In a steady state, buyers request an available seller at a Poisson rate denoted by $\beta_{A}$. On one hand, an individual available seller faces a stochastic request process such that the probability of a new request over the time interval $d t$ is $\beta_{A} d t+o(d t)$. On the other hand, the available sellers jointly face the deterministic arrival process of buyers, where over time interval $d t$ the mass $\beta d t$ of buyers arrives. Uniform assignment of buyers across sellers results in a particularly simple form of the relationship between the two rates (Myerson (2000)):

$$
\beta_{A} \triangleq \frac{\beta}{1-\bar{\rho}} .
$$

A steady-state equilibrium is a market outcome in which the sellers take the request rate $\beta_{A}$ as given and optimize independently, and the busy-available seller flows balance.

Definition 1. The pair $(\sigma, \bar{\rho})$, where $\sigma: S \times Y \rightarrow[0,1]$ is the sellers' strategies and $\bar{\rho} \in[0,1]$ is the mass of busy sellers, constitutes a steady-state equilibrium if the following hold:

1. (Steady state) The average utilization rate $\bar{\rho}$ is given by (2) and (3). The request rate $\beta_{A}$ is given by (4).
2. (Optimality) For all $y$ and every type- $y$ seller, $\sigma(\cdot, y)$ is an optimal strategy given the

request rate $\beta_{A}$ and an information policy $\lambda$.

The platform's objective is to maximize a weighted average of the buyer surplus and the joint seller profits in a steady-state equilibrium. For a given steady-state equilibrium, define by $V(y)$ the seller- $y$ profits, by $V=\int_{Y} V(y) \mathrm{d} G(y)$ the joint seller profits, and by $U$ the buyers' utility. The platform then formally maximizes

$$
\mathcal{J}(\gamma)=\gamma U+(1-\gamma) V .
$$

The general objective $\mathcal{J}(\gamma)$ includes as special cases maximization of total surplus at $\gamma=1 / 2$, maximization of the joint seller profits at $\gamma=0$ and maximization of buyer surplus at $\gamma=1$. As $\gamma$ spans the interval [0, 1] the platform-optimal allocations span constrained-Pareto-efficient allocations.

Assumption Discussion. Before proceeding with the main analysis, we highlight two important features of our model that are motivated by the stylized facts about the online platforms.

First, no two buyers simultaneously contact the same seller. Such coordination frictions in matching markets have been extensively studied in the theoretical literature (Burdett, Shi, and Wright (2001); Kircher (2009)), and digital platforms now usually have good technological means of resolving the simultaneity-driven friction. Instead, we focus on the matching friction that pertains to preference heterogeneity and screening. ${ }^{11}$

Second, the buyers make a single search attempt and pick the seller uniformly at random. The single search attempts can be attributed to rejection intolerance and capture an aspect of real-life matching markets in that rejections are costly to buyers (wasted time, search efforts, bidding costs). ${ }^{12}$ It allows us to not have an endogenous distribution of buyer types.

[^6]The uniform assignment is plausible in settings in which the buyers are indifferent across the sellers or the search costs are prohibitively high. It allows us to analyze the match-quality and match-rate tradeoff in a cleaner model, in which any seller faces the same request rate, $\beta_{A}$, and the same distribution of buyer types, $F$. Altogether, this implies that the buyers are effectively non-strategic and allows us to focus on the sellers' side of the market. We discuss possible implications of buyers' strategic behavior in Section 4.


Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.

Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

## 3 Analysis

We proceed with studying how the platform's information policy affects the equilibrium market outcome. We show that for an arbitrary information policy, an equilibrium exists and is essentially unique. We then show that full disclosure policy aggravates cream skimming and produces an inefficient market outcome. We then characterize optimal information policies in the cases of homogeneous sellers and vertically differentiated sellers.


### 3.1 Equilibrium Existence and Uniqueness

For a given information policy, each seller is facing a dynamic decision problem of maximizing an average profit flow. He is presented with a sequence of buyers arriving at a Poisson rate $\beta_{A}$. For each buyer request, the seller observes the signal realization $s$ and chooses whether to accept the request (see Figure 2). With a small abuse of notation, let $\pi(s, y) \triangleq \mathbb{E}[\pi(x, y) \mid s]$ be the type- $y$ seller's flow profit if he accepts a buyer with signal $s$ and let $V(y)$ be the seller's value function, an average profit flow when he acts optimally. ${ }^{13}$ An opportunity cost of accepting is equal to $\tau V(y)$, and the seller optimization problem corresponds to the following Bellman equation

[^7]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: Seller's dynamic optimization problem with screening and waiting. An available seller receives buyer requests at a Poisson rate $\beta_{A}$. If a request is accepted, the seller becomes busy for time $\tau$, during which he does not receive new requests.

$$
V(y)=\beta_{A} \int \max \{0, \pi(s, y)-\tau V(y)\} \mathrm{d} \lambda(s) .
$$

It is immediate that an optimal seller strategy is cutoff with him accepting all buyers with $s$ such that $\pi(s, y)>\hat{\pi}$ and rejecting all buyers with $s$ such that $\pi(s, y)<\hat{\pi}$. The optimal threshold $\hat{\pi}$ naturally depends on the information policy.

Proposition 1. (Existence and Uniqueness) For an arbitrary information policy $\lambda$, a steadystate equilibrium exists and is unique up to the acceptance of marginal buyers: if $(\sigma, \bar{\rho})$ and $\left(\sigma^{\prime}, \bar{\rho}^{\prime}\right)$ are two steady-state equilibria, then (1) $\bar{\rho}=\bar{\rho}^{\prime}$, (2) for any $y \in Y, \sigma(\cdot, y)$ and $\sigma^{\prime}(\cdot, y)$ coincide except on the set $\{s: \pi(s, y)=\tau V(y)\}$, and (3) the surpluses coincide, $V(y) \equiv V^{\prime}(y), U=U^{\prime}$.

To prove this result, we show that for an arbitrary acceptance rate profile $\alpha(y)$, there is a unique steady-state value of the average utilization rate $\bar{\rho}$. The equilibrium existence and uniqueness roughly follows from the continuity and monotonicity of the reaction curves of $\alpha$ in $\bar{\rho}$ and $\bar{\rho}$ in $\alpha$. On one hand, if the average utilization rate $\bar{\rho}$ increases, then the request traffic to each available seller increases. As a result, sellers become pickier, threshold $\hat{\pi}$ increases, and the acceptance rate $\alpha(y)$ decreases. On the other hand, if $\alpha(y)$ increases,
sellers become less available, and $\bar{\rho}$ strictly decreases. The equilibrium values of $\alpha$ and $\bar{\rho}$ then determine the sellers' strategies up to the acceptance of marginal buyers and, in turn, the sellers' surpluses. The buyer surplus is uniquely determined by the utilization rate as well.


### 3.2 Market Failure Under Full Disclosure

In this subsection, we establish that full disclosure leads to a Pareto inefficient market outcome. The market failure is driven by the sellers' decentralized decision-making and their capacity constraints.

We first define Pareto efficiency in our setting. An outcome $O=(V(\cdot), U)$ is a combination of seller profit profile and buyer surplus. We say that a market outcome is feasible if there is a seller strategy profile under full disclosure that achieves it. A feasible outcome $O$ is Pareto efficient if there is no other feasible $O^{\prime}$ such that $V^{\prime}(y) \geqslant V(y)$ for all $y$ and $U^{\prime} \geqslant U$, and at least one seller type or buyers are strictly better off. The Pareto frontier is the set of all Pareto-efficient outcomes. An outcome $O$ is implementable if there is an information policy that induces it in a steady state equilibrium.

Imagine that the platform starts with full disclosure as its default information policy. Write $\alpha^{F D}(y), V^{F D}(y), U^{F D}$ for equilibrium acceptance rates, seller profits, and consumer surplus. Correspondingly, write $\alpha^{\sigma}(y), V^{\sigma}(y), U^{\sigma}$ for these parameters when a given strategy profile $\sigma$ is played.

Proposition 2. (Market Failure) There exists a strategy profile $\sigma$ under which all market participants match more often and are strictly better off than in a full-disclosure equilibrium. That is for all $y \in Y$ :

$$
\begin{aligned}
\alpha^{\sigma}(y) & >\alpha^{F D}(y), \\
V^{\sigma}(y) & >V^{F D}(y), \\
U^{\sigma} & >U^{F D} .
\end{aligned}
$$

A full proof is given in the Appendix. The basic intuition for the coordination problem is cream skimming: A seller keeps his schedule open by rejecting low-value matches in order to increase his own chances of getting high-value matches. As a result, in equilibrium, sellers spend considerable time waiting for high-value matches. Collectively, this behavior is suboptimal because some low-value matches have to be accepted to maximize the joint seller surplus. The market feature that gives rise to the coordination failure is that collectively, the sellers are not capacity constrained, whereas individually, they are capacity constrained. Notably, unlike price competition that benefits buyers and improves market efficiency, seller competition for better buyers hurts buyers and decreases market efficiency.

It is instructive to highlight the inefficiencies on the seller and the buyer sides of the market as outlined in the introduction. Each seller rejects low-value matches to increase his chances of getting high-value matches. Such rejections directly harm the buyers because the sellers do not internalize their surplus. Such rejections hurt other sellers as well, because the seller remains available on the marketplace and competes with other sellers for incoming buyers.

Can information design mitigate the market failure? To answer this question, it is important to understand competing effects of information provision. Define the match rate $m$ as the number of matches formed on the platform over a unit time interval. Higher $m$ implies that more buyers are served, and the buyer surplus is proportional to it, $U=m u$. Similarly, define the seller average match quality by $q$ so that $V=m q$. Evidently, both surpluses increase in the match rate $m$. However, the match rate and the match quality are in conflict, and the information disclosure affects them through three different channels.

First, information provision has a positive effect on the seller match quality. From an individual seller's point of view, more information increases his set of attainable payoffs. Holding the match rate fixed, he individually benefits from more information about buyers. Second, information provision reduces the match rate by allowing the sellers to reject less valuable buyers. The platform can limit information and induce sellers to accept more
buyers. This effect is reminiscent of static persuasion settings. Third, information provision reduces the match rate through increasing sellers' return on search. With returns on search, the sellers reject more often, and the match rate goes down.

The match quality effect motivates the platform to provide more information, while the effects on match rate work in the opposite direction. The interaction of these effects depends on the primitives of the economic environment and shapes the preferred information policy. In the next section, we show that with homogeneous sellers, information design can fully mitigate the market failure and achieve an efficient allocation.


### 3.3 Optimal Policy with Homogeneous Sellers

We start with the case of homogeneous sellers so that $Y$ is a singleton or, equivalently:

$$
\pi(x, y)=\pi(x) .
$$

It is instructive to think about what seller strategies are Pareto efficient in this environment. It is straightforward to see that any efficient strategy is cutoff-all sellers should accept buyers that generates profits above some threshold. Intuitively, any given acceptance rate $\alpha$ results in the same buyer utility and a cutoff strategy maximizes the seller surplus among all strategies with the same acceptance rate. Conversely, a cutoff monotonically affects the acceptance rate and hence corresponds to some efficient outcome. We show that information control allows the platform to restore efficiency and bring the payoffs to the Pareto frontier.

Theorem 1. (Homogeneous Sellers) Suppose the sellers are homogeneous. Then, for any Pareto-efficient outcome ( $U, V$ ) with $V \geqslant 0$, there is an information policy that sends only two signals that implement $i t$.

