2 for the regulated market. Proposition 2. Fix r, β, c a, c, w, v, u such that Assumptions 1 and 2 hold, and let (m, α ) e the corresponding MFE in the unregulated market. Then for any l 0 there exists a unique mean field equilirium in the (regulated) market with application limit l. If m l, then (m l, α l ) = (m, α ). Otherwise, m l = l and α l is the unique solution to α l A(Q(l, α l )). Also analogous to the unregulated market, we define p l = P(m l, α l ), q l = Q(m l, α l ); then (p l, q l ) is the MFSS corresponding to the MFE of the (regulated) market with application limit l. We let Π l s and Π l e expected seller and uyer payoffs in the mean field equilirium of the (regulated) market with application limit l. In other words, Π l s = Π s (m l, p l ), Πl = Π (αl, q l, m l ). (18) 5 Mean field approximation In this section, we show that our mean field model is (in an appropriate sense) a reasonale approximation to our finite system when the market grows large (i.e., when n ). Formally, we show that the mean field assumptions hold as n, as long as all sellers s choose m s = m, and all uyers choose α = α. Note that once we fix m and α, we have removed any strategic element from the evolution of the n-th system: our desired results then ecome limit theorems aout a certain sequence of stochastic processes. We require the following notation. We let Binomial(n, p) denote the inomial distriution with n trials and proaility of success p, and let Binomial(n, p) k = P(Binomial(n, p) = k). We let Poisson(a) denote the Poisson distriution with mean a, and let Poisson(a) k = P(Poisson(a) = k). In the following theorem, we show that Mean Field Assumption 1 holds as n. In the process, we also show half of Mean Field Assumption 3: that in the limit the numer of applications received y a uyer is Poisson distriuted. Theorem 3. Fix r, β, m, and α. Suppose that the n-th system is initialized in its steady state distriution. Consider any uyer that arrives at t 0. Let R (n) denote the numer of applications received y uyer 13

14 in the n-th system, and let A (n) e the numer of these applicants that are still availale when the uyer screens. Then as n, the pair (R (n), A (n) ) converges in total variation distance to (R, A), where R Poisson(rm), and conditional on R, we let A Binomial(R, q). Analogously, we have the following theorem, where we show that Mean Field Assumption 2 holds as n. In the process, we also show the other half of Mean Field Assumption 3: that in the limit the numer of applications sent y a seller is Poisson distriuted. Theorem 4. Fix r, β, m, and α. Suppose that the n-th system is initialized in its steady state distriution. Consider any seller s arriving at time t s 0. Let T s (n) denote the numer of applications sent y seller s in the n-th system, and let Q (n) s e the numer of these applications that generate offers. Then as n, the pair (T s (n), Q s (n) ) converges in total variation distance to (T, Q), where T Poisson(m), and conditional on T, we let Q Binomial(T, p). In estalishing these results, the fundamental result that we prove is that the evolution of the state of the n-th system satisfies a stochastic contraction condition, and therefore remains close to an appropriately defined fixed point for all time. This asic result allows us to estalish that the mean field assumptions hold asymptotically. 6 Welfare analysis One of the chief advantages of MFE is that they are amenale to analysis. In particular, given the tractale representation we have for MFE, we can gain significant insight into uyer and seller welfare, and the potential value of regulation for each side of the market. In this section, we study uyer and seller welfare in the MFE, and quantify how limiting the numer of applications affects their welfare. Note that we focus on uyer and seller welfare individually, ecause aggregate welfare is a weighted sum of these two that depends on the ratio (v u)/w, which is a free parameter in our model. Throughout, we suppose that Assumptions 1 and 2 hold. 6.1 Preliminaries: An upper-ound on welfare We egin y characterizing uyer and seller welfare in MFE. Proposition 3. For any m and α, we have Π (α, Q(m, α), m) = r(v u)(1 e mp(m,α) c mp(m, α)). (19) 14

