# Inference under Selectively Disclosed Data

Ying Gao I thank Drew Fudenberg and Stephen Morris for their generous advice
and guidance, and Ian Ball, In-Koo Cho, and Wioletta Dzuida for helpful
comments.

###### Abstract

We consider the disclosure problem of a sender who wants to use hard evidence
to persuade a receiver towards higher actions. When the receiver hopes to make
inferences based on the distribution of the data, the sender has an incentive
to drop observations to mimic the distributions observed under better states.
We use a continuum model to predict optimal play when datasets are large and
find that, in the big-data limit, it is optimal for senders to play an
imitation strategy, under which they submit evidence imitating the natural
distribution under some desirable target state. Relative to full information,
the partial pooling equilibrium with voluntary disclosure generates a loss in
welfare to senders who either have little data or a favorable state, who
optimally disclose their full dataset, but a benefit to senders with access to
large datasets, who can profitably drop observations under low states.

##  1 Introduction

Many decision processes – including regulatory approval, policy choices, and
financial decisions – rely on self-disclosed data. The datasets used tend to
be very large, on the order of tens of thousands of trials for drug approval,
and often hundreds of thousands of datapoints about locations and sales in
merger cases. In and of themselves, big datasets may paint an accurate picture
of reality, but the sender can disclose them strategically: it is easier to
verify that the data submitted are real than that they are complete, and even
in the presence of mandatory disclosure rules, deciding which observations are
admissible to include in the dataset is largely at the sender’s discretion.

We want to understand the role data play in strategic communication between
the sender and the decision-maker when receivers have uncertainty about the
underlying dataset from which the sender extracted the submitted data. We
consider the case of a sender with state-independent motives to persuade the
receiver towards a particular action, and a receiver who observes a large
dataset, but interprets it with partial skepticism that the data are worse
than they appear. Equilibrium play between the sender and receiver involves
the sender submitting data as “proof” that the receiver should take a
favorable action, and the receiver evaluating how persuasive the proof is
depending on how likely it is sent by a sender with less persuasive data who
has trimmed some discouraging observations. This can be modeled under the
framework of an evidence game in which senders that have access to datasets
weakly more observations of each outcome can always mimic senders with fewer
observations. A special case, in which senders either have do or do not have
access to a single data point, with probability known to the receiver, is
already well-understood (Dye 1985), and demonstrates that senders can
manipulate the receiver by disclosing nothing when the evidence is
sufficiently poor.

Our innovation is to characterize outcomes of disclosure in the opposite
extreme, when datasets contain many observations. We leverage a continuum
limit model to describe the distribution over potential datasets of the sender
that depend on two things: the true state of the world that generates the
data, and a random variable that describes the amount of data the sender
collects. The continuum assumption captures the fact that empirical
distributions are approximately determistic in the limit with large numbers,
and allows us to eliminate uncertainty over the randomness of draws, therefore
simplifying the analysis of strategies and outcomes through the limit rather
than by directly considering large, finite $N$. We show that the outcome of
communication in finite games approaches the outcome of the limit game as $N$
grows.

The first main observation is that, in an “imitation” equilibrium of the limit
game that yields the receiver-optimal outcome, senders always show the
receiver a dataset that can correspond to a naturally-generated dataset, so on
path the receiver always places positive probability on the event that the
sender is sending all their data. However, the receiver also infers from some
datasets that the sender has with positive probability observed data
corresponding to a different state than the revealed data suggest, but has
dropped observations in order to imitate a more favorable distribution. The
resulting outcome benefits senders under low states with more data at the
expense of senders with less data in better states, since the former pool with
the latter. The extent of pooling depends on the receiver’s uncertainty about
the sender’s data collection capabilities: the greater the variance in the
receiver’s belief about how much data the sender starts out with, the more
senders can profitably imitate other senders, with outcomes converging to the
full-information one as uncertainty vanishes.

Strikingly, the state-contingent experimental outcome distribution affects the
information transmitted only through a handful of key features: what matters
are the observations of outcomes with the greatest likelihood ratio under a
better vs. a worse state. How the distribution of data distinguishes one state
from another depends solely on the relative probabilities of these outcomes,
and a receiver who wants to distinguish a relatively small number of states
with many observations of high-dimensional data can do just as well
restricting the dataset to only retain information about these outcomes.

The other contribution is an algorithm to construct the limit game equilibrium
outcomes that follows a top-down logic: senders with more data receive weakly
greater payoffs, and we can construct the payoff frontiers of the continuous
payoff function by specifying the burden of proof, or how much data of a given
state’s distribution a sender needs, to induce a particular belief in the
receiver. The algorithm is fully generalizable to any number of states and for
distributions with finite, discrete support, and we use it to construct the
equilibrium in examples with 2 and 3 states.

###  1.1 Related literature

Strategic disclosure has been studied since the work of Akerlof (1970),
Grossman (1981) and Milgrom (1981), which showed that full disclosure is the
unique outcome when receivers know the sender wishes to prove the value of a
good is high using verifiable information that they could choose to disclose.
The assumption that receivers know the sender is informed is crucial to this
benchmark, as Dye (1985) and Jung and Kwon (1988) show. They consider a case
in which the sender has access to a single, real-valued piece of evidence with
interior probability $p\in(0,1)$, and shows that only senders for whom the
evidence exceeds a threshold will choose to disclose it, with the rest
withholding it in order to pool with those senders who lack evidence
altogether. Shin (1994, 2003) shows that in the case where senders have an
uncertain endowment of good news and bad news, the fact that senders withhold
bad enough evidence implies a “sanitation equilibrium”, in which all bad news
is disposed of.

We extend these results by considering evidence structures with large,
multidimensonal datasets. In our data-based setting, evidence is neither
exogenously good nor bad, but the receiver draws inferences statistically,
based on knowledge of the relationship between relevant state-parameters and
the distributions of data they generate. The setting we consider both
encompasses the settings above and is a special case of more general evidence
games, of the type considered by Green and Laffont (1986) and Hart et al.
(2017). The main focus in those settings has been on receiver-optimal
mechanisms to induce beneficial disclosures from the sender. The work of Hart
et al. in particular is foundational to this paper, and observes that the
outcome of the optimal mechanism is equivalent to the receiver-optimal
equilibrium outcome when assuming that the receiver’s expected payoffs are
single-peaked, and furthermore that both are equivalent to the unique _truth-
leaning_ equilibrium outcome that captures the sender’s slight preference to
reveal the whole truth, and coincidentally also rules out adverse on- and off-
path beliefs.111The optimal mechanism equivalence result has been noted by
others, in particular Glaezer and Rubinstein (2006), Sher (2011), and Ben-
Porath et al. (2019), show that the fact that commitment is not necessary for
the optimum is robust to other settings, in particular with binary actions and
multiple senders with type-dependent preferences.

Rappoport (2022) and Jiang (2022), like us, use an iterative algorithm to
solve for truth-leaning equilibrium outcomes, but their focus is on games in
which senders have finite amounts of evidence (although the distribution may
be unbounded). Our setting allows for a continuous-data approximation to solve
for outcomes that are difficult to iteratively compute using the existing
algorithm; without explicitly solving for the outcomes of games with large,
finite data spaces, we show, as a rule, that the outcomes from the continuous-
data model describe the big-data limits of outcomes of games with finite data.
Dzuida (2011) is the only other paper we know of that uses a continuous
measure of evidence to solve for communication with verifiable evidence. It
extends the disclosure setting of Shin with positive and negative evidence to
the case in which the sender has a positive probability of being a behavioral,
honest type, and also considers a scenario in which senders may be trying to
persuade the receiver in opposite directions, i.e. either up or down, without
the receiver’s knowledge. In all cases with honest types, it shows that
sending a nonzero amount of negative evidence is part of an otherwise
sanitation-like equilibrium. Our results give a different reason why senders
might send more than observations of one “good” outcome in order to persuade
the receiver: in our setting, there may be no single “good” outcome, since the
meaning of each datapoint is determined endogenously by inferences based on
the state-contingent data-generating distributions. We show that it is without
loss for the sender, in equilibrium, to send an intermediate measure of
observations of many outcomes, and in fact sending interior amounts of some
outcomes may be optimal, while sending either none or as much as possible of
an outcome might both be worse.

We also relate to a broader literature about the optimal collection and
disclosure of evidence, that considers costly (Migrow and Severinov (2022))
and dynamic evidence acquisition (Felgenhauer and Schulte 2014, Henry and
Ottaviani 2019), sender-optimal disclosure mechanisms (Haghtalab et al. 2022),
and discretionary disclosure after test or information design (Shiskin 2022,
Dasgupta et al. 2022). Several papers use a restricted notion of evidence but
are also explicitly concerned with the effect of allowing sample selection:
Fishman and Haggerty (1990) and Di Tillio et al. (2021) study the case in
which only a subset of observations are disclosed, and give conditions under
which it is better that an informant have discretion over which data are
selected. Finally, work in econometrics by Simonsohn et al. (2014), Andrews
and Kasy (2019) and others studies the bias that arises from exogenously
selective reporting, and describes inference procedures that correct for it.

##  2 Model

States and payoffs. There is a sender ($S$), who wishes to communicate to a
receiver ($R$) about an unknown state of the world,
$\theta\in\Theta=\\{\theta_{1},\ldots,\theta_{J}\\}$. The sender and receiver
share a common prior $\beta_{0}(\cdot)$ over $\Theta$. We assume that the
receiver takes an action $a_{r}\in\mathbb{R}$ and that
$\theta_{1},\ldots,\theta_{J}$ are real numbers ordered with
$\theta_{j}\leq\theta_{j+1}$, representing the optimal action for the receiver
under each state, if it was known with certainty. The sender’s payoff is
simply (a monotone function of) $a_{r}$;222Because the receiver will always
play a pure strategy, the sender’s problem is unchanged if their payoffs are
rescaled through a monotone mapping. in short, regardless of their type,they
want to induce the receiver to take the highest action possible.

Finally, we assume the receiver takes the action that matches their
expectation of the value of $\theta$ given their belief $\beta$, that is
$a_{r}(\beta)=\mathbb{E}_{\beta}[\theta]$. The sender’s indirect utility
function is therefore

$u_{s}(\beta)=\mathbb{E}_{\beta}[\theta].$

For example, when the receiver is a policymaker, states can represent the true
optimal policy. While the policymaker might be uncertain, they wish to enact a
policy that matches the optimal policy in expectation, while the sender wishes
for them to take as high an action as possible.

###  2.1 Datasets

Evidence. The private information of the sender comes in the form of hard
evidence about the state of the world. In particular, the sender has access to
a dataset of observations drawn from a finite set of outcomes,
$\mathcal{D}=\\{1,\ldots,D\\}$. The underlying data-generating distribution is
state-contingent: under state $\theta_{j}$, the observations are i.i.d. draws
from distribution $f_{j}$.

Importantly, there is ex-ante uncertainty about how much data the sender has:
the number of datapoints is upper-bounded by $N$, but the sender may have
access to $n<N$ observations as well. Nature’s sequence of moves in drawing
the sender’s dataset is: 1) draw the state, $\theta$, according to prior
$\beta_{0}$; 2) draw the number of observations, $n$, from distribution
$G_{N}$; 3) for each of the $n$ datapoints, draw their realized value i.i.d.
from $f_{\theta}$. Call the disclosure game with these parameters
$\mathcal{G}_{N}(\Theta,D,\beta_{0},\\{f_{j}\\}_{j=1}^{J},G_{N})$. The data
mass distribution is $G_{N}(n)$ supported on some subset of $\\{0,\ldots,N\\}$
that includes $N$ itself. It captures the receiver’s uncertainty about how
much raw evidence the sender has, prior to selecting observations to reveal:
for example, the number of total trials in an experiment, or the number of
trials out of $N$ attempted that completed the whole trial period.

The sender’s dataset is the empirical probability mass function
$t=\frac{1}{N}(t_{1},\ldots,t_{D})$, where $t_{d}$ is the number of
observations of outcome $d$ and $n(t)=\sum_{d=1}^{D}t_{d}$ is the number of
observations they get. $t$ is also their type, and the space of all possible
datasets, $\mathcal{T}_{N}$, is the type space, with type distribution

$q(t)=\frac{n(t)!}{\Pi_{d=1}^{D}t_{d}!}g_{N}(n(t))\sum_{j^{\prime}}\beta_{0}(\theta_{j^{\prime}})\Pi_{d=1}^{D}f_{j^{\prime}}(d)^{t_{d}}.$

Messaging and inference. Senders can choose a subset of observations from
their dataset to submit to the receiver. We assume total flexibility in the
choice of subset:

###### Assumption 2.1

The sender can send any message $m=\frac{1}{N}(m_{1},\ldots,m_{D})$ that is a
_subset_ of their dataset ($m\subseteq t$), where

$m\subseteq t\ \Leftrightarrow\ m_{d}\leq t_{d}\ \forall d\in\mathcal{D}.$

Figure 1: A feasible type and a feasible message.

That is, a sender can drop an arbitrary set of observations from their data,
and then show the remaining ones to the receiver. The message space
$\mathcal{M}_{N}$ includes all sets of at most $N$ draws from $D$, and
$\mathcal{T}_{N}\subseteq\mathcal{M}_{N}$.

Upon observing the sender’s message, the receiver updates their belief about
the sender’s type to $q(t|m)$, and then forms a new belief about the state,

$\beta(\theta_{j}|m)=\frac{\sum_{t\in\mathcal{T}_{N}}q(t|m)\pi(\theta_{j}|t)}{\sum_{t\in\mathcal{T}_{N}}q(t|m)},$
(1)

where

$\pi(\theta_{j}|t)=\frac{\beta_{0}(\theta_{j})\Pi_{d=1}^{D}f_{j}(d)^{t_{d}}}{\sum_{j^{\prime}}\beta_{0}(\theta_{j^{\prime}})\Pi_{d=1}^{D}f_{j^{\prime}}(d)^{t_{d}}}$

is the likelihood of $\theta_{j}$ given that the raw dataset is $t$ –
equivalently, $\pi(\cdot|t)$ is the type-$t$ sender’s belief about the state.

###  2.2 Equilibrium

We first impose that play will be in Perfect Bayesian Equilibrium (Fudenberg
and Tirole 1970), so $\beta^{*}(\cdot|m)$ must be consistent with the sender’s
messaging strategy
$\sigma^{*}:\mathcal{T}_{N}\rightarrow\Delta\mathcal{M}_{N}$, and the sender
must optimize:

$\sigma^{*}(m|t)>0\ \Rightarrow\ m\in\arg\max_{m^{\prime}\subseteq
t}\mathbb{E}_{\beta(\cdot|m^{\prime})}[\theta].$

Call the map from types to payoffs, $u^{*}(t)$, the _outcome_ of the
equilibrium.333This is a departure from the usual definition of an outcome of
an extensive-form game, but consistent with the definition in Hart et al.
(2017) and Rappoport (2022). It describes the action the receiver plays after
communicating with each type, and therefore is sufficient to describe the
consequences of the game. In the perfectly separating outcome, each sender
obtains a payoff $\mathbb{E}_{\pi(\cdot|t)}[\theta]$, and this is what occurs
when $g_{N}$ is a degenerate distribution with only one positive-probability
mass. When $g_{N}$ is nondegenerate, there is (partial) pooling in every PBE.

In PBE there is off-path indeterminacy in the receiver’s beliefs, so there may
be multiple $\beta^{*}$, differing on off-path messages, that are consistent
with $\sigma^{*}$. In addition, the game generically has multiple, non-payoff-
equivalent PBE outcomes, some supported by adverse off-path beliefs. In some
PBE, play is _truth-leaning_ (Hart et al. 2017). Formally, given a base game
$\mathcal{G}_{N}$, for
$\epsilon=(\epsilon_{t},\epsilon_{t|t})_{t\in\mathcal{T}}$, let a game
$\mathcal{G}_{\epsilon}$ be the game with an identical type set and type
distribution, but with two differences. First, type $t$’s payoffs to playing
$t$ are perturbed by $\epsilon_{t}$, so that $t$’s payoff to playing $t$ is
$\mathbb{E}_{\beta(\cdot|t)}[\theta]+\epsilon_{t}$. Secondly, type $t$ plays
$t$ with at least probability $\epsilon_{(t|t)}$ – i.e. with probability
$\epsilon_{t|t}$ a sender with dataset $t$ is a commitment type that plays
their full dataset regardless of whether doing so is optimal, while with
probability $1-\epsilon_{t|t}$ type $t$ is strategic. A truth-leaning
equilibrium is an equilibrium of the base game that can be obtained as a limit
of equilibria of $\epsilon-$perturbed games as $\epsilon\rightarrow 0$.

###### Definition 2.1

An equilibrium messaging strategy $\sigma^{*}$ is _truth-leaning_ for game
$\mathcal{G}$ if it is the limit of a sequence of PBE messaging strategies
$(\sigma^{k})_{k=1}^{\infty}$ in games
$(\mathcal{G}_{\epsilon^{k}})_{k=1}^{\infty}$ with
$\epsilon_{t}^{k}\rightarrow 0$, $\epsilon_{t|t}^{k}\rightarrow k$.

