Compare the incentives to send or imitate a costly signal.
By the end: test both type deviations and identify the beliefs needed for a separating candidate.
Messages are choices with incentives
A fictional supplier has output type H or L, each with prior probability one half. Buyers value H output at 8 and L output at 4 payoff units. A certificate does not alter output. Its cost is effort e for H and 3e for L. The cost difference is stipulated, not inferred from observed credentials.
Consider a candidate where H buys the certificate and L does not. Buyers believe certified means H and uncertified means L, and competition pays the corresponding output values 8 and 4. With both types present, both observations occur on this candidate path, so those beliefs follow from the proposed separating strategies.
A certificate's meaning depends on who chooses it
The signaling problem adds a choice before the observer responds. The supplier knows whether its output is type H or L. The buyer cannot directly see that type but can see whether a certificate was acquired. A separating candidate is a proposed set of strategies in which different types choose different observable actions. Here H certifies and L does not. The word candidate matters because describing this pattern does not prove that either type wants to follow it. A certificate that identifies H under the proposal may also tempt L to imitate H.
Separate the output assumption from the communication assumption. Output is worth 8 for H and 4 for L in the buyer's stipulated valuation. Certification changes neither output value. Instead it changes what the buyer infers about an otherwise hidden type. Under the candidate, observing a certificate identifies H, and observing no certificate identifies L. We assume buyers compete by raising offers up to what the identified output is worth, with no profit left. That response pays 8 to certified suppliers and 4 to uncertified suppliers. These are constructed payoff quantities, not measured wages or a claim that market prices always equal an observer's valuation.
Exact separation makes both observed groups informative
Certificate group
Count
H
50
L
0
Implied H probability
100%
Candidate payment
8
No-certificate group
Count
H
0
L
50
Implied H probability
0%
Candidate payment
4
Paper counts under the candidate strategies. Both groups occur, so their conditional type composition determines the proposed receiver beliefs.
Why must both types have positive prior probability? An observation can support a conditional inference only if it can occur under the proposed strategies. With both types present and deterministic separation, certificates and noncertificates both occur. In a paper population with fifty suppliers of each type, the certificate group contains fifty H and zero L. The noncertificate group contains zero H and fifty L. Counting within each group gives certainty about type. The equal prior fixes group sizes here, but the certainty would remain under any positive proportions if exact separation actually held.
Now keep that payment schedule fixed and ask what happens if one L supplier deviates. The buyer observes its certificate, not its hidden label, and treats it according to the proposed certificate rule. The deviator therefore receives 8 rather than 4. If we instead paid the deviator 4 because the analyst knows it is L, we would give the buyer information the model withholds. The deviation check uses what the observer can see. It does not let the observer detect a deviation simply because the author marked a branch as imitation.
That is why certification must be evaluated as a choice with consequences. The gross benefit of certification under these payments is 4 for either type. H pays effort cost e, while L pays effort cost three times e. Costs here are disutilities, meaning reductions in the supplier's utility. They need not be invoices paid to the buyer. We assume the payment and effort quantities can be combined on the supplier's own utility scale. Neither their numerical difference nor a comparison across suppliers measures society's welfare without further assumptions about a common scale.
At effort two, each type rejects its own deviation
Type
Follow candidate
Deviate alone
Deviation gain
H
Certify: 8 − 2 = 6
No certificate: 4
4 − 6 = −2
L
No certificate: 4
Certify: 8 − 6 = 2
2 − 4 = −2
The buyer observes certification, not the deviator's hidden type. Both choices use the same stipulated payment schedule.
A free statement such as 'my output is H' would have no imitation cost under the same assumed payment response. If it earned the premium, both types would want to say it. A certificate can avoid this failure only when its benefit relative to its cost differs enough across types. Merely observing effort does not prove that condition. We need the cost comparisons and the receiver's response together. This lesson supplies those assumptions, then checks their consequences instead of treating an expensive-looking action as inherently truthful.
Check the imitation incentive
H gains the payment premium 4 but pays cost e. H is willing to certify when e is at most 4. L could imitate, gain the same premium, and pay 3e. L is willing to remain uncertified when 3e is at least 4. Both inequalities are needed.
At e = 2, H gains 2 from certifying while L would lose 2 by imitating. At e = 1, L gains 1 by imitation, so the candidate fails. At e = 5, H loses 1, so it fails for a different reason. The boundary inequalities are weak; equality leaves a type indifferent rather than uniquely selecting an action.