15 Furthermore, Π = r(v u)(1 e m p c m p ). (20) Π s = w(1 (1 + m p )e m p ). (21) The proof takes the definitions of equilirium welfare from (15), and applies three identities. The first is the consistency equation (10) satisfied y any mean-field steady-state; the second is the matching constraint given in (14), and the third is the seller est response function given y (6). We now use Proposition 3 to derive upper-ounds for uyer and seller welfare that hold not only in equilirium of the unregulated market, ut also in the equilirium of any regulated market. In particular (for any l) there holds: Π, Πl r(v u)(1 c + c log c ); (22) Π s, Π l s w min { 1/r, 1 e γ}, (23) The ound (22) for uyers comes directly from maximizing the right side of (19) with respect to mp(m, α), and therefore holds for any MFSS (m, α). The ounds on Π s and Π l s come from two oservations. First, sellers can never match with proaility exceeding 1/r (though this is only meaningful if r > 1). Second, in equilirium uyers must earn nonnegative payoff; it follows from (8) that q c. Because q = g(m p ) (see (10)) and g is decreasing (see Lemma 1), this implies that m p γ, where γ is the unique solution (y Lemma 1 and Assumption 2) to: g(γ) = c. (24) Since the numer of offers received y a seller is Poisson with mean m p, this implies that in equilirium, sellers match with proaility at most 1 e γ. 6.2 Overview of welfare analysis: Three regions In the remainder of the section, we study the ehavior of uyer and seller welfare in oth the regulated and unregulated markets, using the upper ounds (22) and (23) as enchmarks. Intuitively, restricting the numer of applications that can e sent should reduce the numer of matches that form, ut also reduce the application costs paid y sellers and the screening costs paid y uyers (since applicants are less likely to already have another offer). We study these markets in the limit where the application cost c a approaches zero; this simplification is inspired y the fact that in most online markets, application costs are relatively low 8. Furthermore, Lemma 8 in the appendix states that m p is non-increasing in c a. Because (21) is 8 Although we elieve that this is a well-motivated regime, it is not much more difficult to analyze uyer and seller welfare for fixed positive application costs. We are happy to provide figures to this effect upon request. 15

16 1.0 c' A B C Figure 1: A depiction of the three regions identified in this paper. In region A, sellers are scarce or screening costs are high, while in region C screening costs are small and sellers relatively aundant. r increasing in m p, so this implies that seller welfare is non-increasing in c a. Thus, the limit where c a 0 provides an upper ound for seller welfare with any positive c a. We define Π = lim c a 0 Π, Π s = lim ca 0 Π s. (25) Π l = lim c a 0 Πl, Πl s = lim ca 0 Πl s. (26) One key insight is that, we can decompose the parameter space into three regions depending on the value of c and r. Define the functions f 1 (r) and f 2 (r) as follows: We define the following three regions: { 0 if r 1; f 1 (r) = (27) r 1 r if r > 1; { 0 if r 1; f 2 (r) = g(log r r 1 ) if r > 1. (28) A = {(c, r) : c > f 2 (r)}; (29) B = {(c, r) : f 1 (r) < c f 2 (r)}; (30) C = {(c, r) : c f 1 (r)}. (31) These are depicted in Figure 1. Qualitatively, these three regions capture the different ranges of ehavior we can find in the regulated and unregulated markets. Informally, we have the following main oservations. Remark 1 (Main insights). 1. In Region A, the screening costs are relatively high and/or sellers are relatively scarce. In this region, in the unregulated market, uyers otain zero welfare, while in a suitaly regulated market, uyers earn the upper ound (22) on their welfare. 16