Focusing on truth-leaning equilibria lets us specify a particular form for
beliefs consistent with strategy profile $\sigma^{*}$,

$q_{\sigma^{*}}(t|m):=\begin{cases}\frac{q(t)\sigma^{*}(m|t)}{\sum_{t\in\mathcal{T}_{N}}q(t)\sigma^{*}(m|t)}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ for on-path }m,\\\
\mathbbm{1}_{m}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \text{ for off-path }m\in\mathcal{T}_{N}\\\
q_{\sigma^{*}}(t|\underset{t^{\prime}\supseteq m,\
t^{\prime}\in\mathcal{T}_{N}}{\arg\min}\
\mathbb{E}_{\beta(\cdot|\sigma^{*}(t^{\prime}))}[\theta])\ \ \ \ \ \ \text{
for off-path }m\in\mathcal{M}_{N}\setminus\mathcal{T}_{N}.\end{cases}$ (2)

Let $\beta_{\sigma^{*}}(\cdot|m)$ be the receiver’s beliefs about the state
after receiving message $m$, implied by $q_{\sigma^{*}}$ through eq. 1, and

$u^{*}(t)=\underset{m\subseteq
t}{\max}\mathbb{E}_{\beta_{\sigma^{*}}(m)}[\theta].$

This definition imposes 2 restrictions on off-path beliefs. First, any off-
path report that might be honest is believed to be honest, which is necessary
but not sufficient for beliefs to be part of a truth-leaning equilibrium.
Second, we pick a specific form for off-path beliefs for
$m\in\mathcal{M}_{N}\setminus\mathcal{T}_{N}$ that is consistent with
sequential rationality and does not constrain whether a strategy can be
supported in equilibrium by some off-path beliefs, since it imposes that all
off-path messages not consistent with any type’s true dataset yield payoff no
better than another feasible on-path message or off-path message in
$\mathcal{T}_{N}$. Nevertheless, for messages in
$m\in\mathcal{M}_{n}\setminus\mathcal{T}_{N}$, the receiver’s belief is not
too different from what it would be if the sender played some pessimal, nearby
message in $\mathcal{T}_{N}$.

The go-to characterization of the truth-leaning equilibrium is the following
lemma:

###### Lemma 2.2

Every truth-leaning equilibrium strategy profile $\sigma^{*}$ for game
$\mathcal{G}_{N}$ is a best response to beliefs $q_{\sigma^{*}}$, and
satisfies

  * a.

Every on-path message is in $\mathcal{T}_{N}$,

  * b.

Type $t$’s dataset is off-path if
$\mathbb{E}_{\pi(\cdot|t)}[\theta]<\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$

  * c.

Type $t$ is always truthful if
$\mathbb{E}_{\pi(\cdot|t)}[\theta]>\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$.

Furthermore, there always exists an equilibrium
$(\sigma^{*},\beta_{\sigma^{*}})$ satisfying these conditions, and every such
equilibrium has an identical outcome, which is the same as the unique truth-
leaning equilibrium outcome.

Hart et al. (2017) shows that the truth-leaning equilibrium outcome exists and
is unique. The intuition for uniqueness is that when senders’ behavior has the
smallest possible grain of truth, off-path beliefs can’t be too adverse, and
furthermore, the receiver can’t draw adverse inferences about on-path messages
based on the expectation that senders lie even if the truth would be just as
good. We show the rest of the lemma in Appendix B.

In light of this, the conditions of Lemma 2.2 are essentially equivalent to
truth-leaning: The mapping from messages to the action the receiver takes in
response is the same in all of them. Using the weaker set of conditions is
easier and allows us to be conservative about what truth-leaning predicts, at
the cost of ignoring some fine restrictions on truth-leaning equilibrium
messaging strategies. For the remainder of the paper, we will use the term
“truth-leaning” to refer to equilibria that satisfy the necessary conditions
of Lemma 2.2.

While truth-leaning equilibrium strategies capture a sender’s slight bias
towards truth-telling, truth-leaning equilibrium outcomes have desirable
properties in their own right. When the receiver’s expected payoffs are
single-peaked in their action, the truth-leaning equilibrium outcome is also
receiver-optimal, and is the outcome of the optimal mechanism when the
receiver can commit to their response to each message. It is also the only
equilibrium outcome robust to a slightly stronger version of a credible
announcement (Matthews et al. 1991). We say that under equilibrium
$\sigma^{*}$ a coalition $T$ of senders can benefit from an _inclusive_
credible announcement if there is a set of messages such that the coalition is
comprised of every sender that 1) finds some message in the set feasible and
2) weakly benefits from the receiver updating that their type is in $T$ from
the prior, relative to the receiver’s equilibrium inference; and there is at
least one sender in the coalition that strictly benefits.444The departure from
the usual credible announcement is that the coalition must also contain all
senders who are indifferent between participating in the announcement and
their equilibrium payoff. Robustness to such announcements means that the
equilibrium survives even if senders are able to override the receiver’s
beliefs by proposing sensible reinterpretations of messages, and can
coordinate to do so; it rules out, for example, play that is “stuck” in a bad
equilibrium due to immalleable off-path beliefs. For a deeper discussion of
these refinements, see Hart et al. (2017) and Appendix A.

Finally, from Rappoport (2022) we have an algorithm to construct the truth-
leaning equilibrium outcome in evidence games with finite type spaces. Our
data structure fits into the evidence framework as a lattice of types with an
imitation relationship that satisfies transitivity. The algorithm is
iterative: it finds the highest-payoff pool of senders remaining that can
self-separate using a set of messages, and eliminates them at each step until
all senders have been assigned to a pool and a payoff.

###  2.3 Continuous-data limit model

The distribution of datasets converges when $N$ is large, and we can summarize
it in a continuous-data limit model. In addition to capturing the essence of
the disclosure game with big data, a convenience of considering a continuous
mass of data is that it takes advantage of the large-$N$ convergence of
empirical distributions to simplify the type space.

Instead of an upper limit of $N$ datapoints, suppose that the sender’s dataset
has continuous mass, with upper limit $1$. A sender may receive a mass
$\mu\in(0,1)$ of data, and because of the large-$N$ convergence of empirical
distributions to the true generating distribution, the empirical distribution
of data in the limit is $f_{\theta}$ exactly. The sender with a mass $\mu$ of
data in state $\theta_{j}$ therefore receives dataset $\mu f_{j}$ with
probability $1$. In other words, the only true datasets are in
$\mathcal{T}_{\infty}=[0,1]\times\Theta$.

The receiver remains uninformed about the amount of data the sender has, and
their uncertainty is captured by probability density $g$. We assume that $g$
is continuous, infinitely left-differentiable, and supported on an interval
$[a,1]$, with $g(1)=0$.555This assumption implies that $g$ has a vanishing
right tail, and simplifies the equilibrium construction: specifically, it
ensures that the equilibrium payoffs are continuous in $\mu$. It is possible
to construct an imitation equilibrium (see def. 2.2) with discontinuous
payoffs without it, but I will maintain it throughout this paper. This
distribution is the analog of $G_{N}$ in the finite-data setting, with $\mu$
in the place of $\frac{n}{N}$. The type distribution is given by the density

$q(\mu f_{j})=\beta_{0}(\theta_{j})g(\mu)$

and type $\mu f_{j}$ is certain of state $\theta_{j}$, so $\pi(\cdot|\mu
f_{j})=\mathbbm{1}_{\theta_{j}}$.

Recall that $g_{N}$ was a probability mass function representing the
likelihood of obtaining a fraction $\frac{n}{N}$ of the possible data.
Likewise, $g(\mu)$ represents the likelihood of obtaining a fraction $\mu$ of
total potential data. Renormalizing each datapoint in a finite dataset to have
mass $\frac{1}{N}$, we can say that as $N$ increases,
$G_{N}(\frac{n}{N})\rightarrow G(\mu)$, and the type distribution converges to
$q$. We call $\mathcal{G}_{\infty}$ the limit game for a sequence of finite-
data games $\mathcal{G}_{N_{1}},\mathcal{G}_{N_{2}},\ldots$ if their type
distributions converge uniformly to the type distribution in
$\mathcal{G}_{\infty}$, or equivalently, if they share a state space $\Theta$,
outcome space $\mathcal{D}$, priors $\beta_{0}$, and data-generating
distributions $\\{f_{j}\\}_{j=1}^{J}$, and

$G_{N}(\lfloor N\mu\rfloor)\underset{unif.}{\rightarrow}G_{\infty}(\mu).$

Finally, the message space is
$\mathcal{M}_{\infty}=\\{m\in[0,1]\times\Delta\mathcal{D}:\exists t\supseteq
m\\}$. We propose a refinement analogous to truth-leaning for games with
continuous data, by imposing the same set of constraints on beliefs and
strategies. In particular, we define equilibrium beliefs $q_{\sigma^{*}}$
exactly as in the finite-dataset case, by substituting the type space
$\mathcal{T}_{\infty}$ and the message space $\mathcal{M}_{\infty}$,
respectively, and we define an _imitation equilibrium_ to be an equilibrium
satisfying the conditions in 2.2 in the continuous-data setting.

###### Definition 2.2

$(\sigma^{*},q_{\sigma^{*}})$ is an imitation equilibrium in
$\mathcal{G}_{\infty}$ if it satisfies

  1. a.

Every on-path message is in $\mathcal{T}_{\infty}$,

  2. b.

Type $t=\mu f_{j}$’s dataset is off-path if
$\theta_{j}<\max_{m^{\prime}\subseteq
t}\mathbb{E}_{\beta^{*}(\cdot|m)}[\theta]$.

  3. c.

Type $t=\mu f_{j}$ is always truthful if $\theta_{j}>\max_{m\subseteq
t}\mathbb{E}_{\beta^{*}(\cdot|m)}[\theta]$.

This turns out to be the right concept to refine the set of equilibrium
messaging strategies of the limit model to those that capture limit of the
sender’s truth-leaning equilibrium strategy in finite-data games as
$N\rightarrow\infty$, as we show in section 4.

Just as the conditions of 2.2 are necessary for $\sigma^{*}$ to be a truth-
leaning equilibrium in $\mathcal{G}_{N}$, we show in Appendix B that in
$\mathcal{G}_{\infty}$, any truth-leaning equilibrium messaging strategy must
be an imitation equilibrium messaging strategy. While we do not prove the
existence of truth-leaning equilibrium in games with continuous type spaces,
in the next section we show by construction that imitation equilibria exist
and imply a unique equilibrium outcome.

##  3 Characterizing the limit equilibrium

This section characterizes imitation equilibria of the continuous-data limit
game, and shows that their outcome represents the limit of truth-leaning
equilibrium outcomes in games with finite datasets as datasets get large.
Deferring the latter to section 4, here I describe the imitation equilibrium
strategy and outcome, and sketch an algorithm to construct it.

2.2a tells us that the only datasets sent on-path in an imitation equilibrium
are those corresponding to some true dataset in $\mathcal{T}_{\infty}$; the in
which senders never benefit from playing a dataset that the receiver knows for
sure to be incomplete. Thus whenever a sender strategically omits data, their
best strategy is to imitate the full dataset of some other type. In practice,
this means an observer to the interaction between the senders and the receiver
won’t be able to tell if senders are strategically omitting data simply by
looking at the distributions of the published data – some prior about how much
data the sender ought to have is necessary to know if observations are being
dropped. A second observation is that the sender benefits from selective
disclosure if and only if they lie – there are no imitation equilibria that
increase the payoff of truthful senders relative to their payoff when the
receiver is fully informed. On the other hand, truthful senders can suffer –
since other senders can dishonestly imitate them, the receiver is skeptical of
their dataset even if they tell the truth.

All equilibrium outcomes can be described by a vector-valued function
$\hat{u}(\mu)=(u_{j}(\mu))_{j=1}^{J}$, which gives the payoff to $\mu f_{j}$
for each $j$. The imitation equilibrium outcome has an even simpler
description: each sender’s messaging problem can be simplified down to the
choice of a state to imitate, $j\in\\{1,\ldots,J\\}$, and an amount $\mu$ of
that state’s distribution to send, and furthermore, “as much as possible” is
always weakly optimal. Since $\hat{u}$ describes the payoff under every state
to every $\mu$, its inverse $\hat{\mu}$ defined as

$\hat{\mu}_{j}(u)=\min\\{\mu:\hat{u}_{j}(\mu)\geq u\\}$

describes a _burden of proof_ in order to achieve payoff $u$, and what is
necessary is that a type $t$ can provide at least a measure $\mu_{k}(u)$ of
distribution $f_{k}$, where $\theta_{k}\geq u$. Put differently, because
senders can only prove they have at least enough data to send a given
distribution, the best that senders can do is try to show that they have so
much of some favorable state’s distribution that it would be unlikely for them
to produce so much consistent data had the state been different – the sender
floods the receiver with positive-seeming evidence.

The sender’s ability persuade the receiver therefore depends on the mass of
its own dataset, $\mu$, and the extent to which $f_{j}$ resembles $f_{k}$,
given by

$r_{j}(k)=\max_{d\in\mathcal{D}}\frac{f_{k}(d)}{f_{j}(d)}.$

A mass $r_{j}(k)\mu^{\prime}$ of data distributed $f_{j}$ is needed in order
to have a subset that resembles a mass $\mu^{\prime}$ of $f_{k}$. A sharp
prediction in the limit is that the equilibrium depends on the data-generating
distributions $\\{f_{j}\\}_{j=1}^{J}$ solely through $(r_{j}(k))_{j,k\in
1,\ldots,J}$, and the outcome is unchanged by changes in the distribution of
data as long as they don’t affect the likelihood ratios of datapoints that
most distinguish one state from another. For example, an experiment with
large, finite space of outcomes $D>>|\Theta|^{2}$ and a continuum of trials
will generate the same information for the receiver, given strategic
disclosure, as an experiment that only collects data on the
$|\Theta|\times|\Theta-1|/2$ outcomes that maximize
$\frac{f_{k}(d)}{f_{j}(d)}$ for some $k>j$, and a “none of the above” option.

In the imitation equilibrium, a sender that achieves payoff $u$ is either
truthful with $\theta_{j}\geq u$; or they imitate another state $\theta_{k}>u$
in the set

$A_{j}(\mu)=\left\\{\theta_{k}:j\in\arg\max_{j}\hat{u}_{j}\left(\frac{\mu}{r_{j}(k)}\right)\right\\}.$

###### Theorem 3.1

There exists an essentially666Unique up to payoff-irrelevant mixing
probabilities. unique imitation equilibrium, implemented by a vector-valued
burden of proof function
$\hat{\mu}(u):[0,\theta_{J}]\rightarrow\mathbb{R}^{J}$ with inverse
$\hat{u}_{k}(\mu)$ such that

  1. 1.

$\hat{u}_{j}(\mu)$ is continuous and (weakly) increasing in $\mu$ for all $j$.

  2. 2.

$\sigma^{*}(\mu f_{j})$ is supported on
$\\{\mu^{\prime}f_{k}:\mu^{\prime}=\hat{\mu}_{k}(\hat{u}_{k}(\mu/r_{j}(k)))\text{
and }\theta_{k}\in A_{j}(\mu)\\}$.

###  3.1 Construction of the equilibrium

In the Appendix, we give the full step-by-step construction of $\sigma^{*}$;
we sketch the idea here. The algorithm is top-down, noting that from
definition 2.2, fixing a target utility level $u\in(\theta_{j},\theta_{j+1})$,
strategies must consist of senders imitating a state in $j+1,\ldots,J$, so the
burden of proof can be projected down to $J-j$ dimensions. The burden of proof
is then constructed starting with the payoff frontier associated with the
highest possible payoff, $\theta_{J}$.

The construction builds the equilibrium burden of proof iteratively, interval
by interval. It starts by constructing the set of types that achieve the
highest possible payoff, $\theta_{J}$, which consists of those in state
$\theta_{J}$ with enough data that they cannot be imitated by any senders in
other states: the highest payoff frontier is $(1,1,\ldots,\frac{1}{\min_{k\neq
J}r_{k}(J)})$. Fixing a preexisting frontier $\hat{\mu}_{k}(v)$, there are two
cases: either $\hat{u}_{k}(\hat{\mu}_{k}(v))$ is strictly increasing in
$\mu_{k}$ for all $k$, or there is a positive-measure set of types that, when
pooled, generate a payoff $v$ for the sender, so
$\hat{u}_{k}(\hat{\mu}_{k}(v))$ is constant over some range of $\mu_{k}$. In
the former case, the set of types that target some message
$\hat{\mu}_{k}(v)f_{k}$ are those just barely capable of meeting the $k$th
component of $\hat{\mu}(v)$, but that cannot meet any component of
$\hat{\mu}(v)$ with slack: that is, states under which senders are relatively
advantaged in sending $f_{k}$ relative to other components of burden of proof
vector $\hat{\mu}(v)$ are the states that target $\theta_{k}$ to obtain a
payoff of $v$. The construction of the continuation of $\hat{\mu}(v)$ in this
case comes from the fact that, for any collection of target states such that
some senders mix between targeting each of them, the relative rate of change
of $\hat{u}_{k}(\cdot)$ is pinned by mixing types’ indifference. In the latter
case, which corresponds to cases in which payoffs from imitating the greatest
possible volume of some more favorable state are nonmonotone, senders may send
strictly less data distributed $f_{k}$ than they can. Similarly to the finite-
data case, when there exists a positive-measure set of types that attains the
highest payoff out of all such type sets, then it is possible to find a set of
messages and strategies for senders in the positive-measure type set such that
all types included receive the same payoff, which must be $v$.

###  3.2 Properties and extensions

$\mathcal{G}_{\infty}$ is an elaboration of the Dye (1985) disclosure game
with a continuum of data, rather than a single datapoint. Like the Dye
setting, the sender’s ability to manipulate the receiver’s beliefs comes from
the fact that the receiver can never distinguish between senders with more
data that is discouraging (in Dye, a single low-value datapoint) and senders
with less data (e.g. no data at all). When the variance of $G_{\infty}$ is
large, senders can pool with others quite profitably. But, as we prove in
Appendix D, if the receiver can reduce their uncertainty about how much data
the sender generates, then just like the case where the probability of
evidence $p\rightarrow 1$ in the single-datapoint case, the equilibrium
outcome gets closer and closer to the full-information outcome.

