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Exploding offers and unraveling in two-sided matching markets

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Abstract

Many two-sided matching markets tend to unravel in time with transactions becoming inefficiently early. In a two-period decentralized model, this paper shows that when a market culture allows firms to make exploding offers, unraveling is more likely to occur and lead to a less socially desirable matching outcome. A market with a larger uncertainty in early stages is not necessarily more vulnerable to the presence of exploding offers: the conclusion depends on the specific information structure. A market tends to be less vulnerable to exploding offers when there is an excess supply of labor. While a banning policy on exploding offers tends to be supported by high quality firms and workers, it can be opposed by those of low qualities. This explains the prevalence of exploding offers in practice.

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Notes

  1. Roth and Xing (1994) provide a detailed overview of various evidence for market unraveling. See also Mongell and Roth (1991), Haruvy et al. (2006), Avery et al. (2009), Avery et al. (2001), and Fréchette et al. (2007).

  2. For example, see a discussion regarding the US market for new gastroenterologists by Niederle and Roth (2005).

  3. Clearly, this is a simplifying assumption. The similarity of preferences is also a factor that can affect market unraveling. Hałaburda (2010) considers the similarity of firms’ preferences over workers as a comparative statics parameter while having all workers agree on the same ranking of firms. The result shows that similarity of preferences can drive unraveling.

  4. With a linear quality distribution, the main results of the paper do not hinge on a specific quality range. However, some results may not hold for other quality distributions. Quality distribution is also a factor that can affect market unraveling. An example that allows non-linear quality distributions can be found in the working paper version of this paper (http://siqipan.weebly.com/uploads/5/1/4/8/51488789/exploding_offers_workingpaper0801.pdf). It suggests that a market is less likely to be affected by exploding offers if the quality distribution over firms is more convex, or the quality distribution over workers is more concave. With two possible qualities on each side of the market, Niederle et al. (2013) show in a lab experiment that unraveling only occurs when demand and supply are comparable, that is, when there exist excess workers, but a shortage of high quality workers.

  5. The reality in some markets is less stringent than Assumption 2. For example, in the job market for junior economists, although when rejecting an exploding offer, a candidate typically does not consider the possibility that she may receive an offer from the same employer again, the phenomenon of nonbinding rejections is still observed in some situations. In this case, a more realistic setting is to have each firm decide whether to raise its leverage by attaching a commitment of binding rejection when making an early exploding offer. For these markets, although Assumption 2 significantly simplifies the analysis, it can lead to an overestimation of the effects of exploding offers.

  6. For example, exploding offers are publicly discouraged in the US market for new graduate students. The Council of Graduate Schools has published a resolution stating that students are under no obligation to respond to offers of financial support prior to April 15 (http://cgsnet.org/april-15-resolution).

  7. In Sect. 3.3, I provide an example of a different information structure, under which the probability of a ranking gradually decreases as its Kendall \(\tau \) distance to the signal-suggested ranking increases.

  8. Although the cost of rejecting a non-signal-suggested firm \(\frac{1-\alpha }{W}\) increases as the signal becomes less accurate, here the elimination of such a deviation is not the binding condition and does not drive the results. In Sect. 3.3, I provide an example with a different information structure, under which the non-signal-suggested pairs can also influence the binding condition, and thus signal accuracy may have the opposite effect on market unraveling.

  9. Recall that relevant firms include all firms as \(W\ge F\), and only firms \(f_{F}\), \(f_{F-1}\),..., and \(f_{F-W+1}\) as \(W<F\).

  10. See Kendall (1938) and Kemeny (1959).

  11. See Eqs. (14) and (16) in the appendix for the expressions of \(\underline{j^{A}}\) (the lowest quality firm whose early offer is accepted by its signal-suggested type) and \(\overline{j^{O}}\) (the highest quality firm willing to make such an offer given the acceptance) respectively. Signal accuracy \(\alpha \) only enters the former but not the latter.

  12. Admittedly, this paper only provides an example where information structure could alter the comparative statics on signal accuracy. It is not to say that such a conclusion will hold for every market environment.