The proof follows the efficiency approach of Smolin (2017). ${ }^{14}$ Consider an arbitrary seller

[^8]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 3: A sketch of a set of feasible and implementable outcomes in the case of homogeneous sellers. $V$ denotes the seller surplus; $U$ denotes buyer surplus. Any point on the Pareto frontier (thick solid line) can be implemented by some information policy. The full-disclosure outcome is suboptimal.

strategy that under full disclosure achieves a Pareto-efficient outcome with positive seller surplus. Consider an information policy that sends two signals and effectively recommends an action according to the strategy. Under the recommendation policy, there is only one signal with a recommendation to accept, so there is only one type of acceptable buyer. Hence, all profitable buyers are the same, and there is no return to cream skimming. For any Pareto-efficient outcome that delivers a positive surplus to sellers, they are willing to obey the recommendations.

Theorem 1 characterizes at once the range of optimal information policies corresponding to different platform objectives as any point on the Pareto frontier maximizes $\gamma U+(1-\gamma) V$ for some $\gamma \in[0,1]$. Figure 3 illustrates the result. Note that by Proposition 2, the fulldisclosure equilibrium outcome is outside of the Pareto frontier.

Now, note that in the case of homogeneous sellers, Pareto-efficient allocations do not depend on exact parameters $\beta$ and $\tau$. We have an immediate corollary.

Corollary 1. Suppose the sellers are homogeneous. The optimal information policy $\lambda^{*}$ depends only on the objective weights $\gamma$ and not on the buyer traffic intensity $\beta$ or the task
length $\tau$.

This corollary implies that the same information policy would be optimal if sellers became available immediately after accepting a buyer, $\tau=0$. The independence from $\beta$ and $\tau$ is driven by the coarse structure of an optimal policy. The rate of arrival to the available sellers, $\beta_{A}$, matters to sellers only to the extent that a higher request rate magnifies the opportunity cost of accepting. With a binary policy, this opportunity cost is nil because all profitable buyers are the same.

In the next section, we study the setting with heterogeneous sellers and show that the optimal policy is richer and does depend on $\beta$ and $\tau$.


### 3.4 Optimal Policy with Heterogeneous Sellers

We proceed with studying the setting in which the sellers feature vertical payoff heterogeneity. We characterize the optimal information policy and show that, unlike in the case of homogeneous sellers, the Pareto frontier is generally not attainable.

Formally, consider the linear payoff environment in which the seller profit is a linear function of buyer and seller types: $X=[0, \bar{x}], Y=[0, \bar{y}]$, and

$$
\pi(x, y)=y-x .
$$

We interpret $x$ as the level of difficulty of the buyer's task and interpret $y$ as the seller's skill level. In this way, buyers are vertically differentiated in the eyes of the sellers, and high-type sellers can profitably match with more buyers than can low-type sellers. Note that by the profit variability assumption, $\bar{y} \leqslant \bar{x}$.

Similar to the case of a homogeneous sellers, it is easy to see that the Pareto efficient strategies are cutoff-all seller types should accept buyers below a threshold which can now be type-dependent. At the same time, choosing the cutoff profile to maximize the total seller surplus should account for the congestion externalities due to uniform assignment. In
general, this profile depends on the details of seller and buyer type distributions. However, it is clear that the efficient cutoffs are generically different from their types and vary across them.

Finding an optimal information policy in this environment is a challenging problem. First, the sellers' preferences have unobserved heterogeneity and, as discussed above, Pareto efficient strategies differ across sellers. This precludes the effective use of the recommendation policies that provided a solution in the case of homogeneous sellers. Second, the sellers are forward looking, and the information policy determines not only their stage payoff but also their continuation values. As a result, a seller's decision depends not only on the posterior mean of the current buyer type but on the general shape of the information policy.

To solve the problem, we develop a variational approach in which we map information policies to a class of convex functions and then use the calculus of variation to recover a structure of optimal information policies. First, we follow Gentzkow and Kamenica (2016) and show that optimizing $\mathcal{J}$ with respect to information policies is equivalent to optimizing it with respect to a class of convex functions. Namely, for a given information policy $\lambda$, denote the posterior mean of $x$ conditional on signal $s$ by $z(s) \triangleq \int_{X} x \mathrm{~d} s(x)$, the induced distribution of $z(s)$ by $H$, and define the spread function $\Lambda:[0, \infty) \rightarrow \mathbb{R}_{+}$as

$$
\Lambda(z) \triangleq \int_{0}^{z} H(\zeta) \mathrm{d} \zeta
$$

In the linear payoff environment, $\Lambda$ is proportional to the option value of rejecting a buyer with expected cost $z$. Let $\bar{\Lambda}$ be the spread function under full disclosure, $\bar{\Lambda}(z)=\int_{0}^{z} F(\zeta) \mathrm{d} \zeta$, and let $\underline{\Lambda}$ be the spread function under no disclosure, $\underline{\Lambda}(z)=\max \{0, z-\mathbb{E}[x]\}$. The argument of Gentzkow and Kamenica (2016) establishes a one-to-one correspondence between the set of admissible posterior mean distributions and the following set,

$$
\mathcal{L} \triangleq\left\{\Lambda:[0, \infty) \rightarrow \mathbb{R}_{+}, \text {convex, and } \underline{\Lambda} \leqslant \Lambda \leqslant \bar{\Lambda}\right\} .
$$

As a result, maximization over information policies is equivalent to maximization over admissible spread functions $\Lambda \in \mathcal{L}$.

We then build on the analysis of Section 3.1 and show that for any information policy, the seller's optimal strategy has a simple cutoff structure: a type- $y$ seller accepts all buyers with expected difficulty below the cutoff $\hat{z}(y)$ and rejects all buyers above it. The cutoff profile has a simple structure-it is continuously increasing in $y$ and satisfies the following equation

$$
\frac{y-\hat{z}(y)}{\tau \beta_{A}(\Lambda)}=\Lambda(\hat{z}(y)) .
$$

The $\beta_{A}(\Lambda)$ is an equilibrium request rate that is linked to the seller behavior by (3) and (4).
The condition (11) allows easy illustration of cream skimming in Figure 4. Under cream skimming, a type- $y$ seller accepts buyers with expected difficulty $z$ not below his type $y$ but below a cutoff $\hat{z}(y)<y$. The cutoff can be seen as a projection of the type reflected against the spread function along the trajectory with the slope inversely proportional to the request rate $\beta_{A}$ and the task length $\tau$. The cutoff is, hence, increasing with type-higher seller types accept more buyers, cream skimming less. ${ }^{15}$ The higher the request rate, the flatter the slope and the greater the extent of cream skimming. Similarly, the lower the task length, the steeper the slope and the lower the extent of cream skimming. In the limit case as $\tau \rightarrow 0$, the slope is vertical, there is no cream skimming, $\hat{z}(y)=y$, and the problem can be seen as a static persuasion of Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017).

Next, to solve for an optimal spread function, we refer to the calculus of variations and consider effects of small spread-function variations on the platform's objective. Namely, we consider variations $\varepsilon h:[0, \infty) \rightarrow \mathbb{R}_{+}$such that $\Lambda+\varepsilon h \in \mathcal{L}$. We use the equilibrium conditions and calculate a variational derivative of the platform objective at $\varepsilon=0$ :

$$
\frac{d \mathcal{J}}{d \varepsilon}(0)=\int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y)\left(-\frac{g^{\prime}(y)}{g(y)} \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}}\right) g(y) \mathrm{d} y+h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y}) \Psi .
$$

[^9]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 4: Admissible spread functions $\Lambda$ and equilibrium acceptance cutoffs $\hat{z}(y)$.

Here, $\nu(y) \triangleq 1-\rho(y), \bar{\nu} \triangleq 1-\bar{\rho}$ are the equilibrium availability rates under the policy $\Lambda$, and $\Psi>0$ is a positive term that depends on the objective weight $\gamma$ and the aggregate equilibrium behavior. The integral term captures the relative distribution of the platform's surplus across the seller types, whereas the stand-alone term captures its aggregate level.


The behavior of the integrand expression shows the direction in which $\mathcal{J}$ can be locally improved. If $g$ is decreasing, then the integrand is positive everywhere. Any positive admissible variation increases $\mathcal{J}$, pushing the spread function towards full disclosure. If $g$ is log-concave, then the integrand crosses zero from below at most once. ${ }^{16}$ The same thing happens if $g$ is unimodal and the platform maximizes the matching rate, $\gamma=1$. In these cases, an improving deviation pushes towards more disclosure in the upper tail of $x$ and less disclosure in the lower tail. We show that these deviations lead to censorship policies.

Definition. An information policy $\lambda$ is a lower-censorship policy if for some $\hat{x} \in[0,1], \lambda$ fully reveals all $x>\hat{x}$ and pools all $x<\hat{x}$.

Theorem 2. (Heterogeneous Sellers) In the linear payoff environment:

1. If the seller distribution $g$ is log-concave, then a lower-censorship policy is optimal;

[^10]

2. If the seller distribution $g$ is weakly decreasing, then a full disclosure policy is optimal;
3. If the platform maximizes only the matching rate, $\gamma=1$, and the seller distribution $g$ is unimodal then a lower-censorship policy is optimal.

Optimal disclosure depends on the shape of seller type distribution because the optimal policy should satisfy incentive constraints for the sellers. Suppose there are two types of sellers, professionals and amateurs, and two types of tasks, easy and hard. Professionals accrue relatively higher value from tasks than do amateurs; but assume that under full disclosure, both professionals and amateurs accept only easy tasks. Limiting information may cause opposite reactions from them. Consider an information policy that pools easy tasks with hard tasks. A professional's average profit from the pooled task is positive, and so he starts accepting hard tasks. Conversely, an amateur's average profit from the pooled task is negative, and so he starts rejecting even easy tasks. Because the platform does not observe the seller type, the optimal information policy should strike a balance between professionals and amateurs and thus depends on their relative population sizes. If there are sufficiently many professionals, then no disclosure is optimal; if there are sufficiently many amateurs, then full disclosure is optimal.

Theorem 2 extends this binary intuition to the case of many seller types. It shows that for a broad class of type distributions, lower-censorship policies are optimal. These policies pool attractive buyers with infra-marginal buyers and fully reveal the less attractive buyers. They effectively provide no useful information for low-type sellers but bring a lot of informational value to high-type sellers. Under these policies, the worst sellers are excluded from the platform, the medium sellers accept only the pooled buyers, and the best sellers screen among the less attractive buyers. In this way, the platform optimally trades off the buyer and seller surpluses.


## 4 Applications

Strategic information coarsening is relevant in practice and can be found in the designs of many platforms. In this section, we discuss how our model relates to ride-hailing, accommodation, and freelance agencies, and how our analysis can guide their design.

- Ride-Hailing Platforms. Sellers and buyers are, respectively, drivers and passengers. When idle, drivers receive requests from passengers. The driver type captures his home location, preference for long rides, and tolerance for congestion. The rider type captures the destination and ride history. Drivers do not like short rides or rides to remote neighborhoods. ${ }^{17}$ Passengers do not like waiting. ${ }^{18}$ The leading agency Uber applies fixed pricing per mile and per minute that can depend only on aggregate multipliers, such as surge in demand. ${ }^{19}$ The passenger destination is fully concealed and represents the platform information coarsening. ${ }^{20}$
- Accommodation Rental Platforms. Sellers and buyers are, respectively, hosts and guests. Hosts are capacity constrained: once a room is booked, the host cannot accept a better guest. The host type captures his preference for age, race, personality, and daily schedule. The guest type captures his gender, age, socio-economic status, and lifestyle. In the leading agency Airbnb, every host sets a price that applies for all guests, but he may prefer to reject a guest whom he expects to be a bad fit. The host can adopt the "Instant Book" feature, with which the host commits to accepting all guests filtered by a few characteristics (e.g., pets, smoking, infants). The choice of filters acts as an implicit information coarsening.

[^11]

- Freelance Labor Platforms. Sellers and buyers are, respectively, service providers and clients. The service providers are constrained in the number of tasks they can do per week. The provider's type captures his skills and work ethic. The client type captures the task category and difficulty, and the client's expertise and location. In the leading agency TaskRabbit, the service providers set an hourly rate that applies to all tasks in the same category. Pulling many tasks into a single category is a form of information coarsening.

