An imperfect test for a virus can Be worse than No test at all

Whitmeyer, Mark Article — Published Version An imperfect test for a virus can Be worse than No test at all Health Economics Provided in Cooperation with: John Wiley & Sons Suggested Citation: Whitmeyer, Mark (2021) : An imperfect test for a virus can Be worse than No test at all, Health Economics, ISSN 1099-1050, Wiley, Hoboken, NJ, Vol. 30, Iss. 6, pp. 1347-1360, https://doi.org/10.1002/hec.4254 This Version is available at: https://hdl.handle.net/10419/242000 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Received: 22 April 2020 - Revised: 21 January 2021 - Accepted: 28 January 2021 DOI: 10.1002/hec.4254 RESEARCH ARTICLE An imperfect test for a virus can Be worse than No test at all Mark Whitmeyer Hausdorff Center for Mathematics and Institute for Microeconomics, University of Bonn, Bonn, Germany Correspondence Mark Whitmeyer, Hausdorff Center for Mathematics and Institute for Microeconomics, University of Bonn, Bonn, Germany. Email: [email protected] Funding information Deutsche Forschungsgemeinschaft, Grant/Award Number: 390685813 Open Access funding enabled and organized by Projekt DEAL Abstract This note studies the effect of the availability of a test for a virus on the public health of a population. It is shown by example that the existence of a freely available and moderately informative test for a virus may lower society's welfare in comparison to the case where no test exists or access to the test is restricted. In this setting, any test provided to any subset of agents who would find it optimal not to isolate absent the test improves welfare. KEYWORDS coronavirus, COVID‐19, group testing, information design JEL CLASSIFICATION D82, D83, C72, I18 1 | INTRODUCTION In 2020, the virus COVID‐19 swept through the globe. One particular difficulty presented by the virus is that infected individuals may be virtually asymptomatic and carry the virus without knowing it. Moreover, in the early stages of the pandemic, there were well‐publicized shortages of medical tests for the virus, which made it impossible to test everyone, even everyone with symptoms. To get around this, various solutions were proposed, including testing people in groups (Gollier & Gossner, 2020) and testing for inconclusive symptoms like a high temperature. 1 This purpose of this note is to present a simple example that illustrates that the existence (and availability) of a moderately informative test can actually lower social welfare in comparison to the scenario when no such test exists (or is available). This analysis, thus, 1. Supports the regulation of costless tests; 2. Provides a word of caution against moderately informative tests; and 3. Emphasizes the importance of the choice as to whom should be tested. In the model we explore, there is a heterogeneous population of agents with different exposure likelihoods. Each agent has a simple decision: whether to stay home and isolate (or self‐quarantine) or refrain from isolating and instead go out. An agent incurs a reward from not isolating, but possibly suffers a cost as well–she encounters others if out and neither wishes to become infected (if she is not infected) nor wishes to infect others (if she is infected). Thus, the This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2021 The Authors. Health Economics published by John Wiley & Sons Ltd. Health Economics. 2021;30:1347–1360. wileyonlinelibrary.com/journal/hec - 1347 prevalence of infected people who are not isolating is an endogenous equilibrium object determined by the individual isolation decisions of the agents. Each agent's welfare is affected by the decisions of others, and crucially, a change in the infection rate of those out and about affects all of those who are not isolating. Accordingly, the mechanism that generates the possible welfare loss due to a test is the worsening of the participant pool. A false negative from a less than fully informative test can encourage risky people to refrain from isolating, increasing the chances of disease transmission. Although the information provided is itself valuable, the gain in welfare as a result of this information is outweighed by the increased disease prevalence among those an agent encounters. The frequency of false negatives for the COVID‐19 virus is well‐documented. Indeed, the most common test for the virus, the reverse transcriptase polymerase chain reaction (RT‐PCR) test has false negative rates at initial presentation of symptoms that range from 30% to 40% (Ai et al., 2020; Fang et al., 2020; Yang et al., 2020). Moreover, these rates can vary considerably depending on the time since exposure (Kucirka et al., 2020). Computed Tomography (CT) scans may be more effective (Ai et al., 2020; Caruso et al., 2020; Fang et al., 2020), but even those may have high false negative rates in the first few days following the onset of symptoms (Kanne et al., 2020). In this note, the agents are completely rational, yet can be made worse off by the availability of an imperfect test. This is because of the strategic nature of the societal interaction. If this were merely a decision problem for each agent, any test would increase welfare. 