Unraveling Results from Comparable Demand and Supply: An Experimental Investigation

Abstract

Markets sometimes unravel, with offers becoming inefficiently early. Often this is attributed to competition arising from an imbalance of demand and supply, typically excess demand for workers. However this presents a puzzle, since unraveling can only occur when firms are willing to make early offers and workers are willing to accept them. We present a model and experiment in which workers’ quality becomes known only in the late part of the market. However, in equilibrium, matching can occur (inefficiently) early only when there is comparable demand and supply: a surplus of applicants, but a shortage of high quality applicants.

1. Introduction

In many professional labor markets most entry-level hires begin work at around the same time, e.g., soon after graduating from university, or graduate or professional school. Despite a common start time, offers can be made and contracts can be signed at any time prior to the start of employment. Such markets sometimes unravel, with offers becoming earlier and more dispersed in time, not infrequently well over a year before employment will begin.1

Unraveling is sometimes a cause of market failure, particularly when contracts come to be determined before critical information is available.2 Attempts to prevent or reverse unraveling are often a source of new market design in the form of new rules or market institutions.3

It is commonly suggested, both by economists and lay participants in these markets, that unraveling results from competition related to an imbalance of demand and supply. For example, [14], writing about the market for new physicians around 1900, writes

“ The number of positions offered for interns was, from the beginning, greater than the number of graduating medical students applying for such positions, and there was considerable competition among hospitals for interns. One form in which this competition manifested itself was that hospitals attempted to set the date at which they would finalize binding agreements with interns a little earlier than their principal competitors.”

Describing the growth of early admissions to colleges, [15] quotes a 1990 U.S. News and World Report story4

“ Many colleges, experiencing a drop in freshman applications as the population of 18-year-olds declines, are heavily promoting early-acceptance plans in recruiting visits to high schools and in campus tours in hopes of corralling top students sooner.”

There is also no shortage of commentary by market participants linking excess demand for workers to unraveling.5 And indeed, a number of unraveling events seem to have been initiated by a shock to supply or demand. See e.g., [3,18] for a description of just such a development in the market for gastroenterology fellows, in which a shock that produced excess demand for fellows (i.e., a shock that led, for the first time, to fewer fellowship applicants than fellowship positions) led to unraveling that lasted for a decade.

But, when looking at a labor market, it is not uncommon for participants on both sides of the market to be nervous about their prospects, and it can be difficult to be sure which is the short side of the market. We report an experiment in which the supply and demand can be controlled in a detailed way.

The issue is that, even in a market with more applicants than positions, there may be a shortage of the most highly qualified applicants. For example, the market for law clerks has experienced serious unraveling, with positions for new graduates in some recent periods being filled two years before graduation (see [19,20,21]). [22] writes of the judges’ perception of that market as follows:

“ But why the fervent competition for a handful of young men and women when our law schools spawn hundreds of fine young lawyers every year? Very simply, many judges are not looking just for qualified clerks; they yearn for neophytes who can write like Learned Hand, hold their own in a discussion with great scholars, possess a preternatural maturity in judgment and instinct, are ferrets in research, will consistently outperform their peers in other chambers and who all the while will maintain a respectful, stoic, and cheerful demeanor.

... Thus, in any year, out of the 400 clerk applications a judge may receive, a few dozen will become the focus of the competition; these few will be aggressively courted by judges from coast to coast. Early identification of these ” precious few” is sought and received from old-time friends in the law schools - usually before the interview season even begins.”

In just the same way, there are many federal appellate court judges, but only a few dozen are considered to have highly prestigious clerkships.

A similar story can be told about college admissions. For example, [23] says

“ There are more than two thousand four-year colleges in the United States. Only about two hundred reject more students than they accept. The vast majority of American colleges accept eighty per cent or more of those who apply. But among the top fifty there is a constant Darwinian struggle to improve selectivity.”

So both law clerks and college admissions are examples of markets in which participants on each side of the market can think of themselves as being on the long side, i.e., as facing a shortage of appropriate opportunities, if they focus on an elite subset of participants on the other side of the market. In such markets it can be hard to assess the relative balance of demand and supply in field data.

We will argue that such a situation, which we call comparable demand and supply, is in fact characteristic of markets in which we should expect to see costly, inefficient unraveling. The intuition is that unraveling requires both that firms want to make early offers and that applicants want to accept them. If information about match quality evolves over time, and if one side of the market knows that it will be in short enough supply to attract a good match when the good matches become clear, then there will not be unraveling, as one side of the market will want to await the resolution of the uncertainty.6

Unraveling will cause changes in who is matched with whom, but whether this will cause big efficiency losses will depend on whether the demand and supply imbalances are such that some "mismatches" are inevitable (in which case unraveling may just cause redistribution of mismatches), or whether they can largely be avoided by awaiting information on match quality (in which case too-early matches are likely to be inefficient).