Our results shed light on why limiting buyer information can improve performance of these platforms. In an example of ride-hailing platforms, if drivers have too detailed information about passenger requests, they cream skim. This results in low acceptance rates and a suboptimal demand-supply fit. The platform's policy of hiding the passenger destination from drivers can be used to force drivers to accept more rides and avoid the inefficient cream-skimming behavior.

At the same time, we show that providing some information is often beneficial. In particular, we argue that censorship policies that reveal the least valuable buyers and pool the most valuable buyers with intermediate ones can be particularly useful. In the case of ride-hailing platforms, a censorship policy can be implemented as follows. The best driver destinations are pooled with intermediate destinations. The worst destinations, like those to remote places, are flagged and accompanied with a precise description. In equilibrium, the pooled destinations are served by all drivers active on the platform. The flagged destinations are served only by the most willing drivers, depending on the description details. This policy optimally balances between the platform's objectives-it provides sufficient information and surplus to the drivers but insures a high matching rate for the riders.

Before concluding, we discuss two considerations we abstracted away from but that might be important and should be taken into account in some applications: strategic buyers and optimal pricing.

Strategic Buyers. If buyers have heterogeneous match quality and search for better matches, then the platform might face the information design possibilities on both sides of the market. Disclosure to buyers can play several roles, depending on their strategic options.

First, the platform can use the disclosure to direct buyers to the sellers with higher acceptance rates. Imagine buyers observe seller acceptance rates imperfectly. Then they request selective sellers too often compared to what would be efficient. The platform can pool selective and non-selective sellers to shift buyer flow towards non-selective sellers.

Second, the platform can use information disclosure to direct buyers to the sellers who value them the most. Imagine that the prices do not reflect the seller match payoffs perfectly. Then, the information policy that pools matches that are less valuable for buyers and more valuable for sellers with those that are, conversely, more valuable for buyers and less valuable for sellers, will improve overall efficiency. ${ }^{21}$ In particular, one consider a natural extension of our model in which the buyers have differential tastes over sellers. Our analysis can inform this setting as well. If both sides of the market are homogeneous, then the argument analogous to that of Section Section 3.3, can be used to establish that the platform can again fully restore efficiency. If the sides are heterogeneous, then the analysis is more complicated but we expect the variational approach still being useful, even if more tedious. Moreover, we conjecture that the lower-censorship policies would continue to play an important role in analysis.

Optimal Pricing. Throughout the paper, we assumed the participants' payoffs are nontransferable. In practice, a platform may want to complement the information design with the pricing design. Such complementation is straightforward if the pricing is uniform and takes the form of a fixed amount a buyer should pay to a seller if he is accepted. In this case,

[^12]the platform should simply perform backward induction optimization. For any given price, the platform can calculate the matching payoffs and find the optimal information policy and its corresponding payoffs by using our analysis. In the first step, the platform can then choose the price optimal.


In fact, if the sellers are homogeneous, then the uniform pricing is sufficient to implement an efficient outcome that maximizes joint surplus. Indeed, Theorem 1 establishes that any Pareto-efficient outcome can be implemented by some information policy as long as the sellers receive some rents. If the efficient outcome delivers positive rents to the sellers, then no additional price is required. Otherwise, the platform can set the price to subsidize the sellers and give them sufficient rent to participate in the platform. Then, by the same argument as in the main analysis, there exists a recommendation policy that implements the outcome.

If the sellers are heterogeneous, then Theorem 2 shows that the optimal information policy need not even achieve Pareto frontier, so that flexible pricing rules can possibly outperform the uniform pricing. However, most matching platforms adopt simple pricing. There could be several explanations. First, consumers appreciate transparent and simple pricing, both because it is easier to assess the cost of a ride and because complex pricing can feel like price gouging and prompt the risk of price-discrimination litigation. Second, flexible pricing is a non-trivial development task that requires significant resources. Relatedly, it may be impractical to condition on every variable relevant in the transportation market. There is some complexity threshold for the designer beyond which increasing the price complexity is not justified. ${ }^{22}$

As a final remark, we note that often the platforms accompany uniform pricing with a commission proportional to the price of each match. In these cases, maximizing a profit flow is equivalent to maximizing a matching rate. Our results then point out that censorship policies can be useful to bring the most revenue to the platform.

[^13]


## 5 Conclusion

Strategic information disclosure is arguably less intrusive than direct control-and often more palatable than sophisticated pricing schemes. When the two sides of the market are not symmetric in their preferences for match quality and match rate, the more patient and selective side of the market tends to cream skim. In this paper, we provide a case for using information design as a way to mitigate the adverse effects of cream skimming and improve platform performance. Specifically, we highlight the use of censorship policies as a way to balance surpluses on both sides of the market.

For most of this paper, we isolated information design from other tools available to platforms. Flexible pricing, real-time auctions, recommender systems, and queue management can also help improve matching market performance. We believe that studying interaction between these schemes can complement our current insights and constitutes a promising avenue for further research.


## 6 Appendix

Lemma 1. For any information policy $\lambda$ and an equilibrium acceptance rate profile $\alpha(y)$ :

- the average utilization rate $\bar{\rho} \in[0,1]$ is a solution to
$$
1=\int \frac{1}{1-\bar{\rho}+\beta \tau \alpha(y)} \mathrm{d} G(y) ;
$$
- the solution $\bar{\rho}$ exists and is unique; it increases in $\beta, \tau$, and whenever $\alpha(y)$ is increased on a set of strictly positive measure with respect to $G$;
- the utilization rate $\rho(y)$ increases whenever $\alpha\left(y^{\prime}\right)$ is increased on a set of strictly positive measure with respect to $G$.

Proof. By (3):

$$
1-\rho(y)=\frac{1}{1+\tau \beta \alpha(y) /(1-\bar{\rho})}
$$

Integrate both sides with respect to $G(y)$ and rearrange to obtain (13). Its right-hand side is continuously and strictly increasing in $\bar{\rho}$. Evaluated at $\bar{\rho}=0$, it equals $\int \frac{d G(y)}{1+\beta \tau \alpha(y)} \leqslant 1$; evaluated at $\bar{\rho}=1$, it equals $\int \frac{d G(y)}{\beta \tau \alpha(y)} \geqslant 1$, by the aggregate capacity assumption. Therefore, a solution exists and is unique. The monotonicity of $\bar{\rho}$ follows. Finally, consider a perturbation of increasing $\alpha$ on some positive measure of $y^{\prime} \neq y$. By the argument above, $\bar{\rho}$ increases so, by (14), $\rho(y)$ increases. $\square$

Lemma 2. For any given information policy $\lambda$ and utilization rate $\bar{\rho}$, the sellers optimal strategies are cutoff:

$$
\sigma(s, y)= \begin{cases}1, & \pi(s, y)>\hat{\pi}(y) \\ 0, & \pi(s, y)<\hat{\pi}(y)\end{cases}
$$

The cutoff function $\hat{\pi}(y)$ is unique and for all types $\hat{\pi}(y)>0$ and $\alpha(y)<1$.
Proof. Fix $y \in Y$. By (6), the optimal seller strategy is such that all signals from $S_{a}(y) \triangleq$ $\{s: \pi(s, y)>\tau V(y)\}$ are accepted and all signals from $S_{r}(y) \triangleq\{s: \pi(s, y)<\tau V(y)\}$ are rejected. The seller is indifferent between accepting and rejecting signals from $S_{m}(y) \triangleq$ $\{s: \pi(s, y)=\tau V(y)\}$. The cutoff for a type- $y$ seller is $\hat{\pi}(y)=\tau V(y)$. Using (6), the cutoff of the optimal strategy is a solution to

$$
\hat{\pi}(y)=\tau \beta_{A} \int \max \{0, \pi(s, y)-\hat{\pi}(y)\} \lambda(\mathrm{d} s) .
$$

The solution is unique because the left-hand side of (15) is strictly increasing in $\hat{\pi}(y)$ while the right-hand side is decreasing in $\hat{\pi}(y)$. By the variability assumption, $\hat{\pi}(y)>0$. Finally, under the full disclosure policy, $\pi$ is continuously distributed, so $\alpha(y)<1$. $\square$

Proof of Proposition 1. Step 1. Existence. Consider the correspondence $\psi:[0,1] \rightrightarrows[0,1]$ which maps $\bar{\rho}$ to a set of "reaction" $\bar{\rho}$ 's by the following procedure. First, find the unique
cutoff function $\hat{\pi}$ from (15) with $\beta_{A}=\beta /(1-\bar{\rho})$. The cutoff $\hat{\pi}$ does not pin down the acceptance rate profile $\alpha(y)$ uniquely because marginal signals can have positive probability under $\lambda$. The acceptance rates consistent with $\hat{\pi}(y)$ are integrable functions such that

$$
\lambda\left(S_{a}(y)\right) \leqslant \alpha(y) \leqslant \lambda\left(S_{a}(y)\right)+\lambda\left(S_{m}(y)\right), \quad \forall y \in Y .
$$

Denote by $\mathcal{A}$ the set of $\alpha$ consistent with $\hat{\pi}$. For any $\alpha \in \mathcal{A}$, find $\bar{\rho}$ as shown in Lemma 1. Going over all $\mathcal{A}$ will produce the set $\psi(\bar{\rho})$.

By construction, $\mathcal{A}$ is convex. By Lemma 1, $\bar{\rho}$ is continuous in $\alpha$, so $\psi(\bar{\rho})$ is a closed interval in [0, 1]. Moreover, $\psi$ is upper hemicontinuous because $\hat{\pi}$ is continuous in $\beta_{A}$ according to (15). By Kakutani's theorem, $\psi$ has a fixed point, which is a solution.

Step 2. Uniqueness. Towards a contradiction, suppose $\bar{\rho}$ and $\bar{\rho}^{\prime}, \bar{\rho}>\bar{\rho}^{\prime}$ are two distinct fixed points of $\psi$. By uniformity of assignment, $\beta_{A}>\beta_{A}^{\prime}$. By (15) and the variability assumption, $\hat{\pi}(y)>\hat{\pi}^{\prime}(y)$ for any $y \in Y$. Thus, whenever $\pi(s, y) \geqslant \hat{\pi}(y)$ we also have $\pi(s, y)>\hat{\pi}^{\prime}(y)$. But this means that $S_{a}^{\prime}(y) \supseteq S_{a}(y) \cup S_{m}(y)$. Therefore, $\alpha^{\prime}(y) \geqslant \lambda\left(S_{a}^{\prime}\right) \geqslant$ $\lambda\left(S_{a} \cup S_{m}\right) \geqslant \alpha(y)$ for all $y$. By Lemma 1, this implies $\bar{\rho}^{\prime} \geqslant \bar{\rho}$, a contradiction. Moreover, since $\bar{\rho}^{\prime}=\bar{\rho}, U=U^{\prime}$, and by Lemma 2, the strategy cutoffs coincide $\hat{\pi}^{\prime}(\cdot)=\hat{\pi}(\cdot)$. Hence, the seller surpluses coincide as well $V(\cdot)=V^{\prime}(\cdot)$.

Proof of Proposition 2. Rearranging (6),

$$
\begin{aligned}
V(y) & =\frac{\beta_{A}}{1+\tau \beta_{A} \alpha(y)} \int \pi(s, y) \sigma(s, y) \mathrm{d} \lambda(s)= \\
& =\frac{\beta}{1-\bar{\rho}+\beta \tau \alpha(y)} \int \pi(s, y) \sigma(s, y) \mathrm{d} \lambda(s)
\end{aligned}
$$

By Lemma 2, strategy $\sigma^{F D}$ prescribes jobs in $\{x: \pi(x, y)>\tau V(y)\}$ to be accepted with probability 1 and induces acceptance rates $\alpha^{F D}(y)<1$. Consider a deviation $\sigma$ that induces uniformly higher acceptance rates $\alpha(y)=\alpha^{F D}(y)+\Delta$ for some small $\Delta>0$ for all $y \in Y$. As $Y$ is compact, such deviation exists and strictly increases the buyer surplus, $U>U^{F D}$.