  13. Firms \(f_{1},f_{2},\ldots ,f_{F-W}\) are indifferent since they are unmatched in both cases.

  14. There is an abuse of language here since \(\hat{j}\) is not necessarily an integer.

  15. Another boundary case would be that a type-\(\hat{r}\) worker does not accept any offer, which never holds because \(f_{F}\) will always be accepted, that is, \(\underline{j_{1}^{A}}(\hat{r})<F,\forall \hat{r}\).

  16. There is another boundary where \(\overline{j_{1}^{O}}(\hat{r})>F\), which never holds since \(f_{F}\) never wants to deviate.

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Correspondence to Siqi Pan.

Additional information

I would like to thank Paul J. Healy, Yaron Azrieli, Clayton Featherstone, Alex Gotthard-Real, John Kagel, James Peck, Garrett Senney, Wing Suen, Bruce Weinberg, Huanxing Yang, Lixin Ye, seminar participants at the Ohio State University, and audiences at Midwest Economic Theory Conference for helpful comments and inspiration. All remaining errors are mine.

Appendix: Proofs

Appendix: Proofs

1.1 Proof of Proposition 1

Proof

Consider a worker who has received an early open offer from firm \(f_{j}\) (\(j=1,2,\ldots ,F-1\)). As long as a better firm \(f_{j'}\) (\(j'>j\)) moves in a later period in equilibrium, she strictly prefers holding to accepting or rejecting the offer right away.

Firstly, holding is preferred to accepting. By choosing to hold but not to accept, the worker is strictly better off if she receives a better offer in a later period. If not, she is not worse off since she still has the open offer from \(f_{j}\). Secondly, holding is preferred to rejecting. By choosing to hold but not to reject, the worker is not worse off if she receives a better offer in a later period; she is strictly better off otherwise.

Knowing this, the best firm \(f_{F}\) strictly prefers to wait until the last period, so that all workers will stay in the market and the one of the best quality can be perfectly identified. Since no offer is accepted in Period 1, the other firms cannot make themselves better off by moving early; instead, they incur the risk of being rejected in the last period and remaining unmatched.

Hence, there always exist profitable deviations from an equilibrium with partial or full unraveling. Only an equilibrium without unraveling can sustain. It yields the assortative matching according to the true ranking of workers \(\succ \), which is the unique stable matching in the current environment with strict rankings and aligned preferences. \(\square \)

1.2 Proof of Lemma 1

Proof

When making an offer in Period 2, every firm is indifferent because an open offer is equivalent to an exploding offer. Now consider an offer in Period 1. The best firm \(f_{F}\) is still indifferent since neither an exploding offer nor an open offer will be rejected by any worker. However, for other firms, making an open offer is never the strictly best response.

First, when making an exploding offer that has to be accepted within the same period, a firm always knows whether it will be accepted. This is because there is no information asymmetry in the current setting, and the information status remains the same within a period.

Next, if an exploding offer will be accepted, making an exploding offer yields the same or a higher payoff than making an open offer. Suppose a firm makes an open offer in Period 1, there are three possible responses: (i) it is accepted right away; (ii) it is held and accepted in Period 2; (iii) it is held and rejected in Period 2. Compared to an exploding offer, the firm yields the same payoff in cases (i) and (ii), but is strictly worse off in case (iii).

Finally, if an exploding offer will be rejected, waiting yields the same or a higher payoff than making an exploding offer or an open offer. A firm never wants to make an exploding offer knowing it will be rejected because then it cannot make an offer to the same worker again. On the other hand, if a firm makes an open offer in Period 1, there are three possible responses: (i) it is rejected right away; (ii) it is held and accepted in Period 2; (iii) it is held and rejected in Period 2. Compared to waiting, the firm is strictly worse off in cases (i) and (iii), and is weakly worse off in case (ii). \(\square \)

1.3 Proof of Proposition 2

Proof

Below I first calculate the updated beliefs of firms and workers after they observe a signal in Period 1.

