# Unraveling Results from Comparable Demand and Supply: An Experimental Investigation

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## Abstract

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## 1. Introduction

“ 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.”

“ 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.”

“ 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.”

“ 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.”

## 2. The Model

_{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.

_{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 a

_{1}and a

_{2}such that occurs with positive probability. Similarly, the joint distribution of {ε

_{f}} is such that there is no tie in ε

_{f}.

**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

#### 2.1. The Game

^{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.

#### 2.2. Analysis of Unraveling

_{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).

_{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 ).

n_{A} = 6 | n_{A} = 12 | |

n_{F} = 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 |

n_{F} = 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

## 3. Experimental Design

**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).

_{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 {ε

_{1},ε

_{2}} = {−1.2,1.2}, and each permutation, i.e., (ε

_{1},ε

_{2}) = (−1.2,1.2) and (ε

_{1},ε

_{2}) = (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 {ε

_{1},ε

_{2},ε

_{3},ε

_{4}} = {−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 {ε

_{1},ε

_{2},ε

_{3},ε

_{4},ε

_{5},ε

_{6},ε

_{7},ε

_{8}} = {−1.4,−1,−0.6,−0.2,0.2,0.6,1,1.4} and each permutation is chosen with equal probability.

## 4. Experimental Results

#### 4.1. The Analysis of Matches and Unraveling

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

Actual (SPE) % Firms Hiring EarlyIn 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 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.