2 Moreover, it is understood that an imperfect test can encourage sub‐optimal behavior if people misunderstand or if people do not realize that the test is flawed. Here we discover that an imperfect test can be detrimental even when its quality is common knowledge and when agents have no behavioral biases. Although we find that there are some tests and testing protocols that can lower welfare, Proposition 2.2 reveals that any test, provided it is given to people who have a (relatively) low likelihood of exposure–those who would refrain from isolating absent a test–is welfare improving, since it both provides those agents with more information but also ensures that the pool of agents participating in society improves. That is not to say that that group of people is the optimal group to test, merely that such a protocol, according to this framework, cannot reduce welfare. The last statement leads us to the following caveat: in this paper, we do not attempt to characterize optimal testing protocols, nor do we provide a thorough cost‐benefit analysis of the participation/social‐distancing trade‐off. These are both worthwhile concerns, yet the goal of this paper is more modest. Instead, we merely wish to expose a counter‐ intuitive aspect to testing, one that has seemingly been heretofore unmentioned and overlooked. 2 | THE MODEL Let us consider the following formal model. There is a population that consists of a continuum of agents with measure 1. The Bernoulli random variable, Θ, corresponds to the infection status of an agent, where Θ = 1 denotes that an agent is infected and Θ = 0 denotes that an agent is not infected. The only source of heterogeneity is an agent's prior exposure to the disease, which we term her type. That is, an agent's type is her likelihood of infection, μ∈0;1½ �, where μ:¼PΘ¼1ð Þ. We impose that the population distribution of types has an atomless 3 cumulative distribution function Fwith support on [0, 1]. Each agent has a simple choice: either isolate (action I) or participate (action P). If an agent isolates, she obtains a payoff normalized to 0. If an agent participates, then with probability τshe interacts with someone who is infected, where τis the average infection likelihood of those participating. Hence, τis the chance that a agent who is not isolating encounters an infected person. It is important to keep in mind that τis an endogenous equilibrium object, determined by the isolation decisions of the agents in the populace. If an agent of type μparticipates then her payoff is u P;μ;τð Þ¼A−Bμ 1 − τð Þ−Cð1 − μÞτ where A≥0 is her reward from participating, C≥0 is her loss from becoming infected, and B≥0 is her loss from infecting someone else. Note that, in contrast to τ,A,B, and Care exogenous parameters. The solution concept that we use is Nash Equilibrium: given the actions of the other agents, no agent has a (unilateral) profitable deviation. Due to the linearity of the payoff from Pin an agent's type, any equilibrium must be of a particular cut‐off form: all agents whose types are above (or below) a certain threshold will isolate, and all those whose 1348 - WHITMEYER types are below (or above) that threshold will not. Because there is a coordination‐like aspect of the game, there may exist two kinds of equilibria; both those in which high types isolate and low types participate, but also the inverse. Throughout we restrict attention to the first class of equilibria, since it seems to correspond closest to people's behavior during the pandemic–by and large, sick people are encouraged to stay home, not go out. Moreover, the condition below ensures that such an equilibrium exists. We begin by looking at the case in which there is no test, so that we can subsequently compare welfare to the case in which a test is available. We impose the following condition on the parameters: Condition 2.1 There exists some type bμsuch that (i) AþBþCð Þbτ−Bð Þμ−Cbτ≥0;for