Moreover, by (13), $1-\bar{\rho}+\beta \tau \alpha(y)=1-\bar{\rho}^{F D}+\beta \tau \alpha^{F D}(y)$ for all $y$. Yet, the integral strictly increases, because $\hat{\pi}(y)>0$, and so $V(y)>V^{F D}(y)$.

Proof of Theorem 1. Take any Pareto-efficient pair $O=(V, U)$. Since $O$ is feasible, there is a seller strategy $\sigma$ that induces $O$. Consider an information policy $\hat{\lambda}$ that recommends this strategy with $s_{a}$ being the recommendation to accept and $s_{r}$ the recommendation to reject. We need to check that the sellers would follow the recommendations.

From (6) we have $v\left(s_{a}\right)=\pi\left(s_{a}\right)-\tau V, v\left(s_{r}\right)=0$, and $V=\beta_{A}\left(\pi\left(s_{a}\right)-\tau V\right) \hat{\lambda}\left(s_{a}\right)$. The incentive constraints require that $\pi\left(s_{a}\right) \geqslant \tau V$ and $\pi\left(s_{r}\right) \leqslant \tau V$. For the former,


$$
\tau V=\frac{\tau \beta_{A} \hat{\lambda}\left(s_{a}\right)}{1+\tau \beta_{A} \hat{\lambda}\left(s_{a}\right)} \pi\left(s_{a}\right) \leqslant \pi\left(s_{a}\right) .
$$

For the latter, recall that $O$ is Pareto efficient, hence $\sigma$ accepts all profitable jobs. This implies that

$$
\pi\left(s_{r}\right) \leqslant 0 \leqslant \tau V .
$$

Proof of Theorem 2. The sequence of lemmas that follow support and formalize the argument outlined in the main text.

Lemma 3. $\ell \in \mathcal{L}$ if and only if there is $\lambda \in \Delta(S)$ such that $\Lambda(\cdot, \lambda)=\ell$.

Proof. The proof is standard and omitted. Kolotilin et al. (2017) and Gentzkow and Kamenica (2016) describe it in details. $\square$

Lemma 4. In the linear payoff environment, for any information policy $\lambda$, the seller's optimal strategy has a cutoff form such that a type-y seller accepts all buyers with expected difficulty $z<\hat{z}(y)$ and rejects all buyers with $z>\hat{z}(y)$. For a given $\beta_{A}$, the seller payoff $V(y)$ and the cutoff $\hat{z}(y)$ satisfy:

$$
V(y)=\frac{y-\hat{z}(y)}{\tau}=\beta_{A} \Lambda(\hat{z}(y)) .
$$

As a result, $\hat{z}(y)$ is increasing in $y$ and decreasing in $\tau \beta_{A}$.
Proof. The first steps of the proof of Proposition 1 show that in the case of arbitrary $X$ and $Y$, the seller optimal strategy has a cutoff form. They also show that the minimal acceptable profit is $\hat{\pi}(y)=\tau V(y)$ and, moreover, (15) holds. In the linear payoff environment, $\pi(s, y)=y-z(s)$ and $\hat{\pi}(y)=y-\hat{z}(y)$ for some $\hat{z}$. Plug it back into (15) and obtain

$$
y-\hat{z}(y)=\tau \beta_{A} \int_{0}^{\hat{z}(y)}(\hat{z}(y)-z) \mathrm{d} H(z)=\tau \beta_{A} \int_{0}^{\hat{z}(y)} H(z) d z=\tau \beta_{A} \Lambda(\hat{z}(y))
$$

where the second equality follows by the integration by parts. $\square$

Lemma 5. In the linear payoff environment, for any information policy $\lambda$ and for almost all $y$,

$$
\alpha(y)=H(\hat{z}(y))=\Lambda^{\prime}(\hat{z}(y)) .
$$

The acceptance rate $\alpha(y)$ is increasing in $y$ and decreasing in $\tau \beta_{A}$ for any $y \in Y$. Moreover,

$$
\hat{z}^{\prime}(y)=\frac{1}{1+\tau \beta_{A} \alpha(y)}=\nu(y) .
$$

Proof. By Lemma 4, the accepted jobs are those with $z<\hat{z}(y)$. Using the definition of $\Lambda$ from (9), the probability of accepting $\alpha(y)$ must lie between $H(\hat{z}(y)-)$ and $H(\hat{z}(y)+)$. The points of discontinuity of $H$ constitute a set of measure zero and seller types have full support on $[0,1]$. It follows that $\alpha(y)=\Lambda^{\prime}(\hat{z}(y))$ for almost all types $y$.

By (17), $\hat{z}(y)$ is increasing in $y$. Since $\Lambda$ is convex, $\alpha(y)$ is increasing in $y$. Same argument reveals that $\alpha(y)$ is decreasing in $\tau \beta_{A}$.

Finally, using Lemma 4, we have $y-\hat{z}(y)=\tau \beta_{A} \Lambda(\hat{z}(y))$. Differentiating that expression on both sides with respect to $y$ and substituting $\Lambda^{\prime}(\hat{z}(y))$ with $\alpha(y)$, the result follows. $\square$

Consider an initial information policy $\Lambda_{0} \in \mathcal{L}$, and a deviation $h:[0,1] \rightarrow[0,1]$. A deviation is admissible if $\Lambda_{0}+h \in \mathcal{L}$. Since $\mathcal{L}$ is convex, $\Lambda_{0}+\varepsilon h \in \mathcal{L}$ for all $\varepsilon \in[0,1]$. For a functional $\mathcal{I}: \mathcal{L} \rightarrow \mathbb{R}$ we will denote the value of $\mathcal{I}$ under information policy $\Lambda_{0}+\varepsilon h$ by
$\mathcal{I}(\varepsilon)$. The main step in the analysis is to find the variational derivative with respect to $\varepsilon$ at zero:

$$
\mathcal{J}^{\prime}(0)=\gamma U^{\prime}(0)+(1-\gamma) V^{\prime}(0) .
$$

Lemma 6. In the linear payoff environment, for any admissible deviation $h$ :

$$
\begin{aligned}
& U^{\prime}(0)=-u \beta \Psi_{1}\left(\int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y) g^{\prime}(y) \mathrm{d} y-h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y})\right), \\
& V^{\prime}(0)=\Psi_{2} U^{\prime}(0)+\frac{\beta}{\bar{\nu}} \int h(\hat{z}(y)) \nu(y) g(y) \mathrm{d} y,
\end{aligned}
$$

where $\Psi_{1}=1 / \int_{0}^{\bar{y}}\left(\nu^{2}(y)-\tau \nu^{\prime}(y) V(y)\right) g(y) \mathrm{d} y>0$ and $\Psi_{2}=\frac{\tau}{u \bar{\nu}} \int \nu(y) V(y) \mathrm{d} G(y)>0$.

Proof. Consider a variation $\varepsilon h(z)$. The resulting matching rate is by definition and Lemma 5:

$$
m(\varepsilon)=\frac{1}{\tau} \int_{0}^{1} 1-\nu(y, \varepsilon) \mathrm{d} G(y)=\frac{1}{\tau}\left(1-\int \hat{z}^{\prime}(y, \varepsilon) d G(y)\right)=\frac{1}{\tau}\left(1-\hat{z}(\bar{y}, \varepsilon) g(\bar{y})+\int g^{\prime}(y) \hat{z}(y, \varepsilon) \mathrm{d} y\right),
$$

so

$$
m^{\prime}(\varepsilon)=\frac{1}{\tau} \int_{0}^{\bar{y}} g^{\prime}(y) \hat{z}_{\varepsilon}^{\prime}(y, \varepsilon) \mathrm{d} y-g(\bar{y}) \hat{z}_{\varepsilon}^{\prime}(\bar{y}, \varepsilon) .
$$

By Lemma 4,

$$
\tau V(y ; \varepsilon)=y-\hat{z}(y, \varepsilon)=\tau \beta_{A}(\varepsilon)\left(\Lambda_{0}+\varepsilon h\right)(\hat{z}(y, \varepsilon)) .
$$

Finding $U^{\prime}(0)$ : The surplus derivative is $U^{\prime}(0)=u m^{\prime}(0)=u \bar{\rho}_{\varepsilon}^{\prime}(0) / \tau$. Differentiate the second equation in (19) with respect to $\varepsilon$, evaluate at 0 and collect terms:

$$
\begin{aligned}
\hat{z}_{\varepsilon}^{\prime}(y, 0) & =-\left(1+\tau \beta_{A} \alpha(y)\right)^{-1}\left(\tau \beta_{A}^{\prime}(0) \Lambda_{0}(\hat{z}(y, 0))+\tau \beta_{A}(0) h(\hat{z}(y, 0))\right) \\
& =-\nu(y, 0)\left(\tau \beta_{A}^{\prime}(0) \Lambda_{0}(\hat{z}(y, 0))+\tau \beta_{A}(0) h(\hat{z}(y, 0))\right) \\
& =-\nu(y, 0) \tau \beta\left(\frac{\bar{\rho}_{\varepsilon}^{\prime}(0) \Lambda_{0}(\hat{z}(y, 0))}{\bar{\nu}^{2}(0)}+\frac{h(\hat{z}(y, 0))}{\bar{\nu}(0)}\right) .
\end{aligned}
$$

where the second line follows by Lemma 5 and the last line follows as $\beta_{A}(\varepsilon)=\beta / \bar{\nu}(\varepsilon)$.

Substituting (20) into (18) and omitting the $\varepsilon=0$ argument in the function we obtain,

$$
\bar{\rho}_{\varepsilon}^{\prime}=-\int \nu(y) \tau \beta\left(\frac{\bar{\rho}_{\varepsilon}^{\prime} \Lambda_{0}(\hat{z}(y))}{\bar{\nu}^{2}}+\frac{h(\hat{z}(y))}{\bar{\nu}}\right) g^{\prime}(y) \mathrm{d} y+\nu(\bar{y}) \tau \beta\left(\frac{\bar{\rho}_{\varepsilon}^{\prime} \Lambda_{0}(\hat{z}(y))}{\bar{\nu}^{2}}+\frac{h(\hat{z}(y))}{\bar{\nu}}\right) g(\bar{y}) .
$$

Rearranging,


$$
\bar{\rho}_{\varepsilon}^{\prime}=-\tau \beta \Psi_{1}\left(\int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y) g^{\prime}(y) \mathrm{d} y-h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y})\right)
$$

where

$$
\begin{aligned}
\Psi_{1} & \triangleq\left(\bar{\nu}+\tau \beta_{A} \int_{0}^{\bar{y}} \nu(y) \Lambda(\hat{z}(y)) g^{\prime}(y) \mathrm{d} y-\tau \beta_{A} \nu(\bar{y}) \Lambda(\hat{z}(\bar{y})) g(\bar{y})\right)^{-1} \\
& =\left(\bar{\nu}+\tau \int_{0}^{\bar{y}} \nu(y) V(y) g^{\prime}(y) \mathrm{d} y-\tau \nu(\bar{y}) V(\bar{y}) g(\bar{y})\right)^{-1} \\
& =\left(\int_{0}^{\bar{y}} \nu(y) g(y)+\tau \nu(y) V(y) g^{\prime}(y) \mathrm{d} y-\tau \nu(\bar{y}) V(\bar{y}) g(\bar{y})\right)^{-1} \\
& =\left(\int_{0}^{\bar{y}}\left(\nu(y)-\tau \nu^{\prime}(y) V(y)-\tau V^{\prime}(y) \nu(y)\right) g(y) \mathrm{d} y\right)^{-1} \\
& =\left(\int_{0}^{\bar{y}}\left(\nu^{2}(y)-\tau \nu^{\prime}(y) V(y)\right) g(y) \mathrm{d} y\right)^{-1}
\end{aligned}
$$

with the last line following from differentiating the first equation in (19) with respect to $y$ : $\tau V^{\prime}(y)=1-\nu(y)$. The term $\Psi_{1}$ is positive because $\nu^{\prime}(y)<0$.