After observing a signal \(\hat{\succ }\), posteriors on the true state are given by

$$\begin{aligned} \Pr (\hat{\succ }\mid \hat{\succ })= & {} \frac{\left[ \alpha +(1-\alpha )\frac{1}{W!}\right] \frac{1}{W!}}{\left[ \alpha +(1-\alpha )\frac{1}{W!}\right] \frac{1}{W!}+(W!-1)\left[ (1-\alpha )\frac{1}{W!}\right] \frac{1}{W!}}\\= & {} \alpha +\frac{1-\alpha }{W!}. \end{aligned}$$

For any \(\succ ^{\prime }\ne \hat{\succ }\),

$$\begin{aligned} \Pr (\succ ^{\prime }\mid \hat{\succ }) = \frac{1-\frac{\alpha W!+(1-\alpha )}{W!}}{W!-1}=\frac{1-\alpha }{W!}. \end{aligned}$$

Posteriors on types are given by

$$\begin{aligned} \Pr (\hat{r}\mid \hat{r})= & {} \Pr (\hat{\succ }\mid \hat{\succ })+\left[ (W-1)!-1\right] \Pr (\succ ^{\prime }\mid \hat{\succ })\\= & {} \alpha +\frac{1-\alpha }{W}, \end{aligned}$$

and \(\forall r^{\prime }\ne \hat{r}\)

$$\begin{aligned} \Pr (r^{\prime }\mid \hat{r})=\frac{1-\left( \alpha +\frac{1-\alpha }{W}\right) }{W-1}=\frac{1-\alpha }{W}. \end{aligned}$$

Thus, the expected quality of a type-\(\hat{r}\) worker is

$$\begin{aligned} EV(\hat{r})=\left( \alpha +\frac{1-\alpha }{W}\right) \hat{r}+\sum _{r^{\prime }\ne \hat{r}}\frac{1-\alpha }{W}r^{\prime }=\alpha \hat{r}+\frac{\left( 1-\alpha \right) \left( W+1\right) }{2}. \end{aligned}$$

Next, consider the case where \(W>F\). In an equilibrium without unraveling, no actions are taken in Period 1. In Period 2, after \(\succ \) is revealed, \(f_{F}\) makes an offer to \(w_{W}\), \(f_{F-1}\) to \(w_{W-1}\), ..., and \(f_{1}\) to \(w_{W-F+1}\). All offers are accepted.

It is clear that workers do not have any incentive to deviate, nor does firm \(f_{F}\). Given all others are playing the equilibrium strategy, a firm \(f_{j}\) with \(j=1,2,\ldots ,F-1\) will not deviate and make an offer to a different worker in Period 2, since it will not be accepted by a worker better than its current match \(w_{j-F+W}\). I now focus on checking the deviation of \(f_{j}\) in Period 1. Such a deviation involves both sides of the market: a firm should want to make an early exploding offer to a worker who wants to accept it. So a sufficient condition for the existence of an equilibrium without unraveling is that, for each worker type \(\hat{r}\), the firms whose offer would be accepted are not willing to offer.

Suppose \(f_{j}\) deviates by making an early offer to a type-\(\hat{r}\) worker in Period 1, and \(\hat{r}\ne \hat{r}(j)\). The offer is accepted if

$$\begin{aligned} j\ge \left( \alpha +\frac{1-\alpha }{W}\right) \times j(\hat{r})+\frac{1-\alpha }{W}\times \left( \sum _{r^{\prime }=W-F+1}^{W}j(r^{\prime })-j(\hat{r})\right) -\frac{1-\alpha }{W}\times 1, \end{aligned}$$
(6)

or equivalently,

$$\begin{aligned} j\ge \underline{j_{1}^{A}}(\hat{r})\equiv \alpha \times \left( \hat{r}-W+F\right) +\frac{\left( 1-\alpha \right) \left( F^{2}+F-2\right) }{2W}. \end{aligned}$$
(7)

