Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling in Matching Markets Peter Coles Harvard Business School Muriel Niederle Stanford University and NBER

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

‘A Curious Phenomenon’ Colleges, when they decide whether to admit students, look for any information that seems to convey interest (campus visits, legacy...) Colleges, when deciding whom to send application materials and court, gather all possible evidence to get some idea of how interested the student is in coming. To whom should we send one our expensive, glossy application materials?

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

‘Show Them You Care’ “For Kaavya Viswanathan, a high-school senior in Hackensack, N.J., applying to college has involved some serious schmoozing with admissions officials. After narrowing her choices to nine supercompetitive colleges, including Harvard and Yale Universities, she began sending personal e-mail messages and calling admissions representatives at each institution to let them know how serious she was about attending their college. “Beginning last year, she made sure she sent messages to admissions staff members at all nine colleges at least once a month, and she is on a first-name basis with many of them.”

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling “a la Spence” Applicants (students...) have private information about their type (e.g. I am a Stanford student.) The applicant’s type directly affects the utility of the college (employer) (Stanford has a much higher utility from admitting a Stanford student, than, say a Harvard student.) Signaling can help colleges to learn about the students’ types. Signaling can have a positive effect on the final matching.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Outline 1. Introduction 2. Simple Signaling Example 3. Theory 4. AEA Job Market Signaling

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

A Market without Frictions In a simple world, where firms make offers, applicants accept and reject them, and there is no friction: Firms can make offers following the deferred acceptance algorithm. That is, at any point in time, firms make an offer to their most preferable applicant who has not rejected them yet. (And applicants collect offers, keep the one they like the most, and wait for a better offer).

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Path to a Stable Matching Stability: No pair of agents is disatisfied with their outcome to the extent that they would prefer to leave their respective partners and pair with each other. When firms make offers, it may take a long time to get to a stable matching. • Roth & Xing (’97) Market for clinical psychologists. • Segal: Minimal information

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Congestion In practice, we fall short of this goal due to a number of possible frictions, which we term ‘congestion.’ 1. time may run out 2. time may run out in the sense that workers may begin accepting offers from other firms 3. offers may be costly (a) physically costly (b) limited # of offers (c) opportunity cost of offers (1 & 2)

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Offer Tradeoffs in Congested Markets Fundamental tradeoff: Often a lower quality worker may be more likely to accept. While determining the quality of a worker may be a challenge, determining the likelihood of acceptance may be even more difficult.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

This Paper: Preference Signaling Signaling as a way to transmit information about the willingness to accept an offer and hence reduce market congestion. In this paper: Model and analyze a mechanism that allow workers to (credibly) transmit information about their preferences. Information about willingness to accept an offer does not necessarily change the utility a firm has from matching to that worker, but affects the value of making an offer to that worker.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

A Simple Example 2 firms, 2 workers, 1 offer Firms receive wh , wl , 0 Workers receive fh , fl , 0 . . . if they match to first choice, second choice, or not at all. Preferences are strict, but uncorrelated.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

“No Signal” Model 1. Preferences realized 2. Firms make offers 3. Workers accept best offer ⇒ Firms make offer to top choice workers.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling Model 1. Preferences realized 2. Each worker sends a signal to one firm 3. Firms make offers 4. Workers accept best offer

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling Model Observe that when a firm receives 2 signals, no signals, or only a signal from its top ranked worker, then it makes an offer to its top ranked worker. Interesting case: Only signal is from 2nd ranked worker. Tradeoff: Top ranked worker vs “guaranteed” 2nd ranked worker. We only need consider two strategies: respond and ignore.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling Model Let us consider firm 1’s payoffs when he has a signal only from his 2nd choice worker: respond

ignore

respond

wl

wl

ignore

0

.5wh

Note: (ignore, ignore) yields outcomes identical to no signaling case.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Signaling Model respond

ignore

respond

wl

wl

ignore

0

.5wh

(respond, respond) is always an equilibrium. (If firm 2 is responding, firm 1 must respond!) If .5wh > wl , then (ignore, ignore) is also an equilibrium.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Observations from simple example respond

ignore

respond

wl

wl

ignore

0

.5wh

1. “Respond” has a negative externality on other firms 2. Signaling allows for greater # of expected matches 3. Workers are better off 4. Firms may or may not be better off