Finding $V^{\prime}[0]$ : Differentiate the first equation in (19) with respect to $\varepsilon$ and evaluate at zero. Using (20), we have:

$$
\tau V_{\varepsilon}^{\prime}(y, 0)=-\hat{z}_{\varepsilon}^{\prime}(y, 0)=\nu(y, 0) \tau\left(\beta_{A}^{\prime}[0] \Lambda_{0}(\hat{z}(y))+\beta_{A}[0] h(\hat{z}(y))\right) .
$$

Integrating over seller types:

$$
\begin{aligned}
V^{\prime}(0) & =\int V_{\varepsilon}^{\prime}(y, 0) \mathrm{d} G(y) \\
& =\beta_{A}^{\prime}(0) \int \nu(y) \Lambda_{0}(\hat{z}(y)) \mathrm{d} G(y)+\beta_{A} \int \nu(y) h(\hat{z}(y)) \mathrm{d} G(y) \\
& =m^{\prime}(0) \frac{\tau \beta_{A}}{\bar{\nu}} \int \nu(y) \Lambda_{0}(\hat{z}(y)) \mathrm{d} G(y)+\beta_{A} \int \nu(y) h(\hat{z}(y)) \mathrm{d} G(y) \\
& =m^{\prime}(0) \frac{\tau}{\bar{\nu}} \int \nu(y) V(y) \mathrm{d} G(y)+\frac{\beta}{\bar{\nu}} \int \nu(y) h(\hat{z}(y)) \mathrm{d} G(y)
\end{aligned}
$$

The second part of the result follows. $\square$

Lemma 7. $\lambda \in \Delta(S)$ is $x^{*}$-lower-censorship if and only if the corresponding spread function $\Lambda$ has the following form:

$$
\Lambda(z)= \begin{cases}\underline{\Lambda}(z), & z \in\left[0, \mathbb{E}\left[x \mid x<x^{*}\right]\right] \\ \bar{\Lambda}\left(x^{*}\right)+F\left(x^{*}\right)\left(z-x^{*}\right), & z \in\left(\mathbb{E}\left[x \mid x<x^{*}\right], x^{*}\right) \\ \bar{\Lambda}(z), & z \in\left[x^{*}, \bar{x}\right]\end{cases}
$$

Proof. Denote by $z^{*}$ the expected value of $x$ on the pooled part of $X, z^{*}=\mathbb{E}\left[x \mid x>x^{*}\right]$. If $\lambda$ is the $x^{*}$-lower-censorship, then a straightforward application of the spread-function definition obtains (21).

Conversely, suppose $\Lambda$ has a form given in (21). Let $H(z)=\Lambda^{\prime}(z)$, where $\Lambda^{\prime}$ is the right derivative. Function $H(z)$ then equals to $F(z)$ for $z \leqslant x^{*}$, to $F\left(x^{*}\right)$ for $x^{*}<z<z^{*}$, and to 1 for $z \geqslant z^{*}$. Such distribution corresponds to a $x^{*}$-lower-censorship. $\square$

Now let's return to the proof of Theorem 2. Use Lemma 6 to obtain the expression for
the variational derivative of the objective function with respect to $\varepsilon$ :

$$
\begin{aligned}
\mathcal{J}^{\prime}(0) & =\gamma U^{\prime}(0)+(1-\gamma) V^{\prime}(0) \\
& =\int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y)\left(-g^{\prime}(y) \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}} g(y)\right) \mathrm{d} y+h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y}) \Psi \\
& =\int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y)\left(-\frac{g^{\prime}(y)}{g(y)} \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}}\right) g(y) \mathrm{d} y+h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y}) \Psi
\end{aligned}
$$

where $\Psi \triangleq\left(\gamma+\Psi_{2}(1-\gamma)\right) u \beta \Psi_{1}>0$.
If $g$ is weakly decreasing, then $g^{\prime} / g \leqslant 0$, and so the integrand in (22) is positive for all $y \in Y$. Therefore, for any $\Lambda_{0}$ and any $h \geqslant 0, \mathcal{J}^{\prime}(0)>0$. The only spread function that cannot be improved like that corresponds to full disclosure so full disclosure is optimal.

If $g$ is log-concave, then $g^{\prime}(y) / g(y)$ is decreasing. Then for any $\Lambda_{0}$, the integrand in (22) crosses zero from below at most once. If the integrand is always positive, then, again, full disclosure is optimal. If the integrand does cross zero once, then we will show that a lower censorship is optimal. Towards the contradiction, suppose that $\Lambda_{0}$ is not a lower censorship. We will construct a platform-profitable deviation $h$ as follows. Denote by $y^{*}$ the zero of the integrand in (22), and denote by $z^{*}=\hat{z}\left(y^{*}, \Lambda_{0}\right)$ the optimal cutoff of the $y^{*}$-type seller. Let $\tilde{\Lambda} \in \mathcal{L}$ be the lower censorship that passes through the point $\left(z^{*}, \Lambda_{0}\left(z^{*}\right)\right)$, as in (21). Essentially, $\tilde{\Lambda}$ is the policy that discloses as little as possible on the left tail and as much as possible on the right tail but still passes through $\left(z^{*}, \Lambda_{0}\left(z^{*}\right)\right)$. This lower censorship is unique because there is a unique line tangent to $\bar{\Lambda}$ and passing through $\left(z^{*}, \Lambda_{0}\left(z^{*}\right)\right)$. We have $\tilde{\Lambda}(z) \leqslant \Lambda_{0}(z)$ for $z<z^{*}$ and $\tilde{\Lambda}(z) \geqslant \Lambda_{0}(z)$ for $z>z^{*}$; as $\Lambda_{0}$ is not a lower censorship,
for some $z$ one of these inequalities is strict. Consider a deviation $h=\tilde{\Lambda}-\Lambda_{0}$. We have

$$
\begin{aligned}
\mathcal{J}^{\prime}(0)= & \int_{0}^{\bar{y}} h(\hat{z}(y)) \nu(y)\left(-\frac{g^{\prime}(y)}{g(y)} \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}}\right) g(y) \mathrm{d} y+h(\hat{z}(\bar{y})) \nu(\bar{y}) g(\bar{y}) \Psi \\
= & \int_{0}^{y^{*}} \underbrace{h(\hat{z}(y))}_{\leqslant 0} \nu(y) \underbrace{\left(-\frac{g^{\prime}(y)}{g(y)} \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}}\right)}_{\leqslant 0} g(y) \mathrm{d} y \\
& +\int_{y^{*}}^{\bar{y}} \underbrace{h(\hat{z}(y))}_{\geqslant 0} \nu(y) \underbrace{\left(-\frac{g^{\prime}(y)}{g(y)} \Psi+(1-\gamma) \frac{\beta}{\bar{\nu}}\right)}_{\geqslant 0} g(y) \mathrm{d} y+\underbrace{h(\hat{z}(\bar{y}))}_{\geqslant 0} \nu(\bar{y}) g(\bar{y}) \Psi .
\end{aligned}
$$


By the argument above, one of the inequalities is strict, and so $\mathcal{J}^{\prime}(0)>0$, contradicting the optimality of $\Lambda_{0}$.

The same logic applies if the platform maximizes only the matching rate, $\gamma=1$, and the seller distribution $g$ is unimodal.


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[^0]:    *Romanyuk: Harvard University, 1805 Cambridge St, Cambridge, MA 02138, USA (email: gleb.romanyuk@gmail.com); Smolin: University of Bonn, Lennestraße 37, 53113, Bonn, Germany (email: alexey.v.smolin@gmail.com). We thank John Asker, Coeditor, and three anonymous referees for their productive suggestions. This paper builds on Romanyuk's dissertation chapter written at Harvard University, Department of Economics; he is indebted to Susan Athey, Drew Fudenberg, Greg Lewis, Tomasz Strzalecki, as well as to Chiara Farronato and Andrei Hagiu for their guidance and support. We are grateful to Andrey Fradkin, Ben Golub, Divya Kirti, Sergei Kovbasyuk, Jeffrey Picel, as well as seminar participants at Harvard University and MIT for helpful discussions. The views expressed in this article are those of the authors and do not necessarily reflect those of the employing organizations.

[^1]:    ${ }^{1}$ Tadelis and Zettelmeyer (2015) show that in wholesale automobile auctions, information disclosure helps match heterogeneous buyers to cars of varying quality. Einav, Farronato, and Levin (2016) name eliciting and aggregating the dispersed user information as a key objective of peer-to-peer platforms.

[^2]:    ${ }^{2}$ In comparison to standard search models, our market features the congestion externality but not the thick market externality.
    ${ }^{3}$ Kamenica and Gentzkow (2011) and Bergemann and Morris (2016) outline the information design paradigm in static settings.
    ${ }^{4}$ This approach has proven useful in other information design problems as well (e.g., Ely and Szydlowski (2017)).

[^3]:    ${ }^{5}$ Buyer type $x$ captures the payoff-relevant information the platform elicits from the buyer, whether passively from the buyer's cookies and queries or actively by asking questions. For example, on Uber, $x$ would incorporate the rider's destination; on Airbnb, $x$ would incorporate the guest's race, age, and gender.

[^4]:    ${ }^{6}$ Seller type $y$ captures payoff-relevant information that, for whatever reason-costly, unethical, etc.-the platform cannot elicit from the sellers. For example, on Uber, $y$ would incorporate the driver's preference for long rides and traffic; on Airbnb, $y$ would incorporate the host's preference for his guest's age, gender, socio-economic status, race, etc.
    ${ }^{7}$ Many results generalize to the case of heterogeneous buyer utility.
    ${ }^{8}$ The results immediately generalize to the case where the sellers have discount rate $r$ by replacing $\tau$ with

[^5]:    $\tau_{r}=\left(1-e^{-r \tau}\right) / r$.
    ${ }^{9} \Delta(S)$ is the set of Borel probability distributions with the weak-* topology on $\Delta(X)$.
    ${ }^{10}$ Equivalently, we could model an information policy as an arbitrary statistical experiment informative about $x$.


[^6]:    ${ }^{11}$ Fradkin (2015) finds that on Airbnb, seller screening causes around 41\% of all failed matches, whereas simultaneity only around 12\% (the rest is attributed to unavailable vacancies).
    ${ }^{12}$ Fradkin (2015) reports that on Airbnb, an initial rejection decreases by 51\% the probability that the guest eventually books any listing.

[^7]:    ${ }^{13}$ For example, if a seller earns \$1 on each buyer, and the time interval between accepting a pair of consequent buyers is 2 , then $V(y)=1 / 2$.

[^8]:    ${ }^{14}$ The efficiency result also relates to that of Bergemann, Brooks, and Morris (2015). They show that segmentation of a monopolistic market can achieve every feasible combination of consumer and producer surplus

[^9]:    ${ }^{15}$ Thus, the equilibrium exhibits a kind of negative stochastic assorting in terms of Chade (2006).

[^10]:    ${ }^{16}$ The class of log-concave distributions includes normal, logistic, exponential, and uniform distributions, as well as their truncations.