This paper reports an experiment on these issues.7 The control available in the laboratory allows us to precisely vary the conditions of demand and supply.8 Section 2 outlines the elements of a theoretical model, sufficiently to apply it to the experimental design presented in Section 3.

Although we will concentrate here on the predictions about unraveling and inefficiency as a function of demand and supply, we will also note that the theoretical framework we develop enriches the standard model of stable matching in ways that allow for some novel predictions about the effect on one side of the market of adding participants to the other side. In particular we show that, although in the standard model i.e., after uncertainty regarding the applicant quality is resolved, adding more applicants to the market is always good for firms under the Gale-Shapley [43] stable deferred-acceptance rules (cf. [44]) and the subgame perfect equilibrium prediction in our protocol, the SPE predictions of our model can reverse when we allow multiple stages of hiring with uncertainty about the applicant quality being resolved in the middle of the game.

2. The Model

Consider a market with n_F firms and n_A applicants. Every firm f wants to hire up to one applicant, and every applicant a can work for up to one firm. Firms and applicants have one of two possible qualities, high (h) or low (l). A matching is a mapping that (i) maps each agent to a partner on the other side of the market or leaves her unmatched, and (ii) for any firm f and applicant a, maps f to a if and only if it maps a to f. The interpretation is that a firm f and an applicant a are matched if firm f hired applicant a and applicant a works for firm f.

Unmatched firms and applicants receive a payoff of 0. If firm f and applicant a are matched with each other, they receive payoffs that depend upon their own quality, the quality of the matched partner, and a random variable. We assume every agent prefers a high quality partner to a low quality partner, and being unmatched is the least preferred alternative for each agent. For each firm f of quality i, the payoff for being matched with an applicant a of quality j is V_f(a) = u_ij + ε_a, where ε_a is a random variable with mean 0, and u_ij > 0. For each applicant a of quality i, the payoff for being matched with a firm f of quality j is V_a(f) = u_ij + ε_f, where ε_f is a random variable with mean 0. ε_f and ε_a are identically distributed with a marginal distribution G in the interval [−a,a] where a > 0. The joint distribution of {ε_a} is such that there is no tie in ε_a that is, there are no two applicants a₁ and a₂ such that occurs with positive probability. Similarly, the joint distribution of {ε_f} is such that there is no tie in ε_f.

A matching is ex-post stable if there is no firm-applicant pair such that, after all uncertainty is resolved, each prefers one another to her match. A matching is efficient if the sum of the payoffs of the agents is highest among all matchings after all uncertainty is resolved. A matching is qualitywise-efficient if the sum of the payoffs of agents excluding ε’s is highest among all matchings. Since ex-post there are no ties in ε_f and ε_a with probability one, firms and applicants can be strictly rank-ordered with respect to preferences. A matching is assortative if for all k ≤ min{n_F,n_A}, the k^th best applicant is matched with the k^th best firm. The market has a unique assortative matching with probability one. A matching is qualitywise-assortative if there is no firm f of quality i and applicant a of quality j such that f is matched with an applicant of lower quality than j or is unmatched, and a is matched with a firm of lower quality than i or is unmatched.9

We assume that

We choose a (the bound of the ε’s) small enough such that with the above assumptions, an efficient matching is qualitywise-assortative; i.e. no agent ever prefers a low quality partner to any high quality partner. This implies that a matching is qualitywise-assortative if and only if it is qualitywise-efficient. Moreover, there is a unique ex-post stable matching (the unique assortative matching, due to the random variables ε), which is efficient.

We will be mostly interested in qualitywise efficiency or qualitywise inefficiency rather than efficiency or inefficiency, since the values of random {ε} will be rather small and this will cause small welfare losses when a qualitywise-efficient matching is not efficient.11. However, qualitywise-inefficient matchings will lead to comparatively large welfare losses.

While we denote qualities as high and low, it would be misleading to think of those as e.g., Ivy League universities and community colleges. Rather, the analogy would be between the very best colleges and some only slightly less prestigious ones. Note that in the model the firms literally hire from the same pool of applicants. Similarly, each applicant has a chance to be among the very best, or not quite make it (hence they are far from applicants who would have no chance to be accepted to a top university, no matter the final information available about them).

2.1. The Game

Firms and applicants can match in a market with early and late hiring stages. Each stage consists of several hiring periods. There are T^E periods in the early hiring stage, and T^L periods in the late hiring stage. The firms’ qualities, high or low, are common knowledge to all market participants at the beginning of the first period in the early hiring stage. Let be the number of high quality firms in the market and be the number of low quality firms.

The applicants’ qualities are uncertain and revealed only at the first period of the late hiring stage when they become common knowledge. From the beginning, it is common knowledge that a proportion of applicants for some integer will be of high quality, and the remaining applicants, will be of low quality (for example, there might be a special distinction attached to hiring a “ top ten” applicant). To simplify the exposition, and the later experiments, we assume that all applicants have the same ex-ante probability of being of high quality, namely .12 The values of the random variables {ε} are realized at the beginning of the late hiring stage, and become common knowledge. The distribution of {ε} is common knowledge at the beginning of the early hiring stage.

