Probabilistic Verification in Mechanism Design

TL;DR

This paper develops a probabilistic verification model in mechanism design, characterizing when types can be effectively screened and providing a tractable way to optimize mechanisms with verification technology.

Contribution

It introduces a new probabilistic verification framework and characterizes conditions for effective type screening, enabling tractable mechanism design with verification.

Findings

Characterization of when each type has an optimal screening test

Development of a reduced form for verification technology

Mechanism solutions interpolate between no verification and full surplus extraction

Abstract

We introduce a model of probabilistic verification in mechanism design. The principal elicits a message from the agent and then selects a test to give the agent. The agent's true type determines the probability with which he can pass each test. We characterize whether each type has an associated test that best screens out all other types. If this condition holds, then the testing technology can be represented in a tractable reduced form. We use this reduced form to solve for profit-maximizing mechanisms with verification. As the verification technology varies, the solution continuously interpolates between the no-verification solution and full surplus extraction.

Tables

Table 1. Table 1: Taxonomy of verification models

	Reduced form	Microfoundation
partial	message correspondence $M:\Theta \twoheadrightarrow \Theta$ Green and Laffont (1986)	evidence correspondence $E:\Theta \twoheadrightarrow ℰ$ Bull and Watson (2007) Lipman and Seppi (1995)
probabilistic	authentication rate $\alpha :\Theta \times \Theta \to [0,1]$ Our paper Caragiannis et al. (2012)	passage rate $\pi :T\times \Theta \to [0,1]$ Our paper

Equations

$$\pi :T\times \Theta \to [0,1],$$

$$g:\Theta \times T\times \{0,1\}\to \Delta (X).$$

$$a:\Theta \times \Theta \times T\to [0,1],$$

$$\lambda \frac{\pi (\tau |{\theta }^{\prime })}{\pi (\tau |\theta )}+(1-\lambda )\le \frac{\pi (\psi |{\theta }^{\prime })}{\pi (\psi |\theta )}.$$

$$\lambda \frac{1-\pi (\tau |{\theta }^{\prime })}{1-\pi (\tau |\theta )}+(1-\lambda )\ge \frac{1-\pi (\psi |{\theta }^{\prime })}{1-\pi (\psi |\theta )}.$$

$$\alpha ({\theta }^{\prime }|\theta )=\pi (\hat{t}({\theta }^{\prime })|\theta ),\theta ,{\theta }^{\prime }\in \Theta .$$

$$u({\theta }^{\prime }|\theta )=\alpha ({\theta }^{\prime }|\theta )u(g_1({\theta }^{\prime }),\theta )+(1-\alpha ({\theta }^{\prime }|\theta ))u(g_0({\theta }^{\prime }),\theta ),\theta ,{\theta }^{\prime }\in \Theta .$$

$$E[\alpha (\theta |\theta )v(g_1(\theta ),\theta )+(1-\alpha (\theta |\theta ))v(g_0(\theta ),\theta )]$$

$$u(\theta |\theta )\ge u({\theta }^{\prime }|\theta )\vee u(g_0({\theta }^{\prime }),\theta ),\theta ,{\theta }^{\prime }\in \Theta .$$

$$T^{\alpha }=\{{\tau }_{{\theta }^{\prime }}^{\alpha }:{\theta }^{\prime }\in \Theta \},{\pi }^{\alpha }({\tau }_{{\theta }^{\prime }}^{\alpha }|\theta )=\alpha ({\theta }^{\prime }|\theta ).$$

$$\lambda \frac{\alpha ({\theta }_2|{\theta }_1)}{\alpha ({\theta }_2|{\theta }_2)}+(1-\lambda )\le \frac{\alpha ({\theta }_3|{\theta }_1)}{\alpha ({\theta }_3|{\theta }_2)}.$$

$$\lambda \frac{1-\alpha ({\theta }_2|{\theta }_1)}{1-\alpha ({\theta }_2|{\theta }_2)}+(1-\lambda )\ge \frac{1-\alpha ({\theta }_3|{\theta }_1)}{1-\alpha ({\theta }_3|{\theta }_2)}.$$

$$\alpha ({\theta }_3|{\theta }_2)\alpha ({\theta }_2|{\theta }_1)\le \alpha ({\theta }_3|{\theta }_1)\alpha ({\theta }_2|{\theta }_2),$$

$$\alpha ({\theta }^{\prime }|\theta )=\exp \{-d(\theta ,{\theta }^{\prime })\},\theta ,{\theta }^{\prime }\in \Theta .$$

$$\alpha ({\hat{\theta }}^0,{\hat{\theta }}^1|{\theta }^0,{\theta }^1)={\alpha }^0({\hat{\theta }}^0|{\theta }^0),{\theta }^0,{\hat{\theta }}^0\in {\Theta }^0,{\theta }^1,{\hat{\theta }}^1\in {\Theta }^1.$$

$${\theta }_3\in M({\theta }_2)\&{\theta }_2\in M({\theta }_1)⟹{\theta }_3\in M({\theta }_1),$$

$$\alpha ({\theta }^{\prime }|\theta )=\exp \left(-\left|{\int }_{{\theta }^{\prime }}^{\theta }\lambda (\xi )\mathrm{d}\xi \right|\right),\theta ,{\theta }^{\prime }\in \Theta ,$$

$$u(q,t,\theta )=\theta q-t\text{and}v(q,t)=t-c(q),$$

$$\mathrm{maximize}$$

$$\theta q(\theta )-t(\theta )\ge \alpha ({\theta }^{\prime }|\theta )[\theta q({\theta }^{\prime })-t({\theta }^{\prime })],\theta ,{\theta }^{\prime }\in \Theta$$

$$\theta q(\theta )-t(\theta )\ge 0,\theta \in \Theta .$$

$$U(\theta)=U(\underaccent{\bar}{\th})+\int_{\underaccent{\bar}{\th}}^{\theta}q(\xi)\mathop{}\!\mathrm{d}\xi.$$

$$q(\theta )-\lambda (\theta )U(\theta )\le U^{\prime }(\theta )\le q(\theta )+\lambda (\theta )U(\theta ).$$

$$U(\theta)\geq U(\underaccent{\bar}{\th})+\int_{\underaccent{\bar}{\th}}^{\theta}\alpha(\xi|\theta)q(\xi)\mathop{}\!\mathrm{d}\xi.$$

$$\phi (\theta )=\theta -\frac{1}{f(\theta )}{\int }_{\theta }^{\bar{\theta }}\alpha (\theta |\xi )f(\xi )\mathrm{d}\xi .$$

$${\phi }^M(\theta )=\theta -\frac{1}{f(\theta )}{\int }_{\theta }^{\bar{\theta }}f(\xi )\mathrm{d}\xi .$$

$${\phi }^M(\theta )\le \phi (\theta )\le \theta .$$

$$\phi (\theta )=\theta -\frac{1-e^{-\lambda (\bar{\theta }-\theta )}}{\lambda },0\le \theta \le \bar{\theta }.$$

$$\phi (\theta )=\theta -\frac{1}{\gamma +\lambda },\theta \ge 0.$$

$$c^{\prime}(q^{\ast}(\theta))=\varphi(\theta)_{+},\qquad t^{\ast}(\theta)=\theta q^{\ast}(\theta)-\int_{\underaccent{\bar}{\th}}^{\theta}\alpha(\xi|\theta)q^{\ast}(\xi)\mathop{}\!\mathrm{d}\xi.$$

$$t^{\ast }(\theta )={\theta }^{\ast }+{\int }_{{\theta }^{\ast }}^{\theta }(1-\alpha (\xi |\theta ))\mathrm{d}\xi .$$

$$u_i(q,t,\theta )={\theta }_i{q}_i-t_i\text{and}v(q,t)=\sum _{i=1}^n(t_i-c{q}_i).$$

Full text

Keywords: probabilistic verification, ordering tests, evidence.

JEL Codes: D82, D86.

1 Introduction

In the standard paradigm of mechanism design, the principal elicits information from the agents without verifying whether the agents are being truthful. In many applications, however, claims about private information are verified. Sellers have long offered discounts to certain groups such as students, seniors, or veterans, and sellers increasingly make targeted offers to different consumer segments. To verify a buyer’s eligibility for an exclusive offer, many sellers use identity verification platforms such as ID.me and SheerID. If a buyer claims to be eligible for a certain offer, he is directed to a portal, which asks identifying questions or requests documentation, such as a student ID. Depending on the buyer’s responses, the platform’s proprietary algorithm either accepts or rejects the buyer’s claim. In other contexts, lenders verify borrowers’ income reports, and insurers verify the legitimacy of insurance claims. None of these verification systems is perfect—false claims sometimes go undetected. In this sense, verification is probabilistic.

The goal of this paper is to introduce a tractable model of probabilistic verification. The previous literature largely studies models of deterministic verification in which some misrepresentations are detected with certainty and others are never detected. Our probabilistic framework has two main advantages over these deterministic models. First, we can capture applications in which agents recognize that different misrepresentations are detected with different probabilities. Agents therefore trade off the benefit of successful misrepresentation against the risk of detection. In deterministic models, agents consider only misrepresentations that are certain to be successful. Second, in our framework we can use the Myersonian local approach to characterize optimal mechanisms for a range of different verification technologies. In the leading models of deterministic verification, small misrepresentations are not detected, so the local incentive constraints are unaffected by verification. In those models, either verification leaves the solution unchanged, or the local approach fails.

A parsimonious model of probabilistic verification, directly generalizing Green and Laffont’s (1986) deterministic model, would specify for any types $\theta$ and $\theta^{\prime}$ the probability $\alpha(\theta^{\prime}|\theta)$ with which type $\theta$ can “pass” as type $\theta^{\prime}$.111For example, Caragiannis et al. (2012) and Ferraioli and Ventre (2018) take this approach. Call this function $\alpha$ the authentication rate. The problem with this reduced-form model is that the authentication rate is generally endogenous. Whether type $\theta$ can “pass” as type $\theta^{\prime}$ depends on what the principal demands of an agent who claims to be of type $\theta^{\prime}$, e.g., which questions must be answered correctly or which documents must be provided. That is, the authentication rate is induced by the principal’s mechanism.

We model probabilistic verification by endowing the principal with a set of pass–fail tests. A test could be a particular set of questions or a request for certain documentation. We represent a test by its type-dependent passage rate: type $\theta$ can pass test $\tau$ with probability $\pi(\tau|\theta)$. We assume that every type can intentionally fail any test. For example, the agent could leave a question blank (whether or not he knows the answer) or decline to provide the requested documentation (whether or not he has it).

The principal chooses how to utilize the testing technology within a mechanism. Formally, we consider the following protocol. The principal elicits a type report from the agent. Based on the report, the principal selects one test to give the agent. The agent sees the test and privately chooses whether to try on the test or skip it. This choice is costless. If the agent tries, then his passage probability depends on the test and his type, according to the function $\pi$. If the agent skips the test, he fails with certainty. The principal observes whether the agent passes or fails—but not whether the agent tried—and then makes a decision.

Our analysis proceeds in two parts—methodology and then applications.

In the first part of our analysis, we introduce for each type $\theta$ an order on tests. Intuitively, test $\tau$ is more $\theta$-discerning than test $\psi$ if type $\theta$ performs relatively better on test $\tau$ than on test $\psi$, compared to every other type. The formal definition requires that there is a “conversion” from $\tau$-scores to $\psi$-scores that is fair for type $\theta$ but disadvantageous for all other types. This score conversion is similar to a Blackwell garbling of an experiment, but we show that our order neither implies nor is implied by Blackwell’s order.

We use our order on tests to simplify the principal’s implementation problem. Consider two tests, $\tau$ and $\psi$, such that $\tau$ is more $\theta$-discerning than $\psi$. Theorem 1 says that any social choice function that the principal can implement by giving test $\psi$ to type $\theta$ can also be implemented by giving test $\tau$ to type $\theta$. We apply this logic repeatedly to obtain Theorem 2: If each type $\theta$ has an associated test that is most $\theta$-discerning, then there is no loss in assuming that the principal gives each type the associated test. Therefore, this testing technology can be represented by a single authentication rate: type $\theta$ is authenticated as type $\theta^{\prime}$ if and only if he passes the most $\theta^{\prime}$-discerning test. An authentication rate that represents a testing technology in this way is called most-discerning.

Not all authentication rates are most-discerning. We give a condition that characterizes whether an authentication rate is most-discerning. Our condition generalizes the conditions imposed in the literature on deterministic verification (Green and Laffont, 1986) and evidence (Lipman and Seppi, 1995; Bull and Watson, 2007). If an authentication rate is not most-discerning, then taking it as a primitive leads to inconsistencies. For a given testing technology, the testing rule used in some mechanism may induce the given authentication rate, but there exists another implementation problem that requires a different testing rule, which induces a different authentication rate.

For the second part of our analysis, we take a most-discerning authentication rate as a primitive. Consider a seller who can imperfectly verify a potential buyer’s membership in different market segments. It is more difficult for the seller to distinguish buyers who are in market segments with closer valuations. Suppose there is one potential buyer for a single indivisible good. With verification, a posted price is no longer optimal. Instead, the seller charges higher prices to higher-valuation market segments. If a buyer in a higher segment claims to be in a lower segment, there is a chance that he is authenticated and charged a lower price. But there is also a chance that his misreport is detected and he does not receive the good. Under the optimal price schedule, this risk of detection exactly offsets the benefit of a lower price. As the verification technology becomes increasingly accurate, the optimal mechanism converges to perfect price discrimination.