17 2. In Region B, although the screening costs are still relatively high, uyers earn positive welfare in the unregulated market; however, uyers can again earn the upper ound (22) in a suitaly regulated market. 3. In Region C, regulation does not help uyers: r is large enough that availaility is not a pressing concern, so y limiting applications we only hurt uyer welfare. 4. In all regions, we can improve seller welfare y a constant fraction y regulating the market. This is surprising, ecause we are already considering a regime where c a 0. Further, this effect can e quite significant: for example, when r 1.4, we can improve seller welfare y a factor of two y limiting applications. Returning to our discussion of odesk in the Introduction, recall that on odesk the most talented freelancers are scarce, and so invitations often fail due to a lack of availaility. In our model, this corresponds to region A, where the consequences of regulation are perhaps most dramatic: uyer (client) welfare can e raised from zero to the upper ound (22). This is especially surprising, since it is exactly where sellers (freelancers) are relatively scarce. In the following three susections, we descrie in greater detail the performance of the market in these three regions. We summarize our insights through three numerical examples in Figure Region A In this regime, screening costs are relatively high, and/or sellers are relatively scarce. The following proposition summarizes the ehavior of oth the regulated and unregulated market in Region A. Proposition 4. Suppose that (c, r) A. As c a 0 the following limits hold: 1. Buyers. Π = 0, i.e., uyers earn zero surplus in the unregulated market. On the other hand, there exists l such that in the regulated market with application limit l, Π l = r(v u)(1 c + c log c ) (i.e., the upper ound in (22)). 2. Sellers. Π s = w(1 e γ )(1 e γ /c ) in the unregulated market. On the other hand, there exists l such that in the regulated market with application limit l, we have Π l s = w(1 e γ ). From this proposition, we see that in the unregulated market, uyers earn zero welfare. When sellers send many applications, they ecome very unlikely to e availale, and uyer surplus drops to zero (i.e., the uyer individual rationality constraint inds). As a result, some uyers effectively choose not to enter the market. However, y suitaly regulating the market, we are ale to significantly increase uyer welfare all the way to the upper ound given in (22). 17

18 Recall that sellers can match with proaility at most 1 e γ, due to the fact that applicant availaility must e at least c. Nevertheless, in the unregulated market sellers lose an additional constant fraction of e γ /c of their potential surplus in application costs. The magnitude of this loss depends on the screening cost: the fact that uyers are reluctant to screen forces sellers to lose welfare y over-applying. By limiting applications, we can eliminate this welfare loss, so that sellers earn a surplus of w(1 e γ ), matching the upper ound (22). 6.4 Region B In this regime, screening costs are high and/or sellers are scarce, ut not to the same degree as in Region A. The following proposition summarizes the ehavior of oth the regulated and unregulated market in Region B. Proposition 5. Suppose that (c, r) B. As c a 0 the following limits hold: 1. Buyers. In the unregulated market, there holds: Π = (v u) (1 c r ln ) r. (32) r 1 On the other hand, there exists l such that in the regulated market with application limit l, Π l = r(v u)(1 c + c log c ) (i.e., the upper ound in (22)). 2. Sellers. In the unregulated market, there holds: Π s = w r ( ( )) r 1 (r 1) ln. (33) r 1 On the other hand, for any ε > 0, there exists l such that in the regulated market with application limit l, Π l s = w/r ε. From this proposition, we see that in the unregulated market, uyers earn positive surplus this is of course a significant improvement over Region A, and implies that all uyers screen. Nevertheless, we can improve uyer welfare y limiting applications, ecause screening costs and/or seller scarcity are still causing uyers to lose surplus to over-screening. In particular, a suitale choice of l can again increase uyer surplus all the way to the upper ound given in (22). For the sellers, the story is qualitatively similar to region A. Note that sellers can match with proaility no greater than 1/r, since r 1 in this region. In the unregulated market, sellers match with exactly this proaility; however, they over-apply, and do not internalize the externality their applications create on other sellers. In particular, this over-application causes sellers to again lose a constant fraction of their surplus. We can ring their surplus aritrarily close to w/r y suitaly limiting applications. 18