[^11]:    ${ }^{17}$ http://www.forbes.com/sites/harrycampbell/2015/03/24/just-how-far-is-your-uber-driver-willing-totake-you.
    ${ }^{18}$ Cohen, Hahn, Hall, Levitt, and Metcalfe (2016) estimate that an increase in wait time of 1 minute decreases the probability of requesting a ride by 1.7 percent.
    ${ }^{19}$ The surge price coefficient depends on the numbers of riders and nearby drivers who have their Uber app open at that moment: https://newsroom.uber.com/upfront-fares-no-math-and-no-surprises/.
    ${ }^{20}$ Uber drivers can use the destination filter twice a day. If used, the platform filters passenger requests and offers only those aligned with the driver's preferred destination: http://www.ridesharingdriver.com/uberdestination-filter-get-passengers-heading-your-direction/.

[^12]:    ${ }^{21}$ Rayo and Segal (2010) study the last situation in a static match model when both buyers and sellers derive heterogeneous payoffs from matches. They find that in the case of the uniform distribution of receiver reservation values, the optimal information policy pools "non-ordered prospects," where two prospects are non-ordered in the sense described above.

[^13]:    ${ }^{22}$ See Segal (2007) for the study of communication requirements of social choice rules. It demonstrates that in some social choice problems, the space of prices that must be discovered is prohibitively large.


## Citation and provenance

**Authors:** Gleb Romanyuk; Alex Smolin

**Canonical citation:** Romanyuk, Gleb, and Alex Smolin. “Cream Skimming and Information Design in Matching Markets.” American Economic Journal: Microeconomics 11, no. 2 (2019): 250–276.

**Canonical machine-readable version:** https://alexsmolin.com/corpus/papers/cream-skimming-and-information-design-in-matching-markets.md

**Source record:** https://alexsmolin.com/files/cream-skimming-and-information-design-working-paper.pdf

**Published record:** https://doi.org/10.1257/mic.20170154

**Attribution guidance:** Preserve the supplied title, complete author list, citation, and canonical URL when technically practicable.

Provenance metadata: https://alexsmolin.com/corpus/PROVENANCE.txt


> Machine-readable author manuscript.
> Authors: Chiara Margaria; Alex Smolin.
> Canonical citation: Margaria, Chiara, and Alex Smolin. “Dynamic Communication with Biased Senders.” Games and Economic Behavior 110 (2018): 330–339.
> Attribution and provenance: https://alexsmolin.com/corpus/PROVENANCE.txt


# Dynamic Communication with Biased Senders

**Authors:** Chiara Margaria; Alex Smolin

**Manuscript date:** 2017-10-21

#### Abstract

We study dynamic games in which senders with state-independent payoffs communicate to a single receiver. Senders' private information evolves according to an aperiodic and irreducible Markov chain. We prove an analog of a folk theorem-that any feasible and individually rational payoff can be approximated in a perfect Bayesian equilibrium if players are sufficiently patient. In particular, there are equilibria in which the receiver makes perfectly informed decisions in almost every period, even if no informative communication can be sustained in the stage game. We conclude that repeated interaction can overcome strategic limits of communication.

Keywords: Bayesian games, repeated games, communication, folk theorem.
JEL Codes: C72, C73, D82, D83.


## 1 Introduction

Crawford and Sobel (1982) introduced cheap-talk games as a basis for the analysis of strategic transmission of unverifiable information. They considered a one-shot game between a sender who has private information and a receiver who takes an action. In equilibrium, informative communication can be sustained, but misalignment of players' interests limits the amount of information that can be transmitted. In particular, if players' preferences are misaligned, truthful communication of private information cannot be sustained; otherwise, the sender

[^0]would bias reports to induce her preferred outcomes. The strategic considerations restrain communication and translate into inefficiency as both players could benefit from a betterinformed action.

The strategic limits of communication highlighted in a one-shot game provide insights into many economic situations. A buyer relies more on the recommendation of a friend than that of a sales representative; an antitrust legislator takes arguments of firms opposing a newly proposed regulation with a grain of salt; a voter feels skeptical about the promises of a politician running for office. Nonetheless, in many settings in which interests are seemingly misaligned, informative communication is sustained: companies raise funds from investors, government agencies successfully split the state budget, conglomerates allocate resources among different divisions, and so on. In many of these cases, the informed parties have strong biases so that their payoffs are determined solely by the resulting decisions. ${ }^{1}$ Importantly, all of these interactions happen repeatedly over time, with the parties trading off immediate opportunistic gains for the prospect of an ongoing relationship.

To investigate informative communication in these settings, we analyze a dynamic version of an information transmission game in which private information (states) evolve stochastically over time. We allow for many senders but focus on the case in which the senders' payoffs are state independent. In every period, each sender sends a message to the receiver, who then takes a publicly observable action. The "cheap-talk" nature of messages is preserved: no hard evidence can be presented, the sender cannot commit to a communication strategy, and the receiver never observes extraneous information to test the validity of the past messages. No contracts can be written between the players, so at any point in time it must be in players' interests to follow the equilibrium play.

We obtain an analog of a folk theorem-that any feasible and individually rational payoffs can be approximated in a perfect Bayesian equilibrium as the players become patient. ${ }^{2}$ This payoff set, and hence the set of equilibrium payoffs, admits a simple characterization and includes all Pareto efficient payoffs that satisfy the receiver's individual rationality. Specifically, it includes the receiver's largest feasible "complete information" payoff. In equilibrium, the fraction of periods in which the receiver makes perfectly informed decisions can be arbitrarily close to one, even if no informative communication can be sustained in the stage game.

These results contrast with the conventional wisdom that state independence makes it harder to maintain informative communication. Indeed, in this case, each sender has an unambiguous ranking over actions and is willing to report truthfully only if indifferent among the messages she sends. However, what comes as a curse in a one-shot game turns into a blessing when the game is dynamic. If senders' payoffs depend only on actions and not on states, then the receiver fully observes and controls the payoffs. Our equilibrium

[^1]construction actively uses this feature-it targets the senders' total payoffs and ensures they do not depend on the senders' messages. To achieve it, the equilibrium play alternates between communication and adjustment phases. In communication phases, the receiver makes informed decisions relying on the senders' messages. In adjustment phases, the receiver ignores all messages and plays according to a strategy that pulls the senders' payoffs towards the target. Specification of the phases and the transitions between them is tailored to guarantee that the senders' payoffs do not depend on their messages, and thus sustains truth-telling. By the law of large numbers, as players become more patient, the play occurs in the communication phase most of the time and any individually rational payoffs can be achieved by changing the strategy the receiver plays in there.

Related literature The general idea that an ongoing relationship can overcome strategic limits of interaction is the cornerstone of the literature on repeated games as discussed in depth by Mailath and Samuelson (2006). Further, the idea of using players' payoffs as a determinant of equilibrium construction is reminiscent of Abreu et al. (1990)'s recursive technique of equilibrium payoff decomposition. In fact, if individual states are independently and identically distributed, our dynamic game can be viewed as an infinitely repeated game and the Fudenberg et al. (1994)'s standard method can be used to provide an alternative proof of our folk theorem result. However, the standard method cannot be applied to general stochastic sender-receiver games.

At the same time, the idea of linking decisions motivated a strand of mechanism design literature. Jackson and Sonnenschein (2007) showed that Pareto efficient outcomes can be achieved by linking many identical copies of a collective choice problem with private values into a single mechanism. Frankel (2016) extended these ideas into a dynamic setting where the sender has persistent private information. He introduced discounted quota contracts similar to our equilibrium construction and showed their optimality in many environments. The mechanism design setting, however, differs from ours in that it endows the receiver with commitment power and monetary transfers.


Finally, the closest paper to ours is by Renault, Solan and Vieille (2013), who analyzed a dynamic information transmission game between a single sender and a receiver. They considered a more general payoff structure that, however, did not allow the sender's payoff to be state independent. In their setting, they showed that our analog of a folk theorem does not generally hold. In particular, the players' equilibrium payoffs do not necessarily approach the Pareto efficiency frontier as the players become patient.

Neither Renault et al. (2013)'s nor our proof can be directly extended to cover both the cases of state-dependent and of state-independent payoffs. On one hand, their construction is based on the idea of statistical tests that require the sender to match the message distribution with the state distribution. However, in our setting, because payoffs are state independent, statistical tests cannot be used effectively; faced with these tests, the sender would induce
favorable actions earlier in time independently of the realized states, rendering her messages uninformative. On the other hand, our construction is based on the ability of the receiver to adjust the senders' continuation payoffs exactly to the target without knowing the state. ${ }^{3}$ When the payoffs are state dependent, such adjustment must depend on the state and hence rely on the sender's messages running into the same incentive compatibility problem. ${ }^{4}$

The remainder of the paper proceeds as follows: Section 2 introduces the model, Section 3 characterizes the equilibrium payoffs and discusses, and Section 4 concludes. Appendix contains a detailed proof of the main theorem.


## 2 Model

A receiver (he) repeatedly communicates with $n$ senders (she) indexed by $i \in N=\{1, \ldots, n\}$, $n \geq 1$. The stage game is described by the set of states $\Omega \triangleq \times_{i \in N} \Omega_{i}$, their prior distribution $p \in \Delta(\Omega)$, the set of senders' messages $M \triangleq \times_{i \in N} M_{i}$, the set of actions of the receiver $A$, and the stage payoffs of the senders, $v: A \rightarrow \mathbb{R}^{n}$, and of the receiver $u: A \times \Omega \rightarrow \mathbb{R},^{5}$ Denote the stage game by $\Gamma \triangleq(N, \Omega, p, M, A, v, u)$. We assume that the sets $\Omega, M$, and $A$ are all finite. In the stage game, all senders first privately observe their individual states $\omega_{i} \in \Omega_{i}$ and then simultaneously send public messages $m_{i} \in M_{i}$ to the receiver who takes a publicly observable action $a \in A,{ }^{6}$

The stage game is infinitely repeated at times $t=1,2,3, \ldots$ with a common discount factor $\delta$. The state profiles $\omega_{t} \triangleq\left\{\omega_{t 1}, \ldots, \omega_{t n}\right\}$ evolve according to an irreducible and aperiodic Markov chain with a transition kernel $k\left(\omega_{t+1} \mid \omega_{t}\right)$. Hence, the individual states can be arbitrarily correlated across senders but their inter-temporal correlation vanishes as the distance between time periods grows large. Consequently, there exists a unique stationary distribution $p \in \Delta(\Omega)$ that has full support. We assume that the initial state is drawn according to this stationary distribution.

We assume that the validity of senders' messages can never be verified. First, no hard evidence is allowed-the sets of possible messages $M_{i}$ do not depend on the states. Second, the senders cannot commit to communication strategies, in contrast to the Bayesian per-

[^2]suasion literature. Third, the receiver does not observe any additional information besides the senders' messages; for example, he does not observe any additional signals about past states. ${ }^{7}$ We allow the players to perfectly transmit their private information, $M_{i}=\Omega_{i}$, thus concentrating on strategic rather than technological limits of communication. ${ }^{8}$ Denote the message profile by $m_{t} \in M \triangleq \times_{i \in N} M_{i}$. Lastly, we assume that there is a public randomization device that produces a uniformly distributed output $y_{t} \in Y \triangleq[0,1]$ at the beginning of each period independently of states, and all players publicly observe it. We do not require any public randomization at the interim stage, after the senders send their messages but before the receiver takes an action.

Overall timing within each period of the dynamic game is as follows: the public randomization device produces an output $y_{t}$; the state profile $\omega_{t}$ realizes and each sender $i$ privately observes her individual state $\omega_{i t}$; the senders simultaneously send messages $m_{t i}$ to the receiver; the messages are publicly observed and the receiver takes an action $a_{t}$; the action is publicly observed and the game proceeds to the next period. Denote the resulting dynamic game by $\Gamma^{\infty}(\delta)$.