The function \(\underline{j_{1}^{A}}(\hat{r})\) is defined as the lowest ranked firm that is accepted by type \(\hat{r}\). On the other hand, if accepted, the firm is willing to make such an offer if

$$\begin{aligned} j+W-F\le EV(\hat{r}), \end{aligned}$$
(8)

or equivalently,

$$\begin{aligned} j\le \overline{j_{1}^{O}}(\hat{r})\equiv \alpha \hat{r}+\frac{\left( 1-\alpha \right) \left( W+1\right) }{2}-W+F. \end{aligned}$$
(9)

The function \(\overline{j_{1}^{O}}(\hat{r})\) is defined as the highest ranked firm that wants to make an early offer to type \(\hat{r}\). The sufficient condition for no deviation in this case is that for each type, there does not exist a firm that is willing to offer, and is accepted. That is, \(\forall \hat{r}\), we need to have

$$\begin{aligned} \underline{j_{1}^{A}}(\hat{r})\ge \overline{j_{1}^{O}}(\hat{r})\quad \hbox { and }\quad \overline{j_{1}^{O}}(\hat{r})\ge 1, \end{aligned}$$
(10)

or

$$\begin{aligned} \overline{j_{1}^{O}}(\hat{r})<1, \end{aligned}$$
(11)

which solves

$$\begin{aligned} W\ge 2+F. \end{aligned}$$
(12)

Equation (10) ensures that there does not exist a j such that \(\underline{j_{1}^{A}}(\hat{r})\le j\le \overline{j_{1}^{O}}(\hat{r})\). Equation (11) is a boundary condition where no firms are willing to make an offer to a type-\(\hat{r}\) worker.Footnote 15

Now we consider the deviation of a firm \(f_{j}\) to its signal-suggested type \(\hat{r}(j)\). The offer is accepted if

$$\begin{aligned} j\ge & {} \left( \alpha +\frac{1-\alpha }{W}\right) \times j(\hat{r}(j))+\frac{1-\alpha }{W}\nonumber \\&\times \left( \sum _{r^{\prime }=W-F+1}^{W}j(r^{\prime })-j(\hat{r}(j))\right) -\left( \alpha +\frac{1-\alpha }{W}\right) , \end{aligned}$$
(13)

or equivalently,

$$\begin{aligned} j\ge \underline{j_{2}^{A}}\equiv \frac{F^{2}+F-2}{2W}-\frac{\alpha }{1-\alpha }. \end{aligned}$$
(14)

If accepted, the firm is willing to make such an offer if

$$\begin{aligned} j+W-F\le EV(\hat{r}(j)), \end{aligned}$$
(15)

or equivalently,

$$\begin{aligned} j\le \overline{j_{2}^{O}}\equiv \frac{1}{2}+F-\frac{W}{2}. \end{aligned}$$
(16)

The sufficient condition for no deviation in this case is that there does not exist a firm willing to make an offer to its signal-suggested type, and is accepted. That is,

$$\begin{aligned} \underline{j_{2}^{A}}\ge \overline{j_{2}^{O}}\quad \hbox { and }\quad \overline{j_{2}^{O}}\ge 1, \end{aligned}$$

or

$$\begin{aligned} \overline{j_{2}^{O}}<1. \end{aligned}$$

Combining with (12), the two sufficient conditions when \(W>F\) are given by (i) \(W\ge 2F\), or (ii) \(F+2<W\le 2F-1\) and \(\alpha \le \frac{(W-F)^{2}-W+F-2}{(W-F)^{2}+W+F-2}\).

Now I move on to the case where \(W\le F\). In an equilibrium without unraveling, no actions are taken in Period 1. In Period 2, after \(\succ \) is revealed, \(f_{F}\) makes an offer to \(w_{W}\), \(f_{F-1}\) to \(w_{W-1}\), ..., and \(f_{F-W+1}\) to \(w_{1}\). All these offers are accepted.