Derive a lower and an upper bound on effort
Start with H's possible deviation. Following the candidate gives payment 8 minus effort cost e. Dropping the certificate saves e but reduces payment to 4. H weakly prefers the certificate if 8 minus e is at least 4. Subtracting 4 from both sides leaves 4 minus e at least zero, so e must be at most 4. A weak preference permits equality. At exactly 4, H is willing to follow the candidate but is also willing to abandon certification. The inequality rules out a profitable deviation, not every alternative action.
L's comparison points in the other direction. Following the candidate means no certificate and utility 4. Imitating H gives 8 minus three times e. L weakly prefers not to imitate when 4 is at least 8 minus three times e. Moving the cost to the other side gives three times e at least 4, so e must be at least four thirds. To support both proposed choices, effort must satisfy this lower bound and H's upper bound simultaneously. Checking H alone would incorrectly approve very cheap certificates that L wants to acquire too.
Separation requires the overlap of two effort bounds
The exact interval includes both endpoints. Strict incentives exclude the endpoints because one type is indifferent at each boundary.
At effort 2, H's certified utility is 8 minus 2, or 6, compared with 4 without certification. Dropping the certificate loses H 2. L's imitation utility is 8 minus 6, or 2, compared with 4 from remaining uncertified. Imitation loses L 2 as well. These equal margins are accidental consequences of the chosen effort and costs. The important fact is their direction. H's cost of 2 lies below the premium 4, while L's cost of 6 lies above it. Each proposed action is a strict best response under the candidate payments.
Compare two failures. At effort 1, H receives 7 by certifying and prefers that to 4. But L would receive 5 by imitating, better than 4. The candidate fails because its intended nonparticipant wants to join. At effort 5, L's imitation utility is minus 7, so imitation is unattractive. Yet H receives only 3 by certifying, worse than 4 without it. The candidate now fails because its intended participant wants to leave. Making a signal harder to imitate can eventually make it too costly for the type it was meant to identify.
The interval's boundaries need separate treatment. At effort four thirds, L's cost is 4 and its imitation utility is exactly 4. H's certified utility is twenty thirds, about 6.667, so H strictly certifies but L is indifferent. At effort 4, H gets 4 either way, while L would get minus 4 from imitation rather than 4 without certification. The proposed separating actions remain best responses at both boundaries. Strict incentives for both types exist only between them. A pass at equality does not establish a unique outcome or robustness to a small change in costs.
Raising effort strengthens one margin and weakens the other
Here both gains measure following the candidate. The lab instead reports L's imitation gain, with the opposite sign.
Another useful calculation finds the weaker of the two strict incentive margins. H's gain from certifying is 4 minus e. L's gain from staying uncertified rather than imitating is three times e minus 4. At effort 1.5 these margins are 2.5 and 0.5, so L's choice is closer to reversal. At 3.5 they are 0.5 and 6.5, so H's choice is closer. These paper comparisons help diagnose which assumption matters most near a given candidate. They do not give the lab a cost-uncertainty control or prove how real suppliers respond to small incentives.
Compare incentives and outcomes
Check both type incentives
Type H
Action
Net utility
Certificate
6
No certificate
4
Gain from certificate: 2.
Type L
Action
Net utility
Imitate certificate
2
No certificate
4
Gain from imitation: -2.
Candidate passes both weak incentive checks: Yes.
Require both incentive constraints
The payment premium is four utility units. H must not prefer dropping the certificate; L must not prefer imitating it.
Only certificate effort changes. Candidate payments stay at 8 certified and 4 uncertified; costs are e for H and 3e for L. Both types have positive prior probability. Passing checks this candidate’s weak incentive inequalities, not all signaling equilibria or a real credential’s value.
Type at e = 2
Certified net payment
Uncertified payment
Preferred
H
8 − 2 = 6
4
Certify
L
8 − 6 = 2
4
No certificate
Both deviations must be checked, not just the intended high-type choice.
Read both panels when changing certificate effort
Select Cheap to imitate: 1 in the lab. Read the two action utilities in each panel before looking at the pass indicator. H gets 7 with a certificate and 4 without one. L gets 5 by imitation and 4 without one. The intended H choice passes, but the intended L choice fails. The displayed imitation gain is positive 1 because it measures the payoff from copying the certificate minus the payoff from remaining uncertified. It is not the gain from following L's proposed strategy, which would have the opposite sign.
Next select Separates strictly: 2. H's displayed certificate gain becomes 2, while L's displayed imitation gain becomes minus 2. The signs must differ for strict separation under this candidate. Select Too costly for H: 5 and the values become minus 1 and minus 11. Both types now prefer the uncertified payment to their certified net utility. A negative L imitation gain is necessary but insufficient. The combined pass indicator must also require that H's certificate gain is nonnegative. The two panels expose which constraint failed rather than merely labeling a certificate successful or unsuccessful.
Selectable effort values bracket the lower mathematical boundary
Effort e
H certificate gain
L imitation gain
Candidate
1
3
1
Fails L
1.5
2.5
−0.5
Strictly passes
2
2
−2
Strictly passes
4
0
−8
Weakly passes; H ties
4.5
−0.5
−9.5
Fails H
5
−1
−11
Fails H
The half-unit control cannot select four thirds. Its first passing point is 1.5; equality at four still passes weakly.
Use the half-unit slider to move from effort 1 to 1.5. L's certified net utility falls from 5 to 3.5, crossing below the unchanged uncertified value 4. H's certified utility falls from 7 to 6.5 and remains above 4. The exact lower boundary is four thirds, which the half-unit slider cannot select. Therefore 1.5 is the first selectable passing effort, not the mathematical boundary itself. The model's inequality interval and the interface's selectable values are different objects. Reading a sampled control position as an exact threshold would confuse numerical granularity with incentives.
Continue through effort 4 and 4.5. At 4 the H panel shows a tie at utility 4 and the candidate still passes its weak inequalities. At 4.5, H's certified utility is 3.5, so the pass fails even though L's incentive against imitation becomes stronger. The control changes only effort and its two associated costs. It never changes output, candidate payments, the prior, or buyer behavior. A failing result is therefore a rejection of the supplied separating candidate at those inputs, not a simulation showing what replacement market outcome will emerge.
A paper extension can test the role of different costs by replacing L's cost three times e with two times e. H still needs e at most 4, but L now needs twice e at least 4, so the weak interval narrows to 2 through 4. At effort 2, L is indifferent rather than strictly opposed to imitation. If L's cost becomes equal to H's, both types face the same gain 4 minus e. Neither can strictly want the certificate while the other strictly rejects it. This cost comparison is not an editable parameter in the lab.
Reducing the cost difference narrows strict separation
Cost H / cost L
H requirement
L requirement
Both weakly hold
e / 3e
e ≤ 4
e ≥ 4/3
4/3 ≤ e ≤ 4
e / 2e
e ≤ 4
e ≥ 2
2 ≤ e ≤ 4
e / e
e ≤ 4
e ≥ 4
Only e = 4; both tie
Paper alternatives, not editable lab parameters. Equal costs leave only a shared indifference point and no strict separating interval.
The payment premium also matters. In a separate paper scenario, suppose certified payment is 7 while uncertified payment remains 4, leaving premium 3. With the original costs, H requires effort at most 3 and L requires effort at least 1. At effort 1, H strictly wants certification but L is indifferent. With the original premium 4, that same effort instead gives L a strict imitation gain. A conclusion about separation must therefore specify receiver payments as well as costs. Differential cost alone does not identify an effort threshold independent of the benefit being offered.
Do not infer a whole equilibrium from a price schedule
The explorer checks this candidate's type incentives under the stated payments. It does not find all pooling or separating equilibria. Pooling means different types send the same signal. There, an unused signal can require off-path beliefs, and those beliefs affect the incentive to deviate.
A costly signal need not be informative merely because it is costly. Different types must have the relevant incentives, and receivers must respond consistently. Real credentials may also improve skills, filter access, or reflect resources. The model isolates one signaling mechanism; it is not a claim that education or certification is only wasteful signaling.
Check receiver beliefs without claiming every equilibrium
So far we tested supplier deviations while holding fixed the candidate payment schedule. We must also ask why the buyer uses that schedule. Under exact separation, certificate holders really are H and nonholders really are L, so the proposed inferences agree with the proposed strategies. This is a consistency check, not evidence that certification already separates real suppliers. The stipulated competitive response then links those beliefs to payments 8 and 4. A different market structure, buyer objective, or pricing rule would need its own response calculation. The pass indicator does not independently solve those alternatives.