19 Π ~ l Π ~ l s Π ~ * Π ~ * s Application Limit Π ~ l Π ~ l s Π ~ * Π ~ * s Application Limit Π ~ l Π ~ l s Π ~ * Π ~ * s Application Limit Figure 2: Welfare in the regulated market as a function of the application limit l. The horizontal lines represent welfare in the unregulated market. We fix three choices of (r, β, c ), while letting c a In the graph on the left, (r, β, c ) = (0.8, 0.5, 0.2), so that we are in region A. Oserve that we can significantly improve uyer welfare at very little cost to sellers, e.g., y choosing l = 10. On the other hand, regulating the market can also help the sellers, provided we choose a large enough l. In the center graph, (r, β, c ) = (1.05, 0.5, 0.2), so that we are in region B. In this region we can help uyers, though of course not y as much as in Region A. Again, in this region regulation helps sellers, as expected. In the graph on the right, (r, β, c ) = (1.5, 0.5, 0.1), so that we are in region C. In this region we cannot help uyers y limiting applications, ut we can significantly improve seller surplus. In this example, sellers give up approximately half of their match surplus y over-applying in the unregulated market, even as c a Region C In this regime, screening costs are relatively low and there is an adequate supply of sellers, so that availaility never ecomes a pressing concern. The following proposition summarizes the ehavior of oth the regulated and unregulated market in Region C. Proposition 6. Suppose that (c, r) C. As c a 0 the following limits hold: 1. Buyers. In the unregulated market, (32) holds. On the other hand, for any regulated market with application limit l, Π l < Π. 2. Sellers. In the unregulated market, (33) holds. On the other hand, for any ε > 0, there exists l such that in the regulated market with application limit l, we have Π l s > w/r ε. The first part of this proposition states that in the unregulated market, uyer welfare reaches the same limit as in Region B when c a 0. Interestingly, in this case, we cannot improve uyer welfare y limiting seller applications; uyers are already doing as well as they can. To understand this, we return to (20). Maximizing the right hand side with respect to m p yields e m p = c. In other words, to achieve the upper ound in (22), sellers would need to match with proaility 1 c. However, when r > 1, the proaility that a seller matches at most 1/r. Thus if 1 c > 1/r, i.e., c < f 1 (r), we can do no etter than to have sellers match with proaility 1/r. This is exactly what we otain in the unregulated market. 19

20 The second half of this proposition is the same as in Region B: we can improve seller surplus aritrarily close to w/r y limiting their applications appropriately. 7 Discussion We introduced and studied a dynamic model of a decentralized two-sided market where it is costly to identify a compatile match. We focus in particular on the fact that timing effects may cause agents to unknowingly waste effort evaluating potential match partners who are no longer availale. We found that the unregulated market generically performs poorly on agent welfare: The market is in a search-limited regime when there are few sellers or screening costs are high. In this regime uyers have zero welfare. The market operates in a demand-limited regime when there are many sellers and screening costs are low. In this regime the seller welfare is reduced significantly y application costs due to the negative externality that an application y a seller has on other sellers. In either case, placing suitale restrictions on the sending of applications can significantly enhance welfare for agents on at least one (and often oth) sides of the market. There are several aspects of our model that are notaly stylized, with the goal of otaining a tractale model that allows us to study availaility in a meaningful fashion. We riefly comment here on some of these aspects. First, in our model, the proaility of compatiility β is assumed to e constant across uyers and sellers. In practice, we expect that uyers do not look identical to sellers, and that sellers (or the platform) can direct applications to the most promising openings; in this case an application limit encourages sellers to apply to the openings for which they are the est fit. In our model, this could e captured y allowing β to vary (decrease) as m grows. The fact that limiting visiility helps even when applications are sent randomly y uninformed agents is somewhat surprising, and only strengthens our conviction in the potential enefits of this intervention. Second, our model is asymmetric, in the sense that only the sellers sumit applications, and only the uyers screen. Though this seems to capture the operation of many marketplaces, for some contexts it would e more satisfying to allow every agent the option of applying or screening. Our preliminary consideration of such a model suggests that all stale equiliria may involve screening y agents on only one side of the market. The intuition is that if many uyers look for a match via screening whereas few uyers sumit applications, this makes it less attractive for sellers to screen and more attractive for sellers to apply. We leave formalizing this for future work. Third, our model does not endogenize wages. Numerous papers (i.e. [23, 1, 20]) have oserved that endogenously determined wages can guide the market towards efficiency. These results do not extend to our setting directly. For example, in [23, 1], workers apply to a single firm, precluding the possiility that they might e unavailale. In [20], workers apply to multiple firms, ut efficiency crucially relies on the assumption that firms can (costlessly) contact all applicants to find one who is availale. This is roughly 20