Suppose a firm \(f_{j}\) deviates by making an early offer to its signal-suggested type \(\hat{r}(j)=j-F+W\). The offer is accepted if

$$\begin{aligned} j\ge \left( \alpha +\frac{1-\alpha }{W}\right) \times j(\hat{r}(j))+\frac{1-\alpha }{W}\times \left( \sum _{r^{\prime }=1}^{W}j(r^{\prime })-j(\hat{r}(j))\right) -\left( \alpha +\frac{1-\alpha }{W}\right) , \end{aligned}$$
(17)

or equivalently,

$$\begin{aligned} j\ge \underline{j_{3}^{A}}\equiv \frac{2F-W+1}{2}-\frac{1}{W}-\frac{\alpha }{1-\alpha }. \end{aligned}$$
(18)

The firm wants to make such an offer if

$$\begin{aligned} j+W-F\le EV(\hat{r}(j)), \end{aligned}$$
(19)

or equivalently,

$$\begin{aligned} j\le \overline{j_{3}^{O}}\equiv \frac{1}{2}+F-\frac{W}{2}. \end{aligned}$$
(20)

It is easy to show that \(\underline{j_{3}^{A}}<\overline{j_{3}^{O}}\), that is, the sufficient condition for no deviation never holds for \(W\le F\).

Therefore, the equilibrium without unraveling always exists if (i) \(W\ge 2F\), or (ii) \(F+2<W\le 2F-1\) and \(\alpha \le \frac{(W-F)^{2}-W+F-2}{(W-F)^{2}+W+F-2}\). \(\square \)

1.4 Proof of Proposition 3

Proof

For a deviation from the equilibrium without unraveling to occur, there must exist a type of worker and a firm in Period 1 such that, the firm is willing to offer and the worker is willing to accept.

When \(W>F\), for a deviation between a non-signal-suggested pair to exist, we need \(\exists \hat{r}\) such that

$$\begin{aligned} \overline{j_{1}^{O}}(\hat{r})-\underline{j_{1}^{A}}(\hat{r})>1\hbox { if }\underline{j_{1}^{A}}(\hat{r})\ge 0, \end{aligned}$$
(21)

or

$$\begin{aligned} \overline{j_{1}^{O}}(\hat{r})>1\hbox { if }\underline{j_{1}^{A}}(\hat{r})<0. \end{aligned}$$
(22)

Equation (21) ensures that the range between \(\overline{j_{1}^{O}}(\hat{r})\) and \(\underline{j_{1}^{A}}(\hat{r})\) is larger than 1, so that there always exists an integer in between. Equation (22) is a boundary case where \(\underline{j_{1}^{A}}(\hat{r})<0\). Then we need the range to be even larger so that at least \(f_{1}\) is willing to offer.Footnote 16 It is easy to confirm that the two conditions above never hold.

Similarly, for a deviation between a signal-suggested pair to exist, we need

$$\begin{aligned} \overline{j_{2}^{O}}-\underline{j_{2}^{A}}>1\quad \hbox { if }\underline{j_{2}^{A}}\ge 0, \end{aligned}$$
(23)

or

$$\begin{aligned} \overline{j_{2}^{O}}>1\quad \hbox { if }\underline{j_{2}^{A}}<0, \end{aligned}$$
(24)

which solves \(F<W<2F-1\) and \(\alpha >\frac{(W-F)^{2}+W+F-2}{(W-F)^{2}+3W+F-2}\).

When \(W\le F\), for a deviation between a signal-suggested pair to exist, we need

$$\begin{aligned} \overline{j_{3}^{O}}-\underline{j_{3}^{A}}>1\quad \hbox { if }\underline{j_{3}^{A}}\ge 0, \end{aligned}$$
(25)

or

$$\begin{aligned} \overline{j_{3}^{O}}>1\quad \hbox { if }\underline{j_{3}^{A}}<0, \end{aligned}$$
(26)

which solves \(W\le F\) and \(\alpha >\frac{W-1}{2W-1}\).