H_{0} (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** |

#### 4.2. The Analysis of Early Offer and Acceptance Rates

#### 4.3. The Analysis of Efficiency

**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) |

H_{0} (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* |

#### 4.4. Further Consequences: Effects of Changing Labor Supply

**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.

**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.

## 5. Discussion

## Acknowledgements

^{1}See [2] for many examples, including markets other than labor markets in which contracts are fulfilled at around the same time but can be finalized substantially earlier, such as the market for college admissions, or for post-season college football bowls.^{4}Titled ”A Cure for Application Fever: Schools Hook More Students with Early Acceptance Offers.” (April 23).^{6}In any model in which offers are made over time, unraveling will occur at equilibrium only if early offers are made and accepted. The theoretical model closest in spirit to the one explored here is that of [24] in which firms and workers unravel to each insure the other against an outcome that leaves their side of the market in excess supply (in an assignment model in which agents on the long side of the market earn zero). [25] looks at a model with a continuum of agents in which unraveling is driven at equilibrium by the fact that it makes later contracting less desirable because of the difficulty of finding a match when everyone else contracts early. Unraveling as insurance is further explored in [26,27,28]. Other theoretical models have unraveling (under conditions of fixed supply and demand) determined by the competition for workers as determined by how correlated firms’ preferences ([29]), or by how well connected firms are to early information about workers’ qualities ([30]), or by the establishment of certain kinds of centralized matching mechanisms ([2,12,31]), or by strategic unraveling and uncertainty imposed by the negative externality caused by an early offer of an agent ([32]). In prior experimental studies, [13,21,33] look at unraveling as a function of what kinds of centralized market clearing mechanism are available at the time when matching may be done efficiently. [9] looks at unraveling as a function of the rules governing exploding offers in a decentralized market. [34] experimentally studies a stylized model of unraveling that can be interpreted as one in which best replies to others’ decisions about when to hire are early, but not too early, compared to the mean time at which competitors are trying to hire.^{7}There is an evolving literature on laboratory experiments regarding different aspects of matching markets. For example, [13,18,21,33], all report experiments on harmful unraveling and how centralized mechanisms can be used successfully or unsuccessfully to stop unraveling in entry-level professional labor markets. These papers report experiments using various mechanisms including stable ones in different environments. [35,36,37,38,39] report experiments on school choice mechanisms, including some stable ones, for public schools enrollment. New York and Boston school districts adopted stable mechanisms to correct market failures due to other reasons than unraveling. Other experiments on two-sided matching markets include [8,40,41,42] that report experiments on different features of decentralized matching markets.^{8}The usual difficulties of measuring supply and demand in the field are compounded when the supply of workers of the highest quality must be evaluated.^{9}The assortative matching is qualitywise-assortative, although the converse doesn’t hold. Note that the ε’s define an absolute standard of efficiency, but we look at qualitywise efficiency and sorting. This is because the "’s play three roles in our treatment: 1. as a technical assumption to give us uniqueness, 2. to reflect the fact that, in future field work, we expect the data might be able to distinguish quality in the large, but not preferences among applicants with similar observables (so that quality would be observable only up to the ε’s); and 3. to make clear in our model that we are not claiming that just because a market isn’t unraveled it is efficient, rather we are only claiming that avoiding unraveling avoids a large source of inefficiency, qualitywise inefficiency.^{10}That is, high quality firms (and applicants) have a strictly higher marginal expected payoff from increasing their partner’s quality than low quality firms (and applicants). I.e., if the production function regarding a firm and worker is the sum of the pay-offs of the firm and the worker, then it is supermodular.^{11}For example, a qualitywise efficient matching may be inefficient when the applicant with the largest " value remains unmatched while an applicant of the same quality is matched. Qualitywise efficiency or its absence is also what can generally be assessed from evaluating field data such as that obtained by using revealed preferences over choices in marriage or dating markets for estimating preferences over observables, i.e., such estimates can determine how important a potential mate’s education is in forming preferences, but cannot observe which among identically educated potential mates will have the best personal chemistry (see e.g., [45,46,47])^{12}For risk-neutral market participants we can equivalently assume that each applicant has a probability to be of high quality. We instead choose a fixed fraction to reduce variance in the experiment that follows.^{13}There is another asymmetry between cases 2 and 3a, namely that the firms make offers, and applicants accept or reject them. However, this will not be important to determine the SPEs. It may however be important empirically, in the experiment and in the field.^{14}Other results and proofs of this theorem and others regarding the experimental setup are in Appendix A.^{15}In their pioneering theoretical investigation of unraveling, [24] study an assignment market with continuous payoffs in which the supply and demand are assumed to always fall in Case 2. In the early period of their model each worker has a probability of being a productive worker in period 2 (and in period 2 all workers are either productive or unproductive, and only firms matched with a productive worker have positive output). In this context, their assumption that supply and demand fall in Case 2 is that there are more workers than firms, but a positive probability that there will be fewer productive workers than firms. They find, among other things, that inefficient unraveling is more likely “the smaller the total applicant pool relative to the number of positions.” Our framework allows us to see how this conclusion depends on supply and demand remaining in Case 2. When the total applicant pool declines sufficiently, the market enters Case 3a (when the number of workers falls below the number of firms), and inefficient unraveling is no longer predicted. Moreover, we find a characterization of supply and demand conditions necessary for sequentially rational unraveling.^{16}For the graphs and tables, for each session, for each of the markets, we compute the median of the variable in question, in this case the number of firms that hire early. We then compute the median of each market block, for markets 1–5, 6–10, 11–15, 16–20 for each session, and then the median for each market block taking these session medians as data points, to report the final variable.For example, in Session 7 of the thin comparable market treatment, in markets 1 to 5, 100%, 0%, 50%, 0%, and 0% of the high type firms are hiring early, respectively, which results in a 0% median hiring rate. In Sessions 1 to 6, medians are similarly calculated for markets 1–5 as 0%, 50%, 50%, 0%, 50%, and 50%, respectively. The median of these six sessions and Session 7 is 50%, which is marked in the top graph of Figure 1 for markets 1–5.^{17}In Appendix B, we report alternative analyses using means instead of medians. The figures and statistical test results are similar. In general, the mean statistics are noisier than medians due to the fact that the mean takes extreme outcomes into consideration for these samples with relatively small sizes.In the main text, we use medians instead of means for two reasons:First, our statistical test is an ordinal non-parametric median comparison test (i.e., Wilcoxon rank sum test) and not a cardinal parametric mean comparison test. We chose an ordinal test based on the small sample sizes, 7 or 4 for each treatment. Second, many of the empirical distributions are truncated. I.e., even if the efficiency measure is centered around 100%, there will be no observations above 100% while depending on the variance, we will observe lower efficiency levels. Since we use percentages to compare treatments of different size, the appropriate measure of the center of a distribution seems to be a median rather than the mean due to the inevitable skewedness of the empirical distributions.^{18}The mean early firm offer rates in the excess supply treatment are 45.00%, 5.00% 0.00%, and 5.00% within the market groups 1–5, 6–10, 11–15, and 16–20, respectively, and the mean acceptance rates are 100.00% 100.00% and 50.00% in market groups 1–5, 6–10, and 16–20, respectively. For example, 50% acceptance rate means 2.50% of the low firms on average hired early in the last market group on average.^{19}We could also have chosen the welfare of applicants or the sum of applicant and firm payoffs in our efficiency measure. Note that all of these measures will give the same efficiency level, since a firm and a worker who are matched receive exactly the same payoff.^{20}In each treatment, the best full matching is the qualitywise assortative matching, that is, as many high quality firms as possible are matched with high quality applicants, and as many high quality applicants as possible are matched before matching low quality applicants.^{21}The difference in the median of the comparable treatments is due to the small market size of the thin comparable treatment. The mean of both SPE predictions generates 71.38% efficiency. Under a typical matching (i.e., a median SPE matching), in the thick comparable treatment, one low quality firm hires a high quality applicant and the other three low quality firms hire low quality applicants through unraveling, where as in the thin comparable treatment, one low quality firm hires a high quality applicant and the other low quality firm hires a low quality applicant through unraveling.^{22}In our analysis with the averages for the last market block reported in Appendix B, we observe an average of 20–21% of high type firms making early offers which are accepted in the excess demand treatment. SPE predicts that no high quality firms hire early. However, in the excess demand condition, there are only 2 high quality applicants for 4 high quality firms. So, even if 50% of high quality firms were to hire early, efficiency would not be affected given that the other agents follow SPE strategies. And indeed, all seven of our excess demand sessions have close to average 100% efficiency despite of relatively high average of high type firms hiring early.^{23}The firm-optimal stable matching is a stable matching that makes every firm best off among all stable matchings. It always exists in the general Gale-Shapley [43] matching model, and in many of its generalizations (see [48]). In our model, there is a unique stable matching, which is therefore the firm-optimal stable matching.^{24}See [49] for a survey.^{26}This claim is stronger than what we need for the proof of this proposition. However, we will make use of this claim in the proof of Proposition 4.