Firms make offers to applicants, and applicants have to decide whether to accept or reject those offers right away (they cannot hold on to offers). Once an applicant has accepted an offer from a firm, the firm and the applicant are matched with each other: the applicant cannot receive any other offers and the firm cannot make any other offers.

In every period, firms that are not yet matched may decide to make up to one offer to any available applicant. A firm can make an offer again to an applicant who previously rejected its offer. Once all the firms have decided what if any offers to make, the offers become public information. Then, applicants can decide how to respond to offers they have received: whether to accept at most one offer, or reject all offers. After all applicants have decided, their actions become public information.

So the market is a multi-period game in which firms (and applicants) make simultaneous decisions with public actions, and a move by nature in the middle of the game which determines the qualities of the applicants and the {ε} perturbations of preferences over different firms and applicants.

2.2. Analysis of Unraveling

This game has many (Nash) equilibria, including ones in which every match is concluded in period 1 of the early hiring stage. For example, a strategy profile in which for each applicant a there is only one firm f_a whose offer a accepts in period 1, and in which firm f_a only makes an offer in period 1 to applicant a and no other offers are made or accepted, is an equilibrium. However, such equilibria will in general not be subgame-perfect. We restrict our theoretical analysis to subgame perfect equilibria (SPEs).

We partition the space of possible demand and supply relations into 3 different scenarios representing excess supply, comparable demand and supply, and excess demand for labor, respectively. The important variables will be n_F, the total number of firms, , the number of high quality firms, n_A, the total number of applicants, and , the total number of high quality applicants:

Case 1.

: Every firm can be matched with a high quality applicant, some high quality applicants remain unmatched (if ). There is excess supply.

Case 2.

: Excess applicants, but shortage of high quality applicants. There is comparable demand and supply.

Case 3.

n_F ≥ n_A: Each applicant can be matched with a firm. There is excess demand. We analyze excess demand in two subcases:

a.

: Excess firms, but shortage of high quality firms.

b.

: Every applicant can be matched with a high quality firm, some high quality firms remain unmatched (if ).

We will show that the SPEs of this game support the following intuition. In case 1, no firm will wish to make an early offer, because by waiting until applicants’ qualities are known, every firm can hire a high quality applicant. Thus, there is excess supply of labor.

In Case 2, in the early stage, a given applicant does not know if she will be on the long or short side of the market, since her quality is still unknown (and since there will be too few high quality applicants, and too many low quality applicants). This is why we say that there is comparable demand and supply. An applicant may therefore find it attractive to take an early offer from (even) a low quality firm, and making such an offer is profitable for a low quality firm, since this is the only way to possibly hire a high quality applicant. So Case 2 may be subject to unraveling.

Case 3a looks superficially symmetric to Case 2 but with the critical difference that the role of firms and applicants are reversed. Since the qualities of firms are already known in the early stage, no high quality firm is in any doubt that it will be on the short side of the market, and no applicant is in any doubt that she can find a position in the late period. So no applicant will take an early offer from a low quality firm (since such offers will remain available even if the applicant turns out to be low quality), and no high quality firm will make an early offer.13 Case 3b looks like Case 1 with firms and workers reversed, but in this case there is much less difference between early or late matching, since even early matchings cannot be qualitywise-inefficient, as every applicant, regardless of quality, will match to a high quality firm. Thus, we refer to Cases 3a and 3b as excess demand conditions.

We are concerned about the inefficiency (or efficiency) of SPE outcomes. We will mostly focus on “ large scale” inefficiency, namely qualitywise inefficiency. The following result characterizes the demand and supply condition in which there are qualitywise-inefficient outcomes under SPEs:

Table 1. Treatments.

	nA = 6	nA = 12
nF = 4	THIN COMPARABLE MARKET Case 2 (baseline design) SPE Prediction: Low quality firms hire early qualitywise-inefficient outcome	MARKET WITH EXCESS SUPPLY Case 1 SPE Prediction: Late and assortative matching efficient outcome
nF = 8	MARKET WITH EXCESS DEMAND Case 3 SPE Prediction: Late and assortative matching efficient outcome	THICK COMPARABLE MARKET Case 2 SPE Prediction: Low quality firms hire early qualitywise-inefficient outcome

Table 1. Treatments.

	nA = 6	nA = 12
nF = 4	THIN COMPARABLE MARKET Case 2 (baseline design) SPE Prediction: Low quality firms hire early qualitywise-inefficient outcome	MARKET WITH EXCESS SUPPLY Case 1 SPE Prediction: Late and assortative matching efficient outcome
nF = 8	MARKET WITH EXCESS DEMAND Case 3 SPE Prediction: Late and assortative matching efficient outcome	THICK COMPARABLE MARKET Case 2 SPE Prediction: Low quality firms hire early qualitywise-inefficient outcome

Theorem 1 A qualitywise-inefficient early matching is an outcome of a subgame perfect equilibrium only if the market is one of comparable demand and supply (Case 2), i.e., if .14

Note that not all early matching equilibria are qualitywise-inefficient. For example, when , high firms can hire applicants early under SPE as described by Lemma 3 (in Appendix A), but the outcome will not be qualitywise-inefficient. That is the reason we concentrate on qualitywise-inefficient early hiring (as opposed to all early hiring) in Theorem 1, since only qualitywise-inefficient hiring creates large welfare problems.15 Note also that, while qualitywise-inefficient early matching cannot be a perfect equilibrium outcome except under conditions of comparable demand and supply, whether it is depends on the parameter values in a way that will be detailed in Propositions 3 and 4 in Appendix A (which is also where proofs are).