###### Claim 3.2

As $Var[G_{\infty}]\rightarrow 0$,
$u_{\sigma^{*}}(t)\rightarrow\mathbb{E}_{\pi(\cdot|t)}[\theta]$ for all
$t\in\mathcal{T}_{\infty}$.

While the baseline model assumes $\mathcal{D}$ is finite, and this is
convenient for understanding the primitive finite-data games, we might also be
interested in collecting data on continuous variables. The equilibrium
characterization for the limit game does not depend at all on the assumption
that $\mathcal{D}$ is finite: it is fine to let $\mathcal{D}$ be a subset of
$\mathbb{R}^{N}$, as long as the pairwise ratios $r_{j}(k)$ are well-
defined.777Examples of families of distributions with easily-calculated ratios
$r_{j}(k)$ are beta distributions with $\alpha,\beta>1$ and truncated normal
distributions.

Lastly, before proceeding to examples, we have a basic qualitative description
of the equilibrium outcome.

###### Corollary 3.3

Under $\sigma^{*}$, there are thresholds $z^{*}_{j}>z^{**}_{j}$ for each state
such that:

  * •

Whenever the sender’s type is $\mu f_{j}$ with $\mu>z^{*}_{j}$, the sender
masquerades as a higher type, and receives a payoff $u_{\sigma^{*}}(\mu
f_{j})>\theta_{j}$.

  * •

Whenever $\mu\in(z^{**}_{j},z^{*}_{j}]$, the sender is honest and the receiver
knows it upon receiving the data: $u_{\sigma^{*}}(\mu f_{j})=\theta_{j}$.

  * •

Whenever $\mu\leq z^{**}_{j}$, the sender is honest, but the receiver believes
they are a worse type with positive probability, and $u_{\sigma^{*}}(\mu
f_{j})<\theta_{j}$.

###  3.3 Examples

Example 1. Binary state

A seller (the sender) wishes to convince a buyer (the receiver) that a product
is functional using data about the product’s performance. We can model the
state space as $\Theta=\\{L,H\\}$, with both states equally likely ex-ante and
with the sender’s payoff equal to the probability with which the receiver
believes the state is high ($H$) after observing data. Suppose that there are
$3$ possible outcomes, $\mathcal{D}=\\{1,2,3,4\\}$ with the following
distribution of outcomes per state:888For example, the product could be a
weather prediction service, and outcomes $1,2,3,4$ could be (predict rain, no
rain), (predict no rain, rain), (predict no rain, no rain) and (predict rain,
rain), respectively.

| $f_{\theta}(1)$ | $f_{\theta}(2)$ | $f_{\theta}(3)$ | $f_{\theta}(4)$
---|---|---|---|---
H | 0.1 | 0.1 | 0.5 | 0.3
L | 0.2 | 0.2 | 0.4 | 0.2
Table 1: The generating distribution of outcomes under states $H$ and $L$.

A low-state sender is able to use a proportion $\frac{1}{r_{L}(H)}=2/3$ of
their data to imitate the high-state distribution. Outcomes for each type of
sender depend on

$\rho(\mu)=\frac{g(\mu)}{g(\mu)+r_{L}(H)g(r_{L}(H)\mu)},$

the proportion by mass of high-state types out of all types capable of sending
at most measure $\mu$ of distribution $f_{H}$. A preliminary observation, from
Lemma 2.2, is that since there is a single best and a single worst state, all
state-$H$ senders are honest in the imitation equilibrium, and state-$L$
senders all imitate state-$H$ senders, so what remains is to describe the
optimal imitation strategy and resulting payoffs.

1A. Maximal subsets

Suppose the data-mass distribution is a “triangular” distribution so that

$g(\mu)=2-4|x-1/2|,\ \ \ \ \ \ \ \rho(\mu)=\begin{cases}1,\ \ \ \ \ \ \ \ \
\mu\geq 2/3\\\ \frac{4-4\mu}{10-13\mu},\ \ \ \mu\in[1/2,2/3)\\\
\frac{4\mu}{6-5\mu},\ \ \ \ \ \mu\in[1/3,1/2)\\\ \frac{4}{13},\ \ \ \ \ \ \ \
\mu<1/3.\end{cases}$

Because (as plotted in the left panel of Figure 2) $\rho$ is weakly monotone,
the imitation equilibrium is for every type of sender to submit the maximal
subset of their data distributed like $f_{H}$, and in equilibrium, the sender
of type $\mu f_{H}$ receives a payoff $\rho(\mu)$, while type $\mu f_{L}$
receives a payoff $\rho(2\mu/3)$.

In general for the binary case, submitting a maximal measure $\mu$ of $f_{H}$
is optimal and yields payoff $\rho(\mu)$ whenever $\rho(\mu)$ is monotone, and
in turn, a sufficient condition for monotonicity is that $g(\mu)$ is concave
over its support.999Although many common distributions, like truncated
Gaussian and beta, don’t satisfy concavity over their tails, the majority of
their mass lies in a region over which the density is concave, and so as a
benchmark, outcomes are not too far from $\rho(\mu)$ in these cases.

1B. Ironing

Alternatively, consider a case in which $g(\mu)$ is double-peaked – this
perhaps describes a case in which unbeknownst to the regulator, either a lump
sum of funding arrives and the total number of trials attempted by the sender
is high, or it does not arrive and the number of trials is low, and there is
additional randomness in the number of admissible trials out of those
attempted.

$g(\mu)=\begin{cases}2-8|x-1/4|,\ \ \ \ \ x\in[0,1/2]\\\ 2-8|x-3/4|,\ \ \ \ \
x\in(1/2,1],\end{cases}$

so $\rho(\mu)$ is as plotted in the right panel in Figure 2. Clearly,
$\rho(\mu)$ cannot describe the payoffs of the sender with mass $\mu$ under
state $H$, since monotonicity is a minimal requirement for the effect of $\mu$
on payoffs.

Figure 2: Outcomes for 2 different $g(\mu)$ in a binary-state problem.

Instead, the imitation equilibrium payoffs are an _ironed_ version of
$\rho(\mu)$. In this example, there is a single ironing interval. Note that
def. 2.2b restricts some senders in the ironing interval to playing
truthfully, even though they are indifferent between all messages in the
interval. If a sender has a mass $\mu$ of distribution $f_{H}$ where $\mu$
lies in the ironing interval, then under a slight preference for truth-
telling, high-state senders continue to prefer to tell the whole truth, since
the truth must be on-path and will yield the same payoff as any other optimal
lie. On the other hand, a low-state sender has a number of equally good lies
corresponding to submitting different amounts of $f_{H}$, and so may mix
between them. The equilibrium mixed strategies essentially average the mass of
low-state senders over the ironing interval such that all messages in the
interval are, indeed, equally good. The endpoints of the interval are such
that the receiver’s inference from the pool of senders that send any message
in the interval is the same as the inference from $\rho(\mu)$ at the
endpoints.

In general, there may be multiple ironing intervals, but their number is
bounded by the number of intervals on which $\rho(\mu)$ is decreasing. The
construction algorithm is top-down and constructs the necessary ironing
intervals one by one, terminating in finite time as long as the number of
intervals is finite.

Example 2. Drug trials

The relationship between different qualities of a product can also be more
complex. Consider for instance a preventative drug for a potentially deadly
condition that could turn out to be one of 3 things: a very highly effective
“miracle drug” that often completely prevents illness (state $\theta_{H}$); a
reasonably effective drug that sometimes reduces the severity of illness
conditional on infection (state $\theta_{M}$); and a useless placebo that
fails to prevent infection or illness at all relative to the baseline risk
(state $\theta_{L}$). The company collects data about whether those that use
the drug avoid getting infected (outcome 1), get infected but are not
seriously ill (outcome 2), or are infected with serious symptoms (outcome 3).

| $\beta_{0}$ | $f_{\theta}(1)$ | $f_{\theta}(2)$ | $f_{\theta}(3)$
---|---|---|---|---
H | 0.2 | 0.6 | 0.2 | 0.2
M | 0.4 | 0.4 | 0.4 | 0.2
L | 0.4 | 0.4 | 0.2 | 0.4
Table 2: The generating distribution of outcomes under states $H$, $L$ and
$M$.

Here, $r_{L}(H)=3/2$, $r_{M}(H)=3/2$, and $r_{L}(M)=2$. Suppose
$\theta_{L}<\theta_{M}<\theta_{H}$ and that $g(\mu)$ is concave; then we can
picture the imitation equilibrium strategies as in Figure 3.

Figure 3: The ability of senders under each state to imitate $f_{H}$ and
$f_{L}$, and equilibrium imitation strategies (in arrows).

To achieve payoffs above $\theta_{M}$, the sender must be able to send a large
measure of data distributed like $f_{H}$. There is a positive measure of types
under state $M$ that can separate and achieve a payoff of exactly $\theta_{M}$
in equilibrium; the reason is that senders with an ineffective drug but enough
data to be capable of attaining a payoff of $\theta_{M}$ from imitating
$f_{H}$ are not equally capable of imitating the datasets of senders with
medium-effectiveness drugs but with the same amount of distribution $f_{H}$.
Low-data senders with an ineffective drug will optimally mix between imitating
trial outcomes of the miracle drug and of the reasonably effective version,
thereby worsening outcomes for low-data senders with either of those versions.

When does the presence of a third, medium state affect optimal play in the low
state? Observe that in an alternative world in which
$f_{M}(1)=0.5,f_{M}(2)=0.2,f_{M}(3)=0.3$, that is, when the medium-quality
drug is simply less effective at preventing illness than the miracle drug, a
sender’s ability to imitate the miracle drug is a sufficient statistic for
their ability to imitate the medium state. Then the imitation equilibrium
outcome is equivalent to the outcome of an equilibrium in which all senders,
regardless of the true state, imitate $f_{H}$. In this case, there are no
longer medium-state senders who separate themselves entirely from senders in
other states.

##  4 Equilibrium convergence

What’s left to show is that convergence of type spaces to the continuous-data
limit type space implies the convergence of equilibria. To make the
comparison, the notion of an outcome should be extended across type spaces.
There is a global data space $[0,1]\times\Delta\mathcal{D}$, invariant to $N$,
that contains $\mathcal{T}_{1},\ldots$ and $\mathcal{T}_{\infty}$ as long as
they all share a space of observations. Recall that $u_{\sigma^{*}}(t)$, the
outcome of the game for type $t$, is their payoff from the best feasible
message given equilibrium beliefs. If $t\in[0,1]\times\Delta\mathcal{D}$, it
need not also be in the type set for the outcome to be well-defined, since we
can already infer whether $t$ can feasibly send a message from the subset
relation on $[0,1]\times\Delta\mathcal{D}$. The outcome to the hypothetical
type can be understood as a thought experiment: “if the receiver believes we
are playing a game with equilibrium $\sigma^{*}$, and my dataset is $t$, what
is the best payoff I can attain, even if $t$ is inconsistent with the
receiver’s perceived game?”

###### Definition 4.1

A sequence of equilibria $(\sigma_{1},\sigma_{2},\ldots)$ of games
$\\{\mathcal{G}_{N}(\Theta,D,\beta_{,}\\{f_{j}\\}_{j=1}^{J},G_{\infty})\\}_{N=1}^{\infty}$
has outcomes that _converge_ to the outcome of an equilibrium
$\sigma_{\infty}$ of the limit infinite-data game
$\mathcal{G}_{\infty}(\Theta,D,\beta_{,}\\{f_{j}\\}_{j=1}^{J},G_{\infty})$ if
the payoffs $u_{\sigma^{*}_{N}}(t)$ converge uniformly to
$u_{\sigma^{*}_{\infty}}(t)$ over types in $\mathcal{T}_{\infty}$.

###### Theorem 4.1

If $\mathcal{G}_{\infty}$ is the limit game for finite-data games
$\mathcal{G}_{1},\mathcal{G}_{2},\ldots$ with $N=1,2,\ldots$ respectively,
then the truth-leaning equilibrium outcomes in
$\mathcal{G}_{1},\mathcal{G}_{2},\ldots$ converge to the imitation equilibrium
outcome of $\mathcal{G}_{\infty}$.101010Of course, we could just as well have
applied the theorem to any convergent sequence of games
$\\{\mathcal{G}_{N_{i}}\\}_{i=1}^{\infty}$ of increasing dataset size, since
it is possible to complete any such sequence into a convergent sequence of
games $\\{\mathcal{G}_{N}\\}_{N=1}^{\infty}$, for which the theorem holds; in
other words, truth-leaning outcomes converge to the limit-game imitation
equilibrium outcome in any convergent sequence of finite-data games.

We state the theorem in terms of $N=1,\ldots,\infty$ for notational brevity.

Outcomes converge to the limit game outcome if the finite game-implied
outcomes for types in the limit type set converge to the outcomes for those
types in the limit game. They need not converge on
$[0,1]\times\Delta\mathcal{D}\setminus\mathcal{T}_{\infty}$, and indeed they
usually won’t: the way we extend outcomes to $[0,1]\times\Delta\mathcal{D}$ in
the limit game is conservative and does not allow hypothetical types far away
from $\mathcal{T}_{\infty}$ to achieve payoffs above those they can get by
copying some feasible type in $\mathcal{T}_{\infty}$. In contrast, types far
away from $\mathcal{T}_{\infty}$ may do better in finite-data games. However,
with big data, these types are vanishingly improbable. For the majority of
types that are concentrated around the limit type set, convergence of outcomes
gives tight bounds on payoffs.

###### Corollary 4.2

If $\ \sup_{d}||t_{\infty}(d)-\frac{1}{N}t_{N}(d)||\leq\delta$ for some
$t_{\infty}=\mu f_{\theta}$, and
$\sup_{t}||u_{\sigma^{*}_{N}}(t)-u_{\sigma^{*}_{\infty}}(t)||\leq\epsilon$ for
all $t\in\mathcal{T}_{\infty}$, then

$u_{\sigma^{*}_{N}}(t_{N})\in[u_{\sigma^{*}_{\infty}}((\mu-D\delta)f_{\theta})-\epsilon,u_{\sigma^{*}_{\infty}}((\mu+D\delta)f_{\theta})+\epsilon].$

We can briefly describe the proof of the theorem, which is in Appendix E. A
lemma shows that the limit-game outcome $u_{\sigma^{*}_{\infty}}$ is such
that, given any alternative outcome with a different frontier
$\hat{\mu}^{\prime}(v)$, if there are any types above the payoff frontier
$\hat{\mu}(v)$ but below $\hat{\mu}^{\prime}(v)$, then the payoff to the
sender when the receiver believes the sender is one of those types is at least
$v$. Using this, we can show that any equilibrium in a finite-data game
constructed optimally using Rappoport’s iterative algorithm must yield payoffs
not far less than the payoffs from the limit equilibrium for types close to
the limit type set, and that the lower bounds on the difference shrink to $0$
as $N\rightarrow\infty$, and then Bayes plausibility for the information
structure as a whole allows us to conclude an upper bound on the difference as
well.

Outcome convergence shows that it’s reasonable to use the limit game to
describe the distribution of actions the receiver takes after the sender
discloses a large dataset, as well as the mapping from the truth to the
receiver’s inferences. Often this should be enough: when looking to evaluate
how effective using self-disclosed data is, or to design experimental
protocols or a response policy to evidence, the decision-maker cares about
taking actions that correctly match the state, and the outcome describes how
well the decision-maker is able to do so in the optimum. In particular,

  1. 1.

The receiver’s ex-ante expected payoff from $\mathcal{G}_{N}$ approaches
$\mathbb{E}_{\mu,\theta_{j}}[u_{r}(a_{r}(\sigma^{*}_{\infty}(\mu
f_{j})),\theta_{j})]$ as $N\rightarrow\infty$.

  2. 2.

With single-peaked expected utility for the receiver, the limit game outcome
is close to the receiver-optimal outcome with big data, and when the receiver
commits to responding to each message as they would in the limit-game outcome,
the resulting mechanism is nearly optimal for the large, finite-data game.

  3. 3.

With $N>>D$, strategically communicating the outcome of a set of $N$
experiments that only with a large, finite space of outcomes $D>>|\Theta|^{2}$
will yield nearly the same information as an experiment that only collects
data on the $|\Theta|\times|\Theta-1|/2$ outcomes that maximize
$\frac{f_{k}(d)}{f_{j}(d)}$ for some $k>j$.

One might also be curious if _strategies_ converge: do the finite-game
equilibria also approximate an “imitation equilibrium,” that is, do types
close to a type in $\mathcal{T}_{\infty}$ send messages similar to those sent
by a type in the limit model? We can give a partially positive answer.

###### Proposition 4.3

Suppose that $\\{\mathcal{G}_{N}\\}_{N=1}^{\infty}$ converge to
$\mathcal{G}_{\infty}$. Then for all $p^{*},\rho,\eta>0$, there is
$\underline{N}(p^{*},\eta)$ such that for all $N>\underline{N}(p^{*},\eta)$,
conditional on $|u_{\sigma^{*}_{N}}(t)-\theta_{k}|>\eta$ for all $k$, there is
at least probability $1-p^{*}$ that $t$ sends a message with (sup norm)
distance at most $\delta$ from some $t_{\infty}\in\mathcal{T}_{\infty}$ that
is on-path in $\sigma^{*}_{\infty}$ and such that
$|u_{\sigma^{*}_{N}}(t)-u_{\sigma^{*}_{\infty}}(t_{\infty})|\leq\rho$.