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## A. Appendix: Further Theoretical Results, and Proofs

**Lemma 1**Any subgame perfect equilibrium produces assortative matching among the firms and applicants still unmatched at the beginning of the late hiring stage.

**Lemma 2**When (Case 1), and (Case 3a), the unique subgame perfect equilibrium outcome is late, assortative matching.

**Lemma 3**When (Case 3b), the outcome of any subgame perfect equilibrium is qualitywise-efficient.

**Proposition 3**In the case of comparable demand and supply, i.e., (Case 2), all high quality firms hire in the late stage under any subgame perfect equilibrium. At every subgame perfect equilibrium, at least one low quality firm hires early if and only if

**Proposition 4**In the case of comparable demand and supply with (Case 2), if

**Proof of Lemma 1:**First recall that there is a unique strict rank-ordering of firms and applicants with respect to preference with probability one, when all uncertainty is resolved at the beginning of the late hiring stage.

- We first show that in the last period T
^{L}of the late hiring stage, every SPE involves assortative matching among the remaining firms and applicants:- -
- In every subgame starting with applicants’ information sets in period T
^{L}of the late hiring stage, it is a dominant strategy for the applicant to accept the best incoming offer, since otherwise she will either remain unmatched or be matched with a worse firm. - -
- The best remaining unmatched firm, by making an offer to the best applicant, will be accepted by the applicant, and receive the highest possible payoff. The second best remaining firm, will be rejected if she makes an offer to the best applicant, since the applicant will get an offer from the best firm. The highest applicant who will accept the second best firms’ offer is the second best applicant. Similarly, the k
^{th}best remaining unmatched firm maximizes her payoff from making an offer to the k^{th}highest remaining unmatched applicant.

We showed that the outcome of any SPE will involve assortative matching of agents who are available in any last period subgame of the late hiring stage. - Let us assume that we showed for a period t + 1 ≤ T
^{L}that SPE strategies for any subgame starting in period t + 1 involve assortative matching among the remaining unmatched firms and applicants. We now show that this implies that for any subgame starting in period t the SPE involve assortative matching among the unmatched applicants in period t Let us relabel the remaining firms and applicants, such that the remaining firms are f^{1},f^{2},...,f^{m}and available applicants a^{1},a^{2},...,a^{n}such that f^{k}is better than f^{k+1}for any k < m and a^{k}is better than a^{k+1}for any k < n.- -
- We show that at any SPE in period t involves applicant a
^{1}not to accept an offer from firm f^{j}for any j > 1 if firm 1 id not make an offer to any applicant. By rejecting firm f^{j}, applicant a^{1}will be the highest quality remaining applicant in period t+1, and f^{1 }will be the highest quality unmatched firm. That is, by the inductive assumption, a^{1}can expect to be matched to firm f^{1}in the SPE. - -
- We show that at any SPE in period t, firm f
^{1},either makes no offer, or an offer to applicant a^{1}in period t. This guarantees that f^{1}is either accepted by applicant a^{1}in period t, or else both f^{1}and applicant a^{1}are unmatched in period t, in which case firm f^{1}will be matched to applicant a^{1}before the end of the game by the inductive assumption. - -
- We show that at any SPE in period tapplicant a
^{2}does not accept an offer from f^{j}, j > 2 if firms f^{1}and f^{2}both did not make an offer to an applicant a^{l}for any l > 2. As in the case of applicant a^{1}; by rejecting firm f^{j}, a^{2}can expect to be matched to either f^{1}or f^{2}in period t+1, since at least one of the two firms will be unmatched. - -
- We now show that any SPE in period tinvolves firm f
^{2}, not to make an offer to an applicant a^{j}, j > 2. We can follow this line of iterative argument to show that any SPE strategies in any subgame starting in period tinvolve no matches that are not assortative in period t.