3. Experimental Design

The theoretical analysis of this simple market shows that there are a multitude of Nash equilibria. However, restricting attention to SPEs provides clear predictions. Early, qualitywise-inefficient matching can occur only when demand and supply are comparable in the sense that there is initial uncertainty whether a given applicant will be on the long or short side of the market.

However, behavior does not always conform to the predictions of SPE. Consequently we also test the theoretical predictions in laboratory experiments.

Apart from testing the SPE predictions, the experiments also allow us to distinguish these from alternative hypotheses about unraveling. As noted in the introduction, in many markets we have studied, the intuition of market participants is that unraveling results simply from a shortage of applicants.

The experimental markets have two hiring stages each of which consists of four periods, so that there is time to make multiple offers at each stage (the market is uncongested). The qualities of applicants and ε values are determined at the beginning of period 5, i.e., the first period of the late hiring stage.

We consider a design with four treatments in which there are four or eight firms, and six or twelve applicants (in all possible combinations). In each case, half the firms are of high quality, and a third of the applicants are eventually of high quality. It will be helpful to consider the treatment with the fewest firms and applicants as the baseline treatment, to which others will be compared.

We will refer to the baseline treatment as the thin comparable markettreatment, since it satisfies the Case 2 demand and supply condition, comparable demand and supply. We then change the market conditions by increasing the number of firms and workers. Doubling the number of low and high quality applicants yields the treatment referred to as the market with excess supply, satisfying the Case 1 demand and supply condition. Doubling the number of high and low quality firms, keeping the number of applicants at the baseline level, yields the market with excess demand (Case 3, in particular Case3a). The largest treatment is obtained by doubling both the number of high and low quality firms and applicants. When the number of firms and workers are both at their highest levels, we are again in the case of comparable demand and supply (Case 2), so we refer to this treatment as the thick comparable market (see Table 1).

In each market an agent earns points when she gets matched to a partner. We set u_hh = 36 points, u_hl = u_lh = 26 points, and u_ll = 20 points for the preferences, which satisfy that u_hh − u_hl > u_lh − u_ll > 0. We choose the distribution of ε as follows: When there are two agents in a quality type (high quality applicants, high quality firms or low quality firms in various treatments), we set {ε₁,ε₂} = {−1.2,1.2}, and each permutation, i.e., (ε₁,ε₂) = (−1.2,1.2) and (ε₁,ε₂) = (1.2,−1.2), is chosen with equal probability. When there are four agents in a quality type (high quality applicants, low quality applicants, high quality firms or low quality firms in various treatments), we have {ε₁,ε₂,ε₃,ε₄} = {−1.2,−0.4,0.4,1.2}, and each permutation is chosen with equal probability. When there are eight agents in a quality type (such as low applicants in the market with excess supply), we set {ε₁,ε₂,ε₃,ε₄,ε₅,ε₆,ε₇,ε₈} = {−1.4,−1,−0.6,−0.2,0.2,0.6,1,1.4} and each permutation is chosen with equal probability.

At the beginning of each session, participants were randomly assigned their role as applicant and firm, which they kept throughout the session. Participants who were firms also kept their quality types as high quality or low quality throughout each session. In each session, the market game was played 20 consecutive times by generating new ID numbers for applicants in each market. Subjects received a $10 participation fee plus their total earnings in the session except in the treatment with excess supply, where we also gave an additional $5 participation fee to the subjects in the roles of applicants at the end of the session. The experimental points were converted to USD at the exchange rate $1 = 20 points.

The experimental sessions were run at the Computer Laboratory for Experimental Research (CLER) of the Harvard Business School using a subject pool consisting of college students (primarily from Harvard, Boston, Tufts, and Northeastern Universities) and at the Pittsburgh Experimental Economics Laboratory (PEEL) of the University of Pittsburgh using a subject pool consisting of college students (primarily from Pittsburgh, Carnegie Mellon, and Duquesne Universities). Each participant could only participate in one session. The experiment was conducted using z-Tree ([1]). We ran four sessions of the treatments requiring the most subjects, namely the excess supply treatment and the thick comparable market treatment. For the other treatments we collected seven sessions. The instructions for the excess supply market treatment are given in Appendix C. Instructions for the other treatments are similar.