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Observation: Signaling in Congested Markets. Example #1: Clinical Psychologists (Roth & Xing 1997) From 1973 − 1998: uniform regime where offers and acceptances all made in a short time frame - in some years, just one day. An on site visit in 1993: “On the morning of selection day (9 a.m. to 4 p.m.) a program with 5 positions had a ranked order list of 20 acceptable candidates. The codirectors said their general strategy was “don’t tie up offers with people who will hold them all day.” They made an offer to 1,2,3,5, and 12, where 3,5 and 12 had indicated that they would accept an offer immediately. 1,2 were deemed so attractive to be worth taking a chance on.”

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Principles in Play Roth and Xing summarize 3 things to note about this episode, which does not seem to be atypical: 1. Directors’ concern not to make offers which ran the risk of being rejected late in the day 2. Consequent attention they paid to candidates who had indicated they would immediately accept an offer from the program 3. The willingness of candidates to convey such information.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Example #2: Early Admission • Certainly a congested market (waitlists, limited time) • Early admission signals are 1. important to schools (Young ’04) 2. result in more offers (Avery, Fairbanks, Zeckhauser, ’03) But recently early admission has come under fire for privileging the “haves” over the “have nots.” 1. More haves know about the early admission process 2. Early admission reduces competition for financial aid

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Example #3: Regional Preferences When economic graduate students have restrictions in their regional preferences, the letters of advisors are used to convey this information (and their reputation may make such information credible). → Strictly information about likelihood to accept Note: Not all students have well connected advisors who can help to convey such a credible signal.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Theory: Congested Market with Signaling We present a stripped down setting of a labor market that includes 1. Exogenous congestion 2. A signaling mechanism offering scarce signals and we will examine 1. Behavior 2. Welfare consequences 3. Impact of correlation (of preferences)

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Matching Market F = set of firms, W = set of workers, |F | = |W | = N . Θfi = set of all preference lists (or ROLs) for firm i Θwj = set of all preference lists for worker j Θ = Θf1 × . . . × ΘfN × Θw1 × . . . × ΘwN Each firm f has preferences over the workers chosen uniformly and randomly from the set of all strict preference orderings. Worker preferences are analogously chosen.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Match Utility Depends on Rank Firm i with preferences θfi values a match with worker wj as g(θfi , wj ), where g(θfi , ·) is a von-Neumann Morgenstern utility function. • We assume that utility of a match depends on worker rank.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Benchmark: No Signaling Structure Each firm f makes an offer to a worker, and offers are made simultaneously. Workers choose an offer from those available to them.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

The Offer Game Definition 1 The offer game with no signals is given by Γ0 = [F, {Si }, {πi (·)}, ΘF , U (·)] • F is the set of firms. • Message space Si = W for each firm i. • πi : S × Θfi → R yields the payoff to firm i as a function of the message profile and firm i’s type. • ΘF is the type space, the space of firm preference profiles. • U (·) is the uniform distribution over the type space ΘF .

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Equilibrium Definition A strategy for a firm i in this game is a mapping si : Θfi → W . Definition 2 Strategy profile (s1 (·), . . . , sN (·)) is a Bayesian Nash Equilibrium if for all i and θfi we have si (θfi ) ∈ arg max Eθ−fi [π(w, s−i (θ−fi ), θfi ) | θfi ] w∈W

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Anonymous Strategies We focus on strategies that depend on workers’ rank within a firm’s preference list, rather than worker index. Definition 3 Firm i’s strategy si is anonymous if ∀ σ ∈ Σ, θfi ∈ Θfi we have si (σ(θfi )) = σ(si (θfi )). For example, ‘always make an offer to my second ranked worker’ is an anonymous strategy, whereas ‘always make an offer to w2 is not.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Offer Game: Solution Uniformity, anonymous strategies ⇒ unique BNE: si (θfi ) = θf1i for all i, θfi .