To solve for profit-maximizing mechanisms in general quasilinear settings with verification, we derive a new expression for the virtual value that reflects the verification technology. Recall that the virtual value of a given type equals the true value from allocation, net the additional information rent that the allocation generates for higher types. Verification attenuates the effect on information rents by discouraging higher types from deviating downward. As verification ranges from uninformative to perfectly informative, the virtual value increases from the classical virtual value to the true value. Thus, the optimal mechanism continuously interpolates from the classical, no-verification solution to full surplus extraction. In a multi-agent setting with verification, we use our expression for the virtual value to derive the profit-maximizing auction.

The rest of the paper is organized as follows. Section 2 gives an example that illustrates problems with the reduced-form approach. Section 3 presents our model of testing. Section 4 introduces the discernment orders and characterizes whether a single testing rule suffices for all implementation. Section 5 characterizes the class of most-discerning authentication rates. Section 6 solves for revenue-maximizing mechanisms. Section 7 generalizes the model to nonbinary tests. Section 8 compares our model with other models of verification in economics and computer science. The conclusion is Section 9. The main proofs are in Appendix A.

2 A cautionary example

The following example is adapted from Green and Laffont (1986). There is a single agent with three possible types, denoted $\theta_{1},\theta_{2},\theta_{3}$. In Figure 1, the directed graph (shown twice) represents the verification technology: there is an edge from $\theta$ to $\theta^{\prime}$ if type $\theta$ can “pass” as type $\theta^{\prime}$. The principal decides whether to allocate a good to the agent. Every type wants the good. There are no transfers.

Each copy of the graph illustrates an allocation rule. On the left, the rule allocates the good to types $\theta_{1}$ and $\theta_{2}$ (which are shaded). The principal could try to implement this rule by giving the good to the agent if he passes as type $\theta_{1}$ or as type $\theta_{2}$. But then type $\theta_{3}$ would get the good by passing as type $\theta_{1}$. Instead, the principal must give the good to the agent if and only if he passes as type $\theta_{2}$. Types $\theta_{1}$ and $\theta_{2}$ can do so, but type $\theta_{3}$ cannot. On the right, the rule allocates the good to types $\theta_{2}$ and $\theta_{3}$ (which are shaded). Symmetrically, this rule must be implemented by giving the good to the agent if and only if he passes as type $\theta_{3}$. Types $\theta_{2}$ and $\theta_{3}$ can do so, but type $\theta_{1}$ cannot.

According to the directed graph in this example, type $\theta_{1}$ can “pass” as type $\theta_{2}$. But can type $\theta_{1}$ copy type $\theta_{2}$’s equilibrium strategy? The answer depends on the allocation rule and the associated mechanism and equilibrium: yes, in the left mechanism (where type $\theta_{2}$ passes as type $\theta_{2}$); no, in the right mechanism (where type $\theta_{2}$ passes as type $\theta_{3}$).

The issue is that the graph—or the authentication rate $\alpha(\cdot|\cdot)$ in the general case—implicitly (a) introduces a family of tests, and (b) assigns to each type $\theta$ a test, so that “passing” as type $\theta$ means passing the test assigned to type $\theta$. In the example, type $\theta_{2}$ must be given different tests in order to implement different allocation rules. In general, the probability with which one type can “pass” as another is an equilibrium object that may depend on the social choice function being implemented. Further difficulties with this reduced-form approach are discussed in Section 3.3.

3 Model

We model probabilistic verification by explicitly endowing the principal with a testing technology. The principal can commit to use this technology (and communicate with the agent) however she wishes.

3.1 Setting

Principal–agent environment

There are two players: a principal (she) and an agent (he).222We generalize to multiple agents in Section 6.2.3, where we study auctions. The agent has a private type $\theta\in\Theta$, drawn from a commonly known distribution. The principal controls a decision $x\in X$.333If transfers are allowed, they can be included as one component of the decision. The agent and the principal have type-dependent utilities $u(x,\theta)$ and $v(x,\theta)$, respectively. We extend these functions linearly to $\Delta(X)\times\Theta$.

Verification

The testing technology $(T,\pi)$ consists of a set $T$ of pass–fail tests and a passage rate444We implicitly assume that all sets are Polish, $\pi$ is Borel measurable, and mechanisms are universally measurable. The details are in Section B.8.

[TABLE]

where $\pi(\tau|\theta)$ denotes the probability with which type $\theta$ can pass test $\tau$.

The principal can give the agent one test from the set $T$.555If the principal can give multiple tests, then the resulting compound test can be included in $T$. A compound test may have more scores than “pass” and “fail.” Section 7 extends the model to allow for nonbinary tests. The agent observes the selected test and chooses whether to try or skip. If the agent tries, his passage probability is determined by $\pi$. If he skips, he fails with certainty. The test score (“pass” or “fail”) is observed by the principal, but the agent’s try–skip decision is not. Thus, there is moral hazard as well as adverse selection.

Mechanisms and strategies

The principal can commit to an arbitrary dynamic mechanism. We consider protocols of the following form, shown in Figure 2. First the principal elicits a message from the agent. Based on the message, the principal selects a test and then sends a message to the agent. The agent sees the realized test and the message and then privately chooses whether to try on the test or skip it. Nature draws the test score—“pass” (denoted $1$) or “fail” (denoted [math]). The principal observes this score, but not whether the agent tried, and then makes a decision.

Formally, a mechanism is a tuple $(M_{1},M_{2};t,r_{2},g)$ consisting of message spaces $M_{1}$ and $M_{2}$, a testing rule $t\colon M_{1}\to\Delta(T)$, a messaging rule $r_{2}\colon M_{1}\times T\to\Delta(M_{2})$, and an outcome rule $g\colon M_{1}\times T\times M_{2}\times\{0,1\}\to\Delta(X)$. A mechanism induces a dynamic decision problem for the agent. A strategy for the agent consists of a messaging strategy $r_{1}\colon\Theta\to\Delta(M_{1})$ and an action strategy $a\colon\Theta\times M_{1}\times T\times M_{2}\to[0,1]$, which specifies the probability with which the agent tries on the test he is given.

3.2 Implementation

We introduce two social choice objects. A social choice function is a map from $\Theta$ to $\Delta(X)$, which specifies a decision lottery for each type. To keep track of which test is given, we define an extended social choice function to be a map from $\Theta$ to $\Delta(T\times X)$, which specifies for each type a joint lottery over tests and decisions. A mechanism and a strategy together implement an (extended) social choice function $f$ if (i) the strategy is a best response to the mechanism and (ii) the composition of the mechanism and the strategy induces $f$. An (extended) social choice function is implementable if there exist a mechanism and a strategy that implement it.

We will show that it is without loss to focus on a special class of direct mechanisms that induce the agent to (i) report his type truthfully and (ii) try on whichever test he is given. In these mechanisms, the stage of communication from the principal to the agent is omitted. Formally, a canonical mechanism consists of a testing rule $t\colon\Theta\to\Delta(T)$ and an outcome rule

[TABLE]

Given such a mechanism, a strategy for the agent consists of a reporting strategy $r\colon\Theta\to\Delta(\Theta)$ and an action strategy

[TABLE]

which specifies the probability with which the agent tries as a function of his true type, his reported type, and the test. An (extended) social choice function is canonically implementable if it can be implemented by a canonical mechanism $(t,g)$ and a strategy $(r,a)$ in which $r$ is the identity and $a(\theta,\theta,\tau)=1$ for all types $\theta$ and all tests $\tau$ in $\operatorname{supp}t(\theta)$.

Proposition 1** **(Revelation principle)

Every implementable (extended) social choice function is canonically implementable.

The proof has two steps. First, a standard argument (see Myerson, 1986) shows that every implementable social choice function can be implemented by a truthful and obedient mechanism. The second step of the proof is specific to testing. Whenever the truthful, obedient mechanism recommends that the agent skip a test, we modify the mechanism as follows. If the agent passes the test, he gets the same decision as if he had failed. Now the agent is indifferent between trying and skipping, so it is a best response for the agent to try on every test. Thus, there is no need for action recommendations by the principal. Since every type can fail every test, our modification of the mechanism introduces no new deviations.

3.3 Discussion of the model

We discuss two important features of the model: the agent’s unobserved try–skip decision and the principal’s explicit choice of a testing rule.

Try–skip decision

When the principal gives the agent a test, the agent can privately choose to fail that test, no matter his type. For example, if the agent does not provide a requested document, the principal cannot tell whether the agent actually has it. If the agent fails to answer a question, the principal cannot tell whether the agent knows the answer. With this assumption, our model nests deterministic models of hard evidence. In those models, the agent’s choice to present evidence he possesses is an “inalienable action” (Bull and Watson, 2007, p. 76). The agent’s unobserved try–skip decision introduces moral hazard, which naturally makes implementation more difficult. If skipping a test were infeasible (or observable), the principal could implement a weakly larger set of social choice functions.

Example 1** (Skipping matters).**

Consider the problem of allocating a single indivisible good to an agent with two possible types, $\theta_{1}$ and $\theta_{2}$. There is a single test, which type $\theta_{1}$ passes with certainty and type $\theta_{2}$ fails with certainty. In order to allocate the good to type $\theta_{1}$ only, the principal can give the agent the good if and only if he passes the test. In order to allocate the good to type $\theta_{2}$ only, a natural approach is to give the agent the good if and only if he fails the test. But in our model, type $\theta_{1}$ could still get the good by intentionally failing the test. If instead skipping were assumed to be infeasible, then this mechanism would implement the desired allocation rule.

Test choice

We explicitly model the principal’s choice of which test to give the agent. Previous work considers a reduced form-model that specifies an authentication rate as a primitive. The example in Section 2 shows that in that framework, a social choice rule may be implementable but not truthfully implementable. Even if one considers untruthful (mixed) equilibria in that framework, our framework is still more flexible.

• •

Unrestricted tests. Our model allows for an arbitrary test set. In particular, there may be more tests than types. Depending on which social choice function the principal seeks to implement, all of these tests may be needed.

• •

Unrestricted communication. Our model allows for unrestricted communication between the principal and the agent. The reduced-form model conflates communication and verification. Consider the authentication rate from Figure 1. Suppose that the principal would like to allocate different goods to types $\theta_{1}$ and $\theta_{2}$, but nothing to type $\theta_{3}$. In order for types $\theta_{1}$ and $\theta_{2}$ to distinguish themselves from type $\theta_{3}$, they must both pass as type $\theta_{2}$, but then the principal cannot tell them apart (without an additional stage of communication).

• •

Test randomization. In our model, the principal can commit to randomize over which test is given. This is useful if different tests are needed to deter deviations by different types (see Section A.1 for an example). The reduced-form model implicitly prohibits such randomization.

4 Ordering tests

In this section, we introduce a family of orders on tests. We use these orders to identify a smaller class of testing rules that suffices for all implementation.

4.1 Discernment orders

For a fixed type $\theta$, our order captures whether one test is better than another at distinguishing type $\theta$ from all other types.

Definition 1** ($\theta$-discernment).**

Fix a type $\theta$. Test $\tau$ is more $\theta$-discerning than test $\psi$, denoted $\tau\succeq_{\theta}\psi$, if there exist probabilities $k_{1}$ and $k_{0}$ with $k_{1}\geq k_{0}$ such that

1. (i)

$\pi(\tau|\theta)k_{1}+(1-\pi(\tau|\theta))k_{0}=\pi(\psi|\theta)$; 2. (ii)

$\pi(\tau|\theta^{\prime})k_{1}+(1-\pi(\tau|\theta^{\prime}))k_{0}\leq\pi(\psi|\theta^{\prime})$ for all types $\theta^{\prime}$ with $\theta^{\prime}\neq\theta$.

The interpretation is that after test $\tau$ is taken, the score $s_{\tau}\in\{0,1\}$ is converted into a score $s_{\psi}\in\{0,1\}$ according to the transition probabilities $\mathbf{P}(s_{\psi}=1|s_{\tau}=1)=k_{1}$ and $\mathbf{P}(s_{\psi}=1|s_{\tau}=0)=k_{0}$. The inequality $k_{1}\geq k_{0}$ ensures that passing test $\tau$ weakly improves the converted pass probability. Condition i says that this score conversion is fair for type $\theta$. If type $\theta$ tries on test $\tau$ and his score is converted, he is just as likely to pass as if he tries on test $\psi$ directly. Condition ii says that this score conversion is weakly disadvantageous for any other type $\theta^{\prime}$. If type $\theta^{\prime}$ tries on test $\tau$ and his score is converted, he is no more likely to pass than if he tries on test $\psi$ directly. Section 4.3 provides a more intuitive characterization of the discernment orders. We first illustrate, through Theorems 1 and 2, the merits of our definition.

In the language of statistical hypothesis testing, we can think of failing a test as rejecting the null hypothesis. Then our definition requires that test $\tau$ can be used to construct an hypothesis test of the null $\theta$ against the alternative $\Theta\setminus\{\theta\}$ with significance $1-\pi(\psi|\theta)$ that is uniformly more powerful than test $\psi$. The requirement that $k_{1}\geq k_{0}$ preserves incentives, which are not relevant in the statistical framework.

Theorem 1** **(Test replacement)

Fix a type $\theta$ and tests $\tau$ and $\psi$ such that $\tau\succeq_{\theta}\psi$. If a social choice function can be canonically implemented by a mechanism in which $t(\theta)=\psi$, then it can also be canonically implemented by a mechanism in which $t(\theta)=\tau$.

Here is a sketch of the proof. Start with a canonical implementation in which $t(\theta)=\psi$. Adjust the mechanism after the report $\theta$ as follows. The principal gives the agent test $\tau$ and then converts the agent’s score $s_{\tau}$ into a new score $s_{\psi}$ using the transition probabilities $k_{1}$ and $k_{0}$. Then the principal takes the decision that she would have taken in the old mechanism after score $s_{\psi}$ on test $\psi$.

This new mechanism implements the same social choice function. Suppose that type $\theta$ reports truthfully and tries on test $\tau$. By (i), he will get the same decision as in the original mechanism. Suppose another type $\theta^{\prime}$ reports type $\theta$ and tries on test $\tau$. By (ii), he will get a decision that he could have gotten in the original mechanism by reporting type $\theta$ and then skipping test $\psi$ with some probability. Finally, the inequality $k_{1}\geq k_{0}$ ensures that skipping test $\tau$ does not improve the agent’s converted score, and hence is not profitable.

For each fixed type $\theta$, the $\theta$-discernment order $\succeq_{\theta}$ is neither stronger nor weaker than Blackwell’s (1953) order. Blackwell’s order takes the same form as 1 except (a) the inequality $k_{1}\geq k_{0}$ is dropped, and (b) the inequality in (ii) is strengthened to equality. Blackwell’s order is not suited to our setting because it does not consider the agent’s incentives to change his score by skipping the test. Indeed, Blackwell’s order is invariant to relabeling the realizations “pass” and “fail.” Our $\theta$-discernment order is not.

Like Blackwell’s order, each $\theta$-discernment order $\succeq_{\theta}$ is reflexive and transitive but not generally anti-symmetric. Tests $\tau_{1}$ and $\tau_{2}$ are $\theta$-equivalent, denoted $\tau_{1}\sim_{\theta}\tau_{2}$, if $\tau_{1}\succeq_{\theta}\tau_{2}$ and $\tau_{1}\preceq_{\theta}\tau_{2}$. If two tests have the same passage rates, then they are clearly $\theta$-equivalent. We show that the converse holds except in the special case where neither test can screen any other type away from type $\theta$. Formally, type $\theta$ is minimal on test $\tau$ if $\pi(\tau|\theta)\leq\pi(\tau|\theta^{\prime})$ for types all $\theta^{\prime}$.

Proposition 2** **($\theta$-discernment equivalence)

Fix a type $\theta$. Tests $\tau_{1}$ and $\tau_{2}$ are $\theta$-equivalent if and only if (a) $\pi(\tau_{1}|\cdot)=\pi(\tau_{2}|\cdot)$ or (b) type $\theta$ is minimal on $\tau_{1}$ and on $\tau_{2}$.

4.2 Implementation with most-discerning testing

Theorem 1 is particularly useful if there is a single test that can replace every other test for type $\theta$.

Definition 2** (Most discerning).**

A test $\tau$ is most $\theta$-discerning if $\tau\succeq_{\theta}\psi$ for all $\psi$ in $T$. A function $t\colon\Theta\to T$ is most discerning if for each type $\theta$ the test $t(\theta)$ is most $\theta$-discerning.

In the language of hypothesis testing, a most $\theta$-discerning test corresponds to a uniformly most powerful test for the null $\theta$ against the alternative $\Theta\setminus\{\theta\}$. Whether a test is most $\theta$-discerning depends on the feasible set $T$. The only test that is more $\theta$-discerning than every test is the perfect test $\hat{\tau}_{\theta}$ that exactly identifies whether the agent’s type is $\theta$, i.e., $\pi(\hat{\tau}_{\theta}|\theta^{\prime})=[\theta^{\prime}=\theta]$.

To state the main result, we define a decision environment to consist of a decision set $X$ and a utility function $u\colon X\times\Theta\to\mathbf{R}$ for the agent.

Theorem 2** **(Most-discerning implementation)

Fix a type space $\Theta$ and a testing technology $(T,\pi)$. For a testing function $\hat{t}\colon\Theta\to T$, the following are equivalent.

$\hat{t}$* is most discerning.* 2. 2.

In every decision environment $(X,u)$, every implementable social choice function is canonically implementable with $\hat{t}$.

First, consider the implication from condition 1 to condition 2. If a testing function is most-discerning, then it suffices for all implementation. The proof applies the procedure from Theorem 1 to replace any test given to type $\theta$ with test $\hat{t}(\theta)$.

The implication from condition 2 to condition 1 confirms that the most-discerning property is the right one. If $\tau\not\succeq_{\theta}\psi$, then replacing test $\psi$ with test $\tau$ for type $\theta$ introduces a new deviation for some type. The proof constructs a decision environment in which this new deviation is profitable.666Our argument has the same spirit as de Oliveira’s (2018) elegant proof of Blackwell’s theorem using diagrams.

Even if the testing technology does not admit a most-discerning testing function, we can still use the replacement theorem (Theorem 1) to reduce the class of tests that need to be considered. Suppose there is a subset $\hat{T}(\theta)$ of tests with the property that for every test $\psi$ there is some test $\tau$ in $\hat{T}(\theta)$ such that $\tau\succeq_{\theta}\psi$. Then there is no loss in assuming that on type $\theta$ the principal uses only (mixtures over) tests in $\hat{T}(\theta)$. See Section B.1 for a formal statement.

4.3 Characterizing the discernment orders

We now characterize $\theta$-discernment in terms of relative performance.

Proposition 3** **(Relative performance)

Fix a type $\theta$ and tests $\tau$ and $\psi$.777The two (non-exclusive) cases exclude two edge cases. If $\pi(\psi|\theta)=0$, then $\tau\succeq_{\theta}\psi$. If $\pi(\tau|\theta)<\pi(\psi|\theta)=1$, then $\tau\not\succeq_{\theta}\psi$.

Suppose $\pi(\tau|\theta)\geq\pi(\psi|\theta)>0$. We have $\tau\succeq_{\theta}\psi$ if and only if there exists $\lambda\in[0,1]$ such that for all types $\theta^{\prime}$ with $\theta^{\prime}\neq\theta$,

[TABLE] 2. 2.

Suppose $\pi(\tau|\theta)\leq\pi(\psi|\theta)<1$. We have $\tau\succeq_{\theta}\psi$ if and only if there exists $\lambda\in[0,1]$ such that for all types $\theta^{\prime}$ with $\theta^{\prime}\neq\theta$,

[TABLE]

To build intuition, first take $\lambda=1$ in (1) and (2) to get a sufficient condition for $\tau\succeq_{\theta}\psi$ in each case. In the first case, where type $\theta$ is more likely to pass test $\tau$ than test $\psi$, we have $\tau\succeq_{\theta}\psi$ if the relative passage rate of type $\theta^{\prime}$ compared to type $\theta$ is lower on test $\tau$ than on test $\psi$. In the second case, where type $\theta$ is more likely to fail $\tau$ than $\psi$, we have $\tau\succeq_{\theta}\psi$ if the relative failure rate of type $\theta^{\prime}$ compared to type $\theta$ is higher on test $\tau$ than on test $\psi$.

The parameter $\lambda$ controls the difference $k_{1}-k_{0}$ between the transition probabilities $k_{1}$ and $k_{0}$ in 1. In particular, $\lambda=1$ makes $k_{1}-k_{0}$ as large as possible, and $\lambda=0$ makes $k_{1}$ and $k_{0}$ equal. Consider some type $\theta^{\prime}$ who contemplates reporting $\theta$. If $\pi(\tau|\theta)>\pi(\tau|\theta^{\prime})$, then (1) and (2) are easiest to satisfy with $\lambda=1$. Since type $\theta^{\prime}$ performs worse than type $\theta$ on test $\tau$, type $\theta^{\prime}$ is most disadvantaged if the converted $\psi$-score is most sensitive to the true $\tau$-score. If $\pi(\tau|\theta)<\pi(\tau|\theta^{\prime})$, then (1) and (2) are easiest to satisfy with $\lambda=0$. Since type $\theta^{\prime}$ performs better than type $\theta$ on test $\tau$, type $\theta^{\prime}$ is least advantaged if the converted $\psi$-score is insensitive to the true $\tau$-score. In Proposition 3, a single conversion must deter deviations by all types, so a single value of $\lambda$ must work for every deviator $\theta^{\prime}$. The required $\lambda$ may be strictly between [math] and $1$; see Section B.2 for an example.

In view of Proposition 3, a simple sufficient condition for test $\tau$ to be most $\theta$-discerning is that $\tau$ maximizes $\pi(\cdot|\theta)$ and minimizes $\pi(\cdot|\theta^{\prime})$ for all $\theta^{\prime}\neq\theta$. That is, among all tests in $T$, test $\tau$ is one that type $\theta$ is most likely to pass but other types are most likely to fail.

5 Testing in reduced form

In this section, we show how any testing technology that admits a most-discerning testing function can be represented by a primitive authentication rate. We then characterize which authentication rates can be microfounded in this way.

Suppose that the testing technology $(T,\pi)$ admits a most-discerning testing function $\hat{t}$. By Theorem 2, there is no loss of generality in restricting the principal to using $\hat{t}$. With testing function $\hat{t}$, the principal need only select two decisions for each report $\theta^{\prime}$—the decision if the agent passes test $\hat{t}(\theta^{\prime})$, denoted $g_{1}(\theta^{\prime})$, and the decision if the agent fails test $\hat{t}(\theta^{\prime})$, denoted $g_{0}(\theta^{\prime})$. Suppose type $\theta$ reports type $\theta^{\prime}$ and then tries on test $\hat{t}(\theta^{\prime})$. With probability $\pi(\hat{t}(\theta^{\prime})|\theta)$, he passes and gets $g_{1}(\theta^{\prime})$. With probability $1-\pi(\hat{t}(\theta^{\prime})|\theta)$, he fails and gets $g_{0}(\theta^{\prime})$. This simple protocol can be represented by an authentication rate. Formally, the authentication rate induced by $\hat{t}$ is defined by

[TABLE]

For any reduced outcome rule $g=(g_{0},g_{1})\colon\Theta\to\Delta(X)\times\Delta(X)$, define the associated utilities by

[TABLE]

The principal’s problem is to choose a reduced outcome rule $g$ to solve

[TABLE]

The constraints capture truthtelling and obedience. They require that for each type $\theta$, reporting $\theta$ and trying on test $\hat{t}(\theta)$ is weakly preferred to reporting $\theta^{\prime}$ and either trying on test $\hat{t}(\theta^{\prime})$ or skpping it. In particular, taking $\theta^{\prime}=\theta$ ensures that type $\theta$ prefers trying on test $\hat{t}(\theta)$ rather than skipping it. In contrast to models of exogenous lying costs,888In models of lying costs, the agent pays a cost $c(\theta^{\prime}|\theta)$ if he reports $\theta^{\prime}$ when his true type is $\theta$. See, for example, Lacker and Weinberg (1989), Maggi and Rodriguéz-Clare (1995), Crocker and Morgan (1998), Kartik et al. (2007), Kartik (2009), and Deneckere and Severinov (2022). In particular, Kephart and Conitzer (2016) show that, with lying costs, the revelation principle holds if the lying cost function satisfies the triangle inequality. here the effective cost of misreporting is determined jointly by the authentication rate and the principal’s choice of decisions when the agent is not authenticated.

An authentication rate $\alpha$ is most-discerning if there exists a testing technology $(T,\pi)$ with a most-discerning testing function $\hat{t}$ such that (3) holds. To be sure, we believe the right interpretation of any authentication rate $\alpha$ is that the principal has available the testing technology $(T^{\alpha},\pi^{\alpha})$, given by

[TABLE]

But if $\alpha$ is not most-discerning, then the testing function $\theta\mapsto\tau_{\theta}^{\alpha}$ cannot be most-discerning in $(T^{\alpha},\pi^{\alpha})$. It follows from Theorem 2 that for some decision environment and some objective of the principal, the solution of (4) is strictly suboptimal within the class of all mechanisms using the testing technology $(T^{\alpha},\pi^{\alpha})$. If $\alpha$ is most-discerning, then it is easy to check that the testing function $\theta\mapsto\tau_{\theta}^{\alpha}$ is most-discerning in $(T^{\alpha},\pi^{\alpha})$.999If $\alpha$ is induced by some most-discerning testing function $\hat{t}$ in some testing technology $(T,\pi)$, then $\hat{t}$ is also most-discerning in the testing technology $(T^{\prime},\pi^{\prime})$ where $T^{\prime}=\hat{t}(\Theta)$ and $\pi^{\prime}$ is the restriction of $\pi$ to $T^{\prime}\times\Theta$. Observe that $(T^{\prime},\pi^{\prime})$ is isomorphic to $(T^{\alpha},\pi^{\alpha})$ via the map $\hat{t}(\theta)\mapsto\tau_{\theta}^{\alpha}.$ Therefore, the solution of (4) is optimal among all dynamic mechanisms using the testing technology $(T^{\alpha},\pi^{\alpha})$.

Here is the upshot. Our model provides a testing microfoundation for reduced-form models that take a most-discerning authentication rate as a primitive. This approach proves tractable in classical mechanism design settings, as we show in Section 6. There are two caveats. First, not every testing technology can be represented by a single authentication rate. If the available testing technology in a particular application does not admit a most-discerning testing function, then the full protocol from Figure 2 is required.101010Some simplifications are still possible; see Section B.1. Second, not all authentication rates are most-discerning. If an authentication rate is not most-discerning, then treating it as exogenous is not consistent with any model in which the principal can freely commit to use some testing technology.

It is important to be able to determine whether an authentication rate is most-discerning. We now apply Proposition 3 to directly characterize the most-discerning authentication rates.

Proposition 4** **(Authentication rate characterization)

An authentication rate $\alpha$ is most discerning if and only if the following hold for all distinct types $\theta_{2}$ and $\theta_{3}$.

If $\alpha(\theta_{2}|\theta_{2})\geq\alpha(\theta_{3}|\theta_{2})$,111111If $\alpha(\theta_{3}|\theta_{2})=0$, we interpret the right side of the displayed inequality as $\infty$, so that it always holds. then there exists $\lambda\in[0,1]$ such that, for all types $\theta_{1}\neq\theta_{2}$,

[TABLE] 2. 2.

If $\alpha(\theta_{2}|\theta_{2})<\alpha(\theta_{3}|\theta_{2})$,121212If $\alpha(\theta_{3}|\theta_{2})=1$, we interpret the right side of the displayed inequality as $\infty$, so that it never holds. then there exists $\lambda\in[0,1]$ such that, for all types $\theta_{1}\neq\theta_{2}$,

[TABLE]

In some natural cases, a simpler characterization is available. If $\alpha(\theta|\theta)\geq\alpha(\theta|\theta^{\prime})$ for all types $\theta$ and $\theta^{\prime}$, then $\alpha$ is most discerning if and only if

[TABLE]

for all $\theta_{1},\theta_{2},\theta_{3}\in\Theta$. If $\alpha(\theta|\theta)=1$ for all types $\theta$, then this condition reduces to supermultiplicativity: It is more likely that $\theta_{1}$ is directly authenticated as type $\theta_{3}$ than that type $\theta_{1}$ is authenticated as type $\theta_{2}$, and type $\theta_{2}$ is (independently) authenticated as type $\theta_{3}$. We conclude with a few intuitive classes of most-discerning authentication rates.

Example 2** (Semi-metric authentication rate).**

Let $d$ be a semi-metric on the type space $\Theta$. Consider the authentication rate $\alpha$ defined by

[TABLE]

It is easy to check that $\alpha$ is most discerning. The interpretation is that types that are closer in the semi-metric $d$ are more similar and hence are more difficult to distinguish.131313As a special case, suppose $\Theta=\mathbf{R}^{k}$ and $d$ is induced by a norm. The resulting class of functions $\alpha$ is axiomatized in Billot et al. (2008). They interpret these functions as measures of similarity in the context of their model of belief-formation through similarity-weighted averaging. This authentication rate is symmetric because $d$ is symmetric, but most-discerning authentication rates need not be symmetric in general.

Example 3** (Separate verifiable and payoff components).**

Let $\Theta=\Theta^{0}\times\Theta^{1}$, with a generic type denoted by $\theta=(\theta^{0},\theta^{1})$. Let $\alpha^{0}$ be a most-discerning authentication rate on $\Theta^{0}$. We obtain a most-discerning authentication rate $\alpha$ on $\Theta$ by defining

[TABLE]

Think of $\theta^{0}$ as a payoff irrelevant, verifiable attribute, and $\theta^{1}$ as an unverifiable, payoff type. Reuter (2023) considers this structure in a model of partial verification. The prior distribution over $\Theta^{0}\times\Theta^{1}$ controls how informative the agent’s verifiable attribute is about his payoff type. For example, the verifiable attribute could indicate whether the agent is a student, and the population of students may have a different distribution of valuations for a good.

Example 4** (Partial verification).**

Suppose that the authentication rate $\alpha$ is $\{0,1\}$-valued. For each type $\theta$, let $M(\theta)=\{\theta^{\prime}:\alpha(\theta^{\prime}|\theta)=1\}$. Hence, type $\theta$ can pass as any type in $M(\theta)$. Following Green and Laffont (1986), suppose that each type can pass as himself, i.e, $\alpha(\theta|\theta)=1$ for all $\theta$. In terms of $M$, (6) becomes

[TABLE]

which is exactly Green and Laffont’s (1986) nested range condition.

6 Applications: Profit-maximization with verification

As an application of the microfounded reduced-form approach, we solve for profit-maximizing mechanisms with verification in a few classical settings.

6.1 Setting

Authentication rate

The type space is an interval \Theta=[\underaccent{\bar}{\th},\bar{\theta}]. We represent the verification technology by a most-discerning authentication rate $\alpha$. Assume that $\alpha$ takes the exponential form

[TABLE]

for some continuous function \lambda\colon[\underaccent{\bar}{\th},\bar{\theta}]\to\mathbf{R}_{+}. It is easy to check that this function $\alpha$ is most discerning. The exponential authentication rate allows for a cleaner characterization of optimal mechanisms. Section B.3 shows how to extend the analysis to other most-discerning authentication rates.

The value of $\lambda(\theta)$ quantifies the local precision of the verification technology near type $\theta$. The function $\alpha(\theta|\cdot)$ has a kink at type $\theta$ if and only if $\lambda(\theta)>0$. Figure 3 plots the authentication rate when $\lambda(\theta)=1$ for all $\theta$. The plot shows the authentication probability, as a function of the agent’s true type, for two fixed reports $\theta^{\prime}$ and $\theta^{\prime\prime}$.

Quasilinear environment

The agent’s type \theta\in\Theta=[\underaccent{\bar}{\th},\bar{\theta}] is drawn from a distribution function $F$ with strictly positive density $f$. The principal allocates a quantity $q\in Q\subset\mathbf{R}_{+}$ and receives a transfer $t\in\mathbf{R}$.141414The pair $(q,t)$ is the decision $x$ in the general model. In applications, $t$ always denotes transfers (and we make no direct reference to tests). Utilities for the agent and the principal are given by

[TABLE]

for some weakly convex cost function $c\colon Q\to\mathbf{R}_{+}$. For concreteness, we interpret the principal as the seller of a good and the agent as a potential buyer. The agent’s type represents his market segment, which the principal can imperfectly verify. To avoid the known difficulties of multi-dimensional mechanism design, we make the stylized assumption that the agent’s market segment pins down his valuation. With the exponential authentication rate, it is more difficult for the seller to distinguish buyers who are in market segments with closer valuations. An alternative interpretation of this setting is that the principal is a procurer of the good who can imperfectly verify the agent’s production costs.151515Formally, reinterpret $-\theta$ as the agent’s cost, and $-c(q)$ as the principal’s utility from procuring quantity $q$. If the agent misreports his costs, then with some probability (that depends on the true costs and the claim) the audit will produce evidence proving that the agent lied. We assume that if this evidence is produced, the principal can exclude the agent, reducing his utility level to his outside option of [math].

The agent is free to walk away at any time, so we impose an ex post participation constraint.161616In particular, this rules out upfront payments like those used in Border and Sobel (1987). If the principal could impose arbitrarily severe punishments for failed authentication, then probabilistic verification would essentially reduce to perfect verification; see Caragiannis et al. (2012).

Since $\alpha(\theta|\theta)=1$ for all $\theta$, the agent is always authenticated if he is truthful. Therefore, there is no loss in assuming that if the agent is not authenticated, the principal holds him to his outside option utility: the agent does not receive the good and he pays nothing. Formally, we set $g_{0}(\theta)=(0,0)$ for all $\theta$, and we optimize over the decision rule $g_{1}$. Denote the quantity and transfer components of $g_{1}$ by $q$ and $t$.

The principal selects a quantity function $q\colon\Theta\to\mathbf{R}_{+}$ and a transfer function $t\colon\Theta\to\mathbf{R}$ to solve

[TABLE]

The first constraint is incentive compatibility.171717This is the constraint from (4), but it now takes a simple form because $u(g_{0}(\theta^{\prime}),\theta)=0$ for all $\theta^{\prime}$ and $\theta$. The second is ex post participation, conditional upon being authenticated.

6.1.1 Virtual value

We sketch the derivation of the virtual value in this quasilinear setting with verification. In the classical setting without verification, the envelope theorem pins down the derivative of the indirect utility function $U$ in terms of the allocation rule: $U^{\prime}(\theta)=q(\theta)$. Hence,

[TABLE]

With verification, the derivative of the indirect utility function $U$ is no longer pinned down by the quantity function because of the kink in $\alpha(\theta|\cdot)$. Instead, the envelope formula gives the differential inequality181818See Carbajal and Ely (2013) for a general characterization of the incentive compatible indirect utility functions with kinked utility. Carbajal and Ely (2016) apply their characterization in a model of reference-dependent utility.

[TABLE]

The lower bound on the derivative can be shown to imply the bound

[TABLE]

The exponential authentication rate $\alpha$ appears in the integrand as the solution of a linear differential equation, not as part of a global incentive constraint. The envelope formula for $U$ depends only on the local behavior of $\alpha$ around the diagonal, which is captured by the function $\lambda$. For other authentication rates, (9) is more complicated; see Section B.3.

It is optimal to choose $U$ so that U(\underaccent{\bar}{\th})=0 and (9) holds with equality. After substituting this choice of $U$ into the objective and changing the order of integration, the principal’s objective can be expressed as a linear functional in $q$. The coefficient on $q(\theta)$ is the virtual value of type $\theta$:

[TABLE]

Myerson’s virtual value is derived similarly in the no-verification problem, using (8) in place of (9). Myerson’s virtual value can be expressed symmetrically as

[TABLE]

The virtual value of type $\theta$ is the marginal revenue from allocating to type $\theta$. There are two parts. First, the principal can extract the additional consumption utility $\theta$ from type $\theta$. Second, the allocation pushes up the indirect utility of each type $\xi$ above type $\theta$. This marginal effect on type $\xi$, which equals $1$ in (8) and $\alpha(\theta|\xi)$ in (9), is then integrated against the relative density $f(\xi)/f(\theta)$. Comparing the virtual values, we immediately see that

[TABLE]

The virtual value $\varphi(\theta)$ tends toward these bounds in the limiting cases of no verification and perfect verification.

Proposition 5** **(Testing precision)

*As $\lambda$ converges to [math] pointwise, $\varphi(\theta)$ converges to $\varphi^{M}(\theta)$ for each type $\theta$.

As $\lambda$ converges to $\infty$ pointwise, $\varphi(\theta)$ converges to $\theta$ for each type $\theta$.*

Below, we will characterize optimal mechanisms under the assumption that the virtual value $\varphi$ is increasing. This is a joint assumption on the type distribution and the authentication rate. If $\lambda(\theta)=\lambda$ for all $\theta$, then the virtual value has a simple expression for a couple of standard distributions. If the state is uniformly distributed over [\underaccent{\bar}{\th},\bar{\theta}], then

[TABLE]

If $\theta$ is exponentially distributed with intensity parameter $\gamma$, then we have

[TABLE]

In each case, the virtual value $\varphi$ is strictly increasing, for each fixed $\lambda$.

6.2 Optimal mechanisms

We state the optimal mechanism in three classical problems.

6.2.1 Nonlinear pricing

For nonlinear pricing (Mussa and Rosen, 1978), the quantity space is $Q=\mathbf{R}_{+}$. Assume that the principal’s cost function $c$ satisfies $c^{\prime}(0)=0$ and $c^{\prime}$ is strictly increasing with $\lim_{q\to\infty}c^{\prime}(q)>\bar{\theta}$.

Proposition 6** **(Optimal nonlinear pricing)

Suppose that the virtual value $\varphi$ is increasing. The optimal quantity function $q^{\ast}$ and transfer function $t^{\ast}$ are given by191919We refer to the optimal mechanism here and in Proposition 7, even though there is flexibility for a knife-edge set of types.

[TABLE]

The optimal allocation rule has the same form as in the classical case, except the new virtual value appears in place of the classical virtual value. Transfers are determined by the indirect utility function $U$, which is given by the minimal solution of (9).

Each type $\theta$ receives the quantity that is efficient for type $\varphi(\theta)_{+}$. Therefore, quantity is distorted below the efficient level for every type except $\bar{\theta}$. As $\lambda$ increases pointwise, so that the verification technology becomes more precise, downward distortion is attenuated. In the limit of perfect verification, the good is allocated efficiently and the principal extracts the full surplus.

6.2.2 Selling a single indivisible good

For a single indivisible good (Riley and Zeckhauser, 1983), the quantity space is $Q=[0,1]$. Here, quantity is interpreted as the probability of allocating the good. Hence, $c(q)=cq$, where $c$ is the cost of producing a single good. Assume $0\leq c\leq\bar{\theta}$.

Without verification, the profit-maximizing mechanism is a posted price. With verification, the seller charges different prices to consumers in different market segments.

Proposition 7** **(Optimal sale of a single good)

Suppose that the virtual value $\varphi$ is strictly increasing. The optimal quantity function $q^{\ast}$ and transfer function $t^{\ast}$ are given as follows. Define the critical value $\theta^{\ast}=\inf\{\theta:\varphi(\theta)\geq c\}$. If $\theta<\theta^{\ast}$, then $q^{\ast}(\theta)=t^{\ast}(\theta)=0$. If $\theta\geq\theta^{\ast}$, then $q^{\ast}(\theta)=1$ and

[TABLE]

As in the classical solution, the allocation probability takes values [math] and $1$ only—there is no randomization. There is a cutoff type $\theta^{\ast}$ who receives the good and pays his valuation. Each type below the cutoff is excluded. Each type above the cutoff receives the good and pays less than his valuation.

Verification increases the virtual value relative to the classical virtual value, so the cutoff type is lower and more types receive the good. The price is no longer uniform. As long as $\lambda$ is strictly positive, the price is strictly increasing in the agent’s report. Nevertheless, types above the cutoff cannot profit by misreporting downward—the benefit of a lower price is outweighed by the risk of failing to be authenticated and getting nothing.

6.2.3 Auctions

Consider an auction for a single indivisible good (Myerson, 1981). We extend the quasilinear setting to multiple agents. Each agent $i$ has an exponential authentication rate $\alpha_{i}$, with associated continuous precision function $\lambda_{i}$. The solution concept is Bayes–Nash equilibrium. We maintain the ex post participation constraint for each agent. See Section B.4 for the formal extension of our testing model with multiple agents.

Each agent $i$ independently draws his type \theta_{i}\in\Theta_{i}=[\underaccent{\bar}{\th}_{i},\bar{\theta}_{i}] from a distribution function $F_{i}$ with positive density $f_{i}$. The principal allocates the good to agent $i$ with probability $q_{i}\in[0,1]$, where $q_{1}+\cdots+q_{n}\leq 1$. The principal receives transfer $t_{i}$ from agent $i$. Write $q=(q_{1},\ldots,q_{n})$ and $t=(t_{1},\ldots,t_{n})$. The principal’s cost of providing the good is $c$, where $0\leq c\leq\max_{i}\bar{\theta}_{i}$. Utilities are given by

[TABLE]

Proposition 8** **(Optimal auction)

Suppose that each virtual value $\varphi_{i}$ is strictly increasing. An optimal mechanism is given as follows. For each bidder $i$ and each $\theta_{-i}$ in $\Theta_{-i}$, let

[TABLE]

If $\theta_{i}\leq r_{i}(\theta_{-i})$, then $q_{i}^{\ast}(\theta)=t_{i}^{\ast}(\theta)=0$. If $\theta_{i}>r_{i}(\theta_{-i})$, then $q_{i}^{\ast}(\theta)=1$ and

[TABLE]

Proposition 8 describes the particular optimal mechanism that is ex-post incentive compatible. First consider the symmetric case in which $\alpha_{i}=\alpha$ and $F_{i}=F$, hence $\varphi_{i}=\varphi$, for all bidders $i$. Thus, $r_{i}(\theta_{-i})=r^{\ast}\vee\max_{j\neq i}\theta_{j}$, where $r^{\ast}=\varphi^{-1}(c)$. Suppose that at least one bidder’s valuation is above the reserve $r^{\ast}$. Let $i^{\ast}$ be the bidder with the highest valuation. Then bidder $i^{\ast}$ gets the good and pays an amount between $r^{\ast}\vee\max_{j\neq i^{\ast}}\theta_{j}$ and his own valuation $\theta_{i^{\ast}}$. This payment is increasing in the precision of the verification technology.

Next, consider ex ante asymmetric bidders. In the no-verification case, the allocation rule rewards bidders with value distributions that are weaker in the sense of the hazard rate order. With verification, the allocation rule further rewards bidders who can be verified more precisely. Allocating the good to a given type $\theta_{i}$ has a smaller effect on the expected information rent if the bidder is weaker (so that higher types are relatively rarer) or if the bidder’s type is more precisely verifiable (so that higher types find downward deviations less attractive).

7 Beyond pass–fail tests

The main model considers pass–fail tests. In this section, we consider tests that generate scores in a finite score set $S$. The type-dependent performance on each test is given by

[TABLE]

which specifies for each type $\theta$ and test $\tau$ a distribution $\pi_{\tau|\theta}$ over $S$.

To generalize the try–skip decision, we take as primitive a partial order $\succeq$ on $S$. The interpretation is that the agent can reduce his score by changing $s$ to $s^{\prime}$ if $s\succeq s^{\prime}$. As before, the agent’s choice is costless and unobservable to the principal.

Pass–fail testing corresponds to $S=\{0,1\}$ with the usual order $\succeq$. If type $\theta$ mixes between trying and skipping on test $\tau$, he can pass with any probability below $\pi(\tau|\theta)$. With nonbinary tests, type $\theta$ can achieve on test $\tau$ any score distribution $p$ in $\Delta(S)$ satisfying $\pi_{\tau|\theta}\succeq_{\operatorname{st}}p$, where $\succeq_{\operatorname{st}}$ is the stochastic order between measures on a partially ordered space (Kamae et al., 1977). That is, $\mu\succeq_{\operatorname{st}}\nu$ if and only if $\mu(U)\geq\nu(U)$ for every upper set $U$.202020An upper set is a set with the property that if $s$ is in $U$ and $s^{\prime}\succeq s$, then $s^{\prime}$ is also in $U$. The order $\succeq_{\operatorname{st}}$ can be defined dually in terms of lower sets (since a set is an upper set if and only if its complement is a lower set).

Now we define the $\theta$-discernment orders in this setting. A function $k\colon S\to\Delta(S)$ is increasing if $k(s)\succeq_{\operatorname{st}}k(s^{\prime})$ whenever $s\succeq s^{\prime}$. We interpret $k$ as a Markov transition, and we use the following notation from Markov chains. Given $\mu$ in $\Delta(S)$ and $k\colon S\to\Delta(S)$, the measure $\mu k$ on $\Delta(S)$ is defined by $(\mu k)(A)=\sum_{s}\mu(s)k(A|s)$, for $A\subset S$.

Definition 3** ($\theta$-discernment for general tests).**

Fix a type $\theta$. Test $\tau$ is more $\theta$-discerning than test $\psi$, denoted $\tau\succeq_{\theta}\psi$, if there exists an increasing function $k\colon S\to\Delta(S)$ such that

1. (i)

$\pi_{\tau|\theta}k=\pi_{\psi|\theta}$; 2. (ii)

$\pi_{\tau|\theta^{\prime}}k\preceq_{\operatorname{st}}\pi_{\psi|\theta^{\prime}}$ for all types $\theta^{\prime}$ with $\theta^{\prime}\neq\theta$.

In this general setting, the order $\succeq_{\theta}$ remains reflexive and transitive; see Section 7 for a proof. The partial order $\succeq$ determines how the agent can privately control his score. If $s\succeq s^{\prime}$ holds only if $s=s^{\prime}$, then the agent has no influence over his score. In this case, each order $\succeq_{\theta}$ reduces to Blackwell’s order.212121In this case, every Markov transition $k$ is vacuously increasing. Every subset $U$ is vacuously an upper set, so $\succeq_{\operatorname{st}}$ reduces to equality. This shows that the difference between our $\theta$-discernment orders and Blackwell’s order is due to the presence of moral hazard in the form of the agent’s unobservable try–skip decision.

With nonbinary tests, the replacement theorem (Theorem 1) and the forward implication in the main implementation theorem (Theorem 2) go through with essentially the same proofs. Even a most-discerning testing rule, however, cannot be represented by an authentication rate.

8 Related literature on verification

Verification has been modeled in many ways, in both economics and computer science. Here, we focus on costless, imperfect verification.222222In economics, “verification” traditionally means that the principal can learn the agent’s type perfectly by taking some action, e.g., paying a fee or allocating a good. This literature began with Townsend (1979) who studied costly verification in debt contracts. Ben-Porath et al. (2019) connect costly verification and evidence. When monetary transfers are infeasible, costly verification is often used as a substitute; see Ben-Porath et al. (2014); Erlanson and Kleiner (2020); Halac and Yared (2020); Li (2020); Mylovanov and Zapechelnyuk (2017). Our discussion is organized around the taxonomy in Table 1.

Green and Laffont (1986) introduce partial verification.232323A precursor to their work is Postlewaite (1979), which considers exchange mechanisms when endowments are hidden. Each agent can benefit by hiding part of his endowment. They restrict their analysis to direct mechanisms. Verification is represented as a correspondence $M\colon\Theta\twoheadrightarrow\Theta$ satisfying $\theta\in M(\theta)$ for all types $\theta$. Each type $\theta$ can “report” any type $\theta^{\prime}$ in $M(\theta)$. Truthful implementation is without loss if and only if the correspondence $M$ satisfies a form of transitivity called the nested range condition. In our framework, we re-interpret this “failure of the revelation principle” as a consequence of implicitly identifying some type $\theta$ with a test that is not most $\theta$-discerning.242424Strausz (2016) also re-interprets Green and Laffont’s (1986) framework. He models verification as a component of the outcome to recover the revelation principle, much like the first step of our proof of Proposition 1.

Bull and Watson (2004, 2007) and Lipman and Seppi (1995) study hard evidence.252525Evidence was introduced in games (without commitment) by Milgrom (1981) and Grossman (1981); for recent work on evidence games, see Hart et al. (2017), Ben-Porath et al. (2017), and Koessler and Perez-Richet (2019). They introduce an evidence set $\mathcal{E}$ and an evidence correspondence $E:\Theta\twoheadrightarrow\mathcal{E}$. Type $\theta$ possesses the evidence in $E(\theta)$; he can present one piece of evidence from $E(\theta)$ to the principal. The evidence environment is normal if each type $\theta$ has a piece of evidence $e(\theta)$ in $E(\theta)$ that is maximal in the following sense: Every other type $\theta^{\prime}$ who has $e(\theta)$ also has every other piece of evidence in $E(\theta)$. Therefore, type $\theta^{\prime}$ can mimic type $\theta$ if and only if $E(\theta^{\prime})$ contains $e(\theta)$. A normal evidence environment induces an abstract mimicking correspondence that satisfies the nested range condition. In our setting, a deterministic test can be interpreted as a request for a particular piece of evidence. In that case, evidentiary normality is a special case of our most-discerning test condition.

Probabilistic tests can be interpreted as requests for stochastic evidence, where the agent communicates with the principal after he learns his payoff type but before he learns which evidence is available to him. Deneckere and Severinov (2008) study a different kind of stochastic evidence. In their model the agent simultaneously learns his payoff type and his set of feasible evidence messages.

In computer science, Caragiannis et al. (2012) and Ferraioli and Ventre (2018) study a reduced-form model of probabilistic verification in mechanism design. Their analysis is implicitly restricted to direct mechanisms, and they focus on truthful implementation.262626Dziuda and Salas (2018) and Balbuzanov (2019) study a setting without commitment. For some fixed $p\in(0,1)$, the authentication probability $\alpha(\theta^{\prime}|\theta)$ equals $p$ if $\theta^{\prime}\neq\theta$ and $1$ if $\theta^{\prime}=\theta$. This authentication rate satisfies (6) and hence is most discerning. Our paper shows that the restriction to direct, truthful mechanisms is without loss if and only if $\alpha$ is most discerning. Caragiannis et al. (2012) allow the principal to use arbitrarily severe punishments to deter any report that is not authenticated with certainty. In our model, the agent can walk away at any time, so punishment is limited to the agent’s outside option.

Closest to us is the independent paper of Ben-Porath et al. (2021). In their model, the principal does not have the power to control which test is given. Instead, there is an abstract set of pieces of evidence. The agent’s type determines the feasible set of distributions over subsets of pieces of evidence. Our setting fits within their framework if the agent’s choice of evidence-gathering activity is observable.272727In their signal-choice model, suppose each message is a pair $(s,\tau)$ of a score and a test. For each test $\tau$ and type $\theta$, let $\mu_{\tau,\theta}$ be the measure on $S\times T$ that concentrates on the section $S\times\{\tau\}$ with distribution $\pi_{\tau,\theta}$. That is, $\mu_{\tau,\theta}(A)=\pi_{\tau,\theta}(\{s\in S:(s,\tau)\in A\})$ for $A\subset S\times T$. With this notation, test $\tau$ is more $\theta$-discerning than test $\psi$ in our framework if and only if $\mu_{\tau,\theta}$ is more informative than $\mu_{\psi,\theta}$ in theirs. Our papers have different aims. Ben-Porath et al. (2021) study the relationship between the evidence-acquisition and signal-choice models in the most general setting. Our focus is on obtaining a tractable verification framework in which we can apply the first-order approach to solve for optimal mechanisms.

9 Conclusion

We model probabilistic verification as a technology—a family of noisy tests that are available to the principal. The principal chooses a dynamic mechanism with rounds of communication and testing. We characterize whether this complex problem can be reduced to a static problem with relaxed incentive constraints. We solve this reduced problem using the first-order approach in a few classical profit-maximization applications. We believe this first-order approach can be used to solve other mechanism design problems with probabilistic verification.

We have solved for optimal mechanisms for each fixed verification technology in a nonparametric family. We can therefore quantify the value of each technology to the principal. This is the first step towards analyzing a richer setting in which the principal chooses how much to invest in verification technology. We leave this to future research.

Appendix A Proofs

A.1 Insufficiency of mixed strategies in reduced form

Consider a modification of the example in Section 2. The type space and verification technology are as in Figure 1. There are three allocation decisions—nothing, low-quality, and high-quality—with associated type-independent utilities [math], $u_{\ell}$, and $u_{h}$. Assume $0<u_{\ell}<u_{h}$ and $u_{\ell}\geq u_{h}/2$. Consider the social choice function that allocates the high-quality good to type $\theta_{1}$ and the low-quality good to types $\theta_{2}$ and $\theta_{3}$.

We claim that this allocation rule cannot be implemented in the reduced-form model, even if the agent uses a mixed strategy. Type $\theta_{1}$ can pass only as type $\theta_{1}$ or as type $\theta_{2}$. So for some $\theta\in\{\theta_{1},\theta_{2}\}$, the principal must give the high-quality good to the agent if he passes as type $\theta$. But type $\theta_{3}$ can pass as type $\theta_{1}$, and type $\theta_{2}$ can pass as type $\theta_{2}$, so at least one of types $\theta_{3}$ and $\theta_{2}$ has a strictly profitable deviation.

Now replace the authentication rate with a testing technology consisting of three tests, denoted $\tau_{1}$, $\tau_{2}$, $\tau_{3}$. Test $\tau_{i}$ can be passed by those types that can pass as $\theta_{i}$ in the reduced-form model. In this testing model, the principal can implement the desired allocation rule. If the agent reports type $\theta_{2}$ or $\theta_{3}$, he is given the low-quality good. If the agent reports type $\theta_{1}$, then the principal gives either test $\tau_{1}$ or test $\tau_{2}$, each with probability $1/2$. Whichever test is given, the agent gets the high-quality good if he passes and nothing if he fails. If either type $\theta_{2}$ or type $\theta_{3}$ deviates by reporting $\theta_{1}$, then he gets the high-quality good with probability at most $1/2$, and otherwise he gets nothing. This deviation is unprofitable since $u_{\ell}\geq u_{h}/2$.

A.2 Proof of Proposition 1

Let $S=\{0,1\}$. Consider a mechanism $(M_{1},M_{2};t,r_{2},g)$ and a strategy $(r_{1},a)$. For each fixed type $\theta$, the sequence $(m_{1},\tau,m_{2},s,x)$ in $M_{1}\times T\times M_{2}\times S\times X$ is realized according to the following procedure:

• •

Agent sends $m_{1}\sim r_{1}(\theta)$.

• •

Principal selects $\tau\sim t(m_{1})$.

• •

Principal sends $m_{2}\sim r_{2}(m_{1},\tau)$.

• •

Agent tries with probability $a(m_{1},\tau,m_{2})$.

• •

Nature draws $s$ according to $\pi$ and Agent’s try/skip choice.

• •

Principal selects $x\sim g(m_{1},\tau,m_{2},s)$.

This distribution of the sequence $(m_{1},\tau,m_{2},s,x)$ is replicated by the following canonical procedure:

• •

Agent sends $\theta^{\prime}=\theta$.

• •

Principal privately draws $m_{1}\sim r_{1}(\theta^{\prime})$ and then selects $\tau\sim t(m_{1})$.

• •

Agent tries.

• •

Nature draws $s$ according to $\pi$ and Agent’s try/skip choice.

• •

Principal privately draws $m_{2}\sim r_{2}(m_{1},\tau)$; then privately draws $s^{\prime}$ from $s$ according to the transition probabilities $k_{1}=a(m_{1},\tau,m_{2})$ and $k_{0}=0$; and finally selects $x\sim g(m_{1},\tau,m_{2},s^{\prime})$.

We check that the outcome of any deviation by type $\theta$ in the new mechanism can be replicated by a deviation in the old mechanism. It follows that such a deviation cannot be profitable. If type $\theta$ (i) reports $\theta^{\prime}\sim\rho$ and (ii) tries with probability $\alpha(\theta^{\prime},\tau)$, this can be replicated in the old mechanism by (i) drawing $\theta^{\prime}\sim\rho$ and then sending $m_{1}\sim r_{1}(\theta^{\prime})$; and (ii) trying with probability $a(m_{1},\tau,m_{2})\alpha(\theta^{\prime},\tau)$.282828This argument relies in two places on a form of randomization that our model does not technically allow. In the canonical mechanism, the principal remembers her privately drawn $m_{1}$ and uses it to select $x$. In the replicating deviation in the original mechanism, the agent remembers his privately drawn $\theta^{\prime}$ before choosing whether to try. We can replace this memory with fresh draws from the correct conditional distributions. The principal redraws $m_{1}$ conditional on $(\theta^{\prime},\tau)$. The agent redraws $\theta^{\prime}$ conditional on $(\theta,m_{1})$. To construct these conditional distributions, apply disintegration of measures (Kallenberg, 2017, Theorem 1.25, p. 39). This result applies to Borel probability measures, so we first restrict our measures to the Borel $\sigma$-algebra, then apply the theorem, and finally extend the resulting measures to the universal completion.

A.3 Proof of Theorem 1

Let $f$ be a social choice function that is canonically implemented by a mechanism $(t,g)$ in which $t(\theta)=\psi$. Consider a new mechanism $(\hat{t},\hat{g})$ that coincides with $(t,g)$ except for the following modifications. Set $\hat{t}(\theta)=\tau$. Choose $k_{1}$ and $k_{0}$ from the definition of $\tau\succeq_{\theta}\psi$. For each $s=0,1$, set

[TABLE]

Under this new mechanism, if type $\theta^{\prime}$ reports type $\theta$ and tries on test $\tau$ with probability $a$, the resulting decision will be

[TABLE]

where

[TABLE]

From the definition of $\tau\succeq_{\theta}\psi$, we have $p(a|\theta^{\prime})\leq\pi(\psi|\theta^{\prime})$ for all $a$ in $[0,1]$ and all types $\theta^{\prime}$, with equality if $a=1$ and $\theta^{\prime}=\theta$. Therefore, the new mechanism replicates the social choice function $f$ without introducing any new deviations.

A.4 Proof of Proposition 2

Fix a type $\theta$ and tests $\tau_{1}$ and $\tau_{2}$.

One direction is clear. If $\pi(\tau_{1}|\cdot)=\pi(\tau_{2}|\cdot)$, then setting $(k_{0},k_{1})=(0,1)$ shows that $\tau_{1}\succeq_{\theta}\tau_{2}$ and $\tau_{2}\succeq_{\theta}\tau_{1}$. If $\theta$ is minimal on test $\tau_{1}$ then we get $\tau_{2}\succeq_{\theta}\tau_{1}$ by setting $k_{0}=k_{1}=\pi(\tau_{1}|\theta)$. Symmetrically, if $\theta$ is minimal on $\tau_{2}$, then we have $\tau_{1}\succeq_{\theta}\tau_{2}$ by setting $k_{0}=k_{1}=\pi(\tau_{2}|\theta)$.

For the converse, assume $\tau_{1}\sim_{\theta}\tau_{2}$. Choose $(k_{0},k_{1})$ from the definition of $\tau_{1}\succeq_{\theta}\tau_{2}$ and $(k_{0}^{\prime},k_{1}^{\prime})$ from the definition of $\tau_{2}\succeq_{\theta}\tau_{1}$. Suppose type $\theta$ is not minimal on one of the tests, say $\tau_{1}$. Hence there exists $\theta^{\prime}$ such that $\pi(\tau_{1}|\theta)>\pi(\tau_{1}|\theta^{\prime})$.

In Markov transition notation, writing $\pi_{\tau|\theta}$ for the measure that puts probability $\pi(\tau|\theta)$ on $s=1$, the Markov transitions $k$ and $k^{\prime}$ satisfy

[TABLE]

In terms of probabilities, we get the system

[TABLE]

Subtracting we conclude that

[TABLE]

Since $\pi(\tau_{1}|\theta)-\pi(\tau_{1}|\theta^{\prime})>0$, it follows that $(k_{0},k_{1})=(k_{0}^{\prime},k_{1}^{\prime})=(0,1)$, and hence $\pi(\tau_{1}|\cdot)=\pi(\tau_{2}|\cdot)$.

A.5 Proof of Theorem 2

$\eqref{it:stat}\implies\eqref{it:decision}$. Let $\hat{t}$ be a most-discerning testing function. For each type $\theta$ and test $\psi$, select probabilities $k_{0}(\theta,\psi)$ and $k_{1}(\theta,\psi)$ as in the definition of $\hat{t}(\theta)\succeq_{\theta}\psi$; Section B.9 shows that there exists a measurable selection. Let $f$ be an implementable social choice function. By the revelation principle (Proposition 1), there exists a mechanism $(t,g)$ that canonically implements $f$. Define a new mechanism with testing rule $\hat{t}$ and outcome rule $\hat{g}$ satisfying

[TABLE]

for all types $\theta$ and scores $s=0,1$.

Under this new mechanism, if type $\theta^{\prime}$ reports type $\theta$ and tries on test $\hat{t}(\theta)$ with probability $a$, the resulting decision will be

[TABLE]

where

[TABLE]

For each type $\theta$ and test $\psi$, the definition of $\hat{t}(\theta)\succeq_{\theta}\psi$ guarantees that $p(a,\theta,\psi|\theta^{\prime})\leq\pi(\psi|\theta^{\prime})$ for all $a$ in $[0,1]$ and all types $\theta^{\prime}$, with equality if $a=1$ and $\theta^{\prime}=\theta$. Therefore, the new mechanism replicates the social choice function $f$ without introducing any new deviations.

$\eqref{it:decision}\implies\eqref{it:stat}$. Fix a type $\theta$ and a test $\psi$. We will prove that $\hat{t}(\theta)\succeq_{\theta}\psi$.

Construct a decision environment $(X,u)$ as follows. The decision set $X$ consists of three decisions, denoted $\bar{x}$, \underaccent{\bar}{x}, and $x_{0}$. Every type gets utility $1$ from decision $\bar{x}$ and utility [math] from decision \underaccent{\bar}{x}. Each type $\theta^{\prime}$ gets utility $\pi(\hat{t}(\theta)|\theta^{\prime})$ from decision $x_{0}$.

Consider the following mechanism. If the agent reports $\theta^{\prime}\neq\theta$, the principal selects $x_{0}$ (the test and score don’t matter). If the agent reports $\theta$, the principal gives test $\psi$ and then selects $\bar{x}$ if the agent passes and \underaccent{\bar}{x} if the agent fails. Truth-telling and trying is a best response for every type. Denote the induced social choice function by $f$.

By assumption, $f$ can be canonically implemented by a mechanism in which $t(\theta)=\hat{t}(\theta)$. For $s=0,1$, let $k_{s}$ be the probability that the principal selects $\bar{x}$ after the agent reports $\theta$ and receives score $s$ on test $\tau$. We must have $k_{1}\geq k_{0}$, or else type $\theta$ could profitably deviate by skipping the test. Since this mechanism implements $f$, the probabilities $k_{0}$ and $k_{1}$ satisfy (i) in the definition of $\hat{t}(\theta)\succeq_{\theta}\psi$. Since no type $\theta^{\prime}$ can profit from reporting $\theta$ and trying on test $\hat{t}(\theta)$, we get (ii). Therefore, $\hat{t}(\theta)\succeq_{\theta}\psi$, as desired.

A.6 Proof of Proposition 3

Fix type $\theta$ and tests $\tau$ and $\psi$. We have $\tau\succeq_{\theta}\psi$ if and only if there exist probabilities $k_{1}$ and $k_{0}$ satisfying

[TABLE]

such that for all $\theta^{\prime}\neq\theta$, we have

[TABLE]

Taking $k_{0}^{\prime}=1-k_{1}$ and $k_{1}^{\prime}=1-k_{0}$, we can state this condition symmetrically in terms of failure rather than passage. Let $\bar{\pi}=1-\pi$. We have $\tau\succeq_{\theta}\psi$ if and only if there exist probabilities $k_{1}^{\prime}$ and $k_{0}^{\prime}$ satisfying

[TABLE]

such that for all $\theta^{\prime}\neq\theta$ we have

[TABLE]

We split into (overlapping) cases.

If $\pi(\tau|\theta)\geq\pi(\psi|\theta)$ and $\pi(\tau|\theta)>0$, then the solutions of (10) are given by

[TABLE]

Plug these parameterized solutions into (11) and group terms by $\lambda$. We have $\tau\succeq_{\theta}\psi$ if and only if there exists $\lambda$ in $[0,1]$ such that

[TABLE]

Dividing by $\pi(\psi|\theta)$ gives the desired inequality. 2. 2.

If $\bar{\pi}(\tau|\theta)\geq\bar{\pi}(\psi|\theta)$ and $\bar{\pi}(\tau|\theta)>0$, then the solutions of (12) are given by

[TABLE]

Plug these parameterized solutions into (10) and group terms by $\lambda$. We have $\tau\succeq_{\theta}\psi$ if and only if there exists $\lambda$ in $[0,1]$ such that

[TABLE]

Dividing by $\bar{\pi}(\psi|\theta)$ gives the desired inequality.

A.7 Proof of Proposition 4

With testing technology $(T^{\alpha},\pi^{\alpha})$, the function $\theta\mapsto\tau^{\alpha}_{\theta}$ is most-discerning if and only if for all distinct types $\theta_{2}$ and $\theta_{3}$, we have $\tau_{\theta_{2}}^{\alpha}\succeq_{\theta_{2}}\tau_{\theta_{3}}^{\alpha}$. Now apply Proposition 3 to the relation $\tau_{\theta_{2}}^{\alpha}\succeq_{\theta_{2}}\tau_{\theta_{3}}^{\alpha}$ (with the deviating type $\theta^{\prime}$ labelled as $\theta_{1}$) and write out $\pi^{\alpha}$ in terms of $\alpha$.

A.8 Proof of Proposition 5

We apply the dominated convergence theorem. As $\lambda$ converges to [math] pointwise, $\alpha(\theta|\xi)$ converges to $1$ for all $\theta$ and $\xi$ with $\theta\leq\xi$. Hence $\varphi(\theta)$ converges to $\varphi^{M}(\theta)$, for each $\theta$. Likewise, as $\lambda$ converges to $\infty$ pointwise, $\alpha(\theta|\xi)$ converges to [math] for all $\theta$ and $\xi$ with $\theta<\xi$. Hence $\varphi(\theta)$ converges to $\theta$, for each $\theta$.

A.9 Proof of Proposition 6

The equality $U(\theta)=\theta q(\theta)-t(\theta)$ gives a one-to-one correspondence between the transfer function $t$ and the indirect utility function $U$, given the quantity function $q$.

Lemma 1** **(Envelope theorem lower bound)

Let $q$ be a bounded quantity function, and let $U$ be an indirect utility function. If $(q,U)$ is incentive compatible, then for each type $\theta$, we have

[TABLE]

We prove the theorem, taking as given Lemma 1, which is proven in Section A.10. Pick a bounded quantity function $q\colon\Theta\to\mathbf{R}_{+}$. The principal’s objective function can be decomposed as the difference between the total surplus and the agent’s rents:

[TABLE]

Plug in the bound from Lemma 1 and switch the order of integration to obtain the following upper bound on the principal’s objective:

[TABLE]

with equality if (14) holds with equality for every type $\theta$.

The quantity function $q^{\ast}$ from the theorem statement maximizes the expression in brackets pointwise, and $t^{\ast}$ is the corresponding transfer for which $U$ satisfies (14) with equality for every type $\theta$.

We claim that global incentive compatibility holds if the quantity function $q^{\ast}$ satisfies the following monotonicity condition: Whenever $\xi_{1}\leq\xi_{2}\leq\theta$, we have

[TABLE]

This condition is satisfied if the virtual value $\varphi$ is weakly increasing.292929This condition can still hold if $\varphi$ does not decrease too quickly.

The global incentive constraint requires that for all types $\theta$ and $\theta^{\prime}$, we have

[TABLE]

or equivalently,

[TABLE]

Plug in the right side of (14) for $U$ to get

[TABLE]

We separate into cases. If $\theta<\theta^{\prime}$, then (18) holds if

[TABLE]

If $\theta>\theta^{\prime}$, then (18) is equivalent to

[TABLE]

In each case, the inequality is guaranteed by (15).

A.10 Proof of Lemma 1

First, we check that $U$ is absolutely continuous. Fix an incentive-compatible pair $(q,U)$, where $q$ is bounded. Choose $\theta$ and $\theta^{\prime}$ such that $U(\theta^{\prime})\geq U(\theta)$. From (17), we have

[TABLE]

Since $1-e^{-x}\leq x$, it follows that

[TABLE]

where $C=\max\{\bar{\theta},1\}\|q\|_{\infty}$. Since $\lambda+1$ is integrable over [\underaccent{\bar}{\th},\bar{\theta}], we conclude that $U$ is absolutely continuous.

Now we prove (14). Define the auxiliary function $\Delta$ on [\underaccent{\bar}{\th},\bar{\theta}] by

[TABLE]

The function $\Delta$ is absolutely continuous since it is the product of absolutely continuous functions. Let $u(\theta^{\prime}|\theta)=\alpha(\theta^{\prime}|\theta)[\theta q(\theta)-t(\theta)]$. By Theorem 1 in Milgrom and Segal (2002), whenever $U$ is differentiable, we have

[TABLE]

where $D_{2+}u(\theta|\theta)$ denotes the right derivative with respect to the second argument.303030That is, $D_{2+}u(\theta|\theta)=\lim_{h\downarrow 0}h^{-1}(u(\theta|\theta+h)-u(\theta|\theta))$. At each point $\theta$ where all the functions involved are differentiable, which holds almost surely, we have

[TABLE]

Since \Delta(\underaccent{\bar}{\th})=0, the fundamental theorem of calculus implies that $\Delta(\theta)\geq 0$ for all $\theta$, as desired.

A.11 Proof of Proposition 7

The same argument from the proof of Proposition 6 shows that the principal’s maximum value as a function of $q$ can be expressed as

[TABLE]

The quantity function $q^{\ast}$ from the theorem statement maximizes this quantity pointwise, and $t^{\ast}$ is the corresponding transfer function that makes (14) hold with equality. Since $q^{\ast}$ is monotone, (15) is satisfied, so this mechanism is globally incentive compatible.

A.12 Proof of Proposition 8

Applying the same argument from the proof of Proposition 7 agent by agent shows that the principal’s maximum value as a function of $q$ can be expressed as

[TABLE]

The quantity function $q^{\ast}$ from the theorem statement maximizes this quantity pointwise. The induced interim quantity functions are all monotone, so (15) is satisfied for each agent. From (14), the optimal interim utility functions are given by

[TABLE]

We make a particular choice of ex post utility functions consistent with these interim utility functions:

[TABLE]

These ex post utility functions pin down the transfer functions $t_{i}^{\ast}$ in the theorem statement.

Appendix B Online appendix

B.1 Most-discerning correspondences

Definition 4** (Most-discerning correspondence).**

A subset $T_{0}$ of $T$ is most $\theta$-discerning if for each test $\psi$ in $T$ there exists a test $\tau$ in $T_{0}$ such that $\tau\succeq_{\theta}\psi$. A correspondence $\hat{T}\colon\Theta\twoheadrightarrow T$ is most discerning if for each type $\theta$ the set $\hat{T}(\theta)$ is most $\theta$-discerning.

A testing rule $\hat{t}\colon\Theta\to\Delta(T)$ is supported on a correspondence $\hat{T}\colon\Theta\twoheadrightarrow T$ if $\operatorname{supp}\hat{t}(\theta)\subset\hat{T}(\theta)$ for each type $\theta$. The next result says that if a correspondence is most discerning, then we can restrict attention to the testing rules supported on that correspondence.

Theorem 3** **(Implementation with a most-discerning correspondence)

Suppose that the passage rate $\pi$ is continuous. Let $\hat{T}$ be a weakly measurable correspondence from $\Theta$ to $T$ with closed values. If $\hat{T}$ is most discerning, then for every implementable social choice function $f$, there exists a testing rule $\hat{t}$ supported on $\hat{T}$ such that $f$ is canonically implementable with $\hat{t}$.

The proof is essentially the same as the proof of Theorem 2. For each type $\theta$ and test $\psi$, there exists a test $\tau$ in $\hat{T}(\theta)$ such that $\tau\succeq_{\theta}\psi$, but we must check that there is a selection that is measurable; see Section B.9. The additional regularity conditions ensure that such a measurable selection exists. If we can independently construct a measurable selection, then these conditions are not needed.

B.2 Example of $\theta$-discernment with $\lambda\neq 1$

Consider a type $\theta$ and two tests $\tau$ and $\psi$ such that $\pi(\tau|\theta)$ and $\pi(\psi|\theta)$ are equal and nonzero. In this case, test $\tau$ is more $\theta$-discerning than test $\psi$ if and only if there exists $\lambda\in[0,1]$ such that

[TABLE]

The passage rates for these tests are plotted in Figure 4. The type space is an interval, plotted on the horizontal axis. For test $\tau$, the passage rate is an increasing affine function. For test $\psi$, the passage rate is increasing and convex. The dotted line takes the constant value $\pi(\tau|\theta)$, and the dashed line is the average of the passage rate $\pi(\tau|\cdot)$ and the constant $\pi(\tau|\theta)$. From the graph we see that (19) is satisfied with $\lambda=1/2$, so $\tau\succeq_{\theta}\psi$. Moreover, the tangency at the point $(\theta,\pi(\tau|\theta))$ shows that $1/2$ is the only value of $\lambda$ for which (19) holds.

B.3 Beyond exponential authentication rates

Suppose that the verification technology is represented by a Borel measurable most-discerning authentication rate $\alpha\colon\Theta\times\Theta\to[0,1]$ that satisfies the following conditions.

1. (i)

$\alpha(\theta|\theta)=1$ for all types $\theta$. 2. (ii)

For each type $\theta^{\prime}$, the function $\theta\mapsto\alpha(\theta^{\prime}|\theta)$ is absolutely continuous. 3. (iii)

For each type $\theta$, the right and left partial derivatives $D_{2+}\alpha(\theta|\theta)$ and $D_{2-}\alpha(\theta|\theta)$ exist, and the functions $\theta\mapsto D_{2+}\alpha(\theta|\theta)$ and $\theta\mapsto D_{2-}\alpha(\theta|\theta)$ are integrable.

Condition i ensures that the agent is authenticated if he reports truthfully. Conditions ii and iii allow us to apply the envelope theorem. In particular, the exponential authentication rates studied in the main text satisfy these assumptions.

Define the right and left local precision functions $\lambda_{+},\lambda_{-}\colon\Theta\to\mathbf{R}_{+}$ by

[TABLE]

Define the function $\Lambda$ by

[TABLE]

The function $\Lambda$ is determined only by the local behavior of $\alpha$ near the diagonal.

Lemma 2** **(Lower bound)

For all types $\theta$ and $\theta^{\prime}$, we have $\alpha(\theta^{\prime}|\theta)\geq\Lambda(\theta^{\prime}|\theta)$.

For the exponential authentication rates $\alpha$ considered in the main text, we have $\lambda_{+}(\theta)=\lambda_{-}(\theta)=\lambda(\theta)$, and Lemma 2 holds with equality for all types $\theta$ and $\theta^{\prime}$. Therefore, among all most-discerning authentication rates $\alpha$ satisfying $-D_{2+}\alpha(\theta|\theta)=D_{2-}\alpha(\theta|\theta)=\lambda(\theta)$ for each $\theta$, the exponential authentication rate with precision function $\lambda$ makes the global incentive constraints as weak as possible.

In this general setting, Lemma 1 still holds with $\Lambda(\xi|\theta)$ in place of $\alpha(\xi|\theta)$.313131The proof is almost identical to the proof for exponential authentication rates in Section A.10. To establish absolute continuity, apply Lemma 2 and put $\lambda_{+}\vee\lambda_{-}$ in place of $\lambda$. To establish the bound, use $\Lambda$ in place of $\alpha$ in the definition of the auxiliary function $\Delta$. In the main text, we study the relaxed problem in which only the local incentive constraints are required. Then we check that the solution of this relaxed problem is globally incentive compatible, provided that the virtual $\varphi$ is increasing. The solutions given in the main text, with $\Lambda$ in place of $\alpha$, remain correct if (a) the monotonicity condition (15) holds with $\Lambda$ in place of $\alpha$, and (b) the following global bound is satisfied: For $\theta>\theta^{\prime}$,

[TABLE]

See Section B.7 for a proof. Intuitively, the more rapidly the map $\xi\mapsto\Lambda(\xi|\theta)q^{\ast}(\xi)$ increases over the interval $[0,\theta]$, the more slack there is for $\alpha(\theta^{\prime}|\theta)$ to increase above $\Lambda(\theta^{\prime}|\theta)$.

B.4 Testing multiple agents

We extend our model to allow for $n$ agents, labeled $i=1,\ldots,n$. Each agent $i$ knows his type $\theta_{i}\in\Theta_{i}$. The type vector $\theta=(\theta_{1},\ldots,\theta_{n})$ is drawn from a commonly known distribution. Let $\Theta=\prod_{i=1}^{n}\Theta_{i}$. The decision set is still denoted by $X$. We allow interdependent values. Each agent $i$ has utility function $u_{i}\colon X\times\Theta\to\mathbf{R}$. The principal has utility function $v\colon X\times\Theta\to\mathbf{R}$.

For each agent $i$, there is a set $T_{i}$ of tests and a passage rate

[TABLE]

where $\pi_{i}(\tau_{i}|\theta_{i})$ is the probability with which type $\theta_{i}$ can pass test $\tau_{i}$. Let $T=\prod_{i=1}^{n}T_{i}$. After the agents communicate with the principal, the principal simultaneously chooses a test for each agent. Each agent sees his own test, but not the tests of the other agents, and then chooses whether to try or skip. Nature draws each agent’s test score independently. The solution concept is Bayes–Nash equilibrium.

In this multi-agent setting, the revelation principle (Proposition 1) holds. For each agent $i$ and each type $\theta_{i}$ in $\Theta_{i}$, the $\theta_{i}$-discernment order $\succeq_{\theta_{i}}$ over $T_{i}$ is defined as in the baseline model with $\pi_{i}$ in place of $\pi$. Given testing functions $t_{i}\colon\Theta_{i}\to T_{i}$ for each $i$, define the product testing function $\otimes_{i}t_{i}\colon\Theta\to T$ by $t(\theta_{1},\ldots,\theta_{n})=(t_{1}(\theta_{1}),\ldots,t_{n}(\theta_{n}))$. We get the natural generalization of our main result, with a similar proof.

Theorem 4** **(Most-discerning implementation with multiple players)

Fix a type space $\Theta_{i}$ and a testing technology $(T_{i},\pi_{i})$ for each agent $i$. For a testing function $\hat{t}=\otimes_{i}\hat{t}_{i}$, the following are equivalent.

$\hat{t}_{i}$* is most discerning for all $i$.* 2. 2.

In every decision environment $(X,u)$, every implementable social choice function is canonically implementable with $\hat{t}$.

B.5 Nonbinary tests

We first check that $\succeq_{\theta}$ is reflexive and transitive. Reflexivity is immediate by taking $k$ to be the identity, which maps $s$ to the point mass $\delta_{s}$. Transitivity holds because the composition $k_{1}k_{2}\colon S\to\Delta(S)$ defined by $(k_{1}k_{2})(s^{\prime}|s)=\sum_{s^{\prime\prime}}k_{2}(s^{\prime}|s^{\prime\prime})k_{1}(s^{\prime\prime}|s)$ is increasing if $k_{1}$ and $k_{2}$ are increasing.

In the main model, if type $\theta$ tries on a pass-fail test $\tau$ with probability $a$, he passes with probability $a\pi(\tau|\theta)$. Therefore, type $\theta$ can achieve on test $\tau$ any passage probability $p$ satisfying $p\leq\pi(\tau|\theta)$. In the general case, on a nonbinary test $\tau$, type $\theta$ chooses a Markov transition $d\colon S\to\Delta(S)$ that is downward in the sense that $d(s^{\prime}|s)=0$ unless $s\succeq s^{\prime}$. Nature draws the score from the distribution $\pi_{\tau|\theta}d$. By Kamae et al. (1977, Theorem 1, p. 900), type $\theta$ can achieve on test $\tau$ a score distribution $p$ in $\Delta(S)$ if and only if $p\preceq_{\operatorname{st}}\pi_{\tau|\theta}$. In a mechanism, the agent’s performance strategy is a map $d\colon\Theta\times M_{1}\times T\times M_{2}\times S\to\Delta(S)$ such that $d_{\theta,m_{1},\tau,m_{2}}\colon S\to\Delta(S)$ is downward for each $(\theta,m_{1},\tau,m_{2})$.

In this setting with nonbinary tests, the following results go through: the revelation principle (Proposition 1), the replacement theorem (Theorem 1), and the forward implication in the main implementation theorem (Theorem 2).323232The backward implication does not hold in general. If $\tau\not\succeq_{\theta}\psi$, then each mechanism using test $\tau$ in place of test $\psi$ on type $\theta$ will introduce a new deviation, but the type space must be sufficiently rich to ensure that this deviation is profitable for every mechanism. The proofs are virtually identical, with the downward transition $d$ in place of the try–skip choice $a$. The key property is that the composition of downward kernels is downward, which is easy to check.

All of the results on authentication rates and relative performance are specific to the binary setting.333333We could obtain a similar reduction to an authentication function $\alpha\colon\Theta\times\Theta\to\Delta(S)$, but the principal’s problem would generally remain intractable. For Proposition 2, say that type $\theta$ is minimal on test $\tau$ if $\pi_{\tau|\theta}\preceq_{\operatorname{st}}\pi_{\tau|\theta^{\prime}}$ for all types $\theta^{\prime}$. In this case conditions (a) and (b) are each sufficient for $\theta$-equivalence, but they are no longer necessary.

B.6 Proof of Lemma 2

Fix $\theta$ and $\theta^{\prime}$. For each $h$, transitivity gives

[TABLE]

Subtract $\alpha(\theta^{\prime}|\theta)$ from each side to get

[TABLE]

Dividing by $h$ and passing to the limit as $h\downarrow 0$ and $h\uparrow 0$, we see that whenever $D_{2}\alpha(\theta^{\prime}|\theta)$ exists, we have

[TABLE]

Now we use absolute continuity to convert these local bounds into global bounds. Fix a report $\theta^{\prime}$. Define the function $\Delta$ on [\underaccent{\bar}{\th},\bar{\theta}] by

[TABLE]

By construction, $\Delta(\theta^{\prime})=1$. We claim that $\Delta(\theta)\geq 1$ for all $\theta$. Since $\Lambda(\theta^{\prime}|\theta)$ is bounded away from [math], the function $\Delta$ is absolutely continuous. Therefore, the functions $\Delta$, $\alpha(\theta^{\prime}|\cdot)$, and $\Lambda(\theta^{\prime}|\cdot)$ are simultaneously differentiable almost everywhere. If $\theta>\theta^{\prime}$ and these three functions are simultaneously differentiable at $\theta$, we have

[TABLE]

If $\theta<\theta^{\prime}$ and these three functions are simultaneously differentiable at $\theta$, we have

[TABLE]

Since $\Delta$ is absolutely continuous, it follows from the fundamental theorem of calculus that $\Delta(\theta)\geq\Delta(\theta^{\prime})=1$ for all $\theta$.

B.7 Proof of sufficiency of the global bound

Consider an authentication rate $\alpha$ satisfying the conditions in Section B.3. Lemma 2 still holds, with $\Lambda$ in place of $\alpha$. The analogue of (17) is therefore

[TABLE]

We separate into cases. If $\theta<\theta^{\prime}$, then (21) holds if

[TABLE]

which is implied by the monotonicity of the map $\xi\mapsto\Lambda(\xi|\theta^{\prime})q^{\ast}(\xi)$ over $[0,\theta^{\prime}]$. If $\theta>\theta^{\prime}$, then (21) is equivalent to

[TABLE]

Rearranging, we get the global bound:

[TABLE]

B.8 Universal measurability

We begin by introducing universal measurability. For a more detailed discussion with proofs, see Bertsekas and Shreve (1996, Chapter 7). Let $(X,\mathcal{X})$ be a measurable space. Given a probability measure $\mu$ on $(X,\mathcal{X})$, let $\overline{\mathcal{X}}_{\mu}$ denote the $\mu$-completion of $\mathcal{X}$, i.e., the $\sigma$-algebra generated by $\mathcal{X}$ and all $\mu$-null sets of $\mathcal{X}$. The universal completion of $\mathcal{X}$, denoted $\overline{\mathcal{X}}$, is the intersection $\bigcap_{\mu}\overline{\mathcal{X}}_{\mu}$, where the intersection is taken over all probability measures $\mu$ on $(X,\mathcal{X})$. It can be shown that $\overline{\overline{\mathcal{X}}}=\overline{\mathcal{X}}$.

A function from $(X,\mathcal{X})$ to $(Y,\mathcal{Y})$ is universally measurable if it is $(\overline{\mathcal{X}},\mathcal{Y})$-measurable. It can be shown that $(\overline{\mathcal{X}},\mathcal{Y})$-measurability is equivalent to $(\overline{\mathcal{X}},\overline{\mathcal{Y}})$-measurability. Similarly, it can be shown that any probability kernel from $(X,\overline{\mathcal{X}})$ to $(Y,\mathcal{Y})$ can be uniquely extended to a probability kernel from $(X,\overline{\mathcal{X}})$ to $(Y,\overline{\mathcal{Y}})$. Given $(X,\mathcal{X})$ and $(Y,\mathcal{Y})$, a probability kernel from $(X,\overline{\mathcal{X}})$ to $(Y,\overline{\mathcal{Y}})$ is called universally measurable.

On a topological space $X$, the Borel $\sigma$-algebra is denoted by $\mathcal{B}(X)$. For Polish spaces $X$ and $Y$, we have $\mathcal{B}(X\times Y)=\mathcal{B}(X)\otimes\mathcal{B}(Y)$. The left side is the $\sigma$-algebra generated by the product topology on $X\times Y$. The right side is the $\sigma$-algebra generated by all rectangles with Borel-measurable sides.

Now we return to the model. We make the following standing technical assumptions. The sets $\Theta$, $T$, and $X$ are Polish spaces. The function $\pi\colon T\times\Theta\to\Delta(S)$ is Borel measurable (with $\Delta(S)$ viewed as a subset of $\mathbf{R}^{S}$).343434This section uses the nonbinary testing framework, which includes the main model as a special case. In a mechanism, the message spaces $M_{1}$ and $M_{2}$ are Polish, and all maps and probability kernels are universally measurable. Universally measurable sets are convenient because of the following measurable projection theorem (Cohn, 2013, Proposition 8.4.4, p. 264).

Theorem 5** **(Measurable projection)

Let $(X,\mathcal{X})$ be a measurable space, $Y$ a Polish space, and $C$ a set in the product $\sigma$-algebra $\mathcal{X}\otimes\mathcal{B}(Y)$. Then the projection of $C$ on $X$ belongs to $\overline{\mathcal{X}}$.

The definition of $\theta$-discernment imposes an inequality for each type $\theta^{\prime}$. If there are uncountably many types, this can create measurability problems. Using the measurable project theorem, we can show that the score conversion in the definition of $\tau\succeq_{\theta}\psi$ can be selected in a universally measurable way.

B.9 Measurable selection of score conversion

We represent the space of increasing Markov transitions $k\colon S\to\Delta(S)$ as a polytope $\mathcal{K}$ in $\mathbf{R}^{S\times S}$ consisting of vectors $k=(k(s^{\prime}|s))_{s,s^{\prime}\in S}$. Define the subset $G$ of $\Theta\times T^{2}\times\mathcal{K}$ to consist of all tuples $(\theta,\tau,\psi,k)$ such that $k$ satisfies the conditions in the definition of $\tau\succeq_{\theta}\psi$. We will show that $G$ is in $\overline{\mathcal{B}(\Theta\times T^{2})}\otimes\mathcal{B}(\mathcal{K})$. Then the projection of $G$ onto $\Theta\times T^{2}$, which we call $D$, is in $\overline{\mathcal{B}(\Theta\times T^{2})}$ by the measurable projection theorem. The measurable projection theorem also guarantees that the section correspondence $(\theta,\tau,\psi)\mapsto G_{\theta,\tau,\psi}$ on $D$ is weakly measurable, where $D$ is endowed with the restriction of the $\sigma$-algebra $\overline{\mathcal{B}(\Theta\times T^{2})}$.353535That is, the lower inverses of open sets are measurable. To prove weak measurability, take the intersection of $G$ and $\Theta\times T^{2}\times B$, for an open set $B$, and then project onto $X\times T^{2}$. Finally, this correspondence has nonempty, closed values, so we apply the Kuratowski–Ryll-Nardzewski selection theorem (Aliprantis and Border, 2006, 18.13, p. 600) to get the desired universally measurable selection.

Now we check that $G$ is in $\overline{\mathcal{B}(\Theta\times T^{2})}\otimes\mathcal{B}(\mathcal{K})$. On $\Theta\times T^{2}\times\mathcal{K}$, define the real-valued functions $f_{s}$ for each $s$ in $S$, and $g_{U}$ for each upper set $U$ in $S$ by

[TABLE]

The set $G$ is the intersection of $\bigcap_{s}[f_{s}=0]$ and $\bigcap_{U}[g_{U}\leq 0]$. Therefore, it suffices to check that these functions are all $(\overline{\mathcal{B}(\Theta\times T^{2})}\otimes\mathcal{B}(\mathcal{K}),\mathcal{B}(\mathbf{R}))$-measurable. For each function $f_{s}$, this is is implied by the Borel measurability of $\pi$. For each upper set $U$, we check that $g_{U}$ is a Carathéodory function. For each fixed $(\theta,\tau,\psi)$, the function $g_{U}(\theta,\tau,\psi,\cdot)$ is continuous. For each fixed $k$, the function $g_{U}(\cdot,k)$ is $(\overline{\mathcal{B}(\Theta\times T^{2})},\mathcal{B}(\mathbf{R}))$-measurable because the term in brackets, viewed as a function of $(\theta,\tau,\psi,\theta^{\prime})$ is $(\mathcal{B}(\Theta\times T^{2}\times\Theta),\mathcal{B}(\mathbf{R}))$-measurable. Hence, the supremum over $\theta^{\prime}$ is $(\overline{\mathcal{B}(\Theta\times T^{2})},\mathcal{B}(\mathbf{R}))$-measurable by the measurable projection theorem. Therefore, $g_{U}$ is Carathéodory function. By Aliprantis and Border (2006, 4.51, p. 153), $g_{U}$ is $(\overline{\mathcal{B}(\Theta\times T^{2})}\otimes\mathcal{B}(\mathcal{K}),\mathcal{B}(\mathbf{R}))$-measurable

B.10 Measurable selection of more $\theta$-discerning test

Consider the following subsets of $\Theta\times T^{2}$:

[TABLE]

It suffices to check that $A\cap B$ is in $\overline{\mathcal{B}(\Theta\times T)}\otimes\mathcal{B}(T)$. Then the section correspondence $(\theta,\psi)\mapsto(A\cap B)_{\theta,\psi}$ has a $(\overline{\mathcal{B}(\Theta\times T)},\mathcal{B}(T))$-measurable selection by the same argument from Section B.9. We check that $A$ and $B$ are each in $\overline{\mathcal{B}(\Theta\times T)}\otimes\mathcal{B}(T)$. The set $\mathcal{K}$ is compact, and by assumption $\pi$ is continuous, so it is straightforward to check that $A$ is closed. By assumption, $\hat{T}$ has closed values and is weakly $(\overline{\mathcal{B}(\Theta)},\mathcal{B}(T))$-measurable, so the correspondence $(\theta,\psi)\mapsto\hat{T}(\theta)$ has closed values and is weakly $(\overline{\mathcal{B}(\Theta\times T)},\mathcal{B}(T))$-measurable. Its graph, $B$, is therefore in $\overline{\mathcal{B}(\Theta\times T)}\otimes\mathcal{B}(T)$ by Aliprantis and Border (2006, 18.6, p. 596).