Through the experiment the subject received feedback consistent with the modeling of the sequential game. That is, after each offer is made or rejected these offers and rejections were displayed on the screen of the subjects in the next period and on within the same market game. Moreover, who was available and who was not available were also public information on the screen of the subjects throughout the periods of the game. They were acknowledged of their earning in that market in experimental points. To prevent linkage as much as possible between market games, they were not reminded of their earnings of the earlier games between games.

The theoretical properties of the general model apply to the experimental design. Hence, the thin and thick comparable market treatments each fall under Case 2, the treatment with excess supply falls under Case 1, and the treatment with excess demand falls under Case 3a of the demand-supply conditions described in the previous section. It is trivial to check that our comparable market treatments satisfy the conditions of Proposition 4 in the Appendix. Therefore, the SPE predictions for each treatment are given in Table 1 via Theorem 1 and Proposition 4. The SPE predictions are thus that, we should see qualitywise in the baseline - thin comparable market treatment. As we add either only applicants or only firms (the treatments with excess supply and excess demand) efficiency should be restored. Restoring the relative balance (in the thick comparable market treatment), that is adding both firms and applicants, is predicted to bring back qualitywise-inefficient early matching.

Note that this design allows us to distinguish the SPE predictions from the alternative hypothesis that excess demand for workers (i.e., a shortage of workers) causes unraveling, since in that case there should be the most unraveling in the excess demand treatment. The excess supply treatment could well be similar according to this hypothesis, despite the asymmetric roles of firms and workers and how their qualities are revealed. The design also allows us to distinguish the SPE predictions from the congestion hypothesis that increasing the number of firms or of applicants will by itself facilitate unraveling, since the equilibrium predictions are that, moving from cell to cell in the experiment, an increase in the size of one side of the market will in some cases increase and in other cases decrease unraveling. The hypothesis can be summarized as while keeping the number of periods constant, increasing the number of applicants and firms may lead to more congestion even though subgame perfect equilibrium predictions say otherwise.

4. Experimental Results

4.1. The Analysis of Matches and Unraveling

We first analyze when firms and workers match, and whether the market experiences unraveling. In every market, in all cohorts, the maximal number of matches were formed, that is, it was never the case that both a firm and a worker were unmatched. Hence, we can focus the analysis in this section on how many matches were formed in the early hiring stage, the amount of unraveling.

Figure 1 shows over time the number of firms that hire in the early stage, as they gain experience. In Table 2, we report the median of the median percentage of firms that hire early in the last five markets for each treatment.16,17 Especially in the final five markets, the observed behavior in the experiments are similar to the SPE predictions for risk-neutral subjects outlined in , although we do not observe that hiring behavior is fully consistent with SPE.

Figure 1. Median percentage of high and low type firms hiring early across treatments.

Figure 1. Median percentage of high and low type firms hiring early across treatments.

Figure 1 shows that the hiring behavior moves in the direction of the SPE predictions, suggesting that there is learning over time. In the two comparable demand and supply treatments, more and more low quality firms hire early, while more and more high quality firms hire late, approaching the SPE predictions in the last five markets.

Table 2. Median percentage of firms hiring early in the last five markets, with subgame perfect equilibrium predictions in parentheses.

Actual (SPE) % Firms Hiring Early In the last five markets - Medians	THIN COMPARABLE	EXCESS SUPPLY
Low Firms	100% (100% )	25% (0% )
High Firms	0% (0% )	0% (0% )
	EXCESS DEMAND	THICK COMPARABLE
Low Firms	0% (0% )	87.5% (100% )
High Firms	0% (0% )	0% (0% )

Table 2. Median percentage of firms hiring early in the last five markets, with subgame perfect equilibrium predictions in parentheses.

Actual (SPE) % Firms Hiring Early In the last five markets - Medians	THIN COMPARABLE	EXCESS SUPPLY
Low Firms	100% (100% )	25% (0% )
High Firms	0% (0% )	0% (0% )
	EXCESS DEMAND	THICK COMPARABLE
Low Firms	0% (0% )	87.5% (100% )
High Firms	0% (0% )	0% (0% )

We increase excess demand when moving from the thin comparable to the excess demand treatment. The amount of early hiring in the excess demand condition refutes the hypothesis that excess demand, i.e., a shortage of applicants generates unraveling. Furthermore, unraveling in these markets is not driven primarily by congestion, since neither the excess demand or excess supply conditions experience as much early hiring as the thin comparable condition, despite having more participants.

In the excess demand condition, both high and low quality firms approach the SPE prediction of no early hiring. Using the non-parametric two-sample and two-sided Wilcoxon rank sum test, we test whether the median early hiring levels of the excess demand and the two comparable treatments are the same (in the last five markets), taking each session median for the last five markets as an independent observation. The p-values reported in Table 3 confirm that there is significantly more early hiring by low quality firms in the comparable compared to the excess demand treatments.

Similarly, the excess supply treatment also approaches SPE predictions of late hiring by both high and low quality firms (see Figure 1 and Table 3).