Consider a paper pattern that is not exact separation. Of fifty H suppliers, forty certify; of fifty L suppliers, ten certify. The fifty certificate holders then contain forty H and ten L. Their H probability is 80%, not certainty. If buyers pay the group's expected output value under the same competitive assumption, payment is 0.8 times 8 plus 0.2 times 4, or 7.2. The noncertificate group has H probability 20%, producing payment 4.8. Its premium is 2.4 rather than 4. These counts illustrate belief consistency, not a newly solved equilibrium or a lab setting.
Imperfect sorting changes the implied payment premium
Invented paper frequencies imply payments 7.2 and 4.8 under expected-output pricing. These frequencies have not passed an equilibrium check.
The revised payments show why strategies and beliefs cannot be checked independently. At effort 2, the premium 2.4 still exceeds H's cost 2 but falls below L's cost 6. That comparison does not validate the proposed partial-certification frequencies. With these strict incentives, every identical H in this simple model would prefer certifying, not just forty out of fifty. The observed group averages produce one payment calculation, while the individual incentives test whether those frequencies can persist under the assumptions. A pattern can be arithmetically describable without being strategically self-consistent.
Pooling means both types choose the same observed action. If both remain uncertified under the equal prior, the group's expected output is half of 8 plus half of 4, or 6. Under the assumed competitive payment rule, the on-path payment would therefore be 6, not the candidate's 4. A certificate would then be an unused observation, often called off path. The proposed pooling strategies alone do not determine which type the buyer should infer from that unused action. To analyze deviations, we would need to specify those beliefs and the resulting payment before comparing certification costs.
The model also separates an individual incentive from an output improvement. At effort 2, H pays a real utility cost of 2 to obtain a higher payment even though its output remains 8. Comparing two passing efforts, such as 1.5 and 2, leaves the type classification and payments unchanged but raises H's cost by 0.5. H's own net utility falls by 0.5. That within-person comparison is justified by the stipulated utility scale. Calling the difference a measured social loss would require further assumptions about resources, other participants, and how their welfare should be combined.
An unused certificate needs a receiver response
Buyer belief after unused certificate
Certificate payment
H certified net
H stays uncertified
H deviation
Certificate means H
8
8 − 2 = 6
6
Tie
Certificate means L
4
4 − 2 = 2
6
Loses 4
Paper pooling candidate at effort two. Different off-path beliefs change H's incentive; neither row alone proves a complete pooling equilibrium.
For an actual credential, output might improve through learning, costs might reflect available money rather than type, and buyers might already observe other evidence. Each possibility changes the proposed mechanism. Evidence that credential holders earn more cannot by itself distinguish productivity gains, selection of different types, and a payment response to the credential. The transferable analysis is to state a candidate, check both types' deviations, and verify receiver inferences using the candidate's actual behavior. The lab completes the specified supplier incentive checks. It does not identify all equilibria, estimate credential effects, or supply a policy recommendation.
Exercises
Find the allowable effort interval
For the candidate above, what effort values satisfy both weak incentive constraints?
Hint
Combine e ≤ 4 with 3e ≥ 4.
Worked answer
Constraint
Requirement
H prefers certificate weakly
e ≤ 4
L rejects imitation weakly
e ≥ 4/3
Both
4/3 ≤ e ≤ 4
A lower and an upper bound are necessary.
The interval is four thirds through 4, including both boundaries. Strict separation incentives hold only inside the interval. At four thirds L is indifferent; at 4 H is indifferent. Both proposed actions are still best responses at the boundaries, but uniqueness is not established.
Remove the cost difference
Suppose both types pay e for the certificate and the payment premium remains 4. Can both have strict incentives to separate as proposed?
Hint
The two types now face the same premium minus cost.
Worked answer
Common effort cost
H incentive
L incentive
Below 4
Certify
Certify
Exactly 4
Indifferent
Indifferent
Above 4
Do not certify
Do not certify
Costliness alone does not explain separation.
No. Both gain 4 − e by certifying. Below 4 both prefer to certify; above 4 neither does. At 4 both are indifferent, so the proposed separation is not strictly supported. Differential cost was doing essential work in the original mechanism.
Sources and further reading
Selected reference passages support the definitions and algorithms below. The games, explanations, and implementations are original. Terminology and assumptions are stated within each lesson; source review is not an independent audit of this course. Optional links are not runtime dependencies.
Selected candidate-separation incentive checks and the role of different signal costs. Our constructed certificate is not empirical evidence about education or hiring.