**Proof of Lemma 2:**We prove the lemma for Case 1 and Case 3a separately.

- (Case 1): We already established by Lemma 1 that once participants are in the late hiring stage, the unique SPE outcome is assortative matching among the remaining firms and applicants. When , by not hiring any applicant in the early hiring stage each firm guarantees to hire a high quality applicant in the late hiring stage under a SPE. For a firm f of quality i ∈ {h,l}, her expected payoff of hiring an applicant in the early stage is given by
_{ih}. Therefore, under any SPE no firm will make any early hiring, and thus, by Lemma 1 the outcome will be assortative. - (Case 3a)
**:**By Lemma 1, once participants are in the late hiring stage, the unique SPE outcome is assortative matching among the remaining firms and applicants. Since n_{F }° n_{A}, under a SPE, every applicant will at least be matched with a low quality firm by waiting for the late hiring stage. We will show that no matches will occur in the early hiring stage under a SPE by backward iteration.First consider the last period (period T^{E})of the early hiring stage. We will show that no high quality firm is matched under a SPE in this period as long as more applicants than high quality firms are available. We prove this with two claims. Consider an information set Iof applicants located in this period.Claim 1 Under a SPE, no available applicant will accept a low quality firm’s offer in I if there is a high quality firm who did not make an offer in period T^{E}.Proof of Claim 1 Consider a SPE profile and an applicant aavailable in I . Suppose that there is at least one high quality firm that did not make an offer in T^{E}. Then the applicant has a chance to be of high quality and be matched to a high quality firm (of which at least one is available in the late hiring stage), if she is of low quality, she will receive a low quality firm by Lemma 1. Hence her expected payoff from waiting is strictly larger than her expected payoff from accepting a low quality firm offer in period T^{E}. □Consider a subgame starting in the last period (period T^{E}) of the early stage.Claim 2 Under a SPE, no high quality firm makes an offer to an applicant, unless the number of remaining applicants is equal to or smaller than the number of remaining high quality firms (in which case we do not determine the strategies fully).Proof of Claim 2 Suppose there are k^{*}high quality firms left, and k > k^{*}applicants. If a high quality firm f^{h}makes an offer to an applicant, it will hire her and this will be an average quality applicant. If firm f^{h}makes no offer this period, then we have seen that no applicant will accept an offer from a low quality firm. Suppose r_{H}high quality firms are left after the end of the early stage, including the high quality firm f^{h}.This implies that there are r_{A}= k - (k^{*}− r_{H}) > r_{H}applicants unmatched at the end of period T^{E}. Since firm f^{h}has equal chance to be ranked in any place amongst the remaining r_{H}high quality firms, and since by Lemma 1, any SPE matching in the late hiring stage is assortative, firm f^{h}, by not matching early, will match to one of the best r_{H}applicants in the remaining r_{A}, and has an equal chance to match with any of them. The average applicant’s quality amongst the r_{H}(<r_{A})best applicants of all r_{A}is strictly better than the unconditional average applicant quality. Therefore, f^{h}, by not making an offer, is matched with an applicant with an expected quality higher than the average, receiving higher expected earnings in an SPE than by making an early offer. □We showed that in the last period of the early hiring stage no matches will occur, if the number of high quality firms available is smaller than the number of applicants available under a SPE profile. By iteration, we can similarly prove that in period T^{E}- 1 of the early hiring stage no matches will occur, if the number of high quality firms available is smaller than the number of applicants available under a SPE profile. By backward iteration, we conclude the proof that no matches will occur in the early hiring stage under a SPE. Therefore, all hirings occur in the late hiring stage and these hirings are assortative under any SPE by Lemma 1. ■