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Model: Signaling Structure 1. Preferences θ realized 2. Each worker sends a signal to exactly one firm 3. Each firm observes the most preferred signal it has received* 4. Firms make simultaneous offers to workers 5. Workers choose from the set of available offers

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Workers’ Strategic Decisions • Workers will accept the best offer from those available. • Provided workers believe that sending a signal increases (or does not affect) their chance of acceptance, each worker will optimally send a signal to its most preferred firm. We focus on equilibria where these beliefs will be true Other perverse equilibria exist (signal means ‘I hate you’)

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Offer Game with Signals Definition 4 The offer game with signals is given by Γ = [F, {Si }, {πi (·)}, ΦF , p(·)]

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Types and Strategies Firm i’s type space is ΦFi ≡ Θfi × (W ∪ {∅}) . That is, fi ’s type is its preference profile combined with its most preferred signal, if it exists. A strategy for firm i consists of a function si : Θfi × (W ∪ {∅}) → W. Abuse of notation: si (θ).

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Tradeoff: Risky Offer vs. Safe Offer Firms will optimally make an offer to either its top ranked worker (TRW), or else to its top ‘guaranteed’ worker (TGW). Notation T RW : θf1i

T GW : bi (θ)

Note: Depends on uniformity and anonymous strategies assumptions.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Cutoff Strategies Definition 5 Firm fi ’s strategy si (·, ·) is a cutoff strategy if for all θfi , and all w, w0 with w ¹θfi w0 , we have si (θfi , w) = w ⇒ si (θfi , w0 ) = w0 .

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Optimality of Cutoff Strategies Cutoffs seem reasonable. Let’s check. With a higher ranked TGW... 1. Payoff to safe choice (TGW) goes up 2. Payoff to risky choice (TRW) may go up or down. There exist strategies for firm −i such that it is optimal for firm i to use something other than a cutoff strategy.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Cutoffs ⇒ Cutoffs Proposition 1 If firms −i are using cutoff strategies s−i (·), then it is optimal for firm i to use a cutoff strategy.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Junk Signals Define a junk signal for a firm as a signal from a worker to whom the firm does not make an offer.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Junk Signal Lemma Lemma 1 Let firms −i use cutoff strategies s−i (·) and let firm i use strategy si (·). Then for ∀ θfi ∈ Θfi , ∀ w, w0 ∈ W with w ºθf w0 , we have i

Eθ [ Eθ [

πi (si (θ), s−i (θ), θ) | πi (si (θ), s−i (θ), θ) |

θ ∈ Θw ww0

θ∈Θ

]

≤

]

where Θw Θ

ww0

={

θ∈Θ

| θfi = θfi

and

bi (θ) = w

={

θ∈Θ

| θfi = θfi

and

bi (θ) = w

} and

1 θw 0 = fi

}.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Junk Signal Lemma: Intuition You would rather that an opponent receive a junk signal because this reduces competition for your risky offers. 1. A junk signal by definition is not your TRW. 2. Cutoff strategies for −i mean likelihood of making an offer to a worker increases when that worker has signaled to it. 3. Likelihood of −i making offers to your TRW then drops.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Proposition 1 (cutoffs ⇒ cutoffs): proof Sketch Suppose w ranked one higher than w0 on firm i’s list. When TGW is w . . . Case 1. w0 has not sent you a signal ⇒ risky payoff same as when TGW=w0 Case 2. w0 has sent you a signal ⇒ risky payoff lower than when TGW=w0

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Negative Spillover of Signals Firms prefer that other firms not make use of signals. Proposition 2 Suppose that s−i (·) and s0−i (·) vary only in that firm j 6= i has sj (θfj , w) = θf1j while s0j (θfj , w) = w for some pair (θfj , w). Then for all i, si (·), we have Eθ [πi (si (θ), s−i (θ), θ] ≥ Eθ [πi (si (θ), s0−i (θ), θ].

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Negative Spillover: Intuition Offers by firms −i to TGWs provides more competition and stiffer competition to i’s risky choice. When firms −i make more offers to TGWs, 1. These offers will certainly be accepted. 2. These offers are more likely to be made to firm i’s risky choice (as compared to offers by −i to their TRWs )

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Perverse Consequence: Firm Ranking of Multiple Eqa If multiple cutoff equilibria exist, firms all prefer the equilibrium that involves the least use of signals. Proposition 3 Suppose there exist two symmetric pure strategy equilibria with cutoffs j, k where j < k. Then all firms have higher (ex-ante) expected payoffs in the equilibrium with cutoff k than in the equilibrium with cutoff j.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Multiple Eqm: Proof Let s0 and s (less signaling) be symmetric pure strategy cutoff equilibria with cutoffs j and k, respectively, with j < k. Since s is an equilibrium, for any firm i, we have E[πi (si (θ), s−i (θ), θ)] ≥ E[πi (s0i (θ), s−i (θ), θ)]. By proposition 6, we have E[πi (s0i (θ), s−i (θ), θ)] ≥ E[πi (s0i (θ), s0−i (θ), θ)].

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Strategic Complements Proposition 4 Responding to worker signals is a case of strategic complements. That is, if firm j reduces its cutoff point (responds more to signals), firm i will optimally also reduce its cutoff.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Strategic Complements: Intuition By proposition 6, when an opponent of firm i reduces its cutoff, 1. return to i’s risky strategy of making an offer to its TRW is lowered. 2. safe payoff of going for a TGW remains the same. Hence, any tradeoff previously in favor of the TGW remains so, while some additional decisions may now favor the TRW - a lowering of i’s cutoff.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Multiple Equilibria Tarski’s fixed point theorem on lattice now gives us pure strategy equilibria. Strategic complements often yield multiple equilibria.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Number of Matches Proposition 5 In expectation, firms responding to signals increases the number of matches.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Worker Welfare Workers (ex ante) prefer that other firms make use of signals. Proposition 6 Suppose that s−i (·) and s0−i (·) vary only in that firm j 6= i has sj (θfj , w) = θf1j while s0j (θfj , w) = w for some pair (θfj , w). Then for all i, si (·), we have Eθ [πw (si (θ), s−i (θ), θ] ≥ Eθ [πw (si (θ), s0−i (θ), θ].

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Corollaries: Worker Welfare 1. When there are multiple equilibria, workers prefer the equilibrium that involves a higher cutoff (more use of signals). 2. Workers unambiguously prefer the signaling outcome to the no signaling outcome.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Correlation Why is correlation important? • Accurately reflects preferences in many labor markets • Signaling becomes an important strategic choice • Responding to signals becomes an even more complicated task (signaling candidate is no longer a “sure thing”.)

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Perfect Correlation (of worker preferences) In this case, a signaling mechanism adds no value. To see this, observe that the #1 firm will ignore signals, as he is the top choice of all workers. The #2 firm doesn’t care what the firms below him are doing, and has no information about what firm #1 is planning on doing. Hence, firm #2 will ignore signals. And so on....

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Perfect Correlation (of firm preferences) In this case, a signaling mechanism can add value. (see paper for details)

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

“Tiered Correlation” Think of a market as having multiple “tiers,” both of firms and of workers. Participants have idiosyncratic preferences within a tier. If markets operate solely “within tier,” then analysis would be as above. With cross tier signals and offers, many equilibria may exist.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Example #1: Cross Tier Offers

f1 H

w1

f2

w2

f3

w3

L f4

w4

H

L

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Example #2: Cross Tier Signals

f1 H

w1

f2

w2

f3

w3

L f4

w4

H

L

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Conclusions In congested markets with signaling, we find that Behavior 1. We can expect firms to use reasonable (cutoff) strategies. 2. While firms may privately benefit from responding to signals, this has a negative effect on the welfare of other firms, and a positive effect on worker welfare. 3. When a firms opponents respond more to signals, this necessitates the firms to respond more to signals.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Conclusions Welfare. Compared to the no signaling model... 1. May or may not increase the welfare of firms. 2. Unambiguously improves worker welfare. 3. Unambiguously increases the number of matches.

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Signaling in Matching Markets

Peter Coles, Muriel Niederle

Extensions 1. Optimal signaling structure (# of signals, gold/silver stars) 2. “Opting Out” 3. Large vs Small Markets: When is signaling most valuable? 4. Fragmented market (some firms with more resources) 5. Relevance to AEA

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