Signaling Theory

Signaling Theory in Health Care Markets

Signalling theory (spelt signaling in American usage) asks when a costly, observable choice by a hospital, clinician or insurer can tell patients and purchasers something true about a quality they cannot see. Its central result, from Michael Spence's 1973 model of education in the job market, is that a choice carries information only when it costs the better type less than the worse type. Health care fits the model closely, because patients rarely judge clinical quality before treatment, which is the information asymmetry that Kenneth Arrow described between physician and patient. The theory gives health economists a test for accreditation, credentials and public reporting, namely whether each one sorts providers by quality or only adds cost.

Spence's model and its health care reading

Spence starts from an employer who cannot observe a job applicant's productivity and separates observable attributes into indices, which the person cannot alter, such as age, and signals, which the person can alter at a cost, such as education. The employer holds conditional probability beliefs about productivity given what it sees and offers a wage equal to expected productivity. To isolate the informational role, Spence assumes that education does not raise productivity at all.

His critical assumption is that a signal will not distinguish one applicant from another unless its cost is negatively correlated with the hidden attribute. The Royal Swedish Academy of Sciences, explaining the 2001 economics prize to Akerlof, Spence and Stiglitz, put the same point as signalling succeeding only if the cost differs sufficiently among the senders.

In health care the applicant becomes a provider and the employer a patient, commissioner or insurer. The hidden attribute is clinical quality, and candidate signals include voluntary accreditation, board-certified staff and published outcome data. Arrow noted that medical licensing is defended as guaranteeing a minimum of quality. A compulsory licence sets a floor and carries no information above it, so the theory mainly concerns voluntary choices above the legal minimum. The act of signalling has its own page, and signalling theory asks when such acts work and what they cost.

Beliefs, incentive constraints and equilibrium

An equilibrium in Spence's model has two parts. Senders choose the signal that maximises their return net of signalling cost, and receivers hold beliefs about how the signal relates to type. Those beliefs must be self-confirming, so the data they generate do not disconfirm them, and Spence's Nobel lecture traces the possibility of many equilibria to this property. In game theory terms the receiver updates by Bayes' theorem.

$$\mu(H \mid a) = \frac{\Pr(a \mid H)\,\pi_H}{\Pr(a \mid H)\,\pi_H + \Pr(a \mid L)\,\pi_L}$$

where $\mu(H \mid a)$ is the receiver's posterior probability that a provider showing signal $a$ is of high quality $H$, $\pi_H$ and $\pi_L$ are the prior shares of high-quality and low-quality providers $L$, and $\Pr(a \mid H)$ and $\Pr(a \mid L)$ are the probabilities that each type shows the signal.

A separating equilibrium needs two incentive constraints. The low type must prefer not to copy the high type, and the high type must prefer signalling to being treated as low quality.

$$p(s^{*}) - c_L(s^{*}) \le p(0) - c_L(0) \quad \text{and} \quad p(s^{*}) - c_H(s^{*}) \ge p(0) - c_H(0)$$

where $s^{*}$ is the signal level receivers associate with high quality, $p(s)$ is the payment offered to a provider showing level $s$, and $c_H(s)$ and $c_L(s)$ are the costs of that level to each type. Both constraints can hold only if the extra cost of $s^{*}$ over no signal is higher for the low type than for the high type, and models usually assume the stronger single-crossing condition that marginal cost is lower for the high type at every level.

$$\frac{\partial c_H(s)}{\partial s} < \frac{\partial c_L(s)}{\partial s} \quad \text{for all } s > 0$$

where each term is the marginal cost of a further unit of signal for that type. The Academy relates this to the single-crossing, or Mirrlees-Spence, condition.

In a separating equilibrium the types choose different signals and receivers learn each type exactly. In a pooling equilibrium every type chooses the same signal, often none, and receivers pay for average quality, and in a semi-separating outcome some low types mimic, so the signal raises the posterior without settling it. Spence found a continuum of separating equilibria, each with more investment in the signal than full information would need, ordered by the Pareto criterion. The most efficient has the high type buy just enough signal to deter mimicry, and the Academy reports that John Riley showed in 1975 that this is the only signalling equilibrium that survives wage experimentation by employers. Refinements of Nash equilibrium, such as the one in a 1987 paper by Cho and Kreps, often select this least-cost outcome.

Conditions a health care signal must meet

Spence's assumptions translate into practical conditions for a health care quality signal. When one fails, the signal stops sorting providers or costs more than its information is worth.

1. Verifiable. Dranove and Jin count a seller's own disclosure as quality disclosure only if it can be independently verified, which separates it from marketing claims.
2. Cheaper for better providers. The cost, including staff time, must fall with true quality.
3. Noticed. A Romanian study of hospital accreditation found patients largely unaware of their hospital's status.
4. Stable in meaning. If a certifier lowers its standard, low-quality providers obtain the signal and its meaning weakens, as step 6 of the worked example shows.

Worked example with voluntary hospital accreditation

All figures are illustrative. Purchasers buy an elective surgery contract from hospitals they cannot tell apart. A high-quality hospital's contract is worth GBP 2.0 million a year and a low-quality hospital's GBP 1.0 million, and 40% of hospitals are low quality. Purchasers compete, so the price equals expected value given their beliefs. Accreditation at depth $s$ costs a low-quality hospital GBP $s$ million and a high-quality hospital GBP $s/2$ million, since its systems already meet most standards, and accreditation does not change quality.

1. No signal. Purchasers pay for average quality, 0.4 × 1.0 + 0.6 × 2.0 = 1.6, so every hospital receives GBP 1.6 million.

2. Conditions for separation. Purchasers believe accreditation at depth $s^{*}$ means high quality. A low-quality hospital does not copy if 1.0 is at least 2.0 minus $s^{*}$, so $s^{*}$ must be at least 1.0. A high-quality hospital accredits if 2.0 minus half of $s^{*}$ is at least 1.0, so $s^{*}$ can be at most 2.0, and any standard in between supports separation.

3. Least-cost separation. At a depth of 1.0 a high-quality hospital nets 2.0 − 0.5 × 1.0 = 1.5, or GBP 1.5 million, and a low-quality hospital GBP 1.0 million. The average signalling cost across all hospitals is 0.6 × 0.5 = 0.3, or GBP 0.3 million a year.

4. Pooling compared. Without the signal, high-quality hospitals get GBP 1.6 million instead of GBP 1.5 million and low-quality hospitals GBP 1.6 million instead of GBP 1.0 million, so both types prefer pooling. This matches Spence's lecture example, where pooling is preferred by everyone when the low group is under about half the population.

5. When sorting improves care. Suppose knowing types lets purchasers route complex patients to high-quality hospitals, gaining 16 QALYs per high-quality hospital a year. The gain per average hospital is 0.6 × 16 = 9.6 QALYs, so the signalling cost per QALY gained is 300,000 / 9.6 = 31,250, or GBP 31,250. Valued at an illustrative GBP 30,000 per QALY the gain is worth 9.6 × 30,000 = 288,000, a shortfall of 300,000 − 288,000 = 12,000, or GBP 12,000 per hospital a year. An inspection costing GBP 0.1 million per hospital that revealed type directly would cost 100,000 / 9.6 = 10,417 per QALY gained, rounded.

6. A lax scheme. If accreditation became cheap enough that half the low-quality hospitals also held it, Bayes' theorem gives an accredited hospital a posterior of 0.6 / (0.6 + 0.4 × 0.5) = 0.75. Accredited hospitals would receive 0.75 × 2.0 + 0.25 × 1.0 = 1.75, or GBP 1.75 million, and unaccredited ones GBP 1.0 million. For low-quality hospitals to be willing to mix, their accreditation cost must equal 1.75 − 1.0 = 0.75, or GBP 0.75 million.

7. Interpretation. The signal sorts hospitals only because its cost falls with quality, and the sorting is worth paying for only if it changes where patients are treated. Even then, inspection can supply the same information more cheaply.

Evidence from health care markets

Health economists have modelled and tested signalling explanations for several provider choices. Of the two studies below, one is empirical and the other theoretical.

Jones, Propper and Smith asked whether non-profit status signals quality when consumers are poorly informed. After an exogenous increase in consumer information in the US nursing home industry, the share of non-profit homes fell, and the lowest-quality non-profits were the most likely to exit. Chen, Meinecke and Sivey modelled public reporting of hospital waiting times, which has a competition effect and a signalling effect that reveals hospital quality to patients. For the high-quality hospital the effects pull in opposite directions, but competition dominates, so both hospitals' waits fall where maximum waiting time targets are not binding.

Signalling, screening and disclosure compared

Signalling is one of three linked accounts of how hidden information is revealed. They differ in who moves first and what makes information credible.

In screening theory the uninformed party moves first, offering a menu that leads informed parties to sort themselves, as in Rothschild and Stiglitz's insurance model of premiums and deductibles, which underlies much analysis of adverse selection in health insurance. The Academy notes that screening equilibria are typically unique and correspond to the most efficient signalling equilibrium. Disclosure theory concerns information that is credible because it can be verified rather than because it is costly. Grossman and Milgrom showed that costless verifiable disclosure leads all but the worst seller to disclose, since consumers read silence as low quality, and Dranove and Jin set out the strong assumptions behind that result. Mechanism design takes the purchaser's view of which rules to set, and moral hazard concerns hidden actions after a contract, not a hidden type before it.

Using signalling theory in evaluation and policy

For health economic evaluation a signal is a use of resources, and its information has social value only if it changes decisions. Spence's lecture notes that pooling becomes inefficient when an allocation decision is made better with accurate knowledge of type, which in health care means referral, contracting or patient choice. An appraisal of an accreditation or reporting scheme should therefore count compliance costs alongside any health gain from better allocation.

Spence also shows that taxing the signal, with revenue returned as a lump sum, can lower the signalling level without destroying its informational content, provided the tax is costless to administer. Mandated reporting changes incentives too. Dranove and Jin cite evidence that outcome report cards led providers to select patients because risk adjustment was imperfect. They also cite theory that mandatory disclosure can lead sellers to collect less information, such as a drug company limiting studies of side effects.

Misreadings and limits

Several errors recur when the theory is applied to health care. Each one overlooks a condition or result of Spence's model set out above.

1. Treating any costly action as a signal. A cost that is the same for every provider sorts nobody.
2. Assuming signalling is efficient. Every separating equilibrium involves more signalling than full information would need, and pooling can leave every type better off.
3. Expecting a unique answer. Self-confirming beliefs allow several equilibria, so the model alone rarely predicts which one a market reaches.
4. Forgetting productive effects. Spence assumes the signal leaves productivity unchanged, but accreditation work may itself raise quality, which changes how any link between accreditation and outcomes should be read.

Sources

• Arrow KJ. Uncertainty and the welfare economics of medical care. American Economic Review. 1963;53(5):941-973. https://www.jstor.org/stable/1812044
• Chen Y, Meinecke J, Sivey P. A theory of waiting time reporting and quality signaling. Health Economics. 2016;25(11):1355-1371. https://doi.org/10.1002/hec.3222
• Dranove D, Jin GZ. Quality disclosure and certification: theory and practice. Journal of Economic Literature. 2010;48(4):935-963. https://doi.org/10.1257/jel.48.4.935
• Druică E, Wu B, Cepoi V, Mihăilă V, Burcea M. Testing the strength of hospital accreditation as a signal of the quality of care in Romania: do patients' and health professionals' perceptions align? Healthcare (Basel). 2020;8(3):349. https://doi.org/10.3390/healthcare8030349
• Jones DB, Propper C, Smith S. Wolves in sheep's clothing: is non-profit status used to signal quality? Journal of Health Economics. 2017;55:108-120. https://doi.org/10.1016/j.jhealeco.2017.06.011
• Royal Swedish Academy of Sciences. Markets with asymmetric information. Advanced information on the Prize in Economic Sciences 2001. 10 October 2001. https://www.nobelprize.org/uploads/2018/06/advanced-economicsciences2001.pdf
• Spence M. Job market signaling. Quarterly Journal of Economics. 1973;87(3):355-374. https://doi.org/10.2307/1882010
• Spence AM. Signaling in retrospect and the informational structure of markets. Prize Lecture, 8 December 2001. https://www.nobelprize.org/uploads/2018/06/spence-lecture.pdf