Table 3. Testing equivalence of high firm and low firm early hiring percentages in the last five markets. We denote significance regarding the rejection of the null hypotheses at 95% level with * and significance at 99% level with ** after the reported p-values.

H0 (For median % high/low firms hiring early)	sample sizes	p-value: High	p-value: Low
Thin Comparable = Thick Comparable	7,4	1	0.79
Excess Supply = Excess Demand	4,7	0.67	0.76
Thin Comparable = Excess Supply	7,4	0.56	<0.01**
Thin Comparable = Excess Demand	7,7	0.44	<0.01**
Thick Comparable = Excess Supply	4,4	0.43	0.03*
Thick Comparable = Excess Demand	4,7	0.42	<0.01**

Table 3. Testing equivalence of high firm and low firm early hiring percentages in the last five markets. We denote significance regarding the rejection of the null hypotheses at 95% level with * and significance at 99% level with ** after the reported p-values.

H0 (For median % high/low firms hiring early)	sample sizes	p-value: High	p-value: Low
Thin Comparable = Thick Comparable	7,4	1	0.79
Excess Supply = Excess Demand	4,7	0.67	0.76
Thin Comparable = Excess Supply	7,4	0.56	<0.01**
Thin Comparable = Excess Demand	7,7	0.44	<0.01**
Thick Comparable = Excess Supply	4,4	0.43	0.03*
Thick Comparable = Excess Demand	4,7	0.42	<0.01**

As predicted by SPE, we cannot reject the null hypothesis that median early hiring by high quality firms in the last five markets is the same across all treatments under pairwise comparison. On the other hand, the comparable treatments have significantly higher proportions of low quality firms hiring early than the excess demand and excess supply treatments. The comparable treatments do not significantly differ from each other. Neither do the excess demand and supply treatments differ from each other.

Figure 2. Median early offer rates and acceptance rates across treatments.

Figure 2. Median early offer rates and acceptance rates across treatments.

4.2. The Analysis of Early Offer and Acceptance Rates

In what follows, we examine not only the outcomes, but also the observed offer making and acceptance behavior. The top two graphs of Figure 2 show the percentage of high and low quality firms making offers in the early stage across treatments. As before, we represent markets in groups of five and report the medians of the median rates across these five market groups for each treatment. The first median mentioned is obtained across cohorts of data for each treatment. If a firm makes one or more offers in the early stage, it is counted as a firm that is making an early offer. The bottom two graphs of Figure 2 show the acceptance rates by applicants of the early offers. That is, we determine for each early offer whether it was accepted or rejected.We use the same reporting convention as before with one important difference. When there are no early offers by any firm in a market, the acceptance rate is not defined. In our median calculations, we only consider markets in which offers were actually made. This technique results in well-defined acceptance rates, even though the median offer rate is 0% in many instances. (The only exception is markets 11-15 in the excess supply treatment: no low type firms make any offers in any of these markets in any of the cohorts of data, hence we omit that acceptance rate, see the bottom right graph of Figure 2.)

We observe that the number of early offers from high type firms is decreasing and mostly converging to zero in all treatments, and that acceptance rates of high firm offers are 100%. Both of these findings are consistent with the SPE predictions both on and off the equilibrium paths. In the comparable demand and supply treatments, these percentages converge to 0% very fast. However, in the excess supply and excess demand treatments, convergence takes more time, and in the excess supply treatment we do not observe full convergence.

For early offers from low type firms and the applicants’ acceptance behavior we observe different behavior across different treatments. On and off the SPE path, predictions are such that in the comparable demand and supply treatments all low quality firms make early offers and all applicants always accept them, whenever such offers are made. Behavior in the experiment is consistent with the SPE predictions.

In the excess demand treatment, applicants should not accept early offers from low quality firms on the SPE paths. Thus, low quality firms are indifferent between making early offers and not making them at all. However, when applicants make mistakes and accept such offers, we expect low quality firms to make early offers (i.e., on a trembling hand-perfect equilibrium path). Indeed in markets 1-5, 25% of applicants accept an early offer from a low quality firm. This reinforcement seems to cause 100% of low quality firms to make early offers throughout the early markets. However, towards the final markets, early offers from low quality firms are never accepted anymore. This seems to reduce the amount of early offers from low quality firms, only 75% of them make early offers.

In the excess supply treatment, we expect low quality firms not to make early offers under SPE. Yet, off the SPE paths, when they make early offers, we expect the applicants to accept these offers. Initially, this is exactly what happens. Low quality firms almost never make early offers. However, when they make mistakes and make early offers (which is a more frequent mistake in markets 1-10, never happens in market 11-15, and only rarely in markets 16-20), applicants tend to accept those offers (as seen in the bottom right graph of Figure 2). Only in the final markets, markets 16-20, when early offers are very rare, are they accepted only 50% of the time (though this is a very small number phenomenon).18

In summary, in all treatments the behavior of the firms and applicants are mostly consistent not only with SPE predictions, but also with trembling-hand perfection predictions.