all μ≤bμ;and AþBþCð Þbτ−Bð Þμ−Cbτ≤0;for all μ≥bμ; where, bτ¼τbμð Þ :¼∫bμ 0xdFðxÞ Fbμð Þ and (ii) bμ≤C BþC bτis a conditional expectation: it is the average infected likelihood of those participating (those whose type, μ, is less than the cutoff type, bμ). Inequality (ii) ensures that in this equilibrium, the payoff of each type μ∈0;bμ½ �is decreasing in τ, which is realistic: the welfare of the participants gets worse as it becomes more likely that they encounter infected individuals. A necessary condition for Condition 2.1 is that BþCð Þbτ≤B That is, given bτ, an agent's participation payoff is decreasing in her own type. If Condition 2.1 holds, then trivially there exists an equilibrium in which all types μ < bμparticipate and all types μ > bμisolate (indeed the first part of the condition is necessary and sufficient for such an equilibrium to exist). Denote the aggregate payoff from this equilibrium by W,viz., W:¼∫bμ 0AþBþCð Þbτ−Bð Þx−Cbτf gdFðxÞ Clearly, this is an exceedingly simple model: the scenario is static and there are no benefits to testing other than to guide agents' isolation decisions. For instance, there is no contact tracing in this model, nor are there benefits to society from obtaining statistics about the spread, morbidity, or mortality of the disease. These are all important considerations for determining optimal testing policies. Furthermore, outside of their infection likelihoods, agents in this model are homogeneous. We do not distinguish between essential and inessential workers, say, which is another vital component of a thorough cost‐benefit analysis of testing. The model's homogeneity in this dimension also does not allow us to tackle the subtle issue that, in reality, not all agents with the same infection likelihood are the same. Perhaps some high‐likelihood agents are travelers who have recently returned from a virus hot‐spot, whereas others are doctors, who are likely to be sick by virtue of their occupation. It is easy to see how distinguishing between these sorts of agents could be very important when choosing whom to test. 4 Yet another simplification is that the agents who participate interact with each other randomly. They cannot choose the types with whom they interact, and neither the encounter rate of an agent nor whom an agent meets is affected by WHITMEYER - 1349 her infection probability. Again, this is perhaps unrealistic–a doctor is more likely to come into contact with infected individuals than a traveller who has returned home from a hot‐spot, yet our model does not allow for this distinction. Naturally, a model intended to guide policy directly should include many, if not all, of these details that are conspicuously absent from this paper. However, as is noted above, the intent of this paper is not prescriptive. Our goal is to discover and understand the counterintuitive notion that testing may lower welfare. The model is kept deliberately simple in order to clearly illustrate the intuition of the result. We should expect similar incentives to be present in a more complicated model, but they might be hidden or obfuscated by other factors. 2.1 | Testing participants cannot hurt Now let us introduce testing to the scenario and derive our first result. Formally, a test, π, is a stochastic map: π:0;1f g→ΔSð Þ where Sis some (compact) set of signal realizations. In the example that we explore later on, we assume that Sconsists merely of two signal realizations–a positive result and a negative result–but for now, we need not make such a restriction. Then, Proposition 2.2 Let 2.1 hold.Then,any test,π,given to any subset of types in the interval 0;bμ½ � begets an equilibrium that yields society a payoff that is (at least weakly) greater than W. Proof. It suffices to show that the payoff of each type μ∈0;bμ½ � (weakly) increases in expectation. Moreover, recall that the second part of Condition 2.1 implies that the payoff of each type μ∈0;bμ½ � is decreasing in τ. Hence, if the pool of agents in society improves (τdecreases) the payoffs of those agents increase. If the testing protocol begets an equilibrium in which the new average likelihood of participants (τ*) is equal to bτ, then in expectation, the payoff of each type who is tested must weakly improve (this follows from Ramsey, 1990 and Blackwell, 1951). Naturally, this payoff is further improved if τ)≤bτand so we need only establish that there is an equilibrium in which all types μ∈0;μ) ½ �, with μ)≥bμparticipate, and τ)≤bτ. Let Gbe the new distribution over types (beliefs about infection likelihood) as a result of the test. Note that by definition, 5 Gis a mean‐preserving spread of F. 6 There are two cases to consider: Case I, where GðbμÞ¼FðbμÞ; and Case II, where GðbμÞ< FðbμÞ. The first case trivially yields the desired result (the pool of participants and the participation decisions of the agents are unchanged). Let us consider Case II and search for an equilibrium in which all types μ∈0;μ) ½ � participate, where μ)≥bμ. It is straightforward to see footnote 7 7 that bτ > ∫bμ 0xdGðxÞ Gbμð Þ Define τ μð Þ :¼∫μ 0xdGðxÞ G μð Þ which is obviously increasing in μ. Accordingly, since we have assumed that Condition 2.1 holds, there must exist some μ0>bμsuch that τ μ0 ð Þ¼bτ. Clearly, φ μð Þ :¼AþBþCð Þτ μð Þ−Bð Þμ−Cτ μð Þ 1350 - WHITMEYER is continuous in μ. Evidently, φbμð Þis positive and φ μ0 ð Þis negative, so by the intermediate value theorem there exists some μ)∈bμ;μ0 ð Þ, with φ μ) ð Þ¼0 and τ μ) ð Þ<bτ. Consequently, there exists an equilibrium in which all types μ∈0;μ) ½ � participate, where μ)>bμand τ):¼τ μ) ð Þ<bτ. Note that because belief is a martingale, for each agent that is tested, there must be some test result that ensures that her posterior type (remember, this is her posterior belief about her infection likelihood following a test result) is weakly less than bμand hence μ*. 2.1.1 | An example To add intuition to the proof, let us briefly explore an example. Namely, let Fbe the uniform distribution on 0;1½ � and the parameters be such that bμ¼1 2. Consider a binary test, π, that is given to types μ∈1 4;1 2 � �, with πþj1ð Þ¼π−j0ð Þ¼2 3 where the set of test outcomes is S¼ þ;−f g. Accordingly, each type in 1 4;1 2 � �will be “split” into two new types: with probability μþ1 3, outcome + will realize and the posterior belief (new type) of agent μwill be 2μ 1þμ∈2 5;2 3 � �; and with probability 2−μ 3, outcome − will realize and the posterior belief (new type) of agent μwill be μ 2−μ∈1 7;1 3 � �. This splitting is depicted in Figure 1. Furthermore, Figure 2depicts the new cdf of types, G. Also depicted are ∫μ 0GðxÞdx and ∫μ 0FðxÞdx, which illustrates that Gis a mean‐preserving spread of F. 3 | HOW TESTS CAN LOWER WELFARE Let us turn our attention to a simplified version of the previous section's model. We reduce the model as follows: now, the population is inhabited by just two types of agents, ω H (high likelihood) and ω L (low likelihood). Proportion q= 3/4 of the population are ω L . High likelihood agents, ω H , are infected with (prior) probability μ H , where μH:¼PΘ¼1jωH ð Þ¼5=8 Likewise, μ L denotes the prior probability that type ω L is infected: μL:¼PΘ¼1jωL ð Þ¼1=8 Recall that an agent may choose either to isolate (I) or not (P). We impose the following values for the parameters: A, her payoff from participation, equals 5/4; B, her penalty from infecting someone uninfected, is 2; and C, her penalty from becoming infected, is 4. Thus, if she has a belief μthat she is infected, her payoff from not isolating (P) is u P;μ;τð Þ¼5 4− 2μ1 − τð Þ− 4 1 − μð Þτ which simplifies to u P;μ;τð Þ¼5 4− 4τ− 2 − 6τð Þμð1Þ WHITMEYER - 1351 With no testing, the unique equilibrium is that in which agents of type ω H isolate and agents of type ω L participate. Since only agents of type ω L are participating, the likelihood that an agent who is not isolating is infected is merely the likelihood that an agent of type ω L is infected; viz., τ=μ L = 1/8. On path, an agent of type ω L obtains a payoff of 19/32, which is bigger than 0, her payoff from isolating. Should an agent of type ω H deviate and participate, she would obtain a payoff of −1/32, less than her isolation payoff of 0. In fact, an agent of type ω H would always prefer to isolate (since μ H is so high), unless τwere precisely 0, which could never happen at equilibrium. Consequently, this equilibrium is unique. Without testing, the aggregate welfare for society, V, is V¼q5 4− 4τ− 2 − 6τð ÞμL � �¼3 4 19 32 � �¼57 128 ≈:45 FIGURE 1 Example 2.1.1 Splitting of beliefs (types) [Colour figure can be viewed at wileyonlinelibrary.com] FIGURE 2 Example 2.1.1 Old (F, dashed lines) and new (G, solid lines) distributions of types [Colour figure can be viewed at wileyonlinelibrary.com] 1352 - WHITMEYER 3.1 | After the introduction of a test We introduce a binary test to the scenario. The set of signal realizations is S:¼ þ;−f g, corresponding to a positive test and a negative test, respectively. Consequently, πcan be written in terms of the variable p, where p:¼π−j1ð Þ;and π−j0ð Þ¼1 The situation can be conveniently described by the following joint distribution of Sand Θ for an agent with prior μ i , i=H,L: Θ \ S− + PΘð Þ 1μ i p μi1 − pð Þ μ i 0 1 − μi ð Þ 0 1 − μ i PSð Þ μipþ1 − μi ð Þ μi1 − pð Þ 1 Note that in this model there are no false positives. This is done for expositional convenience and to allow us to focus on the effect of the high false negative rates noted in the introduction. Next, we look at the Nash Equilibria of the participation game with testing. As in Section 2, we focus on equilibria in which agents who think their infection likelihood is low participate and agents who think their infection likelihood is high do not. In this example, absent a test, this is the only equilibrium that exists, and we suppose that this is still the equilibrium selected following a test. To put another way, we assume that introduction of the test does not qualitatively alter the equilibrium selected to an (ostensibly less realistic) equilibrium in which low types isolate and high types participate. We consider three cases: in the first, only agents of type ω H have access to the test. 3.1.1 | Case 1 (testing only high likelihood) Here we suppose that only agents of type ω H may take the test. There are three regions of false negative probabilities, p, each of which beget a different equilibrium. If pis sufficiently low (p≤0.73), then agents of type ω L participate and agents of type ω H participate if and only if they get a negative test result. On the other hand, if pis in an intermediate range (0.9 ≥p≥0.73), then such an equilibrium no longer exists. All agents of type ω L continue to participate, but now only a fraction of the types ω H who have received negative results participate. Finally, if pis too high, only agents of type ω L participate. We start by calculating the values of psuch that at equilibrium agents of type ω L participate and agents of type ω H participate if and only if they get a negative test result. The crucial variable is τ, the likelihood of encountering an infected agent while participating. Using the law of total probability, it is τH p¼qμLþ1 − qð ÞμHp qþ1 − qð Þ μHpþ1 − μH ð Þ¼5pþ3 5pþ27 Using Bayes' law, the probability that type ω H is infected after a negative test is μ− H¼μHp μHpþ1 − μH¼5p 5pþ3ð2Þ Then, using Expression 1, agents of type ω H will participate after − if and only if WHITMEYER - 1353 5 4− 4τH p− 2 − 6τH p � �μ− H≥0ð3Þ which simplifies to p≤9 5−12 5ffiffi5 p≈:73. It is easy to verify that agents of type ω L prefer to participate for this range of p. Moreover, clearly, agents of type ω H have no profitable deviation to action Pafter + , since they are sure that they are infected. Should Inequality 1 fail to hold, there is no longer an equilibrium in which all agents of type ω H who have obtained a negative test participate. However, for moderate p, there is also no equilibrium in which none of those agents participate, since the resulting τwould be low enough to entice participation. Instead, fraction σof the agents of type ω H who have seen a negative test participate. Observe that μ− His the same as above, but the likelihood of encountering an infected agent is shaped by σ. This is τ σð Þ¼ qμLþ1 − qð ÞσμHp qþ1 − qð Þσ μHpþ1 − μH ð Þ¼5pσ þ3 5pσ þ3σþ24 ð4Þ Because some of the agents of type ω H isolate after a negative result and others do not, we need them to be indifferent as to whether they participate after −. Thus, using Expression 1, 5 4− 4τ σ) ð Þ− 2 − 6τ σ) ð Þð Þμ− H¼0 Substituting in for τ σ) ð Þ, we obtain σ)¼24 10p− 9ð Þ 5 25p2− 42pþ9ð Þ which is feasible (lies in the interval [0, 1]) provided 0.73 ≤p≤0.9. Substituting σ* into Equation 4we obtain the equilibrium infection likelihood τ σ) ð Þ¼15 1 − pð Þ 48 − 40pð5Þ If p≥0.9, then the only equilibrium yields the same payoff as the scenario without testing. The test is too uninformative to persuade any agents of type ω H and so as in the case without testing, only agents of type ω L participate, yielding a payoff of 0.45. We finish the analysis of Case 1 by inspecting aggregate welfare as a function of the probability p,V H (Hfor “High likelihood”), the details of whose derivation we leave to Appendix A.1: VH¼ 125p2− 1530pþ1917 128 5pþ27ð Þ ;0≤p≤9 5−12 5ffiffiffi5 p −105p− 9 640p− 768;9 5−12 5ffiffiffi5 p≤p≤9 10 57 128;9 10 ≤p≤1 8 > > > > > > > < > > > > > > > : Comparing this to V, aggregate welfare when there are no tests, we see that V H <V= 57/128 for p∈:21; :9ð Þ, and V H =Vfor p∈[0.9, 1]. Moreover, for all p∈:73; :9½ �, there is a Pareto decrease in welfare: agents of type ω H are no better off and agents of type ω L are strictly worse off. 1354 - WHITMEYER