Therefore, the equilibrium without unraveling never exists if (i) \(W\le F\) and \(\alpha >\frac{W-1}{2W-1}\); or (ii) \(F<W<2F-1\) and \(\alpha >\frac{(W-F)^{2}+W+F-2}{(W-F)^{2}+3W+F-2}\). \(\square \)

1.5 Proof of Proposition 4

Proof

Consider the case where \(W>F\).

In an equilibrium with full unraveling, after \(\hat{\succ }\) is revealed in Period 1, \(f_{F}\) makes an offer to type \(\hat{r}=W\), \(f_{F-1}\) to \(\hat{r}=W-1\),..., and \(f_{1}\) to \(\hat{r}=W-F+1\). All offers are accepted. Such an equilibrium can never sustain because after Period 1, there are \(W-F\) workers left in the market. Given all the other firms move early, each firm has an incentive to deviate to Period 2, in which case it becomes the only firm left in the market and can choose the best remaining worker.

Consider the case where \(W\le F\).

In an equilibrium with full unraveling, after \(\hat{\succ }\) is revealed in Period 1, \(f_{F}\) makes an offer to type \(\hat{r}=W\), \(f_{F-1}\) to \(\hat{r}=W-1\), ..., and \(f_{F-W+1}\) to \(\hat{r}=1\). All these offers are accepted. No workers are left in the market after the first period. Therefore, a firm has no incentive to deviate as long as in its deviation, no worker would reject her current offer and become available in Period 2. That is, in the subgame after any firm’s deviation, all workers still accept their offers in Period 1.

Suppose a firm \(f_{j'}\) deviates and waits until Period 2, \(j'=F-W+1,F-W+2,\ldots ,F\). A worker of type \(\hat{r}\) would still accept her current offer if \(j'\le j(\hat{r})\), that is, a worker would never deviate for a firm that is worse than her offer in equilibrium, which is from her signal-suggested firm \(j(\hat{r})\). The binding condition for the existence of an equilibrium with full unraveling then requires type \(\hat{r}=1\) not to unilaterally reject her offer in Period 1 in the deviation of \(f_{F}\), that is,

$$\begin{aligned} F-W+1\ge \frac{1-\alpha }{2}F+\left( \alpha +\frac{1-\alpha }{2}\right) (F-W). \end{aligned}$$
(27)

The RHS of (27) is the worker’s payoff if she accepts her offer in equilibrium. The LHS is the worker’s expected payoff if she rejects. In this case, after the first period, there are two workers (\(\hat{r}=1\) and \(\hat{r}=W\)) and \(F-W+2\) firms (\(f_{F}\), \(f_{F-W+1}\), \(f_{F-W}\), \(f_{F-W-1}\), ..., and \(f_{1}\)) left in the market. Type \(\hat{r}=1\) is matched with \(f_{F}\) if she turns out to have a higher quality than \(\hat{r}=W\), and is matched with \(f_{F-W}\) otherwise since \(f_{F-W+1}\) is no longer available to her after the rejection. Compared to a higher type, the probability of a worker having a higher quality in the true state is given by \(\frac{1-\alpha }{W!}\times \frac{W!}{2}=\frac{1-\alpha }{2}\). Equation (27) solves

$$\begin{aligned} \alpha \ge \frac{W-2}{W}. \end{aligned}$$

Together with the constraint \(W\le F\), an equilibrium with full unraveling exists if \(W\le F\) and \(\alpha \ge \frac{W-2}{W}\).

On the other hand, when \(W\le F\) and \(\alpha <\frac{W-2}{W}\), an equilibrium with full unraveling never exists since type \(\hat{r}=1\) has an incentive to reject her offer in Period 1 in the deviation of \(f_{F}\), which makes \(f_{F}\) strictly prefer to deviate. Combined with the fact that such an equilibrium never exists when \(W>F\), we obtain the sufficient and necessary condition for the existence of an equilibrium with full unraveling: \(W\le F\) and \(\alpha \ge \frac{W-2}{W}\). \(\square \)

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Pan, S. Exploding offers and unraveling in two-sided matching markets. Int J Game Theory 47, 351–373 (2018). https://doi.org/10.1007/s00182-017-0593-7

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