4.3. The Analysis of Efficiency

We conclude our analysis of the experimental results with qualitywise efficiency (i.e., ignoring ε’s). A full matching is an assignment under which the maximum number of firms and applicants are matched. Since for both firms and workers, matching to a high quality partner is worth more than to a low quality partner, we use the welfare of firms to provide a measure of total welfare.19 We aim to measure the proportion of gains achieved compared to the payoffs achieved by a random match that matches as many agents as possible, that is the payoffs achieved by a randomly generated full matching, where each possible full matching is equally likely. We call the sum of these expected firm payoffs the random match payoffs. Normalized (qualitywise) efficiency is defined as the sum of firm payoffs (disregarding ε values) minus the random match payoffs divided by the maximum sum of possible expected firm payoffs minus the random match payoffs (see Table 4). A value of 100% indicates that the matching in the lab achieved all possible gains compared to the average random full matching and is hence the matching that maximizes firm payoffs.20 The random match payoffs and maximum sums of possible expected firm payoffs of a full matching in each treatment are given in Table 4 along with the median of the sum of actual firm payoffs (disregarding ε values) and the normalized efficiency of the last five markets. We also report in the same table the predictions of SPE outcomes in terms of efficiency.

Table 4. Median of the sum of firm payoffs in the last five markets, with random match payoffs and maximum firm payoffs under full matchings in parentheses, and normalized efficiency, with the theoretical SPE prediction in parentheses.

In the last five markets - Medians	THIN COMPARABLE	EXCESS SUPPLY
Actual Firm Welfare [Random, Max.]	108 [102.67, 112]	124 [102.67, 124]
Actual (SPE) N.Efficiency=	57% (57%)	100% (100%)
	EXCESS DEMAND	THICK COMPARABLE
Actual Firm Welfare [Random, Max.]	164 [154, 164]	220 [205.34, 224]
Actual (SPE)N. Efficiency=	100% (100%)	79% (79)

Table 4. Median of the sum of firm payoffs in the last five markets, with random match payoffs and maximum firm payoffs under full matchings in parentheses, and normalized efficiency, with the theoretical SPE prediction in parentheses.

In the last five markets - Medians	THIN COMPARABLE	EXCESS SUPPLY
Actual Firm Welfare [Random, Max.]	108 [102.67, 112]	124 [102.67, 124]
Actual (SPE) N.Efficiency=	57% (57%)	100% (100%)
	EXCESS DEMAND	THICK COMPARABLE
Actual Firm Welfare [Random, Max.]	164 [154, 164]	220 [205.34, 224]
Actual (SPE)N. Efficiency=	100% (100%)	79% (79)

First note that both comparable demand and supply treatments lack full qualitywise efficiency. The median SPE outcomes predict 57% and 79% efficiency for the thin and thick comparable treatment. The outcomes are exactly aligned with the theoretical predictions.21 This reflects the earlier results that both early offers and acceptances, and hence early matches follow SPE predictions (low quality firms hire early, while high quality firms hire late).

Table 5. Testing equality of normalized efficiency in the last five markets.

H0 (For median n. efficiency)	sample sizes	p-value
Thin Comparable = Thick Comparable	7,4	0.12
Excess Supply=Excess Demand	4,7	0.73
Thin Comparable=Excess Supply	7,4	0.33
Thin Comparable=Excess Demand	7,7	0.021*
Thick Comparable=Excess Supply	4,4	0.49
Thick Comparable=Excess Demand	4,7	0.048*

Table 5. Testing equality of normalized efficiency in the last five markets.

H0 (For median n. efficiency)	sample sizes	p-value
Thin Comparable = Thick Comparable	7,4	0.12
Excess Supply=Excess Demand	4,7	0.73
Thin Comparable=Excess Supply	7,4	0.33
Thin Comparable=Excess Demand	7,7	0.021*
Thick Comparable=Excess Supply	4,4	0.49
Thick Comparable=Excess Demand	4,7	0.048*

When we inspected the data, we found that there is less overall early hiring in the excess demand treatment compared to the comparable demand and supply treatments. Furthermore, there is also a significantly higher proportion of efficiency realized, compared to both the thin and thick comparable treatments (the two of which are not significantly different from one another). See Table 5 for p-values of the Wilcoxon rank sum tests of median efficiency.22

In the excess supply treatment, we found that, in the last five markets a median of 25% of high type firms hire early, while low type firms do not hire early. SPE predicts that no high quality firms hire early. In the excess supply treatment, there are four high quality applicants for four firms, including the two high quality firms. A high quality firm that hires early will, however, match with a high type applicant only one third of the time. That is two thirds of the time there will be an efficiency loss of 10 points (when the high type firm only receives 26 points from hiring a low type applicant compared to the 36 from hiring a high type applicant). Since only once in every other market does a high type firm hire early, we expect about one third of all sessions to yield median efficiency below 100%. And indeed, the median efficiencies are 100%, 100%, 100%, and 53.11% (which results from one high type firm hiring a low type worker, and all other firms hiring high type workers). Hence while the median efficiency of the excess supply treatment is 100%, there is some variance, such that the efficiency is not significantly higher in the excess supply treatment compared to the comparable demand and supply treatments (see Table 5).