Strategies The timing and monitoring structure of the game $\Gamma^{\infty}(\delta)$ outlined above result in the following definitions of histories and strategies. A public history $h^{t}$ at time $t$ consists of past actions, messages and the realizations of the randomization device. ${ }^{9}$ Denote the set of all public histories at time $t$ by $H^{t}$. A behavioral strategy of the receiver, $\alpha$, maps past public history, current output of the randomization device and current messages into an action,

$$
\begin{aligned}
h^{t} & \triangleq\left\{y_{s}, m_{s}, a_{s}\right\}_{s=1}^{t-1}, t>1, h^{1} \triangleq \emptyset \\
& \alpha:\left(\bigcup_{t \geq 1} H^{t}\right) \times Y \times M \rightarrow A
\end{aligned}
$$

Senders' private histories $h_{i}^{t}$ contain public histories as well as the individual states they observe. Denote the set of private histories of a sender $i$ by $H_{i}^{t}$. A behavioral strategy of a sender $i, \mu_{i}$, maps her private histories and current output of the randomization device into

[^3]a message
$$
\begin{gathered}
h_{i}^{t} \triangleq h^{t} \cup\left\{\omega_{s i}\right\}_{s=1}^{t} \\
\mu_{i}:\left(\bigcup_{t \geq 1} H_{i}^{t}\right) \times Y \rightarrow M_{i}
\end{gathered}
$$

Payoffs Players' interests are misaligned. The receiver's stage payoff is state dependent, $u: A \times \Omega \rightarrow \mathbb{R}$, so that he generally prefers to take different actions in different states. The senders' stage payoffs, in contrast, depend only on the receiver's actions and not on the states, $v_{i}: A \rightarrow \mathbb{R}$. That is each sender had an optimal action that she prefers to be taken in every state. All players discount the future at a common rate $\delta<1$. Given strategies $\mu \triangleq\left\{\mu_{i}\right\}_{i \in N}$ and $\alpha$, senders' and receiver's expected normalized payoffs (or simply payoffs) can be written as

$$
\begin{aligned}
& V_{i}(\mu, \alpha)=(1-\delta) \mathbb{E}_{\mu, \alpha}\left[\sum_{t=1}^{\infty} \delta^{t-1} v_{i}\left(a_{t}\right)\right], \quad i \in N, \\
& U(\mu, \alpha)=(1-\delta) \mathbb{E}_{\mu, \alpha}\left[\sum_{t=1}^{\infty} \delta^{t-1} u\left(a_{t}, \omega_{t}\right)\right],
\end{aligned}
$$

where the conditional expectations take into account the probability law induced by the strategies of the players, evolution of states, and the randomization device.

The set of feasible payoffs in the dynamic game $\Gamma^{\infty}(\delta), \mathscr{F} \subseteq \mathbb{R}^{n+1}$, with a typical element $(V, U)$ consists of all players' payoffs that can be achieved by some strategies $\mu, \alpha$. Because the first state is drawn from the stationary distribution, the states in all periods are ex ante identically distributed, and the set of feasible payoffs in the repeated game coincides with the set of feasible payoffs in the stage game. This set is a polytope with finitely many vertices and admits a simple characterization: it is equal to the convex hull of the payoffs in the stage game resulting from all pure mappings from states into actions. Moreover, any payoff $(U, V) \in \mathscr{F}$ can be supported by a stage-game strategy a: $\Omega \rightarrow \Delta(A)$ played in every period.


$$
\mathscr{F}=\operatorname{co}\left\{\left(\mathbb{E}_{p} v(\mathrm{a}(\omega)), \mathbb{E}_{p} u(\mathrm{a}(\omega), \omega)\right) \mid \mathrm{a}: \Omega \rightarrow A\right\} .
$$

The set of individually rational payoffs, $\mathscr{F}^{*} \subseteq \mathscr{F}$, in a dynamic game $\Gamma^{\infty}(\delta)$ consists of all feasible payoffs such that all players get at least their minmax payoffs. The senders' minmax payoffs are defined as $\min _{\alpha, \mu_{-i}} \max _{\mu_{i}} V_{i}(\alpha, \mu)$ and the receivers' minmax payoff is defined as $\min _{\mu} \max _{\alpha} U(\alpha, \mu)$. Our payoff structure allows for a particularly tractable characterization of $\mathscr{F}^{*}$. As the receiver can always ignore the senders' messages his minmax payoff is $\underline{U} \triangleq \max _{a \in A} \mathbb{E}_{p}[u(a, \omega)]$. At the same time, because the receiver fully controls the senders' payoffs, their individual rationality is innocuous. As a result,

$$
\mathscr{F}^{*}=\{(V, U) \in \mathscr{F} \mid U \geq \underline{U}\} .
$$

Our equilibrium concept is a perfect Bayesian equilibrium as described by Fudenberg and Tirole (1991). A strategy profile $(\mu, \alpha)$, together with a system of beliefs, is a perfect Bayesian equilibrium if $\mu$ is sequentially rational and Bayes' rule is used to update beliefs whenever possible. ${ }^{10}$ Denote by $\mathscr{E}(\delta) \subseteq \mathbb{R}^{n+1}$ the set of equilibrium payoffs.


## 3 Folk Theorem

Our main result is a characterization of the limit set of equilibrium payoffs as players become arbitrarily patient, $\lim _{\delta \rightarrow 1} \mathscr{E}(\delta)$. We obtain an analog of a folk theorem-any feasible, individually rational payoff can be approximated in an equilibrium if players are sufficiently patient.

A few observations are immediate. First, no payoff outside of $\mathscr{F}^{*}$ can be supported in equilibrium-the receiver can always guarantee at least his minmax payoff. Second, there are fully uninformative equilibria in the dynamic game $\Gamma^{\infty}$ in which all senders babble; that is, they send messages independently of their states, and the receiver plays a myopic best-response. The set of babbling payoffs belongs to the set of equilibrium payoffs for all $\delta$.

Assumption 1. (Valuable Communication) There exists a vector $(V, U) \in \mathscr{F}^{*}$ such that $U>\underline{U}$.

Assumption 1 states that communication is valuable-the receiver can strictly benefit from knowing the individual states. The assumption holds in most relevant economic environments. It is weaker than non-empty interior requirements in existing folk theorems, as it applies to the receiver only. This assumption allows to provide incentives for the receiver to follow the expected equilibrium play by threatening to cease the communication and permanently switch to babbling.

Theorem 1. (Folk Theorem) For any payoffs $(V, U) \in \operatorname{relint}\left(\mathscr{F}^{*}\right)$ there exists $\underline{\delta}<1$ such that for all $\delta>\underline{\delta}$ there is an equilibrium with payoffs $(V, U) .{ }^{11}$ Consequently,

$$
\lim _{\delta \rightarrow 1} \mathscr{E}(\delta)=\mathscr{F}^{*} .
$$

The detailed proof is relegated to the Appendix. Here we briefly outline its main ideas. For any target payoff in $\mathscr{F}^{*}$ we pick the receiver's strategy a : $\Omega \rightarrow \Delta(A)$ that supports it in the stage game and construct an equilibrium in which senders report truthfully and the receiver plays according to a most of the time.

[^4]In particular, the equilibrium play switches between communication and adjustment phases. Transition between phases is determined by the senders' accumulated payoffs, which depend only on past actions and thus are publicly observed. In the communication phase, each sender reports her individual state truthfully, and the receiver plays according to the stage-game strategy a. In the adjustment phase, the receiver plays according to a strategy that brings all senders' discounted payoffs back to the target. The length and strategy of the adjustment phase depend on the senders' payoffs at the end of preceding communication phase and can always be chosen to hit the target as long as the players are sufficiently patient. Any receiver's randomization is done via the public randomization device so his deviations are immediately observable and trigger permanent babbling play.

The exact specification of the phases and their transition is tailored to ensure the players' obedience in following the equilibrium play. The senders are willing to report truthfully because they are guaranteed to obtain the target payoffs irrespectively of their messages. The receiver is effectively deferred from deviating by the assumption of valuable communication if sufficiently patient.

As players become more patient, the length of the communication phase can be increased and we can appeal to the law of large numbers for Markov chains. Hence, on average, there is less adjustment to be made, so the play occurs in the communication phase most of the time. As a result, the equilibrium payoffs approach the target payoffs.

Example 1. (Resource Allocation) We illustrate the setting and the results in a resource allocation example. The receiver is a social planner who decides every period how to allocate an indivisible resource between two ex-ante symmetric regions. The senders are local representatives who privately observe the social values of allocating the resource to their region. They report the values to the planner at a regular meeting. The planner wants to put the resource to the best use and the representatives simply prefer having resource in their region. The social values are independent across regions and positively but imperfectly correlated across periods.

The example fits into our model by setting $n=2, A=\{1,2\}$; for $i \in\{1,2\}, \Omega_{i}=$ $\left\{\omega^{0}, \omega^{1}\right\}, v_{i}(a)=\mathbf{1}(a=i), k\left(\omega_{i}, \omega_{j} \mid \omega_{i}, \omega_{j}\right)=\rho^{2}$, and $k\left(\omega_{i}, \omega_{j}^{\prime} \mid \omega_{i}, \omega_{j}\right)=\rho(1-\rho)$, for $\omega_{j}^{\prime} \neq$ $\omega_{j}$. Normalizing $\omega_{i}^{0}=0$ and $\omega_{i}^{1}=1, u(a, \omega)=\omega_{a}$. Positive but imperfect correlation implies $\rho \in(1 / 2,1)$ and corresponds to a uniform stationary distribution. The set of feasible and individually rational payoffs can be calculated by (1) and (2), and is presented in Figure 1. The set does not depend on the degree of correlation $\rho$ and has an empty interior since the feasibility of allocation requires $V_{1}+V_{2}=1$.

The assumption of valuable communication is satisfied. In the absence of additional information the planner is indifferent between allocating the resource to one region or another and obtains a payoff $\underline{U}=1 / 2$. However, the resource allocation matters to representatives. The "babbling" payoffs are generically Pareto inefficient and constitute a set

$$
\mathscr{B}=\left\{(V, U) \mid V_{1}+V_{2}=1, U=1 / 2\right\} .
$$

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 1: The set of feasible payoffs, $\mathscr{F}$, with a typical element $\left(V_{1}, V_{2}, U\right)$. Feasibility implies that the set $\mathscr{F}$ lies on the plane $V_{1}+V_{2}=1$. The shaded area indicates individually rational payoffs $\mathscr{F}^{*}$.

If the stage game were played only once, then strategic considerations would restrain communication and the babbling payoffs would be a unique equilibrium payoffs. The argument is standard. If the planner were to rely on a representative's report then the representative would communicate the report that maximizes a probability of getting the resource irrespectively of its social value. This would in turn make the report uninformative and preclude payoffs outside of $\mathscr{B}$.


However, in a dynamic game any payoffs in $\mathscr{F}^{*}$ can be approximated in an equilibrium if the players are sufficiently patient as shown in Theorem 1. According to our equilibrium construction the planner should simply ensure the representatives of the total discounted allocation to their regions irrespectively of their reports. It can be achieved by infrequently shutting down the meetings and bringing the total discounted allocation back to the targets. In this way, the resource is allocated efficiently in a fraction of periods arbitrarily close to one. $\square$

To recapitulate, in the absence of any contract enforcement or message verification, almost fully informed decision-making can be sustained in a dynamic game even if no informative communication can be sustained in the stage game. Moreover, the result does not require the payoff set to have a non-empty interior in contrast to most of existing folk theorems.


## 4 Discussion

In this section we discuss two important features of our model that are necessary for our equilibrium construction: ergodicity of state transitions and state-independence of senders' payoffs.

Ergodicity It is crucial for our equilibrium construction that the Markov chain according to which states evolve is ergodic, that is irreducible and aperiodic. Both of these properties are important and ensure that even though the states can be correlated across periods, this correlation vanishes as the distance between them grows. It gives the game its recurrent structure and allows to split the equilibrium play in "almost independent" blocks. The following example shows that the folk theorem result can fail if the chain is not irreducible.