**Proof of Lemma 3:**Let . By Lemma 1, every applicant guarantees to be matched with a high quality applicant by waiting for the late hiring stage under a SPE. Therefore, no applicant will accept an offer from a low quality firm in the path of a SPE. Therefore, all applicants will be matched with high quality firms in every SPE. Every such matching is qualitywise-efficient. ■

**Proof of Proposition 3:**Let the market be of comparable demand and supply with . Let σ be a SPE profile. We will prove the proposition using three claims:

_{F}high firms left, then there will be r

_{A}> r

_{F}applicants left (since n

_{A}> n

_{F}), and the high firm, instead of receiving the average quality of all r

_{A}receives the average quality of best r

_{F}out of r

_{A}applicants, which is strictly better. □

**Proof of Proposition 4:**Let the market be of comparable demand and supply with . Let σ be a SPE profile. By Claim 1 of Proposition 3, no high quality firm will hire early under σ. Since , no low quality firm will go late under σ as long as their offers are accepted early by Claim 3 of Proposition 3.

_{A}> n

_{F}by Lemma 1. Suppose that the best offer is from a low quality firm in period t and suppose that she rejects all offers in I under σ. Let Γ be the subgame such that information set I is located at the beginning of Γ. Let be the number of high quality firms available, be the number of low quality firms available, and r

_{A}be the number of applicants available in the late hiring stage under σ restricted to Γ. By Lemma 1, applicant a will be matched with a high quality firm if she turns out to be one of the best applicants, she will be matched with a low quality firm if she turns out to be one of the applicants who are not among the best applicants. Under σ restricted to Γ, applicant a’s expected payoff is

_{h}is the expected payoff a gets by being matched with a high quality firm and w

_{l}is the expected payoff she gets by being matched with a low quality firm. The lowest upper-bound for w

_{h}is u

_{hh}, and the lowest upper-bound for w

_{l}is u

_{hl}. Thus, v

^{σ}is bounded above by

_{A}, satisfies , we have

_{A}, we obtain that

**Proof of Proposition 1:**By Lemma 1, there is a unique SPE outcome that is assortative among available firms and applicants in the late hiring stage. When there are more available applicants and not less high quality applicants, each firm’s partner quality weakly increases. Hence, each firm’s expected payoff weakly increases. ■

**Table 6.**Average percentage of firms hiring early in the last five markets, with subgame perfect equilibrium predictions in parentheses.

Actual (SPE) % Firms Hiring EarlyIn the last five markets - Averages | THIN COMPARABLE | EXCESS SUPPLY |

Low Firms | 77.14%
(100%) | 2.5%
(0%) |

High Firms | 7.14% (0%) | 20% (0%) |

EXCESS DEMAND | THICK COMPARABLE | |

Low Firms | 6.43% (0%) | 87.5% (100%) |

High Firms | 20.71% (0%) | 1.25% (0%) |

**Table 7.**Testing equivalence of high firm and low firm early hiring percentages in the last five markets.

H_{0} (For average % high/low firms hiring early) | sample sizes | p-value: High | p-value: Low |
---|---|---|---|

Thin Comparable = Thick Comparable | 7,4 | 1 | 0.33 |

Excess Supply = Excess Demand | 4,7 | 0.86 | 0.64 |

Thin Comparable = Excess Supply | 7,4 | 0.56 | <0.01** |

Thin Comparable = Excess Demand | 7,7 | 0.085 | <0.01** |

Thick Comparable = Excess Supply | 4,4 | 0.43 | 0.029* |

Thick Comparable = Excess Demand | 4,7 | 0.15 | <0.01** |

**Proof of Proposition 2:**We prove the proposition with an example. Let n

_{F}= 4, , n

_{A}= 4, , u

_{hh}= 36, u

_{hl}= u

_{lh}= 26, u

_{ll}= 20. Since , this market satisfies demand and supply condition Case 1, and therefore by Lemma 2, all SPE outcomes are assortative. So each high quality firm’s expected payoff is u

_{hh}. Suppose that the number of applicants increases to and number of high quality applicants does not change. The new market is the thin comparable market treatment in our experiment,i.e., a market of comparable demand and supply (Case 2). By Proposition 3 since , and , all low quality firms hire early under a SPE, causing that a high quality firm gets matched with a low quality applicant with positive probability. This expected payoff of a high quality firm is lower than u

_{hh}in the new market under any SPE. ■

## B. Appendix: Alternative Analyses with the Means

**Table 8.**Average 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 - Averages | THIN COMPARABLE | EXCESS SUPPLY |

Actual Firm Welfare [Random, Max.] | 109.31 [102.67, 112] | 121.2 [102.67, 124] |

Actual (SPE)N. Efficiency = | 71.21% (71.38%) | 86.87% (100%) |

EXCESS DEMAND | THICK COMPARABLE | |

Actual Firm Welfare [Random, Max.] | 163.66 [154, 164] | 219.8 [205.34, 224] |

Actual (SPE) N.Efficiency= | 96.57% (100%) | 77.49% (71.38%) |

H_{0} (For average n. efficiency) | sample sizes | p-value |
---|---|---|

Thin Comparable = Thick Comparable | 7,4 | 0.65 |

Excess Supply=Excess Demand | 4,7 | 0.94 |

Thin Comparable=Excess Supply | 7,4 | 0.28 |

Thin Comparable=Excess Demand | 7,7 | 0.014* |

Thick Comparable=Excess Supply | 4,4 | 0.31 |

Thick Comparable=Excess Demand | 4,7 | 0.012* |

## C. Appendix: Instructions of the Experiment

#### WELCOME

- When a HIGH quality firm hires a HIGH quality applicant, on average the firm and the applicant each get 36 points.
- When a HIGH quality firm hires a LOW quality applicant, OR a LOW quality firm hires a HIGH quality applicant, on average the firm and the applicant each get 26 points.
- When a LOW quality firm hires a LOW quality applicant, on average the firm and the applicant each get 20 points.
- If you end up unmatched you do not earn any points.

**Exactly how are applicants’ qualities determined?**

**How is the exact ranking among HIGH and LOW quality firms and applicants determined?**

#### Making and accepting offers

**Firms’ decisions in each period:**

**Applicants’ decisions in each period:**

#### The Information on the Screen of Applicants and Firms:

#### The Information on the Screen of Firms:

**Firm screen: Period 5–8**

#### The Information on the Screen of Applicants:

**Applicant screen: Periods 5–8:**

#### Payment

#### Summary

- Periods 1–4: Each applicant has equal chance to be a HIGH quality applicant (4 of the 12 that is 1 in 3 applicants will be of HIGH quality) and equal chance to be a LOW quality applicant (8 of the 12 that is 2 in 3 applicants will be of LOW quality).
- Periods 5–8: At the beginning of period 5, four applicants become HIGH quality and the remaining eight applicants become LOW quality applicants.
- Periods 5–8: Firms and applicants are assigned small fit scores. A HIGH quality partner with the lowest fit is still much more desirable than a LOW quality partner with the highest fit. But the higher the fit score, the more desirable that partner is.

- In each period, each firm that has not yet hired an applicant, has to decide whether to make an offer and, if so, to which applicant. Each firm can only make one offer in each period, and only to applicants who have not accepted an offer yet.
- In each period, applicants who receive offers have to decide whether to accept or reject the offer.
- Once an applicant accepted an offer, he cannot accept another offer in the same market, and will no longer receive offers.
- Firms and Applicants that are not matched by the end of period 8 in a market remain unmatched and earn zero points.
- In a HIGH quality-HIGH quality match, firms and applicants each earn 36 points plus the fit scores of their partner.
- In a HIGH quality-LOW quality match, firms and applicants each earn 26 points plus the fit scores of their partner.
- In a LOW quality-LOW quality match, firms and applicants each earn 20 points plus the fit scores of their partner.
- After period 8, a completely new market begins, and everyone is free to try to match once again.
- The experiment has 20 consecutive markets each with 8 periods length.

© 2013 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Niederle, M.; Roth, A.E.; Ünver, M.U.
Unraveling Results from Comparable Demand and Supply: An Experimental Investigation. *Games* **2013**, *4*, 243-282.
https://doi.org/10.3390/g4020243

**AMA Style**

Niederle M, Roth AE, Ünver MU.
Unraveling Results from Comparable Demand and Supply: An Experimental Investigation. *Games*. 2013; 4(2):243-282.
https://doi.org/10.3390/g4020243

**Chicago/Turabian Style**

Niederle, Muriel, Alvin E. Roth, and M. Utku Ünver.
2013. "Unraveling Results from Comparable Demand and Supply: An Experimental Investigation" *Games* 4, no. 2: 243-282.
https://doi.org/10.3390/g4020243