4.4. Further Consequences: Effects of Changing Labor Supply

The experiment confirms the theoretical prediction that, under some circumstances, the addition of agents on one side of the market can cause unraveling, or prevent it. This gives us an opportunity to revisit a robust result in the literature of static matching models about the welfare of agents on one side of the market when the number of agents on the other side of the market increases or decreases. Using the firm-optimal stable matching as the solution concept for centralized models of matching markets, [44] showed that increasing the number of applicants in the market while keeping the firm preferences unchanged within the old applicant set cannot make any firm worse off (Proposition 2, p. 264).23 We can state a similar result on our domain, if we confine our attention to the late hiring stage.

Proposition 1 For a given set of firms that remain unmatched until the late hiring stage, if the number of available applicants increases and the number of available high quality applicants does not decrease, no available firm will be ex-ante worse off under any subgame perfect equilibrium.

However, if we consider our full two-stage matching game this conclusion no longer holds. An increase in the number of applicants can move the market conditions from Case 3a, when all equilibria yield late matching, to Case 2, and an unravelled market, and this can harm some firms (a change in the number of applicants could also simply increase the amount of unraveling within Case 2. Note that Proposition 1 implies that firms can become worse off from an increase in the number of applicants only when this leads to a decrease in the number of high quality applicants available in the late hiring stage, through unraveling).

Proposition 2 It is possible that, for a given set of firms, if the number of applicants increases and the number of high quality applicants does not decrease, a firm can be ex-ante worse off at a subgame perfect equilibrium outcome.

Note also that a decrease in the number of high quality applicants can hurt firms in two ways. First, as in static models (and as in Proposition 1) a reduction in the number of potential high quality partners hurts firms in the usual way. Second, if the decrease in the number of applicants moves the market from Case 1 to Case 2, it may also change the SPE outcome from late matching to one with some unraveling, and this may further hurt the high quality firms by decreasing the amount of assortative matching. (Something like this happened in the market for gastroenterologists in the late 1990s.24 )

5. Discussion

It has been known at least since [2] that many markets unravel, so that offers become progressively earlier as participants seek to make strategic use of the timing of transactions. It is clear that unraveling can have many causes, because markets are highly multidimensional and time is only one dimensional (and so transactions can only move in two directions in time, earlier or later). So there can be many different reasons that make it advantageous to make transactions earlier.25

Thus the study of factors that promote unraveling is a large one and a number of distinct causes have been identified in different markets or in theory, including instability of late outcomes (which gives blocking pairs an incentive to identify each other early), congestion of late markets (which makes it difficult to make transactions if they are left until too late), and the desire to mutually insure against late-resolving uncertainty. There has also been some study of market practices that may facilitate or impede the making of early offers, such as the rules and customs surrounding “ exploding” offers, which expire if not accepted immediately.

In this paper we take a somewhat different tack, and consider conditions related to supply and demand that will tend to work against unraveling, or to facilitate it. There seems to be a widespread perception, in markets that have experienced it, that unraveling is sparked by a shortage of workers.

But for inefficient unraveling to occur, firms have to be willing to make early offers and workers have to be willing to accept them. Our experiment supports the hypothesis that a shortage of workers is not itself conducive to unraveling, since workers who know that they are in short supply need not hurry to accept offers by lower quality firms. Instead, in the model and in the experiment, it is comparable supply and demand that leads to unraveling, in which attention must be paid not only to the overall demand and supply, but to the supply and demand of workers and firms of the highest quality. An important feature of our model and experiments is that when there is inefficient unraveling, this is due to low type firms, but not high types, hiring early.

In our design, all market participants, in the model and in the experiment, know which of the treatments they are in, i.e., they know the supply and demand conditions. If we introduced uncertainty about the supply and demand conditions, theoretically we can likely emulate similar predictions as our model. Though, it would be interesting to see how these would reflect in an experimental environment to the behavior of subjects.

We emphasize again that the qualities of workers and firms in our model should not be taken literally as “ low” and “ high” while mapping our predictions to real markets. Being hired in our model can refer to being hired by one of the elite firms in a real market, some better than the others, and being unmatched can refer to being hired by one of the ordinary firms in a secondary market. For example, clerkships for federal appellate judges are elite positions for law school graduates. Yet, there is unraveling in this market and it seems to start in the 9th Circuit, whose judges are a little disadvantaged compared to the East Coast circuits.

Our results seem to reflect what we see in many unraveled markets, in which competition for the elite firms and workers is fierce, but the quality of workers may not be reliably revealed until after a good deal of hiring has already been completed. This observation not only helps us understand which markets unravel, but also casts new light on the welfare effects of changes in supply and demand.