This is a partial characterization of large-$N$ equilibrium strategies, saying
that among types that obtain payoff bounded away by an arbitrarily small
amount from the rewards to certainty about any particular state, the
likelihood of playing a message close to their optimal message under limit-
game beliefs is very high when there is plentiful access to data. In other
words, these types play imitation-like strategies. This follows from the
convergence theorem, since in truth-leaning equilibrium a type that receives
payoff less than its full-information payoff always discloses its full
dataset, so types with
$u_{\sigma^{*}_{N}}(t)<<\mathbb{E}_{\pi(\cdot|t)}[\theta]$ tell the truth;
convergence of outcomes implies that they receive payoffs similar to those
obtained by nearby types in $T_{\infty}$ under $\beta_{\sigma^{*}_{\infty}}$,
and convergence of the type distribution implies that most such senders are
indeed near some type in $T_{\infty}$. In aggregate a similar set of imitators
must pool with such senders as the set of imitators pooling with better-state
senders in $\sigma^{*}_{\infty}$, which means that types with
$u_{\sigma^{*}_{N}}(t)>>\mathbb{E}_{\pi(\cdot|t)}[\theta]$ play messages close
to $\mathcal{T}_{\infty}$ with high probability.

The caveat is that when
$|\mathbb{E}_{\beta_{\sigma^{*}_{N}}(\cdot|m)}[\theta]-\theta_{k}|$ is small
for some $k$, then there may be no significant mass of senders playing $m$ to
earn a payoff much greater or much less than their full-information payoff,
which makes it hard to apply the technique of matching imitators to the
imitated, though we do not have a counterexample for this case. From Corollary
3.3, we know that there is a positive-measure set of types that receive
payoffs close to their full-information payoffs in $\sigma_{\infty}^{*}$, and
the proposition does not pin down the large-$N$ limit of equilibrium
strategies of types close to them, but generically, besides these, the set of
types excluded from the proposition is measure-0.111111Genericity here can be
with respect to perturbations in $\beta_{0}$ or
$\theta_{1},\ldots,\theta_{J}$.

##  5 Conclusion

Selective disclosure means that data-driven inferences depend on an
understanding of the sender’s motives and strategy. We model the sender’s
voluntary disclosure problem through an evidence game with large datasets, and
find that their optimal strategy approximates a simple procedure: claim a
state that is at least as good as the truth, and provide as much data as
possible imitating its distribution. The combination of the receiver’s
skepticism and their inability to verify missing or undisclosed data benefits
senders for whom data are plentiful, and it hurts senders with limited data
access, regardless of the state of the world.

There is an open question about how to integrate endogenous, costly data
acquisition with the incentives we have presented for data collection and
storage. Having more data benefits senders strategically regardless of its
informational value, suggesting that over-collection of data might result if
senders have the choice. A separate, interesting open technical question is
whether the necessary conditions we give for truth-leaning messaging
strategies are also sufficient, and to what extent they apply to general
evidence games.

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##  Appendix A Inclusive announcement-proofness and receiver optimality

Here, I discuss alternative ways of selecting the truth-leaning equilibrium
outcome. First, as Hart et al. (2017) showed, under the assumption that the
receiver’s expected payoffs,

$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[u_{r}(a,\theta)],$

are single-peaked in their action $a$, and that higher expected states
correspond to higher optimal actions, the truth-leaning outcome maximizes the
sender’s expected welfare amongst all equilibrium outcomes, and furthermore
coincides with the outcome of the sender-optimal mechanism when the sender can
commit to a policy $a^{*}(m)$.

Next, I formally define a credible inclusive announcement and show that the
truth-leaning equilibrium outcome in $\mathcal{G}_{N}$ is also the unique
outcome immune to credible inclusive announcements.

###### Definition A.1

Given an outcome $u_{\sigma^{*}}$, a set of types $T$ has a _credible
inclusive announcement_ that they will play a strategy $\hat{\sigma}_{M}$
supported over message set $M$ for payoff $v$ if

  * •

$\hat{\sigma}_{M}:M\times T\rightarrow\mathbb{R}$ is such that $\sum_{t\in
T}\hat{\sigma}_{M}(m|t)=1$ for all $m\in M$, $\sum_{m\in
M}\hat{\sigma}_{M}(m|t)=1$ for all $t\in T$, and
$\mathbb{E}_{\beta_{\hat{\sigma}_{M}}(\cdot|m)}[\theta]=v$ for all $m\in M$.

  * •

$T=\\{t:u_{\sigma^{*}}(t)\leq v\ \text{and}\ \exists m\in M\text{ s.t.
}m\subseteq t\\}$, and there is some $t\in T$ with $u_{\sigma}(t)<v$.

A very closely-related notion is the original idea of a credible announcement,
from Matthews et al. (1991), but there is a subtle difference, which is that
in a credible announcement, $T=\\{t:u_{\sigma^{*}}(t)\leq v\ \text{and}\
\exists m\in M\text{ s.t. }m\subseteq t\\}\bigcup S$ where
$S\subseteq\\{t:u_{\sigma^{*}}(t)=v\text{ and }\exists m\in M\text{ s.t.
}m\subseteq t\\}$. Thus what we use is an “inclusive” notion of a credible
announcement in that the set of announcing types must include all who weakly
prefer to participate; it is stronger to claim there exists an credible
inclusive announcement than that there exists a credible announcement, and
correspondingly, inclusive announcement-proofness is weaker than announcement-
proofness. In fact, there may exist no announcement-proof equilibrium at all
in the game we study, while there always exists exactly one inclusive
announcement-proof equilibrium outcome.

To show that the truth-leaning equilibrium outcome is inclusive announcement-
proof, I construct it, using the algorithm from Rappoport, which I summarize
here. In short, the equilibrium is constructed by iteratively choosing a
frontier of types such that the set of types “above” the frontier, in the
sense of being able to imitate some frontier type, yields as favorable a
belief as possible.

Algorithm (Finite $N$). First, define for any type set $T$ the subset of types
$T^{+}(M)$ as the set of types in $T$ that are capable of sending some message
in message set $M$, and define $u_{pool}(T)$ to be the payoff to the sender if
the receiver knows only that their type must be in $T$.

  1. 1.

Let $T_{1}=\mathcal{T}_{N}$, and find the set of messages $M_{1}\subseteq
T_{1}$ that maximizes the payoff to a pool consisting of the set of senders in
$T_{1}$ who can send at least one message in it:

$M_{1}\in\arg\max_{M\subseteq T_{1}}u_{pool}(T_{1}^{+}(M)).$

If there are multiple such pools, then we take their union, which is also such
a pool.

  2. 2.

For $s=2$ onwards, restrict the set of types to $T_{s}=T_{s-1}\setminus
T_{s-1}^{+}(M_{s-1})$, and find (the union of)

$M_{s}\in\arg\max_{M\in T_{s}}u_{pool}(T_{s}^{+}(M)).$

  3. 3.

Continue until $T_{s}\setminus T_{s}^{+}(M_{s})=\emptyset$. Given each set
$M_{s}$, there always exists a mixed strategy profile $\sigma_{pool}^{M}$
defined over types in $T_{1}^{+}(M)$ such that each message in $M$ yields the
same payoff under the receiver’s induced beliefs from
$\sigma_{pool}^{M}$.121212Otherwise, the worst possible payoff to particular
message in $M$ over all strategy profiles over $M_{s}$ is better than the best
possible payoff to some other message; then there always exists $M\subset
M_{s}$ such that $T_{s}^{+}(M)>T_{s}^{+}(M_{s})$. Define $\sigma^{*}$ by
$\sigma^{*}(m|t)=\hat{\sigma}_{pool}^{M_{s}}(m|t)$ where $M_{s}$ is the pool
containing $m$.

  * Proof. (Unique credible inclusive announcement-proof outcome)

By construction, there is no credible inclusive announcement, since such an
announcement would constitute a better set of types than the one constructed
at some step of the algorithm; this violates the optimality of the pool of
types constructed in each step. No other outcome is immune: if
$u_{\sigma^{*}_{alt}}\neq u_{\sigma^{*}}$, then there exists a $v$ such that
the set of pools achieving a payoff greater than $v$ is identical in
$u_{\sigma^{*}_{alt}}$ and $u_{\sigma^{*}}$, but the pool of types $T$
achieving payoff $v$ under $u_{\sigma^{*}}$ is a strict superset of that under
$u_{\sigma^{*}_{alt}}$. Then types in $T$ can make a credible inclusive
announcement that they will play as they do in $\sigma^{*}$.

##  Appendix B Necessary conditions for truth-leaning equilibrium

Here we prove the truth-leaning characterization of lemma 2.2, as well as the
necessity of the conditions in def. 2.2 for a truth-leaning equilibrium in
$\mathcal{G}_{\infty}$.

  * Proof of Lemma 2.2

First, we prove that every truth-leaning equilibrium messaging strategy
$\sigma$ satisfies the necessary conditions of the lemma.

Hart et al. (2017)’s proof of their Proposition 1 shows that in every truth-
leaning equilibrium, if $t$’s dataset is off-path then the receiver plays a
best response to the belief $\mathbbm{1}_{t}$ upon seeing t. This suffices to
show that every truth-leaning equilibrium messaging strategy $\sigma$ is a
best response to $q_{\sigma}$.

For part a), note that if a message $m$ is on-path in $\sigma$, then there
exists $K_{1}$ such that for all $k>K_{1}$, $m$ is on-path in $\sigma^{k}$.
For every $k$, however, all on-path messages are in $\mathcal{T}_{N}$, since
if $m$ is on-path and $m\not\in\mathcal{T}_{N}$, then there is a highest-value
type $t$ that plays $m$, and there cannot be any type $t^{\prime}$ that
imitates $t$, otherwise
$\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}[\theta]\geq\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|m)}[\theta]$,
so
$\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}[\theta]+\epsilon_{t}>\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|m)}[\theta]$
and $t$ would not find it optimal to play $m$. But then
$\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}[\theta]=\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|m)}[\theta]$,
leading, again, to a contradiction. Hence, all on-path $m$ must be in
$\mathcal{T}_{N}$.

To prove that a truth-leaning equilibrium messaging strategy satisfies c),
suppose there is $t$ such that
$\mathbb{E}_{\pi(\cdot|t)}[\theta]>\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma}(\cdot|m)}[\theta]$ but $\sigma(t|t)<1$.

We will show that there is no sequence of perturbations
$\\{\epsilon^{k}\\}_{k=1}^{\infty}\to 0$ such that equilibria of games
$\mathcal{G}_{\epsilon^{k}}$ converge to $\sigma$. Start by supposing for the
sake of contradiction that there is. We know that if $t$ is off-path under
$\sigma$, then $\mathbb{E}_{\pi(\cdot|t)}[\theta]=\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma}(\cdot|m)}[\theta]$, so $t$ must be on path in
$\sigma$. If $\sigma^{k}$ is an equilibrium of game
$\mathcal{G}_{\epsilon^{k}}$ with $\epsilon^{k}_{t}>0$, there cannot
$t^{\prime}\neq t$ such that $\sigma^{k}(t|t^{\prime})>0$, otherwise
$\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}[\theta]\geq\max_{t^{\prime}\subset
t}\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t^{\prime})}[\theta]$ and so
$\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}[\theta]+\epsilon_{t}^{k}>\max_{t^{\prime}\subset
t}\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t^{\prime})}[\theta]$ and we would have
to have $\sigma^{k}(t|t)=1$. Then, likewise, in the limit $\sigma$, we must
have $\sigma(t|t^{\prime})=0$ for all $t^{\prime}$. Since $t$ is on-path in
$\sigma$, it must be that $\sigma(t|t)\in(0,1)$.

Take a type $t^{\prime\prime}\neq t$ such that $\sigma(t^{\prime\prime}|t)>0$.
We know that there exists $K$ such that for all $k>K$,
$\sigma^{k}(t^{\prime\prime}|t)>0$ as well. Then whenever $k>K$,
$\mathbb{E}_{\pi(\cdot|t)}+\epsilon^{k}_{t}=\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t^{\prime\prime})}[\theta]$.
Because $\sigma^{k}\rightarrow\sigma$, we have that

$\lim_{k\rightarrow\infty}\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t^{\prime\prime})}[\theta]=\mathbb{E}_{\beta_{\sigma}(\cdot|t^{\prime\prime})}[\theta]=\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma}(\cdot|m)}[\theta].$

But this contradicts that $\mathbb{E}_{\pi(\cdot|t)}[\theta]>\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma}(\cdot|m)}[\theta]$ and

$\lim_{k\rightarrow\infty}\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t^{\prime\prime})}[\theta]=\lim_{k\rightarrow\infty}\mathbb{E}_{\pi(\cdot|t)}+\epsilon^{k}_{t}=\mathbb{E}_{\pi(\cdot|t)}.$

To show that b) holds, note that for any $k$, if $t$ is on-path and played by
some $t^{\prime}\neq t$, then $\sigma^{k}(t|t)=1$. By c),
$\mathbb{E}_{\pi(\cdot|t^{\prime})}\leq\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}$,
but if $t$ also plays $t$ and
$\mathbb{E}_{\pi(\cdot|t)}<\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}$, then the
receiver is not Bayesian. On the other hand, if $t$ is on-path and only $t$
plays $t$, then we must have
$\mathbb{E}_{\pi(\cdot|t)}=\mathbb{E}_{\beta_{\sigma^{k}}(\cdot|t)}$.

The second part of the lemma shows that every equilibrium
$(\sigma,q_{\sigma})$ that has on-path messages in $\mathcal{T}_{N}$, and
satisfies a), b), and c) has the same outcome as a truth-leaning equilibrium.
To do so, we use the result from Hart et al. (2017) that every equilibrium
that satisfies the two conditions

  1. (A0)

For every type $t$, if $t\in\arg\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$, then
$\sigma^{*}(\cdot|t)=\mathbbm{1}_{t}$.

  2. (P0)

For every message $m\in\mathcal{T}_{N}$ with $\sigma^{*}(m|t)=0$ for all $t$,
the receiver’s belief is $\beta^{*}(\cdot|m)=\mathbbm{1}_{\theta(m)}$.

is truth-leaning.

From the definition of $q_{\sigma}$, we already have that
$(\sigma,q_{\sigma})$ satisfies (P0). We must show there is an outcome-
equivalent equilibrium that also satisfies (A0). To do so, we first show that
(A0) is equivalent to saying that

$\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]\ \ \rightarrow\ \
\sigma(t|t)=1.$ (3)

To show that (A0) implies eq. 3, we show that if
$\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$ then
$t\in\arg\max_{m\subseteq t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$:
if no other types imitate $t$, then
$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]=\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$ and so
$t\in\arg\max_{m\subseteq t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$,
while if some other types imitate $t$, then we know directly from their
incentives that $t\in\arg\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$.

To show that eq. 3 implies (A0), note that under the statement, the only types
that might imitate $t$ are types $t^{\prime}$ for which
$\mathbb{E}_{\pi(\cdot|t^{\prime})}[\theta]<\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]$,
which means that
$\mathbb{E}_{\pi(\cdot|t^{\prime})}[\theta]<\mathbb{E}_{\pi(\cdot|t)}[\theta]$,
which in turn means that
$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]<\mathbb{E}_{\pi(\cdot|t)}[\theta]$
if there are any types $t^{\prime}\neq t$ that imitate $t$. So, if
$t\in\arg\max_{m\subseteq t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$
then either no other types imitate $t$, in which case
$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]=\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$, or some other types
imitate $t$, in which case
$\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$.

Because c) is weaker than eq. 3, c) is also weaker than (A0). Note that under
c), $\mathbb{E}_{\pi(\cdot|t)}[\theta]=\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$ implies that
$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]=\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$ because there would be
imitators only if
$\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|t)}[\theta]\geq\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$, but any imitator
$t^{\prime}$ must have
$\mathbb{E}_{\pi(\cdot|t^{\prime})}[\theta]\leq\mathbb{E}_{\pi(\cdot|t)}[\theta]$.
Thus any equilibrium $(\sigma,q_{\sigma})$ that satisfies c) can be turned
into an equilibrium satisfying (A0) and (P0) by taking the set of types for
which $\mathbb{E}_{\pi(\cdot|t)}[\theta]=\max_{m\subseteq
t}\mathbb{E}_{\beta_{\sigma^{*}}(\cdot|m)}[\theta]$, and setting
$\sigma(t|t)=1$ for only those types while keeping the rest of the messaging
strategy unchanged. This also leaves the receiver’s inference function
$\beta_{\sigma}$ unchanged, and does not change the equilibrium outcome.

Existence and outcome-uniqueness of an equilibrium satisfying these conditions
follows from the fact that a truth-leaning equilibrium exists, and every
equilibrium satisfying (A0) and (P0) has the same outcome.

###### Claim B.1

Any truth-leaning equilibrium of $\mathcal{G}_{\infty}$ has a messaging
strategy profile $\sigma$ that is a best response to $q_{\sigma}$, and
satisfies

  1. a.

Every on-path message is in $\mathcal{T}_{\infty}$,

  2. b.

Type $t=\mu f_{j}$’s dataset is off-path if
$\theta_{j}<\max_{m^{\prime}\subseteq
t}\mathbb{E}_{\beta^{*}(\cdot|m)}[\theta]$.

  3. c.

Type $t=\mu f_{j}$ is always truthful if $\theta_{j}>\max_{m\subseteq
t}\mathbb{E}_{\beta^{*}(\cdot|m)}[\theta]$.

  * Proof

The proof is essentially the same as the proof of necessity for
$\mathcal{G}_{N}$.

##  Appendix C Construction of limit equilibrium

  * Proof of Theorem 3.1

Recall that $\hat{u}_{k}(\mu)$ is the equilibrium payoff to sending the
message $\mu f_{k}$.

We construct $\hat{u}_{k}(\mu)$ that is monotone increasing in $\mu$ – this
implies that it must be almost-everywhere differentiable. Since it is also
continuous, it is completely determined by its derivative over the points at
which the derivative exists. To avoid confusion, we focus on the left
derivative of $\hat{u}_{k}$, which we denote by $\hat{u}_{k}^{-}$ and,
analogously to the top-down construction of the finite-data equilibrium, we
construct the payoff function starting from the top down, starting from the
frontier $v=\theta_{J}$.

Recall that $r_{j}(k)=\max_{d\in\mathcal{D}}\frac{f_{k}(d)}{f_{j}(d)}$ is the
ratio of the amount of data necessary to imitate a certain amount of $f_{k}$
under state $j$ to the amount necessary under state $k$, and

$A_{j}(\mu)=\left\\{\theta_{k}:j\in\arg\max_{j}\hat{u}_{j}\left(\frac{\mu}{r_{j}(k)}\right)\right\\}.$

is the set of states that type $\mu f_{j}$ finds it weakly optimal to target
given $\hat{\mu}$.

The range of $\hat{u}_{k}(\mu_{k})$ is $[0,\theta_{k}]$ since no type of
higher state ever targets state $\theta_{k}$, so payoffs to targeting
$\theta_{k}$ cannot exceed $\theta_{k}$ itself.

Define

$S(v):=\\{\theta_{k}:\theta_{k}>v\\}$

to be the set of states under which the receiver optimally takes an action
that yields the sender a payoff greater than $v$. Then
$\hat{\mu}_{k}(v)<\infty$ iff $\theta_{k}\in S(v)$, and since play is
supported on $\\{\hat{\mu}_{k}(u_{k}(\mu/r_{j}(k)))f_{k}:\theta_{k}\in
A_{j}(\mu)\\}$ and $\sigma(\hat{\mu}_{k}(u)f_{k}|\hat{\mu}_{k}(u)f_{k})=1$,
$S(v)$ is exactly the set of states that are targeted by some type under
$\sigma$ to obtain a payoff of $v$.

Given a burden of proof vector $\hat{\mu}(v)=(\hat{\mu}_{k}(v))_{\theta_{k}\in
S(v)}$, the associated _frontier_ consists of all types that are just able to
meet some component of $\hat{\mu}(v)$ with no slack, that is, all types
$\tilde{\mu}_{j}f_{j}$ such that

$r_{j}(k)\tilde{\mu}_{j}=\hat{\mu}_{k}(v)\text{ for some }\theta_{k}\in
S(v),\text{ and }\not\exists\theta_{k^{\prime}}\in S(v)\text{ s.t.
}r_{j}(k^{\prime})\tilde{\mu}_{j}>\hat{\mu}_{k^{\prime}}(v).$ (4)

Given a particular burden of proof function $\hat{\mu}$, the implied frontier
for payoff $v$ is
$\tilde{\mu}[\hat{\mu}(v)]=(\tilde{\mu}_{1},\ldots,\tilde{\mu}_{l-1},\hat{\mu}_{l}(v),\ldots,\hat{\mu}_{J}(v))$
if $S(v)=\\{\theta_{l},\ldots,\hat{\theta}_{J}\\}$ where
$\tilde{\mu}_{1},\ldots,\tilde{\mu}_{l-1}$ satisfy eq. LABEL:frontier.

Let the set of states under which some type of sender obtains payoff $v$ and
finds it weakly optimal to target state $\theta_{k}$ be

$\tau_{\hat{\mu}}^{opt}(\theta_{k},v)=\\{\theta_{j}:f_{k}\in A_{j}(\mu)\text{
for some }\mu\\}$

and let the set of states such that some type of sender obtains payoff $v$ by
targeting a state $\theta_{k}$ with strictly positive probability under
$\sigma$ be

$\tau_{\hat{\mu}}^{supp}(\theta_{k},v)=\\{\theta_{j}:\hat{\mu}_{k}(v)f_{k}\in\text{supp
}\sigma(\cdot|\mu f_{j})\text{ for some }\mu\\}.$

Of course,
$\tau_{\hat{\mu}}^{supp}(\theta_{k},v)\subseteq\tau_{\hat{\mu}}^{opt}(\theta_{k},v)$.

For convenience of notation, we extend the definitions of these set-valued
functions to any set of inputs (rather than a single input) by letting the
function of the set be the union of the function applied to each individual
element of the input set: thus for every set $S$ of states,
$\tau_{\hat{\mu}}^{opt}(S,v)=\bigcup_{\theta_{k}\in
S}\tau^{opt}_{\hat{\mu}}(\theta_{k},v)$ and
$\tau_{\hat{\mu}}^{supp}(S,v)=\bigcup_{\theta_{k}\in
S}\tau_{\hat{\mu}}^{supp}(\theta_{k},v)$, and for every set
$\omega\subseteq[0,1]$, we let
$A_{j}(\omega)=\bigcup_{\mu\in\omega}A_{j}(\mu)$.

Additionally, we define the expectation of the state under the (receiver’s)
belief that the the sender is a type that receives $v$ under $\hat{\mu}$ and
finds it weakly optimal to target a state in $S$ as follows.

$V_{\hat{\mu}}(S,\hat{\mu}(v))=\dfrac{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(S,v)}\beta_{0}(\theta_{j})\theta_{j}g(\tilde{\mu}_{j}[\hat{\mu}(v)])\frac{d\tilde{\mu}_{j}[\hat{\mu}(v)]}{dv}}{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(S,v)}\beta_{0}(\theta_{j})g(\tilde{\mu}_{j}[\hat{\mu}(v)])\frac{d\tilde{\mu}_{j}[\hat{\mu}(v)]}{dv}}.$

In contrast, the expectation of the state under the receiver’s true belief
over $\theta$ conditional on knowing that the sender has sent some message
that yields payoff $v$ and targets a state in $S$ is

$W_{\hat{\mu}}(S,v|\sigma):=\dfrac{\sum_{\theta_{j}\in\tau^{opt}_{\hat{\mu}}(S,v)}\beta_{0}(\theta_{j})\theta_{j}g(\tilde{\mu}_{j}[\hat{\mu}(v)])\frac{d\tilde{\mu}_{j}[\hat{\mu}(v)]}{dv}\sigma(\\{\hat{\mu}_{k}f_{k}\\}_{\theta_{k}\in
S}|\tilde{\mu}_{j}[\hat{\mu}(v)]f_{j})}{\sum_{\theta_{j}\in\tau^{opt}_{\hat{\mu}}(S,v)}\beta_{0}(\theta_{j})g(\tilde{\mu}_{j}[\hat{\mu}(v)])\frac{d\tilde{\mu}_{j}[\hat{\mu}(v)]}{dv}\sigma(\\{\hat{\mu}_{k}f_{k}\\}_{\theta_{k}\in
S}|\tilde{\mu}_{j}[\hat{\mu}(v)]f_{j})}=v.$ (5)

For any partial strategy $\hat{\sigma}$ that gives mixing probabilities
between the messages $\hat{\mu}_{k_{i}}(v)\mathbf{f}$, the payoff
$W_{\hat{\mu}}(S,v|\hat{\sigma}(v))$ is always weakly greater than
$V_{\hat{\mu}}(S,v)$. The two are equal exactly when all types obtaining
payoff $v$ that find it weakly optimal to target a state in $M$ do so with
probability $1$.

Fix a frontier $\hat{\mu}(v)$, where $\theta_{l-1}<v\leq\theta_{l}$. It will
be useful to define an undirected graph $H(v)$ on $S(v)$ by adding an edge
between $\theta_{k}$ and $\theta_{k^{\prime}}$ if and only if
$\tau^{opt}_{\hat{\mu}}(\theta_{k},v)\bigcap\tau^{opt}_{\hat{\mu}}(\theta_{k^{\prime}},v)\neq\emptyset$,
that is, if there is some type that finds it optimal to target either state
$\theta_{k}$ or state $\theta_{k^{\prime}}$, and is indifferent between the
two. Let $C$ be the collection of connected components of $H(v)$.

We use the following algorithm to partition $S(v)$ at a given frontier
$\hat{\mu}(v)$.

Algorithm: This algorithm calculates the payoffs to targeting a state in
$S(v)$ at frontier $\hat{\mu}(v)$ when all types that do not obtain higher
payoffs than $v$ and who can target some $\hat{\mu}_{k}(v)f_{k},\theta_{k}\in
S(v)$ target the highest-payoff of these messages among those that they can,
and assigns states $\theta_{k}$ to the same partition element if, across them,
$\hat{\mu}_{k}(v)f_{k}$ must result in the same payoff, and for $\alpha$ close
to $1$, $\alpha\hat{\mu}_{k}(v)f_{k}$ must also result in the same payoff, so
that for states under which types at the frontier are indifferent between such
messages, they remain so for nearby frontiers.

First, note that if $\sigma$ is such that, when there is a collection of
states $\Sigma\subseteq S(v)$ such that, over an interval of payoffs, there
always exists between any 2 states in $\Sigma$ a path of other states in
$\Sigma$ such that there are types that mix with interior probability between
any two successive states, then for all
$\theta_{k},\theta_{k^{\prime}}\in\Sigma$,

$\frac{r_{j}(k)}{r_{j}(k^{\prime})}=\frac{\hat{\mu}_{k}(u)}{\hat{\mu}_{k^{\prime}}(u)}=\dfrac{\frac{d\hat{\mu}_{k}(u)}{du}}{\frac{d\hat{\mu}_{k^{\prime}}(u)}{du}}\
\
\left(=\dfrac{\frac{d\hat{u}_{k^{\prime}}(\hat{\mu}_{k^{\prime}}(u))}{d\mu}}{\frac{d\hat{u}_{k}(\hat{\mu}_{k}(u))}{d\mu}}\right)$
(6)

for all $u$ in the interval of payoffs and for all $j$ that target some state
in $\Sigma$ at the frontier $\hat{\mu}(u)$.

We define

$\Delta_{n}(\Sigma,\hat{\alpha}):=\frac{d^{n}}{d\alpha^{n}}\dfrac{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(\Sigma,v)}\beta_{0}(\theta_{j})\theta_{j}g\left(\alpha\tilde{\mu}_{j}[\hat{\mu}(v)]\right)\tilde{\mu}_{j}[\hat{\mu}(v)]}{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(\Sigma,v)}\beta_{0}(\theta_{j})g\left(\alpha\tilde{\mu}_{j}[\hat{\mu}(v)]\right)\tilde{\mu}_{j}[\hat{\mu}(v)]}\Bigg{|}_{\alpha=\hat{\alpha}}.$

This is equal to the $n$th derivative of the payoff to the set of senders in
states that target a state in $\Sigma$ with positive probability at frontier
$\hat{\mu}(v)$, that have an amount
$\hat{\alpha}\tilde{\mu}_{j}[\hat{\mu}(v)]$ of data, when we assume that eq. 6
holds over $\Sigma$.

Start with a collection of assigned partition elements,
$\mathcal{A}_{0}=\emptyset$, and a collection of sets of unassigned states,
$\mathcal{C}_{0}=C$. Given $\mathcal{A}_{n}$ and $\mathcal{C}_{n}$, initialize
$\mathcal{A}_{n+1}=\mathcal{C}_{n+1}=\emptyset$, and, taking each set
$S\in\mathcal{C}_{n}$ sequentially, proceed as follows:

  1. 1.

Take all subsets $\Sigma\subseteq S$ and calculate $\Delta_{0}(\Sigma,1)$.
Tiebreak any with the same value by
$\Delta_{1}(\Sigma,1),\Delta_{2}(\Sigma,1),\ldots$, successively, and take the
largest subset $\Sigma$ that is maximal. Label it with
$\tau_{\hat{\mu}}^{supp}(\Sigma,v)$, and add it to $\mathcal{A}_{n+1}$.

Note that this implies that
$\frac{du_{k}(\hat{\mu}_{k}(v))}{d\mu_{k}}=\frac{\Delta_{1}(\Sigma,1)}{\hat{\mu}_{k}(v)}$
when equation 6 holds for $\theta_{k},\theta_{k^{\prime}}\in\Sigma$ over
$[v-\epsilon,v]$, $\epsilon>0$.

  2. 2.

Take $S\setminus\Sigma$, and let $C(S)$ be the collection of connected
components of the graph on $S$ constructed analogously to $H(v)$. Add $C(S)$
to $C_{n+1}$ (i.e. augment $C_{n+1}$ as the union of itself and $C(S)$).

  3. 3.

Repeat on $\mathcal{A}_{n+1}$ and $\mathcal{C}_{n+1}$ until
$\mathcal{C}_{n+1}=\emptyset$.

Putatively, if senders of types $\alpha\tilde{\mu}_{j}[\hat{\mu}(v)]f_{j}$ for
some $\theta_{j}\in\tau_{\hat{\mu}}^{supp}(v)$ pooled with each other, then
payoffs are equal to

$\hat{u}_{k}(\alpha\hat{\mu}_{k}(v))=v_{\Sigma}(\alpha,\hat{\mu}(v)):=\dfrac{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(\Sigma,v)}\beta_{0}(\theta_{j})\theta_{j}g\left(\alpha\tilde{\mu}_{j}[\hat{\mu}(v)]\right)\tilde{\mu}_{j}[\hat{\mu}(v)]}{\sum_{\theta_{j}\in\tau^{supp}_{\hat{\mu}}(\Sigma,v)}\beta_{0}(\theta_{j})g\left(\alpha\tilde{\mu}_{j}[\hat{\mu}(v)]\right)\tilde{\mu}_{j}[\hat{\mu}(v)]}\Bigg{|}_{\alpha=\hat{\alpha}},$

which is continuous in $\alpha$ because $g$ is continuous everywhere in
$(0,1]$.131313It is important for continuity that $g(1)=0$, since this ensures
that $g$ is continuous at $\mu=1$. The burden-of-proof function for
$\underline{v}\leq v$ is then given by

$\mu_{\Sigma}^{put}(\underline{v}):=\\{v_{\Sigma}^{-1}(\underline{v},\hat{\mu}(v))\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\in
S(v)},$

where $v_{\Sigma}^{-1}(\underline{v},\hat{\mu}(v))$ is the inverse of
$v_{\Sigma}(\cdot,\hat{\mu}(v))$.

The reason that a partition element is a subset of targetable states in which
all messages must achieve the same payoff at the is that, since $\Sigma$ is a
maximal highest-value subset over those that do not already have a higher
value, it is either partitionable into smaller subsets, each of which also
achieves the same value, or not; but in either case, in each minimal subset
that achieves the maximal value, there is a path of messages between any two
messages in the subset such that, in the targeting strategy, some type mixes
with strictly positive probability between any two adjoining messages. The
reason for this is that, for any smaller subset
$\Sigma^{\prime}\subset\hat{\Sigma}$, we have that
$V_{\hat{\mu}}(\Sigma^{\prime},\hat{\mu}(v))<V_{\hat{\mu}}(\hat{\Sigma},\hat{\mu}(v))$
if $\Sigma$ is a minimal subset that achieves the maximal value. Since the
expectation of the state conditional on knowing the message played is in
$\hat{\Sigma}$ is at least $V_{\hat{\mu}}(\hat{\Sigma},\hat{\mu}(v))$, there
must be some message that yields payoff at least
$V_{\hat{\mu}}(\Sigma,\hat{\mu}(v))$. But since there is no message, and
indeed no proper subset of messages in $\hat{\Sigma}$ that achieve payoff
$V_{\hat{\mu}}(\Sigma,\hat{\mu}(v))$ if all types that can play one of them
do, it must be that for any subset, there is a type that can play some message
in the subset but plays a message outside the subset with positive
probability.

The reason the same holds true in frontiers to the left of $\hat{\mu}(v)$ is
that, if $\Delta_{0}(\Sigma,1)$ is uniquely maximal, then
$\Delta_{0}(\Sigma,\alpha)$ is still greater than
$\Delta_{0}(\Sigma^{\prime},1)$ for any $\Sigma^{\prime}$ and $\alpha$
sufficiently close to 1. So, in any state under which senders target a state
in $\Sigma$ at $\hat{\mu}(v)$, it remains optimal for them to do so for
$\alpha$ close to 1, assuming the putative payoffs above. In addition, the
putative payoffs are feasible, because every subset of $\Sigma$ has lower
value. If tiebroken by $\Delta_{1},\Delta_{2},$ and so on, then although
$\Delta_{0}(\Sigma,1)$ is not uniquely maximal, $\Sigma$ does maximize
$\Delta(\cdot,1)$ immediately to the left of $\hat{\mu}(v)$.

We will use the partition constructed by the algorithm to construct the
equilibrium in chunks. For consistency, we want the following condition:

Condition 1. The value of each partition element constructed using the
algorithm is the same, and is equal to $v$.

Under this condition, there is a partial strategy $\hat{\sigma}$ on each
partition element such that $W_{\hat{\mu}}(\theta_{k},v|\hat{\sigma})=v$ for
all states $\theta_{k}$ in the partition element, and furthermore, there is no
partial strategy on a subset of messages in that partition element such that
all messages in the subset result in the same payoff that is greater than $v$.

If Condition 1 holds at $\hat{\mu}(v)$ and $\Sigma$ is the partition
constructed using the algorithm at $\hat{\mu}(v)$, then there exists some
$\epsilon>0$ such that, for all $\underline{v}\in[v-\epsilon,v]$, Condition 1
holds for the frontier
$\\{v_{\Sigma}^{-1}(\underline{v},\hat{\mu}(v))\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\in
S(v)}$. To show this, observe the following claim, which follows directly from
statement of the condition and from continuity of
$v_{\Sigma}(\alpha,\hat{\mu}(v))$:

###### Claim C.1

Let the set of types that target a state in $\Sigma$ and achieve a payoff of
$\underline{v}$ under $\mu_{\Sigma}^{put}$ be
$\tau_{\Sigma}^{put}(\underline{v})$.

If Condition 1 holds at $\hat{\mu}(v)$, then if there exists no
$v^{\prime}\in(\underline{v},v]$ such that either

  1. 1.

There is a type $t\in\tau_{\Sigma}^{put}(v^{\prime})$ such that $t$ can
imitate a higher-value state, i.e. there exists partition element such that
$\Sigma^{\prime}$
$v_{\Sigma^{\prime}}^{-1}(v^{\prime\prime},\hat{\mu}(v))\hat{\mu}_{k}(v)f_{k}\subseteq
t$ for some $v^{\prime\prime}>v^{\prime}$

  2. 2.

There is a partition element $\Sigma$ with a subset
$\Sigma^{\prime}\subseteq\Sigma$ such that
$v_{\Sigma^{\prime}}(v_{\Sigma}^{-1}(v^{\prime},\hat{\mu}(v)),\hat{\mu}(v))>v^{\prime}$,

then Condition 1 continues to hold at $\hat{\underline{v}}$.

Note that, because for any partition element $\Sigma^{\prime}\neq\Sigma$
either
$v_{\Sigma^{\prime}}^{-1}(v,\hat{\mu}(v))\hat{\mu}_{k}(v)f_{k}\not\subseteq
t$, or $\Delta_{n}(\Sigma^{\prime},1)<\Delta_{n}(\Sigma,1)$ for some $n$ such
that $\Delta_{i}(\Sigma^{\prime},1)=\Delta_{i}(\Sigma,1)$ for all $i<n$, the
continuity of $v_{\Sigma}(\alpha,\hat{\mu}(v))$ implies that for
$\underline{v}$ close to $v$ (1) cannot not hold. Again by continuity, (2)
cannot hold for $\underline{v}$ close to $v$ because for all
$\Sigma^{\prime}\subseteq\Sigma$,
$v_{\Sigma^{\prime}}(v_{\Sigma}^{-1}(v,\hat{\mu}(v)),\hat{\mu}(v))\leq v$ and
$\Delta_{n}(\Sigma^{\prime},1)<\Delta_{n}(\Sigma,1)$ for some $n$ such that
$\Delta_{i}(\Sigma^{\prime},1)=\Delta_{i}(\Sigma,1)$ for all $i<n$.

We will use this to construct the equilibrium in segments over which Condition
1 holds, and re-construct partitions using the algorithm in at most countably
many points at which either (1) or (2) holds. For every reasonable example we
can think of, the number of such points (and thus steps in the construction)
is not just countable, but finite.

Now we turn to constructing larger pooling sets when there is a positive-
measure set of types that can achieve the frontier payoff. Given that types
support their play on
$\\{\hat{\mu}_{k}(u_{k}(\mu/r_{j}(k)))f_{k}:\theta_{k}\in A_{j}(\mu)\\}$, and
$\hat{u}_{k}(\mu_{k})$ is increasing, all types capable of sending a message
in $\hat{\mu}(v)\cdot\mathbf{f}$ achieve a payoff of at least $v$. We define
the set of types that are incapable of sending a message in
$\hat{\mu}(v)\cdot\mathbf{f}$, but capable of sending a message in set $M$, as
$T(v,M)$. We will denote the payoff to the sender of the receiver knowing they
are one of a set of types that has positive probability measure under the
receiver’s prior as $U(T)$, and in particular,

$U(T(v,M))=\frac{\sum_{j=1}^{J}\beta_{0}(\theta_{j})\theta_{j}\max(\max_{k\geq
l}(G(\frac{\hat{\mu}_{k}(v)}{r_{j}(k)}))-\min\\{G(\mu):\exists m\in M\text{
s.t. }m\subseteq\mu
f_{j}\\},0)}{\sum_{j=1}^{J}\beta_{0}(\theta_{j})\max(\max_{k\geq
l}(G(\frac{\hat{\mu}_{k}(v)}{r_{j}(k)}))-\min\\{G(\mu):\exists m\in M\text{
s.t. }m\subseteq\mu f_{j}\\},0)}.$

Note that $\sup_{M}U(T(v,M))\geq v$, because $\lim_{\alpha\rightarrow
1}U(T(v,\alpha\hat{\mu}))=v$. If there is a positive-measure type set $T(v,M)$
that achieves the value $\sup_{M}U(T(v,M))$, then take the largest such set
and call it $\hat{T}_{\hat{\mu}}^{max}(v)$. Then the following hold:

  1. 1.

If there exists a set $T(v,M)$ that achieves the value $\sup_{M}U(T(v,M))$,
then there is a unique largest set that does so, and so
$\hat{T}_{\hat{\mu}}^{max}(v)$ is well-defined.

  2. 2.

Whenever $\hat{T}_{\hat{\mu}}^{max}(v)$ exists, there exist
$\mu_{l},\ldots,\mu_{J}$ such that
$\hat{T}^{\max}_{\hat{\mu}}(v)=T(v,\\{\mu_{l}f_{l},\ldots,\mu_{J}f_{J}\\})$.

  3. 3.

Whenever $\hat{T}_{\hat{\mu}}^{max}(v)$ exists, there exists a partial
strategy $\hat{\sigma}:\hat{T}^{\max}_{\hat{\mu}}(v)\rightarrow
M=\\{\mu_{l}f_{l},\ldots,\mu_{J}f_{J}\\}$ such that the payoff to any message
$m\in M$ given that senders in $\hat{T}^{\max}_{\hat{\mu}}(v)$ play according
to $\hat{\sigma}$ is $\hat{U}_{\hat{\mu}}(v)$.

The first point follows from the fact that, unless the union of two such sets
yields payoff at least $\hat{U}_{\hat{\mu}}(v)$, then their intersection –
which corresponds to the pool of types implemented by a different message set
– yields strictly greater payoff. To see the 2nd point, simply take $\mu_{k}$
to be the minimum amount of data distributed $f_{k}$ such that the dataset
still contains a message in $M$, for each $k\geq l$, and note that the
resulting set of types is a subset of $T(v,M)$ that has a smaller mass of
types $\theta_{j}$, $j<l$ but the same mass of types $\theta_{k}$, $k\geq l$.
Since $U(T(v,M))\geq v\geq\theta_{l-1}$, this can only improve the payoff to
the pool. The last point comes from the fact that, if
$\hat{T}^{\max}_{\hat{\mu}}(v)$ is a maximum-payoff pool, then for each subset
$S\subseteq M$, the payoff to the pool implemented by $S$ is no greater than
$U(\hat{T}^{\max}_{\hat{\mu}}(v))$, which is sufficient to ensure that
$\hat{\sigma}$ exists. In addition, $U(T(v,M))$ is absolutely continuous with
respect to every component of $\hat{\mu}(v)$ and each $\mu_{k}$.

###### Lemma C.2

If $\hat{T}^{\max}_{\hat{\mu}}(v)$ exists, then Condition 1 is satisfied by
the burden of proof vector $M=\\{\mu_{l}f_{l},\ldots,\mu_{J}f_{J}\\}$ such
that $\hat{T}^{\max}_{\hat{\mu}}(v)=T(v,M)$.

  * Proof

Suppose not; then one of two cases is true:

    1. 1.

There is a collection of states $\Sigma\subset S(v)$ such that
$V_{\hat{\mu}}(\Sigma,M)>v$.

Then, since $V_{\hat{\mu}}(\Sigma,\alpha(\mu_{k}f_{k})_{k=l}^{J})$ is
continuous in $\alpha$, there is $\underline{\alpha}<1$ such that
$V_{\hat{\mu}}(\Sigma,\alpha(\mu_{k}f_{k})_{k=l}^{J})>v$ for all
$\alpha\in[\underline{\alpha},1]$. Consider an alternative type set,
$T(M_{\underline{\alpha},\Sigma},v)$ where $M_{\underline{\alpha},\Sigma}$
includes the messages $\mu_{k}f_{k}$ for $\theta_{k}\in S(v)\setminus\Sigma$,
and the messages $\underline{\alpha}\mu_{k}f_{k}$ for $\theta_{k}\in\Sigma$.

For $\underline{\alpha}$ small enough, the set of types in
$T(M_{\underline{\alpha},\Sigma},v)\setminus T(M,v)$ includes exactly those in
frontiers $(\alpha M)_{\alpha=\underline{\alpha}}^{1}$ that find it weakly
optimal to target a state in $\Sigma$. So, the expectation of the state given
that the sender’s type is in $T(M_{\underline{\alpha},\Sigma},v)\setminus
T(M,v)$ exceeds $v$, and so $T(M_{\underline{\alpha},\Sigma},v)$ is higher-
payoff than $T(M,v)$, contradicting that
$T(M,v)=\hat{T}^{\max}_{\hat{\mu}}(v)$.

    2. 2.

There is a element of the partition, $\Sigma^{\prime}\subset S(v)$, such that
$V_{\hat{\mu}}(\Sigma^{\prime},M)>v$.

Then WLOG let $\Sigma^{\prime}$ be the lowest-value element of the partition.
Similarly to the above, since
$V_{\hat{\mu}}(\Sigma,\alpha(\mu_{k}f_{k})_{k=l}^{J})$ is continuous in
$\alpha$, there is $\bar{\alpha}>1$ such that
$V_{\hat{\mu}}(\Sigma,\alpha(\mu_{k}f_{k})_{k=l}^{J})<v$ for all
$\alpha\in[1,\bar{\alpha}]$. Consider an alternative type set,
$T(M_{\bar{\alpha},\Sigma^{\prime}},v)$ where
$M_{\bar{\alpha},\Sigma^{\prime}}$ includes the messages $\mu_{k}f_{k}$ for
$\theta_{k}\in S(v)\setminus\Sigma^{\prime}$, and the messages
$\bar{\alpha}\mu_{k}f_{k}$ for $\theta_{k}\in\Sigma^{\prime}$.

For $\bar{\alpha}$ small enough, the set of types in $T(M,v)\setminus
T(M_{\bar{\alpha}},\Sigma^{\prime})$ includes exactly those in frontiers
$(\alpha M)_{\alpha=1}^{\bar{\alpha}}$ that find it weakly optimal to target a
state in $\Sigma$. Then the expectation of the state given that the sender’s
type is in $T(M,v)\setminus T(M_{\bar{\alpha}},\Sigma^{\prime})$ is less than
$v$, so the expectation given that the type is in
$T(M_{\bar{\alpha}},\Sigma^{\prime})$ exceeds $v$, contradicting that
$T(M,v)=\hat{T}^{\max}_{\hat{\mu}}(v)$.

Since neither case is possible, $M$, taken as the payoff frontier
corresponding to $v$, must satisfy Condition 1.

The iterative algorithm to construct the equilibrium of 3.1 starts from the
highest-potential-payoff senders and creates payoff frontiers that satisfy
Condition 1. It proceeds as follows:

  1. 1.

Start with $l=J$ and $\hat{\mu}_{J}(\theta_{J})=1$.

  2. 2.

For each $l$, construct frontiers $\hat{\mu}_{k}(v)$ as follows:

    1. (a)

Start at $v=\theta_{l}$ and burden-of-proof vector $\hat{\mu}(\theta_{l})$, as
constructed from the previous step. For all $v>\theta_{l}$, let $\hat{\mu}(v)$
be as already constructed. Define

$\check{\mu}_{l}(\theta_{l})=\max\\{\mu:\exists j<l\text{ s.t.
}\tilde{\mu}_{j}[\hat{\mu}(\theta_{l})]\geq\mu\\},$

and rewrite
$\hat{\mu}(\theta_{l})=(\check{\mu}_{l}(\theta_{l}),\hat{\mu}_{l+1}(\theta_{l}),\ldots,\hat{\mu}_{J}(\theta_{l}))$.
Proceed as below to rewrite $\hat{\mu}(v)$ for $v<\theta_{l}$:

    2. (b)

Fix $S=\\{\theta_{k}\\}_{k=l}^{J}$. Given the frontier $\hat{\mu}(v)$, check
if $\hat{T}_{\hat{\mu}}^{max}(v)$ exists, and if so, find
$M=\\{\mu_{l}f_{l},\ldots,\mu_{J}f_{J}\\}$ that implements
$\hat{T}_{\hat{\mu}}^{max}(v)$ and rewrite $\hat{\mu}(v)=M$.

    3. (c)

At $\hat{\mu}(v)$, using the algorithm, partition $S$ into subsets of states,
and calculate $v_{\Sigma}(\alpha,\hat{\mu}(v))$ for all $\alpha\in[0,1]$ for
each subset. Take the lowest-value frontier, $\hat{\mu}(v^{\prime})$, under
putative payoffs $v_{\Sigma}(\alpha,\hat{\mu}(v))$ such that the conditions of
Claim C.1 are satisfied and such that
$\hat{T}_{\hat{\mu}}^{max}(v^{\prime\prime})$ does not exist for any
$v^{\prime\prime}\in(v^{\prime},v]$, and assign strategies according to
Algorithm 2 between $\hat{\mu}(v)$ and the new frontier
$\hat{\mu}(v^{\prime})$.

    4. (d)

Set $v=v^{\prime}$ and set $\hat{\mu}(v^{\prime})$ as the new frontier, and
repeat the above 2 steps until $v^{\prime}=0$.

  3. 3.

Repeat the above steps for each $l$ in descending order until $l=1$, and fix
the resulting $\hat{\mu}$.

The existence of an imitation equilibrium, and the monotonicity of
$\hat{u}_{k}$, follow directly from this construction. Continuity of
$\hat{u}_{k}$ also follows from this construction. The value of $\hat{u}_{k}$
is defined on series of closed intervals on each of which it is continuous –
$v_{\Sigma}(\alpha,\mu)$ is continuous in $\alpha$, and $\hat{u}_{k}(\mu)$ is
constant for $\mu f_{k}\in T(v,M)$. Together, these cover the domain of
$u_{k}$, that is, $[0,1]$, and they overlap only at their endpoints, at which
they coincide.

The proof that the imitation equilibrium outcome is unique relies on the
following lemma, which we prove first.

###### Lemma C.3

If $M$ is a collection of messages and
$(\underline{\mu}_{1}f_{1},\ldots,\underline{\mu}_{i}f_{i};\underline{\mu}_{i+1}f_{i+1},\ldots,\underline{\mu}_{J}f_{J})$
is the frontier of types achieving a payoff of at least $v$ under
$\sigma^{*}$, where $\theta_{i}<v\leq\theta_{i+1}$, then

$\mathbb{E}_{\beta_{\sigma^{*}}}[\theta|t\in\mathcal{T}_{\infty}^{+}(\\{\underline{\mu}_{j}f_{j}\\}_{j=1}^{J})\setminus\mathcal{T}_{\infty}^{+}(M)]\geq
v$

whenever
$\mathcal{T}_{\infty}^{+}(\\{\underline{\mu}_{j}f_{j}\\}_{j=1}^{J})\setminus\mathcal{T}_{\infty}^{+}(M)$
is nonempty.

  * Proof of Lemma

Denote
$T(v,M)=\mathcal{T}_{\infty}^{+}(\\{\underline{\mu}_{j}f_{j}\\}_{j=1}^{J})\setminus\mathcal{T}_{\infty}^{+}(M)$.
Let
$(\bar{\mu}_{1},\ldots,\bar{\mu}_{i};\bar{\mu}_{i+1},\ldots,\bar{\mu}_{J})$ be
the minimum masses of data distributed like
$f_{1},\ldots,f_{i};f_{i+1},\ldots,f_{J}$, respectively, necessary to send
some message in $M$. Then

$\mathbb{E}_{\beta_{\sigma^{*}}}[\theta|t\in
T(v,M)]=\frac{\sum_{j=1}^{J}\beta_{0}(\theta_{j})\theta_{j}(G(\bar{\mu}_{j})-G(\underline{\mu}_{j}))}{\sum_{j=1}^{J}\beta_{0}(\theta_{j})(G(\bar{\mu}_{j})-G(\underline{\mu}_{j}))}.$

If
$(\bar{\mu}_{i+1},\ldots,\bar{\mu}_{J})\leq(\underline{\mu}_{i+1},\ldots,\underline{\mu}_{J})$
pointwise, then $T(v,M)$ is empty. Otherwise, let the states
$j_{1},\ldots,j_{A}$ be the maximal set such that
$(\bar{\mu}_{j_{1}},\ldots,\bar{\mu}_{j_{A}})>(\underline{\mu}_{j_{1}},\ldots,\underline{\mu}_{j_{A}})$
pointwise. Call the set of types that send $\mu^{\prime}f_{j_{a}}$ with
positive probability under $\sigma^{*}$ by
$\tau_{\sigma^{*}}^{supp}(\mu^{\prime}f_{j_{a}})$, and let $\theta(t)$ refer
to the state corresponding to the distribution of dataset $t$. Denote by
$\hat{\sigma}_{v}$ the partial strategy, restricting to types in $T(v,M)$,
where those types play as they do in $\sigma^{*}$, and assume that the
receiver knows the sender is in $T(v,M)$ and playing according to this
strategy.

Let $\phi_{\sigma^{*}}$ be a joint density over types and messages induced by
$\sigma^{*}$, so that for type $t=\mu f_{j}$ and message
$m=\tilde{\mu}f_{\tilde{j}_{a}}$,

$\phi(t,m):=g(\mu
r_{j}(\tilde{j}))\sigma^{*}(m|t)\beta_{0}(\theta_{j})r_{\theta_{j}}(j^{\prime})$

is the density on the event that the sender is type $t$ and plays message $m$,
when $t$ plays $m$ with positive probability. In the case when payoffs under
$u_{\sigma^{*}}$ are strictly increasing at $\mu^{\prime}f_{j_{a}}$, each
sender who plays $\mu^{\prime}f_{j_{a}}$ is randomizing between at most a
finite number of messages in their mixed strategy, one corresponding to each
state that is weakly optimal for them to imitate. Thus, they play each message
in the support of their strategy with strictly positive probability, rather
than randomizing with some density over a continuum of messages; $\phi$
therefore fully captures the distribution of play for senders playing
$\mu^{\prime}f_{j_{a}}$.

When payoffs are strictly increasing at $\mu^{\prime}f_{j_{a}}$, we know that
for every $\mu^{\prime}f_{j_{a}}$ that is in $T(v,M)$ and is on-path in
$\sigma^{*}$, the receiver’s inference when they know the sender’s type is in
$T(v,M)$ in addition to knowing they played message $\mu^{\prime}f_{j_{a}}$ is
weakly better than if they only know $\mu^{\prime}f_{j_{a}}$ was the message
played. Formally,

$\begin{split}\mathbb{E}_{\hat{\sigma}_{v}}[\theta|\mu^{\prime}f_{j_{a}}]&=\frac{\sum_{t\in\tau_{\sigma^{*}}^{supp}(\mu^{\prime}f_{j_{a}})\bigcap
T(v,M)}\theta(t)\phi(t,\mu^{\prime}f_{j_{a}}))}{\sum_{t\in\tau_{\sigma^{*}}^{supp}(\mu^{\prime}f_{j_{a}})\bigcap
T(v,M)}\phi(t,\mu^{\prime}f_{j_{a}})}\\\
&\geq\frac{\sum_{t\in\tau_{\sigma^{*}}^{supp}(\mu^{\prime}f_{j_{a}})}\theta(t)\phi(t,\mu^{\prime}f_{j_{a}})}{\sum_{t\in\tau_{\sigma^{*}}^{supp}(\mu^{\prime}f_{j_{a}}))}\phi(t,\mu^{\prime}f_{j_{a}})}\\\
&\geq v\end{split}$ (7)

where the first inequality comes from the fact that $\theta_{j_{a}}\geq
v>\theta(t)$ whenever $\theta(t)\neq\theta_{j_{a}}$, and
$\mu^{\prime}f_{j_{a}}\in T(v,M)$ only if all types that play it under
$\sigma^{*}$ are also in $T(v,M)$.

Since, of course, payoffs under $u_{\sigma^{*}}$ may not be strictly
increasing at every $\mu^{\prime}f_{j_{a}}$ in $T(v,M)$, we have to separately
consider the case in which they are constant, i.e. the case where there are
positive-measure pools $T$ of senders achieving the same payoff $v^{\prime}>v$
under $\sigma^{*}$ with $T\bigcap T(v,M)$ nonempty. Then let $M^{\prime}$ be
the set of messages that implements the pool, and

$\mathbb{E}_{\hat{\sigma}_{v}}[\theta|m\in
M^{\prime}]=\mathbb{E}_{\hat{\sigma}_{v}}[\theta|t\in T\bigcap T(v,M)].$

The value of $T\setminus T(v,M)$ is equal to the value of
$T\bigcap\mathcal{T}_{\infty}^{+}(M)$, which is no more than $v^{\prime}$
since $T=\hat{T}^{max}_{\hat{\mu}}(v^{\prime})$, and so it contains no subsets
of higher value. Therefore, $\mathbb{E}_{\hat{\sigma}_{v}}[v(\theta)|t\in
T\bigcap T(v,M)]\geq v^{\prime}\geq v$.

Then, taking the total expectation over both cases, the expectation of
$\theta$ given that the sender’s type is in $T(v,M)$ is a weighted average of
$\mathbb{E}_{\hat{\sigma}_{v}}[\theta|\mu^{\prime}f_{j_{a}}]$ over on-path
messages $\mu^{\prime}f_{j_{a}}$ in $T(v,M)$ in which the payoff is strictly
decreasing; and the value over positive-measure sets of equal payoff. We have
shown that each component is no less than $v$, and so the weighted average is
also at least $v$.

To prove uniqueness using the lemma, suppose for the sake of contradiction
that there are equilibria $\sigma^{*}$ and $\sigma^{*^{\prime}}$ with 2
distinct imitation equilibrium outcomes, given by distinct payoff frontiers
$\hat{u}$ and $\hat{u}^{\prime}$ respectively (identically, burden-of-proof
vectors $\hat{\mu}$ and $\hat{\mu}^{\prime}$). There must then be some $v$
such that
$\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq
v})\setminus\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v})$ is nonempty. We know from the lemma that

$\mathbb{E}[\theta|t\in\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq
v})\setminus\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v})]\geq v.$

Define

$I(\hat{\mu},\hat{\mu}^{\prime},v)=\\{\mu
f_{k}\in\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq
v})\setminus\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v}):\theta_{k}\geq v\\}.$

From Lemma LABEL:lem:cts_truth_leaning_reduction, we know that for all $\mu
f_{k}\in I(\hat{\mu},\hat{\mu}^{\prime},v)$, it must be that
$\sigma^{*^{\prime}}(\mu f_{k}|\mu f_{k})=1$, since we have
$u_{\sigma^{*^{\prime}}}(\mu f_{k})<v$. We know that
$I(\hat{\mu},\hat{\mu}^{\prime},v)$ is nonempty, since the payoff frontier for
$v$ can lag behind in $\sigma^{*}$ relative to $\sigma^{*^{\prime}}$ only if
$\hat{\mu}_{k}(v)>\hat{\mu}_{k}^{\prime}(v)$.

We also know that the set of types that imitate some type in
$I(\hat{\mu},\hat{\mu}^{\prime},v)$ under $\sigma^{*^{\prime}}$ is a subset of
$\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq
v})\setminus\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v})$, since the types in $I(\hat{\mu},\hat{\mu}^{\prime},v)$ are strictly
suboptimal to imitate for any types in
$\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v})$ if the receiver forms inferences using $\beta_{(}\sigma^{*^{\prime}})$;
and they cannot be imitable by any types not in
$\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq v})$.

Let $m_{\sigma^{*^{\prime}}}(t)$ be the realization of the message played by
$t$ under $\sigma^{*^{\prime}}$, that is, a random variable with distribution
given by $\sigma^{*^{\prime}}(\cdot|t)$ when taken as a message distribution.
Let $\mathcal{F}_{\sigma^{*^{\prime}}}$ be the marginal distribution over
realized messages played, induced by $\sigma^{*^{\prime}}$.

When we apply Lemma C.3, we get:

$\begin{split}v&\leq\mathbb{E}[\theta|t\in\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}(v)f_{k}\\}_{\theta_{k}\geq
v})\setminus\mathcal{T}_{\infty}^{+}(\\{\hat{\mu}_{k}^{\prime}(v)f_{k}\\}_{\theta_{k}\geq
v})]\\\ &\leq\mathbb{E}[\theta|m_{\sigma^{*}}(t)\in
I(\hat{\mu},\hat{\mu}^{\prime},v)]\\\ &\leq\frac{\sum_{k:\theta_{k}\geq v,\
\hat{\mu}_{k}(v)>\hat{\mu}_{k}^{\prime}(v)}\int_{\mu=\hat{\mu}_{k}^{\prime}(v)}^{\hat{\mu}_{k}(v)}\mathbb{E}_{\beta_{\sigma^{*^{\prime}}}(\cdot|\mu
f_{k})}[\theta]\mathcal{F}_{\sigma^{*^{\prime}}}}{\sum_{k:\theta_{k}\geq v,\
\hat{\mu}_{k}(v)>\hat{\mu}_{k}^{\prime}(v)}\int_{\mu=\hat{\mu}_{k}^{\prime}(v)}^{\hat{\mu}_{k}(v)}\mathcal{F}_{\sigma^{*^{\prime}}}},\end{split}$
(8)

but, as a contradiction, the last term has to be strictly less than $v$, since
$\mathbb{E}_{\beta_{\sigma^{*^{\prime}}}(\cdot|\mu f_{k})}[\theta]<v$ for all
$\mu f_{k}\in I(\hat{\mu},\hat{\mu}^{\prime},v)$.

##  Appendix D Proof of convergence to full-information outcome

###### Corollary D.1

Given any sequence of densities $g_{1},g_{2},\ldots$ on $[0,1]$ with a fixed
mean and variances $Var_{1},Var_{2},\ldots\rightarrow\infty$, the payoff to a
sender conditional on the state, in a sequence of otherwise identical games
with mass distributions $g_{1},\ldots$, converges in probability to their
full-information payoff.

  * Proof

We show that for any $\delta$ and $\epsilon$, there exists $L$ such that for
all $l\geq L$, the distribution $g_{l}$ is such that
$Pr[u_{\sigma^{*}}(\mu,\theta_{k})<\theta_{k}-\delta]<\epsilon$ under every
state.

Define the mean of $\mu$ to be $\bar{\mu}$, and

$B=\max_{j\neq k}\frac{1}{r_{j}(k)}$

so that for any two states $j$ and $k$, the difference between the amount of
the state-$k$ distribution that the mean type under state $k$ has and the
amount the mean type under state $j$ has is $\bar{\mu}(1-B)$.

Suppose that the variance of $\mu$ under density $g_{L}$ is less than
$\Delta^{2}\epsilon^{2}$, where $\Delta>0$ is an arbitrary parameter. Then
there can be at most a probability $\epsilon^{2}$ that the state is $k$ and
the sender has less than $\bar{\mu}-\Delta$ data distributed like $f_{k}$. A
sender under state $j$ has more than $\frac{\bar{\mu}-\Delta}{B}$ data with
probability no more than
$\frac{\Delta^{2}\epsilon^{2}B^{2}}{(\bar{\mu}(1-B)-\Delta)^{2}}$.

Recall that whenever $u_{\sigma^{*}}(\mu,\theta)<\theta$, the type with
dataset $\mu f_{\theta}$ is truthful in equilibrium. So, if under state
$\theta_{k}$ we have
$Pr[u_{\sigma^{*}}(\mu,\theta)<\theta-\delta]\geq\epsilon$, then the type with
$\mu=G^{-1}(\epsilon)$ must obtain payoff less than $\theta-\delta$, and so
must all types with less data, and all such types must be truthful. But the
total mass of all types _not_ in state $k$ that can pool with types with
$\mu\in[G^{-1}(\epsilon^{2}),G^{-1}(\epsilon)]$ cannot exceed

$(J-1)(1-\beta_{0}(\theta_{k}))\frac{\Delta^{2}\epsilon^{2}B^{2}}{(\bar{\mu}(1-B)-\Delta)^{2}}$

and so the payoff to type $G^{-1}(\epsilon)f_{k}$ cannot be less than

$\frac{\epsilon(1-\epsilon)\theta_{k}}{\epsilon(1-\epsilon)+(J-1)(1-\beta_{0}(\theta_{k}))\frac{\Delta^{2}\epsilon^{2}B^{2}}{(\bar{\mu}(1-B)-\Delta)^{2}}}$

which, for small enough $\Delta$, must be at least $\theta_{k}-\delta$. Since
there is always $L$ large enough that $Var_{L}<\epsilon^{2}\Delta^{2}$, we are
done.

All that remains is to note that, since the ex-ante expected payoff must
always be $\mathbb{E}_{\beta_{0}}[\theta]$, this lower bound on the
probability of payoffs less than the full-information payoffs implies a
corresponding upper bound on payoffs exceeding the full-information payoffs,
and so we obtain convergence of the distribution of payoffs, state-by-state,
to those in the outcome where the receiver knows the truth.

##  Appendix E Proof of convergence to continuum limit

  * Proof

The proof of theorem 4.1 uses Lemma C.3 to establish that, for any set of
messages $M$, when the set of all types in
$\mathcal{T}_{\infty}\setminus\mathcal{T}^{+}_{\infty}(M)$ that attain a
payoff of at least $v$ in $\sigma^{*}_{\infty}$ is nonempty, their payoff when
they form a pool is at least $v$. Using this, we show that
$u_{\sigma^{*}_{\infty}}$ is a lower bound on payoffs for types in
$\mathcal{T}_{\infty}$ in the limit. Then, Bayes plausibility implies that

$\lim_{N\rightarrow\infty}\sum_{j=1}^{j}\beta_{0}(\theta_{j})\int_{\mu=0}^{1}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu=\mathbb{E}_{q_{\infty}}[\mathbb{E}_{\beta(\cdot|\sigma(t))}[\theta]|t]=\mathbb{E}_{\beta_{0}}[\theta],$

which in conjunction with the lower bound implies that in the limit outcomes
must coincide exactly with $u_{\sigma^{*}_{\infty}}$ for types in
$\mathcal{T}_{\infty}$.

Before proceeding to construct bounds on payoffs in the finite games, it is
helpful to define a neighborhood of $\mathcal{T}_{\infty}$ as the set of types
in each finite game with datasets distributed similarly to the underlying
distribution in some state. For $\eta\in(0,1]$ and $k\in[0,1]$, define

$S_{N}(\eta,k)=\\{t\in\mathcal{T}_{N}:|t|\geq k\text{ and }\exists\theta\text{
s.t. }\sup_{d}|t(d)-|t|f_{\theta}(d)|\leq\eta\\}.$

Fix an integer $n$. Conditional on $|t|=n$ and the true state being
$\theta_{j}$, the Glivenko-Cantelli theorem states that there is a bound on
the probability that $\sup_{d}|\sum_{x=1}^{d}t(d)-\frac{n}{N}F_{j}(d)|>\eta$
that decreases to $0$ for large $n$, irrespective of $N$. Because data have a
discrete distribution, this implies a similar bound on the empirical
probability mass function: if $|t|=n$ and $\theta_{j}$ is the true state, the
probability that $\sup_{d}|t(d)-\frac{n}{N}f_{j}(d)|>\eta$ is at most
$b_{=}(n,\eta)$, with $\lim_{n\rightarrow\infty}b_{=}(n,\eta)=0$ for all
$\eta>0$. If the true state is $\theta_{j^{\prime}}\neq\theta_{j}$ and
$|t|=n$, then the probability that $\sup_{d}|t(d)-\frac{n}{N}f_{j}(d)|>\eta$
is at least $b_{\neq}(n,\eta)$, with $\lim_{\eta\rightarrow
0}\lim_{n\rightarrow\infty}b_{\neq}(n,\eta)=1$.

When $N$ and $k$ are large, the proportion of types that lie in
$S_{N}(\eta,k)$ is close to 1, for all $\eta$. In particular,
$\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}q_{N}(S_{N}(\eta,k))=1$, since:

    * –

With probability decreasing to $0$ as $k\rightarrow 0$, $|t|<k$.

    * –

For fixed $k$ and $\eta$, the probability that there does not exist $\theta$
such that $\sup_{d}|t(d)-|t|f_{\theta}(d)|\leq\eta$ given that $|t|\geq k$
decreases to $0$ as $Nk\rightarrow\infty$.

We may further subdivide $S_{N}(\eta,k)$ into a set of types associated with
each state,

$S^{j}_{N}(\eta,k)=\\{t\in
S_{N}(\eta,k):\sup_{d}|t(d)-|t|f_{j}(d)|\leq\eta\\}.$

A further consequence of the convergence of empirical distributions is that,
when $Nk\rightarrow\infty$ and $\eta\rightarrow 0$, the sets
$(S^{j}_{N}(\eta,k))_{j=1}^{J}$ are disjoint. Additionally, for all $t\in
S_{N}^{j}(\eta,k)$, there is a uniform lower bound on the probability that the
state is $\theta_{j}$ given that the sender is of type $t$, which we call
$w(k,\eta,N)$, with $\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}w(k,\eta,N)=1$.

In addition, we can lower-bound $q_{N}(\\{t\in
S_{N}^{j}(\eta,k):\underline{\mu}f_{j}\subseteq t\subseteq\bar{\mu}f_{j}\\})$
for all $k<\underline{\mu}<\bar{\mu}$. Let $\Delta(N)$ be a bound on
$\sup_{d}|(\sum_{x=1}^{d}g_{N}(x))-G(d)|$ that goes to $0$ as
$N\rightarrow\infty$. Observe that if
$\underline{\mu}+\eta<|t|<\bar{\mu}-\eta$ and $t\in S_{N}^{j}(\eta,k)$, then
$\underline{\mu}f_{j}\subseteq t\subseteq\bar{\mu}f_{j}$, so a lower bound is

$q_{N}(\\{t\in S_{N}^{j}(\eta,k):\underline{\mu}f_{j}\subseteq
t\subseteq\bar{\mu}f_{j}\\})\geq\beta_{0}(\theta_{j})(1-b_{=}(Nk,\eta))(G(\bar{\mu}-D\eta)-G(\underline{\mu}+D\eta)-\Delta(N)).$
(9)

Similarly, there is an upper bound on $q_{N}(\\{t\in
S_{N}^{j}(\eta,k):t\not\subseteq\underline{\mu}f_{j}\text{ and
}\bar{\mu}f_{j}\not\subseteq t\\})$:

$q_{N}(\\{t\in S_{N}^{j}(\eta,k):t\not\subseteq\underline{\mu}f_{j}\text{ and
}\bar{\mu}f_{j}\not\subseteq
t\\})\leq\beta_{0}(\theta_{j})(G(\bar{\mu}+D\eta)-G(\underline{\mu}-D\eta)+\Delta(N))+(1-\beta_{0}(\theta_{j}))b_{\neq}(kN,\eta).$
(10)

Now we proceed to construct a lower bound for
$u_{\sigma_{N}}(\hat{\mu}f_{\hat{j}})$. First, recall that $u_{\sigma_{N}}(\mu
f_{j})\geq\max_{\\{f\in\mathcal{T}_{N}:t\subseteq\mu
f_{j}\\}}u_{\sigma_{N}}(t)$. Observe that there exists a dataset
$\hat{t}=\frac{1}{N}(\lfloor
N\hat{\mu}f_{\hat{\theta}}(1)\rfloor,\ldots,\lfloor
N\hat{\mu}f_{\hat{\theta}}(k)\rfloor)$ in $\mathcal{T}_{N}$ and that
$u_{\sigma_{N}}(\hat{\mu}f_{\hat{j}})\geq u_{\sigma_{N}}(\hat{t})$.

For a given $N$, suppose $\hat{t}$ belongs to the $m$th upper pool under the
algorithm that constructs $\sigma_{N}$. Denote by $\hat{M}_{N}(m-1)$ the set
of messages that implement the upper pools in step $1,\ldots,m-1$, and fix
${\mathcal{T}}_{N,m}=\mathcal{T}^{+}_{N}(\hat{M}_{N}(m-1))$ to be the set of
remaining types at the start of the $m$th step of the algorithm that
constructs $\sigma_{N}$; therefore, $\hat{t}$ belongs to
${\mathcal{T}}_{N,m}$.

Let $\underline{M}_{\infty}(\epsilon,N)$ be the set of on-path messages that
result in a payoff of $u_{\sigma}((\hat{\mu}-\epsilon)f_{\hat{j}})$ under
infinite data. We see that the set of types in
$\mathcal{T}_{N,m}^{+}(\underline{M}_{\infty}(\epsilon,N))$ includes $\hat{f}$
when $N$ is large enough. From Lemma C.3, there is an upper pool in
$\mathcal{T}_{\hat{N},m}$ that achieves a payoff of at least
$u(\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon,N)))$, so
$u_{\sigma_{N}}(\hat{\mu}f_{\hat{j}})$ is lower-bounded by
$u(\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon,N)))$.

Let $(\underline{\mu}_{1}(\epsilon,N),\ldots,\underline{\mu}_{J}(\epsilon,N))$
be a vector that gives the minimum mass of data under distributions
$f_{1},\ldots,f_{J}$, respectively, such that the dataset contains some
message in $\underline{M}_{\infty}(\epsilon,N)$, and let
$(\bar{\mu}_{1}(N),\ldots,\bar{\mu}_{J}(N))$ be the maximum mass of data under
each distribution such that there does not exist $t\in\mathcal{T}_{\hat{N},m}$
such that $t\subseteq\bar{\mu}_{j}f_{j}$. All
$t\in\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon))$ satisfy
$t\not\subseteq\underline{\mu}_{j}(\epsilon,N)f_{j}$ and
$\bar{\mu}_{j}(N)f_{j}\not\subseteq t$, and all $t$ satisfying
$\underline{\mu}_{j}(\epsilon,N)f_{j}\subseteq
t\subseteq\bar{\mu}_{j}(N)f_{j}$ for some $j$ are in
$\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon))$.

We may rewrite

$u(\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon)))=\frac{\sum_{j=1}^{J}\sum_{t\in\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon))}q_{N}(t)\theta_{j}\pi_{N}(\theta_{j}|t)}{\sum_{t\in\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon))}q_{N}(t)}.$
(11)

Let the numerator be $Q(N,\hat{\mu}f_{\hat{j}},\epsilon)$ and the denominator
be $R(N,\hat{\mu}f_{\hat{j}},\epsilon)$. Analogously to eq. 9, a lower bound
for $Q(N,\hat{\mu}f_{\hat{j}},\epsilon)$ is

$\underline{Q}(N,\hat{\mu}f_{\hat{j}},\epsilon)=\sum_{j}\beta_{0}(\theta_{j})\theta_{j}[G(\bar{\mu}_{j}(N)-\eta
D)-G(\max(\underline{\mu}_{j}(\epsilon,N)+\eta
D,k))-\Delta(N)]w(k,\eta,N)(1-b_{=}(k,\eta)),$ (12)

and it follows from eq. 10 that an upper bound for $R$ is

$\begin{split}\bar{R}(N,\hat{\mu}f_{\hat{j}},\epsilon)=&\left(\sum_{j}\beta_{0}(\theta_{j})[G(\bar{\mu}_{j}(N)+\eta
D)-G(\underline{\mu}_{j}(\epsilon,N)-\eta D)+\Delta(N)]\right)\\\
&+J(1-b_{\neq}(k,\eta))+(1-q_{N}(S_{N}(\eta,k))).\end{split}$ (13)

We have

$\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim\inf_{N\rightarrow\infty}\underline{Q}\geq\lim\inf_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\theta_{j}(G(\bar{\mu}_{j}(N))-G(\underline{\mu}_{j}(\epsilon,N)))$

and

$\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim\inf_{N\rightarrow\infty}\bar{R}\leq\lim\inf_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})(G(\bar{\mu}_{j}(N))-G(\underline{\mu}_{j}(\epsilon,N))).$

Both of the RHS are finite and strictly positive for all $N$ and $\epsilon>0$;
therefore,

$\begin{split}\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim\inf_{N\rightarrow\infty}\frac{\ \underline{Q}\ }{\ \bar{R}\
}&\geq\lim\inf_{N}\frac{\sum_{j=1}^{J}\beta_{0}(\theta_{j})\theta_{j}(G(\bar{\mu}_{j}(N))-G(\underline{\mu}_{j}(\epsilon,N)))}{\sum_{j=1}^{J}\beta_{0}(\theta_{j})(G(\bar{\mu}_{j}(N))-G(\underline{\mu}_{j}(\epsilon,N)))}\\\
&=\lim\inf_{N}\mathbb{E}[\theta|t\in
T(u_{\sigma_{\infty}}((\hat{\mu}-\epsilon)f_{\hat{j}}),\hat{M}_{N}(m-1))]\\\
&\geq u_{\sigma_{\infty}}((\hat{\mu}-\epsilon)f_{\hat{j}}),\end{split}$ (14)

where the last inequality follows from Lemma C.3.

Because $k$ and $\eta$ are arbitrary variables used to obtain the bound, it
follows from this that
$\lim_{N\rightarrow\infty}u(\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon)))\geq
u_{\sigma_{\infty}}((\hat{\mu}-\epsilon)f_{\hat{j}})$. Finally, because
payoffs are continuous, taking a sequence of bounds as $\epsilon\rightarrow 0$
implies that
$\lim\inf_{N\rightarrow\infty}u_{\sigma_{N}}(\hat{\mu}f_{\hat{j}})\geq\lim_{\epsilon\rightarrow
0}\lim\inf_{N\rightarrow\infty}u(\mathcal{T}_{\hat{N},m}^{+}(\underline{M}_{\infty}(\epsilon)))\geq
u_{\sigma_{\infty}}(\hat{\mu}f_{\hat{j}})$.

The last step is to show that

$\lim_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{1}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu=\mathbb{E}_{\beta_{0}}[\theta].$

Since we know already that

$\lim_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{1}u_{\sigma_{\infty}}(\mu
f_{j})g(\mu)d\mu=\mathbb{E}_{\beta_{0}}[\theta]$

and $\lim\inf_{N\rightarrow\infty}u_{\sigma_{N}}(\mu f_{j})\geq
u_{\sigma_{\infty}}(\mu f_{j})$ for all $\mu f_{j}\in\mathcal{T}_{\infty}$,
this additional fact suffices to ensure that
$u_{\sigma_{N}}(\cdot)=u_{\sigma_{\infty}}(\cdot)$ over
$\mathcal{T}_{\infty}$.

The proof comes from dividing $\mu\in(k,1)$ into $X$ chunks, with the $x$th
chunk given by $(\mu_{x-1},\mu_{x}]$ where $\mu_{x}=x\frac{1-k}{X}+k$.

Consider types $t\in S^{j}_{N}(\eta,k)$ such that $\mu_{x-1}f_{j}\subseteq
t\subseteq\mu_{x}f_{j}$: their payoff under $\sigma_{N}$ has to be in
$[u_{\sigma_{N}}(\mu_{x-1}f_{j}),u_{\sigma_{N}}(\mu_{x}f_{j})]$. This implies
that

$\begin{split}\underline{V}_{N}(k,\eta,X)&=\sum_{j=1}^{J}\beta_{0}(\theta_{j})\sum_{x=1}^{X}u_{\sigma_{N}}(\mu_{x}f_{j})[G(\mu_{x+1}-\eta
D)-G(\mu_{x}+\eta D)-\Delta(N)](1-b_{=}(k,\eta))\\\
&\leq\mathbb{E}_{\beta_{0}}[\theta],\end{split}$ (15)

since $\underline{V}_{N}(k,\eta,X)$ is a lower bound for the total
probability-weighted sum of payoffs under $\sigma_{N}$ over
$t\in\mathcal{T}_{N}\bigcup S_{N}(\eta,k)$, while
$\mathbb{E}_{\beta_{0}}[\theta]$ is equal to the total probability-weighted
sum of payoffs under $\sigma_{N}$ of all types in $\mathcal{T}_{N}$.

Finally, the difference between
$\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{1}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu$ and $\underline{V}_{N}(k,\eta,X)$ vanishes as
$X\rightarrow\infty$, $k\rightarrow 0$, $\eta\rightarrow 0$, and
$N\rightarrow\infty$. To see this, observe that if $c$ is an upper bound on
$g$ (which exists because $g$ is continuous on compact interval $[0,1]$),

$\begin{split}V_{N}(k,\eta,X)\geq\sum_{j=1}^{J}\beta_{0}(\theta_{j})\Big{(}\sum_{x=1}^{X}&u_{\sigma_{N}}(\mu_{x}f_{j})[G(\mu_{x+1})-G(\mu_{x})]\\\
&-(\theta_{J}b_{=}(k,\eta)[G(\mu_{x+1})-G(\mu_{x})]+2c\eta
D+\Delta(N))\Big{)}\\\
\geq\sum_{j=1}^{J}\beta_{0}(\theta_{j})\sum_{x=1}^{X}&u_{\sigma_{N}}(\mu_{x}f_{j})[G(\mu_{x+1})-G(\mu_{x})]\\\
-JX\theta_{J}(b_{=}&(k,\eta)+2c\eta D+\Delta(N)).\end{split}$ (16)

Then, for any $\epsilon$ and $j$, define $\xi_{N}^{j}(\epsilon,X)$ to be the
set of values of $x$ such that
$u_{\sigma_{N}}(\mu_{x+1}f_{j})-u_{\sigma_{N}}(\mu_{x}f_{j})>\epsilon$. The
size of $\xi_{N}^{j}(\epsilon,X)$ is at most $\frac{\theta_{J}}{\epsilon}$.
For all $x\not\in\xi_{N}^{j}(\epsilon,X)$, we have the bound
$\int_{\mu_{x}}^{\mu_{x+1}}u_{\sigma_{N}}(\mu f_{j})g(\mu)d\mu-
u_{\sigma_{N}}(\mu_{x}f_{j})[G(\mu_{x+1})-G(\mu_{x})]<\epsilon[G(\mu_{x+1})-G(\mu_{x})]$.
So,

$\begin{split}\sum_{j=1}^{J}\beta_{0}(\theta_{j})&\int_{\mu=0}^{2}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu-\underline{V}_{N}(k,\eta,X)\\\
\leq&\left(\sum_{j=1}^{J}\beta_{0}(\theta_{j})\sum_{x=1}^{X}\left(\int_{\mu_{x}}^{\mu_{x+1}}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu-
u_{\sigma_{N}}(\mu_{x}f_{j})[G(\mu_{x+1})-G(\mu_{x})]\right)\right)\\\
&+JX\theta_{J}(b_{=}(k,\eta)+2c\eta D+\Delta(N)+(1-q_{N}(S_{N}(\eta,k))))\\\
\leq&\sum_{j=1}^{J}\left(\beta_{0}(\theta_{j})\left(\sum_{x\not\in\xi_{N}^{j}(\epsilon,X)}\epsilon[G(\mu_{x+1})-G(\mu_{x})]\right)+\left(\sum_{x\in\xi_{N}^{j}(\epsilon,X)}\theta_{J}[G(\mu_{x+1})-G(\mu_{x})]\right)\right)\\\
&+JX\theta_{J}(b_{=}(k,\eta)+2c\eta D+\Delta(N)+(1-q_{N}(S_{N}(\eta,k))))\\\
\leq&\epsilon+J\frac{c(1-k)}{X}\frac{\theta_{J}^{2}}{\epsilon}+JX\theta_{J}\Big{(}b_{=}(k,\eta)+2c\eta
D+\Delta(N)+(1-q_{N}(S_{N}(\eta,k)))\Big{)}\end{split}$ (17)

since
$\sum_{x\in\xi_{N}^{j}(\epsilon,X)}[G(\mu_{x+1})-G(\mu_{x})]\leq\frac{c(1-k)}{X}\frac{\theta_{J}}{\epsilon}$.
Then

$\lim_{\epsilon\rightarrow 0}\lim_{X\rightarrow\infty}\lim_{k\rightarrow
0}\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{2}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu-\underline{V}_{N}(k,\eta,X)=\lim_{\epsilon\rightarrow
0}\lim_{X\rightarrow\infty}\epsilon+J\frac{c(1-k)}{X}\frac{\theta_{J}^{2}}{\epsilon}=0.$

Again, since $\epsilon$, $X$, $k$, and $\eta$ were all constructed variables,
this implies that

$\lim_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{2}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu=\lim_{\epsilon\rightarrow
0}\lim_{X\rightarrow\infty}\lim_{k\rightarrow 0}\lim_{\eta\rightarrow
0}\lim_{N\rightarrow\infty}\underline{V}_{N}(k,\eta,X)\leq\mathbb{E}_{\beta_{0}}[\theta].$

As it is already clear from the lower bound on $u_{\sigma_{N}}(\mu f_{j})$
that
$\lim_{N\rightarrow\infty}\sum_{j=1}^{J}\beta_{0}(\theta_{j})\int_{\mu=0}^{2}u_{\sigma_{N}}(\mu
f_{j})g(\mu)d\mu\geq\mathbb{E}_{\beta_{0}}[\theta]$, equality obtains.

##  Appendix F Proof of strategic convergence

Here, we prove proposition 4.3.

Define $m_{\sigma^{*}_{N}}(t)$ to be the realization of the message played
when the sender’s type is $t$ – formally, $m_{\sigma^{*}_{N}}(t)$ is a random
variable with outcomes in $\mathcal{M}_{N}$ whose distribution is given by the
equilibrium strategy $\sigma^{*}_{N}(\cdot|t)$.

Define $A_{N}(x,\Delta;\epsilon)$ to be the set of types
$t\in\mathcal{T}_{N}\bigcap T(\epsilon)$ such that
$u_{\sigma^{*}_{N}}(t)>\mathbb{E}_{\pi(\cdot|t)}[\theta]$, and
$u_{\sigma^{*}_{N}}(t)\in(x,x+\Delta]$.

Define $B_{N}(x,\Delta;\epsilon)$ to be the set of types
$t^{\prime}\in\mathcal{T}_{N}\bigcap T(\epsilon)$ with
$u_{\sigma^{*}_{N}}(t^{\prime})<\mathbb{E}_{\pi(\cdot|t)}[\theta]$, and
$u_{\sigma^{*}_{N}}(t^{\prime})\in(x,x+\Delta]$.

Define
$X(\eta,\xi,\omega):=\\{x\in\bigg{[}\max_{j}\hat{u}_{j}(\xi)+\omega,u_{\sigma^{*}_{\infty}}(f_{\theta_{J}})\bigg{)}:\min_{k}|u_{\sigma^{*}_{N}}(t)-\theta_{k}|>\eta\\}$.

For small enough $\eta$, $\xi$ and $\omega$, $X(\eta,\xi,\omega)$ is nonempty.
On the other hand, $A_{N}(x,\Delta;\epsilon)$ and $B_{N}(x,\Delta;\epsilon)$
may be empty, in particular for small $N$. However, for $x\in
X(\eta,\xi,\omega)$, there is large-enough $\underline{N}^{*}$ so that they
are nonempty for all $N>\underline{N}^{*}$. Continuity of
$u_{\sigma^{*}_{\infty}}(\mu f_{j})$ ensures there is positive-measure set of
types in $\mathcal{T}_{\infty}$ with
$u_{\sigma^{*}_{infty}}(t)\in[x,x+\Delta]$; the bound away from $\theta_{k}$
for all $k$ ensures that some such types have
$u_{\sigma^{*}_{infty}}(t)-\mathbb{E}_{\pi(\cdot|t)}[\theta]\geq\eta$ and some
have $u_{\sigma^{*}_{i}nfty}(t)\leq\mathbb{E}_{\pi(\cdot|t)}[\theta]<-\eta$,
and so there is a positive-measure set of types nearby with the same
properties under $\sigma^{*}_{N}$ in $\mathcal{T}_{N}$ for large-enough $N$.

We first prove the claim that if $\\{\sigma^{*}_{N}\\}_{N=1}^{\infty}$ are
truth-leaning equilibria of games $\mathcal{G}_{N}$ that converge to limit