Example 2. (Reducible Chain) Consider a game with a single sender, two states $\Omega=$ $\left\{\omega^{0}, \omega^{1}\right\}$, and two actions $A=\{0,1\}$. The payoff functions are $v(a)=a$, and $u(a, \omega)=$ $\mathbf{1}\left(\omega=\omega^{a}\right)$ : the sender always prefers action 1 whereas the receiver wants to match the state. In contrast to the previous analysis, the state is perfectly persistent, that is $k(\omega \mid \omega)=1$. The initial state $\omega_{1}$ is drawn according to the distribution ( $1-p_{0}, p_{0}$ ), with $p_{0}<1 / 2$.

We argue that the babbling payoff is the unique equilibrium payoff of the dynamic game for any $\delta<1$. In fact, we show that it is the unique equilibrium payoff even if the receiver could commit to his strategy. If the receiver can commit to his strategy, by the revelation principle of Myerson (1986) any equilibrium payoff can be implemented by a direct mechanism. In the mechanism, the sender truthfully announces the initial state and the receiver follows a pre-committed dynamic strategy $a: \Omega \rightarrow A^{\infty}$. For a given (incentive compatible) mechanism, the players' expected payoff are

$$
\begin{aligned}
& V=\left(1-p_{0}\right)(1-\delta) \sum_{t=0}^{\infty} \delta^{t} \operatorname{Pr}\left[a_{t}\left(\omega^{0}\right)=1\right]+p_{0}(1-\delta) \sum_{t=0}^{\infty} \delta^{t} \operatorname{Pr}\left[a_{t}\left(\omega^{1}\right)=1\right], \\
& U=\left(1-p_{0}\right)(1-\delta) \sum_{t=0}^{\infty} \delta^{t} \operatorname{Pr}\left[a_{t}\left(\omega^{0}\right)=0\right]+p_{0}(1-\delta) \sum_{t=0}^{\infty} \delta^{t} \operatorname{Pr}\left[a_{t}\left(\omega^{1}\right)=1\right]
\end{aligned}
$$

Since the sender's payoffs are state-independent, incentive compatibility implies that she must be indifferent between reporting states $\omega_{0}$ and $\omega_{1}$. Substituting into to the formula for the receiver's payoff we obtain that the mechanism is incentive compatible only if

$$
U=1-p_{0}-V\left(1-2 p_{0}\right) .
$$

The unique feasible individually rational payoff that satisfies this equation is the babbling payoff $\left(0,1-p_{0}\right)$. It follows the babbling payoff is also the unique equilibrium payoff of the dynamic game in which the receiver cannot commit. $\square$

This example illustrates that when the state chain is aperiodic but reducible, the sender's incentive compatibility and the receiver's individual rationality alone can preclude equilibrium communication. We conjecture that when the chain is irreducible but periodic, the folk theorem result still fails with the receiver's incentives playing an important role. For this reason, constructing a specific counterexample in that case seems to be more complicated.

State-independent payoffs Another important assumption of our model is that the senders' payoffs do not depend on the state. One might think that our equilibrium construction still work if the payoffs are "almost" state-independent. Unfortunately, this is not the case. In our equilibrium, the senders are indifferent between any history of messages and report truthfully in equilibrium. If the payoffs are even slightly state-dependent then the senders' best-response can be far from truthtelling.

In fact, for a generic payoff perturbation one can apply the results of Renault et al. (2013) to characterize the limit set of equilibrium payoffs, at least for the case of a single sender and Markov chain satisfying their Assumption A. They show that the equilibrium payoffs must satisfy senders' "incentive compatibility"-she shouldn't be able to benefit from permuting her reports. This constraint is ordinal and even small payoff perturbations can drastically restrict the equilibrium payoff set.


## 5 Conclusion

We analyzed dynamic information transmission when senders' payoffs are state independent. We show that the strategic limits of communication prevalent in a stage-game disappear when the interaction is repeated. Any individually rational payoffs can be approximated in equilibrium if the players are sufficiently patient. This result complements the existing results for state-dependent payoffs and provides a rationale for informative communication to be sustained in a variety of dynamic economic settings between players with seemingly misaligned interests. In fact, our equilibrium construction delivers a clear message how to do so-to induce truthtelling the receiver should track the senders' payoff and adjust it whenever they are doing too good or too bad. This ensures the senders of their payoffs irrespective of their reports and eliminates incentives to lie.

There are many opportunities for further research. First, the limit analysis for statedependent payoffs not captured by the previous literature should be completed. Second, one can analyze the joint limit of state independency and patience. Finally, an important open question is the general analysis of equilibrium payoffs when players are impatient. We suspect that in this case finer details of Markov transition and not just its stationary distribution will come into play.


## Appendix


## Proof of Theorem 1

Without loss of generality, we normalize players' payoffs so that $\min _{A} v_{i}(a)=0, \forall i \in N$, and $\min _{(a, \omega) \in A \times \Omega} u(a, \omega)=0$. Let $\bar{v}_{i} \triangleq \max _{a \in A} v_{i}(a)$, and $\bar{u} \triangleq \max _{(a, \omega) \in A \times \Omega} u(a, \omega)$. Recall that

$$
\mathscr{F}^{*}=\{(V, U) \in \mathscr{F} \mid U \geq \underline{U}\} .
$$

where $\underline{U}=\max _{a \in A} \mathbb{E}_{p}[u(a, \omega)]$. Given any feasible and individually rational payoffs, we construct an equilibrium that achieves payoffs arbitrarily close to it provided that $\delta$ is sufficiently high. In particular, define

$$
\mathscr{F}^{* *} \triangleq\left\{(V, U) \in \operatorname{relint}\left(\mathscr{F}^{*}\right)\right\} .
$$

and let $|\cdot|$ denote the Euclidean norm. For any $(V, U) \in \mathscr{F}^{* *}$ there exists $\eta>0$ such that $\forall V^{\prime}:\left|V^{\prime}-V\right| \leq \eta, V^{\prime} \in \mathscr{F}^{* *}$. We will show that for any $\varepsilon>0$ and any $(V, U) \in \mathscr{F}^{* *}$, there exists $\underline{\delta}<1$ such that for all $\delta>\underline{\delta}$ we can construct an equilibrium $\left(\mu^{*}, \alpha^{*}\right)$ of $\Gamma^{\infty}(\delta)$ with payoffs $\left(V^{*}, U^{*}\right)$, where $V^{*}=V$ and $\left|U^{*}-U\right|<\varepsilon$. Since we allow for a public randomization device, the result of the theorem will follow.

In what follows, fix $\delta<1, \eta>0$, the target payoff vector $(V, U) \in \mathscr{F}^{* *}$, and the corresponding supporting strategy a: $\Omega \rightarrow \Delta(A)$.

Strategies We first describe the equilibrium strategies ( $\mu^{*}, \alpha^{*}$ ). Fix some $T_{c} \in \mathbb{N}$ as a function of $\delta$ such that $\lim _{\delta \rightarrow 1} \delta^{T_{c}(\delta)}=1, \lim _{\delta \rightarrow 1} T_{c}(\delta)=\infty .{ }^{12}$ The equilibrium play is divided into consecutive blocks, each starting with a communication phase of length $T_{c}$ followed by an adjustment phase of (random) length $T_{a}$.

On the equilibrium path, the behavior within each block is described by strategies $\left(\mu_{1}, \alpha_{1}\right)$. According to $\mu_{1}$, the senders always report truthfully. According to $\alpha_{1}$, the receiver's behavior depends on the current phase within a block. In the communication phase, he plays according to the strategy a. In the adjustment phase, he ignores the senders' reports and plays according to an adjustment strategy that depends on the profile of senders' normalized discounted payoffs at the end of the communication phase,

$$
\bar{v}_{c} \triangleq \frac{1-\delta}{1-\delta^{T_{c}}} \sum_{t=1}^{T_{c}} \delta^{t-1} v\left(a_{t}\right),
$$

and the target payoff vector $V$. In particular, define

$$
\lambda\left(\bar{v}_{c}\right) \triangleq \frac{\bar{v}_{c}-V}{\left|\bar{v}_{c}-V\right|}, \quad V_{a}\left(\bar{v}_{c}\right) \triangleq V-\lambda\left(\bar{v}_{c}\right) \eta .
$$

[^5]

> [Figure omitted from this text-only corpus; refer to the source manuscript.]
Figure 2: The choice of adjustment payoffs $V_{a}$ given the realized senders' payoff at the end of the communication phase $\bar{v}_{c}$ and the target equilibrium payoffs $V$. Schematic illustration for the case of two senders, $n=2$.

Let $\hat{T}_{a} \in \mathbb{R}_{+}$satisfy

$$
\delta^{T_{c}}\left(1-\delta^{\hat{T}_{a}}\right) \eta=\left(1-\delta^{T_{c}}\right)\left|\bar{v}_{c}-V\right| .
$$

Notice that $\hat{T}_{a}$ is well defined as long as $\eta>\left(1-\delta^{T_{c}}\right) \delta^{-T_{c}} \sum_{n=1}^{N} \bar{v}_{i}$, which, in light of our choice of $\delta$ and $T_{c}(\delta)$, is verified for $\delta$ high enough. Define $T_{a} \triangleq\left\lfloor\hat{T}_{a}\right\rfloor+1$, and let $0 \leq r\left(\bar{v}_{c}\right)<\eta$ solve

$$
\frac{1-\delta}{1-\delta^{T_{c}+T_{a}}}\left(\sum_{t=1}^{T_{c}} \delta^{t-1} v\left(a_{t}\right)+\sum_{t=T_{c}+1}^{T_{c}+T_{a}-1} \delta^{t-1} V_{a}\left(\bar{v}_{c}\right)+\delta^{T_{c}+T_{a}-1}\left(V-\lambda\left(\bar{v}_{c}\right) r\left(\bar{v}_{c}\right)\right)\right)=V,
$$

The adjustment phase lasts $T_{a}$ periods. In the first $T_{a}-1$ periods, the possibly random action $\mathrm{a}_{a} \in \Delta A$ is played with $V_{a}\left(\bar{v}_{c}\right)=\mathbb{E} v\left(\mathrm{a}_{a}\right)$. In the last period, the possibly random action $\mathrm{a}_{r} \in \Delta A$ is played with $V-\lambda\left(\bar{v}_{c}\right) r\left(\bar{v}_{c}\right)=\mathbb{E} v\left(\mathrm{a}_{r}\right)$.

If the receiver ever deviated, then $\left(\mu^{*}, \alpha^{*}\right)$ prescribes the senders to babble; that is, to send messages irrespectively of their individual states, and the receiver to play an action that is optimal given no information.

As for the deviation of each sender, because individual states are correlated, some deviations during the communication phase are detectable as they lead to an inconsistent sequence
of messages. During the communication phase, whenever the current messages $m_{t}$ are inconsistent with the messages from the previous period $m_{t-1}$ the receiver plays a after replacing the reports with an artificially generated state $\hat{\omega} \in \Omega$ consistent with the previous messages and the underlying Markov chain. The strategy $\mu^{*}$ prescribes each sender to keep playing according to $\mu_{1}$ after any detected deviation of another sender, and if she ever privately deviated.

Finally, transitions between and within blocks are as follows. Each block starts with a communication phase that lasts for $T_{c}$ periods and is followed by the adjustment phase, which lasts for $T_{a}$ periods according to (3). At period $T_{a}+1$, a new block starts and the timer is reset to $t=1$.

Payoffs By construction, the equilibrium strategies deliver expected continuation payoffs $V$ to the senders at the beginning of each block and, in particular, at the beginning of the game. We show that for sufficiently high $\delta$, the equilibrium strategies deliver the expected continuation payoff within $\varepsilon$ of the target payoff $U$ to the receiver after any public history.

First, we show that the expected continuation payoff of the receiver is within $\varepsilon$ of the target payoff $U$ at the beginning of each block. Consider the average realized payoffs within each block at the end of communication and adjustments phases: