# Persuasion in Veto Bargaining111We are grateful to Nageeb Ali, Navin Kartik,
John Patty, Maggie Penn, Carlo Prato, Ian Turner and seminar audiences at
Emory, University of Pittsburgh, and UC Riverside for helpful comments.
Kyungmin Kim is supported by the Ministry of Education of the Republic of
Korea and the National Research Foundation of Korea (NRF-2020S1A5A2A03043516).

Jenny S. Kim Kyungmin Kim Richard Van Weelden Department of Political Science,
Emory University, Email<EMAIL_ADDRESS>of Economics, Emory
University, Email<EMAIL_ADDRESS>of Economics, University
of Pittsburgh<EMAIL_ADDRESS>
###### Abstract

We consider the classic veto bargaining model but allow the agenda setter to
engage in persuasion to convince the veto player to approve her proposal. We
fully characterize the optimal proposal and experiment when Vetoer has
quadratic loss, and show that the proposer-optimal can be achieved either by
providing no information or with a simple binary experiment. Proposer chooses
to reveal partial information when there is sufficient expected misalignment
with Vetoer. In this case the opportunity to engage in persuasion strictly
benefits Proposer and increases the scope to exercise agenda power.

Keywords: agenda-setting, veto-bargaining, persuasion

##  1 Introduction

There are many settings in which an agenda setter makes a proposal and other
player(s) can decide whether to approve or reject (veto) it. In almost all
such cases, before the approval decision, the proposer attempts to persuade
the veto player to approve the proposal they put forward. For example, if a
committee makes a recommendation (such as to hire a particular candidate) it
will also present the case to get its recommendation approved. Similarly, a
legislator who introduces a bill will make a speech urging their colleagues to
vote for it. And, if the bill is passed by Congress, the Speaker or other
Congressional leaders will make the case to the public, and the President,
that the bill should not be vetoed. In this paper we explore how the
possibility of persuasion augments agenda-setting power and influences the
proposals that will be made.

Following the seminal contribution of Romer and Rosenthal (1978), a large
literature has explored the consequences of being able to set the
agenda.222See Cameron and McCarty (2004) for an excellent survey of the
literature. We extend the model to allow Proposer to also engage in Bayesian
persuasion (Kamenica and Gentzkow, 2011) to convince Vetoer to accept the
proposal. We assume that Vetoer is uncertain of his optimal policy, and
Proposer can design a Blackwell experiment to reveal information about it to
Vetoer. We consider both the case in which Proposer first engages in
persuasion and then makes a proposal after observing the results, and the case
in which Proposer makes a proposal without observing the outcome of
persuasion.

As a leading example, consider an interest group who both develops legislation
and lobbies for its passage. Beyond campaign contributions and financial
favors, interest groups can influence legislation both by providing a
legislative subsidy (Hall and Deardorff, 2006), and through informative
lobbying (Austen-Smith, 1993; Potters and Van Winden, 1992; Schnakenberg,
2017; Schnakenberg and Turner, 2021, 2023; Awad, 2020; Ellis and Groll, 2020;
Hirsch et al., 2023; Dellis, 2023). We interpret the former as the interest
group using its resources and expertise to draft legislation and the latter as
the interest group using information design to persuade the decisive
policymaker—which could be the median legislator, the relevant committee
chair, the executive, or even a critical bureaucrat.333Prato and Turner (2023)
also consider persuasion of a bureaucrat. In their model the bureaucrat can
choose from an exogenous set of alternatives; in our framework the bureaucrat
would choose between implementing new legislation or dragging their feet in
which case the status quo is maintained. Policymakers act on a large number of
issues, and outside of the central, highly salient ones, an individual
policymaker may not know much about the issue at hand or have a developed
position. The interest group however can engage in informational lobbying
which reveals (partial) information about the costs and benefits to the
policymaker or their constituents, allowing them to learn about their most
preferred policy.444Our informational environment is similar to that of Hirsch
et al. (2023), in which a policy generates a known benefit to an interest
group but an uncertain benefit to the policymaker. Relevant applications they
discuss include tariff policy, travel restrictions to a particular country,
and domestic environmental regulation. Our paper differs in that the interest
group can choose both the content of the proposal and the persuasion, whereas
in their case the policy is fixed and the policymaker can learn about it at a
cost. Based on this they decide whether or not to pass the
legislation.555Policy development is costly, and so policymakers cannot simply
adjust the policy content of the legislation (Hirsch and Shotts, 2015), but
rather can decide whether to approve or reject it. In this setting, the
interest group is the proposer and the policymaker the veto player.

We consider bargaining between a proposer and a single veto player over a one
dimensional policy. There is a status quo policy $0$ and Proposer’s ideal
policy is $1$. Vetoer’s ideal point is
$\theta\in[\underline{\theta},\bar{\theta}]$, which is initially unknown to
both players, but can be learned about from an experiment designed by
Proposer. This can be interpreted as Vetoer’s ideal action depending on an
unknown state of the world, about which Proposer can reveal information. In
the bulk of our analysis we focus on the case in which Vetoer’s loss is
quadratic in the distance between his bliss point, $\theta$, and the
implemented policy. In this case, Vetoer is willing to accept a proposal of
$p$ if and only if it is closer to his expected bliss point than the status
quo of $0$. Exploiting this property and applying the state-of-the-art
verification technique by Dworczak and Martini (2019), we fully characterize
Proposer’s optimal proposal and experiment.

We show that Proposer can achieve her maximal payoff by either providing no
information or designing a simple binary experiment. When Vetoer’s bliss point
is always positive ($\underline{\theta}\geq 0$), and so Proposer and Vetoer
are always on the same side of the status quo, Proposer’s payoff is maximized
by providing no information. This is because the resulting policy is
proportional to Vetoer’s expected bliss point, so when Proposer’s utility is
concave over policy she never benefits from revealing information. When
Vetoer’s bliss point can be negative $(\underline{\theta}<0)$, however,
Proposer may benefit from revealing whether $\theta$ is above or below some
threshold. The key is that Proposer’s agenda-setting authority ensures the
resulting policy to be at least the status quo regardless of Vetoer
preferences. Consequently, by revealing when Vetoer’s bliss point is on the
opposite side of the status quo, Proposer can increase the expected policy.

We study two versions of the problem: the first in which Proposer runs an
experiment prior to making a proposal (persuasion first) and the other where
Proposer cannot condition on the experiment outcome (proposal
first).666Whether persuasion and the proposal happen simultaneously or
sequentially is not important; what matters is whether Proposer is able to
observe what Vetoer learns prior to making her proposal. Whether persuasion
precedes the proposal or not depends on the application. In some cases,
Proposer may be able to commission and reveal the outcome of a study or an
opinion poll prior to engaging in policy bargaining. In others—such as when
proposing a particular candidate to be hired or confirmed—the proposal would
necessarily come first.

Under persuasion first, Proposer observes the realization of the experiment
and can tailor her proposal to this realization, in contrast to proposal-first
in which the realization is Vetoer’s private information. Therefore, Proposer
always does weakly better under persuasion first than proposal first. However
we show that Proposer does not strictly benefit from persuasion first; rather,
the Proposer’s solution is the same in both settings. As Proposer and Vetoer
are symmetrically informed under persuasion first, Proposer proposes twice the
conditional expectation of Vetoer’s bliss point (or the status quo if
negative) which is then accepted. Analogous to the optimality of no
information when Vetoer’s bliss point is always positive, Proposer cannot
benefit from inducing more than one strictly positive proposal: she could
always do at least as well by pooling the signals and making a proposal based
on the mean. Hence, Proposer optimally generates a binary experiment, with the
status quo maintained after one signal realization. Notice that an identical
outcome can be achieved under proposal first by making the same take-it-or-
leave-it proposal followed by the same binary experiment. Thus, the ability to
tailor the proposal to the realization of the experiment provides no benefit
to Proposer.

The optimal binary experiment is informative whenever the lower bound on
Vetoer’s bliss point is sufficiently negative relative to its expected value.
Intuitively, when Vetoer’s expected bliss point is high, Proposer can get
close to her ideal point without providing information and so chooses not to
gamble by revealing information. How much proposer is willing to reveal
depends on her risk preferences and the distribution. As Proposer’s utility
function becomes more concave she compromises more, resulting in a lower
policy proposal and fewer vetoes. A likelihood ratio increase in the
distribution of Vetoer bliss points also results in fewer vetoes but with a
higher policy proposal.

When Vetoer does not have quadratic loss, proposal-first and persuasion-first
are not always equivalent, and Proposer may do strictly better under
persuasion-first. Further, because more than just the expectation matters, it
may be optimal to provide information even when Vetoer’s bliss point is known
to be positive. We illustrate these points by considering a version in which
Vetoer has a linear loss function, focusing on the case in which his ideal
point could take one of two positive values.

Our paper belongs to a large literature on veto bargaining (Romer and
Rosenthal, 1978, 1979; Matthews, 1989; Cameron, 2000; Cameron and McCarty,
2004; Kartik, Kleiner and Van Weelden, 2021; Ali, Kartik and Kleiner, 2023).
To the best of our knowledge, none of the previous papers have allowed the
agenda-setter to make the case for her proposal. Thus the previous literature
underestimates the proposer’s power: Proposer’s ability to engage in
informative lobbying enhances her agenda-setting power (strictly when no
information is suboptimal) and allows her to introduce and pass more favorable
legislation. We show that the benefit of persuasion is particularly pronounced
when Proposer and Vetoer are initially expected to be quite misaligned, in
which case absent persuasion the benefits of agenda-setting authority are
muted.

Conceptually and methodologically, our model is closely related to the
monopoly (price setting) model in which the seller chooses how much (and what)
product information to provide for the buyer.777See Lewis and Sappington
(1994), Che (1996), and Anderson and Renault (2006) for some seminal
contributions. Roesler and Szentes (2017) analyze a related problem where the
buyer chooses how much information to acquire in order to influence the
seller’s pricing. Cotton and Li (2018) consider a policymaker acquiring
information to increase the contributions raised from competing lobbyists. In
the price setting model, the seller can extract full surplus by providing
either perfect information (if she can condition the price on the buyer’s
information) or no information (if she does not observe the buyer’s
information). In contrast, we show that, whether Proposer can observe Vetoer’s
information or not, it is often the case that neither perfect nor no
information is optimal. The difference is driven by the partial alignment
between Proposer’s and Vetoer’s (single-peaked) preferences, namely that,
depending on his policy preferences, Vetoer may (or may not) prefer Proposer
to make a more aggressive proposal; in contrast, in the price setting model,
raising a price is harmful to the buyer, regardless of his valuation.

This paper proceeds as follows. Section 2 introduces the formal model, and
Section 3 studies the optimal veto decision. Section 4 illustrates how
information affects policy bargaining and Section 5 solves for the optimal
experiment and proposal when Vetoer has quadratic loss. Section 6 considers
specific examples and explores comparative statics. Section 7 examines how the
results are changed when Vetoer has linear loss. Section 8 concludes.

##  2 Model

We build upon the canonical incomplete-information veto bargaining model.
There are two players, a proposer (she) and a vetoer (he). Proposer makes a
policy proposal $p\in\mathcal{R}_{+}$, and Vetoer decides whether to accept or
veto it. If $p$ is accepted by Vetoer then it is implemented. If it is vetoed
then the status quo policy, normalized to $0$, remains in effect.888We assume
that Proposer makes a single proposal instead of a menu of options. Unlike in
a price setting model, if Vetoer has private information at the time of his
decision a single proposal is generally not an optimal mechanism (Kartik,
Kleiner and Van Weelden, 2021). However, as we illustrate in Section 5.3,
under the optimal information structure a single proposal will be optimal in
our baseline model.

Let $a\in\mathcal{R}$ denote the policy chosen, either the policy $p$ proposed
by Proposer or the status quo policy $0$. Vetoer’s ex post payoff is given by
$v(a,\theta)=-\left(\theta-a\right)^{2}$, while Proposer’s ex post payoff is
given by $u(a)=-c\left(|1-a|\right)$, where $c$ is a twice continuously
differentiable, strictly increasing, and convex function with $c(0)=0$. In
other words, Vetoer’s ideal point is $\theta$, Proposer’s ideal point is $1$,
Vetoer incurs utility losses that are quadratic in $|\theta-a|$, and
Proposer’s losses are convex in $|1-a|$.999The analysis would be unchanged if
Proposer’s utility were increasing in $a$ but with $1$ the maximum possible
action. As illustrated shortly, the assumption on Vetoer’s quadratic
preferences provides a key technical advantage for our model.

There is imperfect information about Vetoer’s ideal point $\theta$.
Specifically, $\theta$ is drawn from the compact interval
$\Theta:=[\underline{\theta},\overline{\theta}]$ according to the distribution
$F$. For a clean analysis, we assume that $F$ has continuous and strictly
positive density $f$ on $\Theta$. If $\overline{\theta}\leq 0$ then the
problem becomes trivial, as Vetoer would never accept $p>0$. In what follows,
we maintain the assumption that $\overline{\theta}>0$.

Importantly, neither player observes Vetoer’s ideal point $\theta$; that is,
unlike in the standard incomplete-information veto bargaining, $\theta$ is
_not_ Vetoer’s private information. However, Proposer can choose to produce
information about $\theta$. That Vetoer is not certain of his most-preferred
policy is critical for persuasion to play a role and, as highlighted in the
Introduction, is reasonable in many applications of interest. To focus on
Proposer’s strategic incentives, as in the recent literature on Bayesian
persuasion and information design, we endow Proposer with full flexibility in
her choice of information; that is, Proposer can provide any type (amount) of
information. Formally, Proposer can choose any ordered set $S$ and joint
distribution $\pi$ over $\Theta\times S$. Vetoer observes the realization
$s\in S$ and Bayesian updates his belief. We let $F_{s}$ denote the
distribution that represents Vetoer’s posterior belief following the
realization $s\in S$.

We consider the following two versions of the problem.

  * •

Persuasion first: Proposer publicly runs an experiment and makes a proposal
after observing the experiment outcome. In other words, Proposer can condition
his proposal $p$ on the signal realization $s$ (equivalently, Vetoer’s
posterior belief $F_{s}$).

  * •

Proposal first: This is when Proposer’s proposal $p$ cannot condition on the
experiment outcome. This case is relevant if Proposer makes a proposal $p$
first and then runs an experiment; in what follows, we adopt this
interpretation. However, this case also arises if Proposer chooses $p$ and $F$
simultaneously; and if Proposer runs an experiment first but only Vetoer
observes its outcome.

For each case, we study weak perfect Bayesian equilibria of the given
sequential-move game. This ensures that Vetoer takes the sequentially rational
decision in the “subgame” after the proposal and outcome of the experiment
have been revealed. For convenience, we refer to weak perfect Bayesian
equilibria simply as equilibria.

##  3 Vetoer’s Acceptance Decision

We begin by analyzing the last stage of the game in which Vetoer, with updated
information about $\theta$, decides whether to accept Proposer’s proposal $p$
or veto it.

Let $V(a,G)$ denote Vetoer’s expected payoff from policy $a$ when his
(posterior) belief about $\theta$ can be represented by the distribution
function $G$; formally,

$\displaystyle
V(a,G):=\mathbb{E}_{G}\left[-\left(a-\theta\right)^{2}\right]=-\int\left(a-\theta\right)^{2}dG(\theta).$

Vetoer is willing to accept $p$ if and only if

$\displaystyle
V(p,G)=-\mathbb{E}_{G}\left[\left(p-\theta\right)^{2}\right]\geq
V(0,G)=-\mathbb{E}_{G}\left[\theta^{2}\right]\Leftrightarrow
p\left(p-2\mathbb{E}_{G}[\theta]\right)\leq 0.$

This leads to the following simple but important result.

###### Lemma 1

Given his posterior belief $G$, Vetoer accepts $p>0$ if and only if $p\leq
2\mathbb{E}_{G}[\theta]$.

Since her bliss point is $1$ and she can always induce the status quo policy
$0$, Proposer would never offer $p<0$. This means that Lemma 1 provides an
effectively necessary and sufficient condition for Vetoer’s acceptance
decision. As illustrated in Section 5, this, together with the fact that $G$
affects Vetoer’s decision only through its expectation, greatly simplifies the
technical analysis, thereby allowing us to study the general problem.

If Vetoer is perfectly informed about $\theta$ then he will accept $p(\geq 0)$
if and only if $p$ is closer to $\theta$ than the status quo policy $0$, that
is, $p-\theta\leq\theta$. Lemma 1 can be seen as generalizing this result to
the case where there is uncertainty about $\theta$. It should be noted,
however, that the result crucially depends on Vetoer’s quadratic preferences;
as we formally show in Section 7.1, if Vetoer’s loss function is linear then
comparing $p$ to $2\mathbb{E}_{G}[\theta]$ does not suffice to determine
Vetoer’s decision.

One immediate implication of Lemma 1 is that the problem becomes trivial when
$\mathbb{E}[\theta]\geq 1/2$:101010In what follows, we do not add subscript
$F$ to $\mathbb{E}$ if the expectation is with respect to the prior
distribution $F$, that is, $\mathbb{E}[\theta]=\mathbb{E}_{F}[\theta]$. In
that case Vetoer is willing to accept Proposer’s ideal policy $1$ with his
prior $F$. Then, it is certainly optimal for Proposer to propose her ideal
policy $1$ while providing no information. In what follows, we maintain the
following assumption.

###### Assumption 1

The unconditional expectation of $\theta$ is strictly less than $1/2$, that
is, $\mathbb{E}[\theta]<1/2$.

##  4 Illustration: Value of Persuasion

This section considers a simple example in order to illustrate how Proposer
can fruitfully utilize her persuasion ability. Specifically, we consider the
case where Proposer’s loss function is also quadratic (i.e.,
$c(|1-a|)=(1-a)^{2}$) and $F$ is a uniform distribution over
$[\underline{\theta},1]$ for some $\underline{\theta}<0$.

#### No information.

Suppose Vetoer receives no additional information about his bliss point
$\theta$; he only knows that $\theta$ is distributed according to
$U[\underline{\theta},1]$. In this case, his expected bliss point is
$\mathbb{E}[\theta]=\frac{\underline{\theta}+1}{2}$. If
$\mathbb{E}[\theta]\leq 0$ then he will never accept a positive proposal (see
Section 3). Then, the status quo policy $0$ will be implemented, in which case
Proposer earns $-c(1)=-1$. If $\mathbb{E}[\theta]>0$ then Vetoer accepts
$p(>0)$ as long as $p\leq 2\mathbb{E}[\theta]$. In this case, Proposer’s
optimal proposal is $2\mathbb{E}[\theta]=\underline{\theta}+1$, resulting in a
payoff of $-c(1-2\mathbb{E}[\theta])=-\underline{\theta}^{2}$. To sum up, when
Vetoer obtains no information, Proposer’s indirect payoff as a function of
$\underline{\theta}$ is given by

$\displaystyle
U_{NO}(\underline{\theta}):=-\min\\{\underline{\theta}^{2},1\\}=\left\\{\begin{array}[]{ll}-1&\text{if
}\underline{\theta}<-1.\\\ -\underline{\theta}^{2}&\text{if
}\underline{\theta}\in[-1,0).\end{array}\right.$

#### Full information in the proposal-first model.

Now suppose Vetoer privately observes his bliss point $\theta$; this
corresponds to the standard incomplete-information veto bargaining model. In
this case, a proposal $p(\geq 0)$ will be accepted if and only if
$p-\theta\leq\theta-0\Leftrightarrow\theta\geq p/2$. This implies that
Proposer’s problem is to maximize

$\displaystyle-F\left(\frac{p}{2}\right)c(1)-\left(1-F\left(\frac{p}{2}\right)\right)c(1-p)=-\frac{p/2-\underline{\theta}}{1-\underline{\theta}}-\frac{1-p/2}{1-\underline{\theta}}(1-p)^{2}.$

The solution to this problem is $p=2/3$, and the resulting expected payoff of
Proposer is

$\displaystyle
U_{FL,1}(\underline{\theta}):=-\frac{1/3-\underline{\theta}}{1-\underline{\theta}}-\frac{2/3}{1-\underline{\theta}}\frac{1}{9}=-\frac{11-27\underline{\theta}}{27(11-\underline{\theta})}.$

As depicted in Figure 1, $U_{NO}$ crosses $U_{FL,1}$ once from below. With no
information, Vetoer believes that his ideal action may be very low and so is
unwilling to accept a policy much above the status quo. Full information
allows Proposer to avoid this problem as now she can target only high Vetoer
types. However, it comes at the cost of yielding some information rent to
Vetoer. The fact that the cost associated with no information declines fast as
$\underline{\theta}$ rises illustrates the particular single-crossing pattern.

$-2$$0$$\underline{\theta}$$U$$0$$-1$$U_{NO}$$-1$$U_{FL,1}$$U_{FL,2}$$U_{BI}$$-\frac{1}{3}$
Figure 1: This figure depicts Proposer’s indirect expected payoffs as a
function of $\underline{\theta}$ under four different information regimes.

#### Full information in the persuasion-first model.

What if Proposer also observes Vetoer’s bliss point $\theta$, that is,
$\theta$ becomes public before Proposer chooses a proposal? In that case,
Proposer offers $0$ if $\theta\leq 0$, $2\theta$ if $\theta\in(0,1/2)$, and
$1$ if $\theta\geq 1/2$. The resulting payoff of Proposer is

$\displaystyle U_{FL,2}(\underline{\theta})$
$\displaystyle:=-\int_{\underline{\theta}}^{0}c(1)dF(\theta)-\int_{0}^{1/2}c(1-2\theta)dF(\theta)-\int_{1/2}^{1}c(0)dF(\theta)$
$\displaystyle=-\int_{\underline{\theta}}^{0}1\frac{d\theta}{1-\underline{\theta}}-\int_{0}^{1/2}\left(1-2\theta\right)^{2}\frac{d\theta}{1-\underline{\theta}}=-\frac{1}{1-\underline{\theta}}\left(\frac{1}{6}-\underline{\theta}\right).$

As shown in Figure 1, $U_{FL,2}$ is uniformly above $U_{FL,1}$. This holds
simply because Proposer has more flexibility under persuasion first than under
proposal first; she can always mimic her proposal-first strategy under
persuasion first by choosing to not condition her proposal on the experiment
outcome, but not vice versa. It is more intriguing that $U_{FL,2}<U_{NO}$ when
$\underline{\theta}$ is relatively close to $0$, at which point the resulting
action with no information is close to Proposer’s bliss point. Note that the
current full-information persuasion-first case corresponds to the perfect
price discrimination in the price setting model (which yields the first-best
outcome to the seller). The comparison result is driven by the following two
facts: (i) Vetoer (still) obtains information rents when $\theta\in(1/2,1]$,
and (ii) Proposer is risk averse and full information results in a random
outcome (depending on $\theta$).

#### Optimal binary signal.

Consider a simple binary signal that produces $0$ if $\theta<s_{\ast}$ and $1$
if $\theta\geq s_{\ast}$ for some
$s_{\ast}\leq[\max\\{\underline{\theta},-1\\},0]$. If the signal realization
is $0$ then Vetoer’s posterior belief is uniform over
$[\underline{\theta},s_{\ast})$. In this case,
$\mathbb{E}[\theta|0]=\frac{\underline{\theta}+s_{\ast}}{2}<0$, so Proposer
has no choice but to induce the status quo policy $0$. If the realization is
$1$ then Vetoer’s posterior belief is uniform over $[s_{\ast},1]$. In this
case, $\mathbb{E}[\theta|1]\geq 0$, so Proposer’s optimal proposal is
$2\mathbb{E}[\theta|1]=s_{\ast}+1$. This implies that Proposer chooses
$s_{\ast}$ to maximize:

$\displaystyle-F(s_{\ast})c(1)-(1-F(s_{\ast}))c\left(1-2\mathbb{E}[\theta|1]\right)=-\frac{s_{\ast}-\underline{\theta}}{1-\underline{\theta}}-\frac{(1-s_{\ast})s_{\ast}^{2}}{1-\underline{\theta}}.$

The solution to this problem is $s_{\ast}=\max\\{\underline{\theta},-1/3\\}$,
and the resulting indirect utility of Proposer is

$\displaystyle
U_{BI}(\underline{\theta}):=\left\\{\begin{array}[]{ll}-\frac{1}{1-\underline{\theta}}\left(-\frac{5}{27}-\underline{\theta}\right)&\text{if
}\underline{\theta}<-\frac{1}{3}\\\ -\underline{\theta}^{2}&\text{if
}\underline{\theta}\geq-\frac{1}{3}.\end{array}\right.$

Notice that our derivation is based on the persuasion-first interpretation. It
is straightforward, however, that Proposer can obtain the same outcome under
proposal first; it suffices that she employs the same binary experiment and
proposes $2\mathbb{E}[\theta|1]$. Note that this equivalence holds because—and
whenever—Proposer makes only one strictly positive proposal under persuasion
first.

#### Comparison.

As depicted in Figure 1, this optimal binary information uniformly dominates
all the above cases. This is because a binary signal offers a simple but
powerful way to avoid the problems of both no information and full
information. Unlike no information, it enables Proposer to exclude Vetoer
types below $s_{\ast}$ and so be aggressive in her proposal; binary
information strictly dominates no information unless $\underline{\theta}$ is
sufficiently large. Meanwhile, unlike full information, it leaves no rent to
Vetoer and induces a single positive policy. Can Proposer do better than
$U_{BI}$ by choosing a more sophisticated experiment? Our subsequent general
analysis shows that the answer is no, that is, the optimal binary signal is an
optimal signal among all possible ones.

##  5 Main Characterization

This section provides a full characterization for our general model.

###  5.1 Persuasion-First Model

We first consider the persuasion-first model. Recall that given the
realization $s\in S$ (or Vetoer’s posterior belief $F_{s}$), Vetoer is willing
to accept any $p\leq 2\mathbb{E}[\theta|s]=2\mathbb{E}_{F_{s}}[\theta]$ (Lemma
1). Combining this with the fact that Proposer can always choose to induce the
status quo policy $0$ and her bliss point is $1$, it follows that Proposer’s
optimal proposal is given by

$\displaystyle p^{\ast}(s)=\left\\{\begin{array}[]{ll}0&\text{if
}\mathbb{E}[\theta|s]\leq 0\\\ 2\mathbb{E}[\theta|s]&\text{if
}\mathbb{E}[\theta|s]\in(0,1/2)\\\ 1&\text{if }\mathbb{E}[\theta|s]\geq
1/2.\end{array}\right.$

The corresponding indirect utility of Proposer is

$\displaystyle U(s):=u(p^{\ast}(s))=\left\\{\begin{array}[]{ll}-c(1)&\text{if
}\mathbb{E}[\theta|s]\leq 0\\\ -c(1-2\mathbb{E}[\theta|s])&\text{if
}\mathbb{E}[\theta|s]\in(0,1/2)\\\ 0&\text{if }\mathbb{E}[\theta|s]\geq
1/2.\end{array}\right.$

See the blue solid curve in Figure 2 for the representative shape of $U$.

#### Reformulation in terms of the distribution of posterior means.

Crucially, Proposer’s indirect utility depends only on the posterior mean
$\mathbb{E}[\theta|s]$, not on other aspects of $F_{s}$; without loss of
generality, we use $s$ to denote the mean of the posterior distribution
following realization $s$. This implies that we can focus on the distribution
over _posterior means_ , not the distribution over _posterior distributions_.
Then, we can utilize a powerful fact that given $F$ there exists an experiment
that produces a distribution $G$ of posterior means if and only if $F$ is a
mean-preserving spread of $G$.111111See, e.g., Blackwell (1953); Gentzkow and
Kamenica (2016); Kolotilin (2018). Intuitively, $G=F$ if the experiment is
perfectly informative, while $G$ is degenerate at $\mathbb{E}[\theta]$ if the
experiment carries no information. Any intermediate level of information
produces a distribution between these two extremes. This allows us to write
Proposer’s optimal persuasion problem as

$\displaystyle\max_{G\in MPC(F)}\int U(s)dG(s),$ (1)

where $MPC(F)$ denotes the set of all mean-preserving contractions of $F$.

$\underline{\theta}$$0$$\frac{1}{2}$$1$$s$$U(s)$$-c(1)$$0$$s_{\ast}$$\phi(s)$$s^{\ast}$
Figure 2: This figure depicts Proposer’s indirect utility function $U(s)$
(blue solid) and a convex function $\phi(s)$ (red translucent) that can be
used to apply Dworczak and Martini’s results. In this figure, $c$ is
quadratic.

To solve (1), we apply a recent theoretical development by Dworczak and
Martini (2019).121212No elementary method can be used to analyze (1), because
it is an infinite-dimensional programming problem with a complex constraint
($G\in MPC(F)$). They provide a simple geometric verification tool for a class
of Bayesian persuasion problems that can be written in the form of (1). To
apply their technique, we first plot $U(s)$; see the blue solid curve in
Figure 2. Then, we consider a smallest convex function $\phi$ that uniformly
stays above $U$ and identify the set of values such that $\phi(s)=U(s)$; in
Figure 2, the red translucent curve represents such a convex function, and
$\\{s:\phi(s)=U(s)\\}=[\underline{\theta},s_{\ast}]\cup\\{s^{\ast}\\}$. The
remaining task is to see whether there is a distribution $G$ such that
$supp(G)\subseteq\\{s:\phi(s)=U(s)\\}$ and
$\int\phi(s)dF(s)=\int\phi(s)dG(s)$. In our problem, due to the piece-wise
linear structure of $\phi$, this last requirement reduces to

$\displaystyle s^{\ast}=\mathbb{E}[\theta|\theta\geq
s_{\ast}]=\int_{s_{\ast}}^{\overline{\theta}}\theta\frac{dF(\theta)}{1-F(s_{\ast})}.$
(2)

Note that there is a one-to-one mapping between the cutoff $s_{*}$ and the
expectation $s^{*}$. Applying this procedure to our problem yields the
following result.

###### Proposition 1

In the persuasion-first model, no information is optimal to Proposer if and
only if

$\displaystyle U(\underline{\theta})\leq
U(\mathbb{E}[\theta])+U^{\prime}(\mathbb{E}[\theta])(\underline{\theta}-\mathbb{E}[\theta]).$
(3)

If this condition fails then the binary signal that reveals only whether
$\theta\geq s_{\ast}$ or not is optimal to Proposer, where $s_{\ast}(\leq 0)$
is the value such that for all $s\in[s_{\ast},\overline{\theta}]$,

$\displaystyle U(s)\leq U(s_{\ast})+\frac{U(\mathbb{E}[\theta|\theta\geq
s_{\ast}])-U(s_{\ast})}{\mathbb{E}[\theta|\theta\geq
s_{\ast}]-s_{\ast}}(s-s_{\ast}).$ (4)

Most notable in Proposition 1 is that the proposer-optimal outcome can be
achieved with either no information or a simple binary experiment. This
emerges because Proposer is risk averse over policy outcomes and, as shown
above, the implemented policy $p^{\ast}(s)$ is linear in the posterior mean
$\mathbb{E}[\theta|s]$ over the relevant region $[0,1]$. To see this clearly,
suppose there are two signal realizations that lead to two different positive
policies. If Proposer pools the two realizations into one then she can achieve
the same expected policy with no risk, which is always beneficial to (risk-
averse) Proposer. Meanwhile, Proposer seeks to maximize the expected mean
after a positive message. This explains why the optimal binary signal takes a
simple cutoff structure.131313The optimality of an experiment with at most two
messages is reminiscent of the classical result in Matthews (1989) on size-two
equilibria with cheap talk and veto threats. In addition to the very different
environments—Matthews (1989) considers cheap talk by a privately informed veto
player—the logic is quite different. The key to a binary cheap talk message is
that Proposer always offers her bliss point if not receiving a veto threat,
whereas a binary message emerges in our setting due to Proposer’s risk
aversion over policy outcomes.

Since $U(s)=u(p^{\ast}(s))$ and $p^{\ast}=2\mathbb{E}[\theta|s]$ whenever
$\mathbb{E}[\theta|s]\in(0,1/2)$, (3) can be rewritten as

$2u^{\prime}(2\mathbb{E}[\theta])(\mathbb{E}[\theta]-\underline{\theta})\leq
u(2\mathbb{E}[\theta])-u(0).$

To understand this condition, suppose $\underline{\theta}<0$ and
$\mathbb{E}[\theta]>0$,141414If $\underline{\theta}\geq 0$ then the experiment
only creates uncertainty in the resulting policy, without changing the
expected policy. Therefore, the experiment is never beneficial to Proposer. If
$\mathbb{E}[\theta]\leq 0$ then no information is obviously not optimal. and
compare no information with an experiment which reveals if $\theta$ is in a
small neighborhood of $\underline{\theta}$ or not (i.e., whether
$\theta<\underline{\theta}+\varepsilon$ or not for $\varepsilon$ sufficiently
small). This experiment raises the induced positive policy (from
$\mathbb{E}[\theta]$ to
$\mathbb{E}[\theta|\theta\geq\underline{\theta}+\varepsilon]$), but now
results in policy $0$, instead of $2\mathbb{E}[\theta]$, when
$\theta<\underline{\theta}+\varepsilon$. The left-hand side of the above
inequality captures the former marginal benefit (of revealing whether
$\theta<\underline{\theta}+\varepsilon$), while the right-hand side represents
the corresponding marginal cost. In other words, (3) coincides with the
condition under which Proposer has no incentive to reveal $\theta$ even when
it is arbitrarily close to $\underline{\theta}$. Note that it follows
immediately that if $\underline{\theta}$ is sufficiently negative relative to
$\mathbb{E}[\theta]$, then Proposer necessarily benefits from providing some
information. This is because when $\theta<\underline{\theta}+\varepsilon$ is
revealed the resulting action is $0$, so when $\underline{\theta}$ is
sufficiently low providing some information increases the expected action
significantly.

When no information is suboptimal, the optimal binary experiment is given by
revealing whether or not $\theta\geq s_{*}$, defined by (4). Note that when
$u$ is differentiable (i.e., no kink at Proposer’s bliss point), this
inequality can only be satisfied for all $s$ if

$U^{\prime}(s_{*})=\frac{U(\mathbb{E}[\theta|\theta\geq
s_{\ast}])-U(s_{\ast})}{\mathbb{E}[\theta|\theta\geq s_{\ast}]}$

or equivalently

$2u^{\prime}(2E[\theta|\theta\geq s_{*}])(E[\theta|\theta\geq
s_{*}]-s_{*})=u(2E[\theta|\theta\geq s_{*}])-u(0).$

This condition determines the $s_{*}$ for which (3) holds with equality when
$\underline{\theta}=s_{*}$. In particular, this implies that $s_{*}\leq 0$:
$s_{*}=0$ maximizes the expected action, but a risk averse Proposer has an
added incentive to reduce vetoes. When $u$ has a kink at $a=1$ it is also
possible to get a “corner” solution in which $s^{*}=1/2$ and so the resulting
proposal is either $0$ or $1$; see Section 6.1.

Finally, we note that when (3) fails, there exist multiple optimal
experiments; any experiment that reveals whether $\theta\geq s_{\ast}$ or not,
but provides no further information above $s_{\ast}$ is optimal. For example,
it is optimal to fully reveal $\theta$ below $s_{\ast}$ but provide no other
information. This multiplicity arises because for $\theta<s_{\ast}$, the
status quo policy is implemented and so the fine detail of the experiment is
inconsequential for the outcome.

$\underline{\theta}$$0$$\overline{\theta}=\frac{1}{2}$$s$$U(s)$$-c(1)$$0$$\underline{\theta}$$0$$s^{\ast}=\frac{1}{2}$$\overline{\theta}$$s$$U(s)$$-c(1)$$0$$s_{\ast}$
Figure 3: Both panels depict Proposer’s indirect utility function $U(s)$ (blue
solid) and the smallest convex function $\phi(s)$ above $U$ when $u$ is convex
($c$ concave). They differ in that $\overline{\theta}\leq 1/2$ in the left
panel, while $\overline{\theta}>1/2$ in the right panel.

#### Optimality of full information.

As is well known, full information is optimal if and only if $U$ is convex
over $[\underline{\theta},\overline{\theta}]$. Intuitively, the agent is “risk
loving” and so wishes to disperse the distribution $G$ of posterior means as
much as possible. In $MPC(F)$, the most dispersed distribution is $F$ itself.
In our problem, this condition holds if and only if, as depicted in the left
panel of Figure 3, $\overline{\theta}\leq 1/2$ and $u$ is _convex_ ($c$ is
_concave_) over $[0,2\overline{\theta}]$; the former ensures that $U$ does not
have the second flat region (above $1/2$), while the latter is necessary and
sufficient for $U$ to be convex over $[0,\overline{\theta}]$. Note that if
$\overline{\theta}>1/2$ then, even if $u$ is convex, full information is
suboptimal: Proposer will never reveal that $\theta>1/2$ as she would prefer
to pool such vetoers with some below $1/2$, securing her ideal action of $a=1$
more often. See the right panel of Figure 3.

###  5.2 Proposal-First Model

We now turn to the proposal-first model where Proposer cannot tailor her
proposal to the outcome of the experiment. In this case, one can first find
the optimal signal for each proposal $p$ and then optimize over $p$; this is
the approach we take for the linear Vetoer case in Section 7.3. For our main
quadratic case, we take a different (and arguably more elegant) approach. By
definition, Proposer has more flexibility under persuasion-first than under
proposal-first. Therefore, the optimal persuasion-first outcome provides an
upper bound for her expected payoff in the proposal-first model. We show that
the optimal persuasion-first outcome in Proposition 1 can be implemented in
the proposal-first model.

###### Proposition 2

The optimal experiments in Proposition 1 remain optimal in the proposal-first
model. Proposer optimally proposes $2\mathbb{E}[\theta]$ if (3) holds (i.e.,
no information is optimal) and $2\mathbb{E}[\theta|\theta\geq s_{\ast}]$
otherwise.

Proposition 2 holds because the proposer-optimal in the persuasion-first model
involves only one positive proposal; Proposer can achieve the same payoff as
under proposal-first by making that positive proposal and accepting a veto
otherwise. The optimality of a single positive proposal holds whenever $u$ is
concave. If $u$ is strictly convex and $\overline{\theta}\in(0,1/2)$, however,
the optimal signal under persuasion-first should fully reveal all $\theta$’s
above $0$. Such an outcome clearly cannot be implemented in the proposal-first
model. Proposition 2 also relies on the assumption on Vetoer’s quadratic
preferences. As we formally show in Section 7, if Vetoer incurs linear losses
then Proposer can often do strictly better under persuasion-first than under
proposal-first.

###  5.3 Discussion of Assumptions

A novel theoretical implication of Proposition 2 is that a single take-it-or-
leave-it offer is optimal even in the class of all mechanisms in our model. In
the persuasion-first model, it is natural because Proposer and Vetoer are
symmetrically informed when the proposal is made and, therefore, Proposer can
always offer the highest policy Vetoer would accept. In the proposal-first
model, however, Vetoer acquires some private information prior to his
decision, so in principle Proposer could benefit from offering a menu of
alternatives (Kartik, Kleiner and Van Weelden, 2021). Proposition 2 shows that
(with quadratic loss) such mechanism design is of no use to Proposer in our
model where she engages in optimal information design; Proposer can obtain the
optimal persuasion-first outcome even with a single take-it-or-leave-it offer.

Furthermore, achieving the persuasion-first outcome does not depend on
Proposer being able to commit not to revise her proposal in the event of a
veto. While in general Proposer may want to do so (Cameron, 2000; Ali, Kartik
and Kleiner, 2023), under the optimal experiment vetoes occur only after
Vetoer’s bliss point is revealed to be below the status quo, in which case no
mutually preferred policy exists. As such, Proposer commitment is immaterial.

However, Proposer’s ex ante commitment to an experiment (and truthfully
revealing its outcome) is crucial in implementing her optimal outcome.
Specifically, Proposer cannot necessarily obtain her optimal outcome if she
privately observes Vetoer’s ideal point $\theta$ and communicates with Vetoer
through cheap talk à la Crawford and Sobel (1982) or voluntary disclosure à la
Grossman (1981) and Milgrom (1981). Cheap talk cannot be effective because
Proposer’s preferences are independent of $\theta$. Voluntary disclosure, in
which Proposer can disclose any set containing $\theta$ but cannot lie, can
implement Proposer’s optimal outcome if and only if
$s^{\ast}=\mathbb{E}[\theta|s\geq s_{\ast}]=1/2$, so that the induced policy
is either the status quo $0$ or Proposer’s ideal one $1$; this is because
Proposer has an incentive to reveal $\theta\geq 1/2$, which is necessary and
sufficient to trigger familiar unraveling, if and only if
$s^{\ast}<1/2$.151515This discussion assumes Proposer privately knows $\theta$
at the beginning of the game. If the players are symmetrically informed at the
beginning of the game, and Proposer only privately learns $\theta$ after
committing to a proposal, then as in Titova (2022), there is an equilibrium in
which Proposer only reveals whether or not $\theta\geq s_{*}$ in the
disclosure stage. In this way proposal-first can outperform disclosure-first,
provided the proposal is made prior to Proposer becoming informed.

##  6 Examples and Comparative Statics

This section applies the above characterization to two leading examples where
Proposer’s loss function $c$ is either linear or quadratic. It also studies
how Proposer’s optimal strategy and the resulting outcome depend on the
characteristics of the underlying veto bargaining problem.

###  6.1 Linear Proposer

$\underline{\theta}=0$$\frac{1}{2}$$\overline{\theta}$$U(s)$$0$$-c(1)$$s$

$\underline{\theta}$$0$$s_{\ast}$$\frac{1}{2}$$\overline{\theta}$$U(s)$$0$$-c(1)$$s$

$\underline{\theta}$$0$$s_{\ast}$$\frac{1}{2}$$\overline{\theta}$$U(s)$$0$$-c(1)$$s$

Figure 4:  This figure depicts Proposer’s indirect utility function $U(s)$
(blue solid) and the smallest convex function $\phi(s)$ (red translucent)
above $U$ when $c$ is linear. The left panel considers the case where
$\underline{\theta}\geq 0$, while the other two panels represent the case
where $\underline{\theta}<0$. In the middle panel,
$\mathbb{E}[\theta|\theta\geq 0]>1/2$, while $\mathbb{E}[\theta|\theta\geq
0]<1/2$ in the right panel.

Suppose Proposer’s loss function is linear, that is, $c(|1-a|)=|1-a|$. In this
case, as depicted in Figure 4, $U$ is a piecewise linear function. Clearly,
$U$ is concave over $[\underline{\theta},\overline{\theta}]$ if
$\underline{\theta}\geq 0$ (the left panel) but not if $\underline{\theta}<0$
(the two right panels). This implies that no information is optimal if and
only if $\underline{\theta}\geq 0$. In addition, if $\underline{\theta}<0$
then an optimal signal structure is a binary experiment with cutoff
$\min\\{0,s_{\ast}\\}$ such that

$\displaystyle s^{\ast}=\mathbb{E}[\theta|\theta\geq s_{\ast}]=\frac{1}{2}.$

To interpret the cutoff, note that if $2\mathbb{E}[\theta|\theta\geq 0]<1$
then Proposer is risk neutral on the relevant range and seeks to maximize the
expected action, which is achieved by setting $s_{*}=0$. Otherwise, an optimal
experiment provides just enough information that Vetoer is willing to accept
Proposer’s ideal policy $1$ upon learning that $\theta\geq s_{\ast}$.

Clearly, conditional on $\underline{\theta}<0$ and
$\mathbb{E}[\theta|\theta\geq 0]>1/2$, Proposer’s optimal (positive) proposal
($2\mathbb{E}[\theta|\theta\geq s_{\ast}]=1$) is independent of the details of
$F$. Vetoer accepts $1$ if and only if $\theta\geq s_{\ast}$, whose
probability is equal to $1-F(s_{\ast})$. To see how this probability depends
on $F$, consider a simple case where $F$ is a uniform distribution over
$[\underline{\theta},\overline{\theta}]$ (and
$\underline{\theta}<0<\overline{\theta}$). In that case,
$\mathbb{E}[\theta|\theta\geq s_{\ast}]=\frac{s_{\ast}+\overline{\theta}}{2}$,
and thus

$\displaystyle
1-F(s_{\ast})=\frac{\overline{\theta}-s_{\ast}}{\overline{\theta}-\underline{\theta}}=\frac{2\overline{\theta}-1}{\overline{\theta}-\underline{\theta}}.$

It is easy to see that this probability is increasing in both
$\underline{\theta}$ and $\overline{\theta}$. If $\underline{\theta}<0$ and
$\mathbb{E}[\theta|\theta\geq 0]<1/2$, then Proposer’s optimal (positive)
proposal depends on $F$. When $F$ is a uniform distribution, the proposal
$2\mathbb{E}[\theta]=\overline{\theta}$ is increasing in $\overline{\theta}$
and independent of $\underline{\theta}$. The probability that Vetoer accepts
is $1-F(0)$, which is also increasing in both $\underline{\theta}$ and
$\overline{\theta}$.

###  6.2 Quadratic Proposer

Now suppose Proposer’s loss function is quadratic, that is,
$c(|1-a|)=(1-a)^{2}$. In this case, (3)—a necessary and sufficient condition
for no information to be optimal—reduces to

$\displaystyle-1\leq-(1-2\mathbb{E}[\theta])^{2}+(4-8\mathbb{E}[\theta])(\underline{\theta}-\mathbb{E}[\theta])\Leftrightarrow-\underline{\theta}\left(1-2\mathbb{E}[\theta]\right)\leq\mathbb{E}[\theta]^{2}.$

If $F$ is uniform over $[\underline{\theta},\overline{\theta}]$ then the
condition further simplifies to161616Observe that if $\overline{\theta}=1$
then $\kappa(\overline{\theta})=-1/3$, which is consistent with our derivation
at the end of Section 4.

$\displaystyle\underline{\theta}\left(1-2\frac{\underline{\theta}+\overline{\theta}}{2}\right)\leq\left(\frac{\underline{\theta}+\overline{\theta}}{2}\right)^{2}\Leftrightarrow\underline{\theta}\geq\kappa(\overline{\theta}):=-\frac{\overline{\theta}^{2}}{2-\overline{\theta}+2\sqrt{1-\overline{\theta}+\overline{\theta}^{2}}}.$

That no information is optimal if, and only if, $\underline{\theta}$ is not
too negative is clear from Figure 2 and the discussion following Proposition
1. Through rather tedious calculus, it can be shown that $\kappa$ is strictly
decreasing in $\overline{\theta}$. Then, the above condition also suggests
that no information is more likely to be optimal the higher
$\overline{\theta}$ is. Note that this result does not immediately follow from
our characterization in Section 5.

Suppose the above condition fails and, therefore, an optimal signal must
reveal whether $\theta\geq s_{\ast}$ or not for some
$s_{\ast}\in(\underline{\theta},0]$. In this case, (4)—the condition
characterizing the optimal $s_{\ast}$—reduces to

$\displaystyle\frac{1-(1-2\mathbb{E}[\theta|\theta\geq
s_{\ast}])^{2}}{\mathbb{E}[\theta|\theta\geq
s_{\ast}]-s_{\ast}}=4(1-2\mathbb{E}[\theta|\theta\geq s_{\ast}]).$

Applying $\mathbb{E}[\theta|\theta\geq
s_{\ast}]=\frac{s_{\ast}+\overline{\theta}}{2}$ to this equation, we obtain a
quadratic equation for $s_{\ast}$, whose unique negative solution is
$s_{\ast}=\kappa(\overline{\theta})$. This implies that if
$\underline{\theta}<\kappa(\overline{\theta})$ then it is optimal for Proposer
to reveal whether $\theta\geq\kappa(\overline{\theta})$ or not. It follows
that Proposer’s optimal proposal is $2\mathbb{E}[\theta|\theta\geq
s_{\ast}]=\kappa(\overline{\theta})+\overline{\theta}$. This value is clearly
independent of $\underline{\theta}$ and can be shown to be increasing in
$\overline{\theta}$. It also can be shown that, as in the previous linear
case, the probability that Vetoer accepts
$\kappa(\overline{\theta})+\overline{\theta}$ increases in both
$\underline{\theta}$ and $\overline{\theta}$.

###  6.3 General Comparative Statics

Comparing the two examples, there is a sense in which Proposer wishes to
provide less information in the quadratic case than in the linear case. To
begin with, no information is optimal in the linear case only when
$\underline{\theta}\geq 0$, but it can be in the quadratic case even when
$\underline{\theta}$ is strictly negative (as long as
$\underline{\theta}\geq\kappa(\overline{\theta})$). Furthermore, when no
information is suboptimal in both cases (i.e.,
$\underline{\theta}<\kappa(\overline{\theta})$), the relevant cutoff
$s_{\ast}$ is always larger in the linear case than in the quadratic case;
that is, $1-\overline{\theta}\geq\kappa(\overline{\theta})$. Since a binary
signal can be interpreted as to fully reveals $\theta$ if $\theta<s_{\ast}$
but no information if $\theta\geq s_{\ast}$, this also means that Proposer
provides less information in the quadratic case. Equivalently, since a
positive proposal is made and accepted when $\theta\geq s_{\ast}$, Vetoer is
less likely to learn information which causes them to reject the status quo
when Proposer has quadratic loss. Note that maintaining the status quo is also
less likely when the bounds on the support shift right. Our general
comparative statics result below generalizes these findings and illustrates
the driving force.

We consider changes in $F$ and the concavity of Proposer’s utility function
(convexity of $c$). We define more risk averse as a higher co-efficient of
absolute risk aversion,171717That is, if
$-\frac{u_{1}^{\prime\prime}(a)}{u_{1}^{\prime}(a)}\leq-\frac{u_{2}^{\prime\prime}(a)}{u_{2}^{\prime}(a)}$
for all $a\in[0,1]$, then Proposer is more risk averse under $u_{2}$ than
$u_{1}$. and show that a more risk averse proposer chooses lower $s_{*}$ (and
$s^{*}$) to reduce vetoes. Similarly, if the distribution of $\theta$ shifts
right, for any $s_{*}$ a more favorable proposal, $s^{*}$, would result.
Proposer would then have an incentive to decrease $s_{*}$ to reduce vetoes,
resulting in a higher proposal and fewer vetoes. The next result establishes
this formally.

###### Proposition 3

  1. (i)

$s_{*}$ and $s^{*}$ are decreasing in Proposer’s risk aversion.

  2. (ii)

A rightward likelihood ratio shift in the distribution of $\theta$ decreases
$s_{*}$ and $F(s_{*})$ and increases $s^{*}$.

Note that likelihood ratio dominance is stronger than needed to get the second
result. In fact, the result holds as long as $\mathbb{E}[\theta|\theta\geq s]$
becomes larger for any $s$; this is weaker than likelihood ratio dominance but
not implied by first order stochastic dominance.

##  7 Vetoer Linear Loss

So far, we have assumed that Vetoer has a quadratic loss function (i.e., his
ex post payoff is $-(\theta-a)^{2}$ when his ideal point is $\theta$ and the
chosen policy is $a$). This section studies the extent to which our results
depend on this specification. In particular, we characterize Proposer’s
optimal policies when Vetoer has a linear loss function (i.e., his ex post
payoff is given by $-|\theta-a|$).

###  7.1 Vetoer’s Acceptance Decision

As in Section 3, suppose Vetoer’s posterior belief about $\theta$ is
represented by the distribution $G$. When his ex post payoff is $-|\theta-a|$,
he is willing to accept proposal $p$ if and only if

$\displaystyle V(p,G)=-\int|\theta-p|dG(\theta)\geq
V(0,G)=-\int|\theta|dG(\theta),$

which can be rewritten as

$\displaystyle\int_{\underline{\theta}}^{p}(p-\theta)dG(\theta)+\int_{p}^{\overline{\theta}}(\theta-p)dG(\theta)\leq\int_{\underline{\theta}}^{\max\\{\underline{\theta},0\\}}(-\theta)dG(\theta)+\int_{\max\\{\underline{\theta},0\\}}^{\overline{\theta}}\theta
dG(\theta).$

Unlike in our baseline quadratic case, this condition cannot be reduced to a
more tractable and easy-to-interpret condition.

To proceed, we focus on the case where $\theta$ is either $\ell$ or $h$, and
$0\leq\ell<h$.181818This latter assumption is not necessary for the subsequent
analysis; with a linear loss, $\ell<0$ and $\ell=0$ are equivalent. However it
reduces the number of cases to consider and, more importantly, highlights
differences from our baseline quadratic case. Letting $\mu$ denote Vetoer’s
belief that his ideal point is $h$, the above acceptance condition becomes

$V(p,\mu)=-(1-\mu)|p-\ell|-\mu|p-h|\geq V(0,\mu)=-(1-\mu)|\ell|-\mu|h|,$

which can be simplified to

$\mu\geq\phi(p):=\frac{p-2\ell}{2\left(\min\\{p,h\\}-\ell\right)}.$

If $p$ is around $2\ell$ then $\phi(p)$ is close to $0$, so this inequality
necessarily holds. In contrast, it fails if $p$ is sufficiently large.

$p$$\overline{p}=1$$0$$\mu,\phi(p)$$1$$\ell$$2\ell$$h$$\phi(1)$$\phi(h)$
Figure 5: This figure shows $\phi(p)$—the lowest belief level at which Vetoer
is willing to accept $p$—in the binary linear vetoer case.

Figure 5 depicts the representative shape of the function $\phi$. Particularly
notable is that $\phi$ has an upward kink at $h$, that is, $\lim_{p\to
h-}\phi^{\prime}(p)<\lim_{p\to h+}\phi^{\prime}(p)$.191919Note that this is
relevant only when $h>2\ell$; otherwise, $\phi$ is a linear function above
$2\ell$. It stems from the single-peakedness of Vetoer’s preferences. To see
this more clearly, consider a marginal increase of Proposer’s proposal
$p\in[2\ell,1)$. If $p\geq h$ then the increase is detrimental to Vetoer,
regardless of his ideal point $\theta$. If $p<h$, however, it is beneficial to
Vetoer when his ideal point is $h$, although it is still harmful when his
ideal point is $\ell$. This implies that as $p$ rises, a relatively smaller
increase of $\mu$ is sufficient for Vetoer to stay indifferent between
accepting and rejecting $p$ when $p<h$ than when $p>h$. The kink at $h$
reflects this change around $h$. In fact, the same effect is present in our
baseline quadratic case. However, in that case, Vetoer’s ex post payoff is
smooth around his bliss point, so it does not result in a kink at $h$.

It is convenient to define the inverse of the function $\phi$. Formally, for
each $\mu\in[0,1]$, we let

$\psi(\mu):=\max\\{p\in[0,1]:\phi(p)\leq\mu\\}.$

Since $\phi$ is increasing, $\psi(\mu)$ represents the highest proposal below
$1$ that Vetoer would accept with belief $\mu$. In other words, Vetoer accepts
$p$ with belief $\mu$ if and only if $p\leq\psi(\mu)$.

###  7.2 Persuasion-First Model

As in Section 5, we first consider the persuasion-first model. As illustrated
above, given his belief $\mu$, Vetoer accepts $p$ if and only if
$p\leq\psi(\mu)$. Then, clearly, Proposer’s optimal proposal is $\psi(\mu)$,
and Proposer’s indirect utility associated with posterior $\mu$ is given by

$\widehat{U}(\mu)=-c\left(1-\psi(\mu)\right).$

If Proposer chooses a distribution $H$ of posteriors then her expected payoff
is given by $\mathbb{E}[\widehat{U}(\mu)]=\int\widehat{U}(\mu)dH(\mu)$. This
implies that Proposer’s optimal persuasion problem can be written as

$\max_{H\in\Delta([0,1])}\int\widehat{U}(\mu)dH(\mu)\text{ subject to }\int\mu
dH(\mu)=\mu_{0},$

where $\Delta([0,1])$ denotes the set of all distributions over $[0,1]$. As
observed by Kamenica and Gentzkow (2011), this linear programming problem can
be analyzed by the method of concavification (Aumann and Maschler, 1995).
Specifically, the optimal value, denoted by $\overline{U}(\mu_{0})$, coincides
with the value of the smallest concave function uniformly above $\widehat{U}$;
see the red translucent curves in Figure 6. Any distribution $H$ such that
$supp(H)\subseteq\\{\mu:\widehat{U}(\mu)=\overline{U}(\mu)\\}$, $\int\mu
dH(\mu)=\mu_{0}$, and $\int\widehat{U}(\mu)dH(\mu)=\overline{U}(\mu_{0})$ is
an optimal strategy.

$\mu$$1$$0$$\widehat{U}(\mu)$$-(1-2\ell)$$-(1-2h)$$0$$\phi(h)$$\mu$$1$$0$$\widehat{U}(\mu)$$0$$\phi(h)$$\phi(1)$
Figure 6: Both panels show Proposer’s indirect utility function
$\widehat{U}(\mu)$ (blue solid) and its smallest concave closure
$\overline{U}(\mu)$ (red translucent) when $c$ is linear. In both panels,
$\ell=0.1$. In the left panel $h=0.45(<1/2)$, while in the right panel
$h=0.7(>1/2)$.

#### No information.

We begin by studying the condition under which Proposer does not wish to
provide any information.

###### Proposition 4

Consider the persuasion-first model in which Vetoer’s loss function is linear
and his bliss point is either $\ell$ or $h$. Then no information is optimal if
$\mu_{0}\geq\max\\{0,\phi(h)\\}$. Conversely, there exists
$\overline{\ell}(>0)$ such that for any $\ell<\overline{\ell}$, no information
is sub-optimal if $\mu_{0}$ is sufficiently small.

To understand this result, recall that $\psi$ is convex below $\phi(h)$ and
piece-wise linear and concave above $\phi(h)$; see the blue curves in Figure
6. Combining this with the fact that $\widehat{U}(\mu)=-c(1-\psi(\mu))$, it
follows that, although $\widehat{U}$ may not be convex or concave below
$\phi(h)$, it is unambiguously concave above $\phi(h)$. The result is
straightforward from this shape of $\widehat{U}$. Intuitively, above
$\phi(h)$, there is no preference alignment between Proposer and Vetoer; an
increase of $p$ benefits the former but hurts the latter. Then, Vetoer’s
linear (risk-neutral) preferences imply that Proposer cannot increase the
expected value of the chosen policy. She then prefers the “safest” strategy
inducing a deterministic outcome to any other non-degenerate strategy.

The second result follows from the fact that if $\ell$ is close to $0$ then
$\widehat{U}(\mu)=-c(1-\psi(\mu))$ is convex in the small neighborhood of
$\mu$, regardless of convexity of $c$. To see this more clearly, observe that

$\widehat{U}^{\prime}(\mu)=c^{\prime}(1-\psi(\mu))\psi^{\prime}(\mu)>0.$

If $\ell$ is close to $0$ then
$\widehat{U}^{\prime}(0)=c^{\prime}(1-2\ell)2\ell\approx 0$. It then follows
that that $\widehat{U}^{\prime}(\mu)$ must be increasing—that is,
$\widehat{U}(\mu)$ is convex—around $0$.

#### Optimal persuasion.

In general, a necessary and sufficient condition for no information to be
optimal (equivalently, for persuasion to have value) and the structure of an
optimal signal depend on the detailed shape of $c$. We illustrate this with
the simplest example in which Proposer’s loss function is also linear. The
analysis can be easily extended for other loss functions.

######  Example 1 (Linear loss)

If $c$ is linear (i.e., $c(|1-a|)=|1-a|)$ then

$\widehat{U}(\mu)=-1+\psi(\mu)=-1+\left\\{\begin{array}[]{ll}2\ell\frac{(1-\mu)}{1-2\mu}&\text{if
}\mu\leq\phi(h)\\\ 2\left((1-\mu)\ell+\mu h\right)&\text{if
}\mu\in\left[\phi(h),\min\\{1,\phi(1)\\}\right]\\\ 1&\text{if
}p\in[\phi(1),1].\end{array}\right.$

As depicted in Figure 6, $\widehat{U}$ is strictly convex below $\phi(h)$ and
so coincides with $\overline{U}$ only above $\phi(h)$. This implies that
$\mu\geq\phi(h)$ is necessary and sufficient for no information to be optimal.
In addition, if $\mu<\phi(h)$ then any signal such that the support of
posteriors belongs to $0\cup[\phi(h),\min\\{\phi(1),1\\}]$ is optimal. For
example, the binary signal that induces posterior $0$ or $\phi(h)$ is optimal.

Note that the optimal persuasion mechanism can be achieved with a binary
message as in the quadratic model. However this emerges only because $\theta$
can take on only two values. As illustrated in Appendix B, more than two
messages may be necessary to achieve the optimal when Vetoer has linear loss.

###  7.3 Proposal-First Model

In our baseline quadratic case, the persuasion-first optimal outcome could
always be implemented in the proposal-first model. The result no longer holds
in the current linear environment. Hence, we provide a separate analysis and
compare the resulting outcome to the persuasion-first one.

Suppose Proposer has chosen $p\in[\underline{p},\overline{p}]$. To solve for
her optimal persuasion strategy, recall that Vetoer accepts $p$ if and only if
$\mu\geq\phi(p)$. This implies that, as depicted in Figure 7, Proposer’s
utility as a function of the induced posterior $\mu$ is a step function, with
a jump at $\phi(p)$ (the black solid curve). In this case, the following is
immediate from the standard concavification argument:

  * •

If $\mu_{0}\geq\phi(p)$ then it is optimal for Proposer to provide no
information.

  * •

If $\mu_{0}<\phi(p)$ then a binary signal that leads to posterior $0$ or
$\phi(p)$ is optimal.

$\mu$$1$$0$$\widetilde{U}(\mu;p)$$\phi(p)$$-c(1-p)$$-c(1)$$\mu_{0}$ Figure 7:
This figure illustrates the optimal information structure given $p$. In the
figure, the black solid curve denotes Proposer’s indirect utility as a
function of posterior $\mu$, while the red translucent curve represents the
smallest concave closure of the indirect utility function.

The best-case scenario for Proposer is that Vetoer accepts
$\overline{p}:=\min\\{2h,1\\}$ with probability $1$. The above result implies
that Proposer obtains this best outcome if and only if

$\mu_{0}\geq\phi(\overline{p})=\frac{\overline{p}-2\ell}{2(h-\ell)}.$ (5)

Notice that $\phi(\overline{p})<1$ if $h>1/2$, but $\phi(\overline{p})=1$ if
$h\leq 1/2$. This suggests that if $h>1/2$ then Proposer can implement
$\overline{p}$ with probability $1$ for $\mu_{0}$ sufficiently large; while if
$h\leq 1/2$ then $\overline{p}=2h$ cannot be implemented with probability $1$.
Intuitively, Vetoer is indifferent between $0$ and $\overline{p}=2h(\leq 1)$
if he is certain that his ideal point is $h$. If he assigns a positive
probability to $\ell$ then he prefers $0$ to $2h$. From now on, we focus on
the case where (5) fails.

In general, Proposer’s (indirect) expected utility as a function of proposal
$p$ is given by

$\widetilde{U}(p)=:-\min\left\\{\frac{\mu_{0}}{\phi(p)},1\right\\}c(1-p)-\max\left\\{\frac{\phi(p)-\mu_{0}}{\phi(p)},0\right\\}c(1).$

Since $\phi(p)\leq\mu_{0}\Leftrightarrow p\leq\psi(\mu_{0})$, $\widetilde{U}$
can be rewritten as

$\widetilde{U}(p)=\left\\{\begin{array}[]{ll}-c(1-p)&\text{if
}p\in[\underline{p},\psi(\mu_{0}))\\\
-c(1)+\frac{\mu_{0}}{\phi(p)}(c(1)-c(1-p))&\text{if
}p\in[\psi(\mu_{0}),\overline{p}].\end{array}\right.$

#### Optimal proposal.

It suffices to find $p$ that maximizes $\widetilde{U}(p)$. Clearly,
$\widetilde{U}$ is increasing below $\psi(\mu_{0})$. In addition, it is
decreasing after $\max\\{\psi(\mu_{0}),h\\}$.202020To see this, observe that
for $p>\max\\{\psi(\mu_{0}),h\\}$,  $\displaystyle\widetilde{U}^{\prime}(p)$
$\displaystyle=$
$\displaystyle\frac{\mu_{0}}{\phi(p)^{2}}\left[c^{\prime}(1-p)\phi(p)-(c(1)-c(1-p))\phi^{\prime}(p)\right]=\frac{\mu_{0}}{\phi(p)^{2}2(h-\ell)}\left[c^{\prime}(1-p)(p-2\ell)-(c(1)-c(1-p))\right]$
This expression is strictly negative because $\ell\geq 0$ and $c$ is convex,
so
$c^{\prime}(1-p)(p-2\ell)<c^{\prime}(1-p)p=c^{\prime}(1-p)\left(1-(1-p)\right)\leq
c(1)-c(1-p).$ For $p\in[\psi(\mu_{0}),h]$, $\widetilde{U}(p)$ is given by

$\widetilde{U}(p)=-c(1)+\mu_{0}(c(1)-c(1-p))\left(1+\frac{\ell}{p-2\ell}\right).$

This function may not be monotone, as $c(1)-c(1-p)$ is strictly increasing in
$p$, while $\frac{\ell}{p-\ell}$ is strictly decreasing in $p$. We make a mild
technical assumption on $c$ that $\widetilde{U}(p)$ is quasi-convex when
extended over $(2\ell,\infty)$; this is plausible, because the value of this
function grows arbitrarily large as $p$ tends to $2\ell$ or $\infty$, but its
precise shape depends on the detail of $c$. It is easy to see that this
assumption always holds if $c$ is linear or quadratic.

###### Proposition 5

Consider the proposal-first model in which Vetoer’s loss function is linear
and Vetoer’s bliss point is either $\ell$ or $h$. Suppose
$(c(1)-c(1-p))(p-\ell)/(p-2\ell)$ is quasi-convex in $p(>\underline{p})$ and
(5) fails.

  * •

Proposing $h$, with a binary signal that leads to posterior $0$ or $\phi(h)$,
is optimal if

$-c(1-\psi(\mu_{0}))\leq-c(1)+\frac{\mu_{0}}{\phi(h)}(c(1)-c(1-h)).$

  * •

Otherwise, proposing $\psi(\mu_{0})$, with no information, is optimal.

$p$$1$$2\ell$$\underline{p}$$h$$-0.7$$-0.6$(a)
$\mu_{0}=0.2$$p$$1$$2\ell$$\underline{p}$$h$$-0.7$$-0.42$(b)
$\mu_{0}=0.3$$p$$1$$2\ell$$\underline{p}$$h$$-0.7$$-0.2$(c) $\mu_{0}=0.45$
Figure 8: This figure shows Proposer’s indirect utility $\widetilde{U}(p)$ for
different values of $\mu_{0}$ when $c$ is linear. The parameter values used
for this figure are $\ell=0.15$ and $h=0.7$.

Figure 8 illustrates Proposition 5. The optimal proposal is either
$\underline{p}$ or $h$. It is certainly intuitive that $\psi(\mu_{0})$—the
maximal proposal Vetoer would accept with no information—can be optimal. The
potential optimality of $h$ is similar to the reason why Proposer never
induces $\mu\in(0,\phi(h))$ in Example 1: When Vetoer’s ideal point is $h$, an
increase of $p$ is beneficial to him if $p<h$ but not if $p>h$. As reflected
in the kinks of $\widetilde{U}(p)$ at $h$, this introduces a discontinuity in
Proposer’s incentive to raise $p$ at $h$, which makes $h$ a particularly
tempting option.

It is intriguing that $h$ is optimal—so vetoes occur with a positive
probability—only when $\mu_{0}$ is neither too small nor too large. For
$\mu_{0}$ sufficiently large, the result simply follows from the fact that
$\underline{p}>h$, that is, Vetoer would be willing to accept even a higher
proposal than $h$ with no information. To see why $\mu_{0}$ should not be too
low, consider $p\in[2\ell,h)$. As explained above, an increase of $p$ is
beneficial to a vetoer with ideal point $h$. However, it is still harmful if
it is $\ell$. Therefore, Proposer’s overall incentive to increase $p$ depends
on the relative probability of ideal point $h$. If $\mu_{0}$ is sufficiently
small then Proposer’s incentive to raise $p$ is small and so $\underline{p}$
is optimal. Otherwise, Proposer has a strong incentive to raise $p$, so $h$
can be optimal.

###  7.4 Proposal First vs Persuasion First

As is clear from Figure 9, in the current linear environment Proposer’s
expected payoff can be strictly larger under persuasion-first (red dashed)
than under proposal-first (blue solid). To see why this difference arises,
recall that in the proposal-first model, vetoes occur with a positive
probability whenever Proposer chooses to provide information (with proposal
$h$). In the persuasion-first model, Proposer can avoid vetoes even if she
provides information and so induces a non-degenerate distribution of
posteriors; if the realized posterior is $0$ then she can propose $2\ell$.
This is not only directly beneficial to Proposer, but also enables her to be
more aggressive with her information provision.

$\mu_{0}$$1$$0$$\overline{U}(\mu_{0})$$0$$1-2\ell$$\ell/h$$\phi(h)$$\phi(1)$
Figure 9: This figure shows Proposer’s expected payoffs as a function of the
prior $\mu_{0}$ when $c$ is linear, $\ell\geq 0$, and $h>1/2$. Blue solid: the
proposal-first model. Red dashed: the persuasion-first model. Brown dash-
dotted: no information.

##  8 Conclusion

A central element of agenda-setting is the opportunity to persuade others to
accept the proposal. In this paper we provide the first analysis of the effect
of persuasion in veto bargaining. With quadratic loss we give a full
characterization of Proposer’s optimal strategy, showing that a simple binary
experiment strictly increases the value of proposal power whenever the players
are sufficiently misaligned.

We have analyzed one particular setting: when both players are initially
symmetrically informed about Vetoer’s optimal action. An important avenue for
future research is to understand how information provision affects bargaining
in different informational environments. For example, in many settings the
Proposer may be an expert and so may have information Vetoer lacks.
Alternatively, in a private values environment Vetoer’s preferences are likely
known only to him. In parallel work we are exploring the effect of information
provision in the latter setting.

##  Appendix A Proofs

###### Proof of Proposition 1.

Consider a tangent line to $U$ at
$(\mathbb{E}_{F}[\theta],U(\mathbb{E}_{F}[\theta]))$—the right-hand side of
(3). By Theorems 1 and 2 of Dworczak and Martini (2019), the degenerate
distribution $\delta_{\mathbb{E}_{F}[\theta]}$—so, no information—is optimal
if and only if the tangent line stays uniformly above $U$. Due to the
structure of $U$, this latter condition is equivalent to (3).

If (3) fails then there exists a unique value of
$s_{\ast}\in[\underline{\theta},0]$ that satisfies (4); we establish this
below. Given $s_{\ast}$ satisfying (4), by Theorems 1 and 2 of Dworczak and
Martini (2019), a distribution $G$ is optimal if and only if
$G(s_{\ast})=F(s_{\ast})$, $G$ puts all probability mass above $s_{\ast}$ on
$\mathbb{E}[\theta|\theta\geq s_{\ast}]$, and the line that passes
$(s_{\ast},U(s_{\ast}))$ and $(\mathbb{E}[\theta|\theta\geq
s_{\ast}],U(\mathbb{E}[\theta|\theta\geq s_{\ast}])$ is uniformly larger than
$U$ above $s_{\ast}$. The binary signal reveals whether $\theta\geq s_{\ast}$
or not satisfies all these properties and so is optimal to Proposer.

Finally, we prove that there exists a unique value of
$s_{\ast}\in[\underline{\theta},0]$ that satisfies (4). For each $s_{\ast}<0$,
let $s^{\ast}\in[0,1/2]$ be the value such that

$\displaystyle U(s)\leq
U(s_{\ast})+\frac{U(s^{\ast})-U(s_{\ast})}{s^{\ast}-s_{\ast}}(s-s_{\ast})\text{
for all }s\geq[s_{\ast},\overline{\theta}].$

The structure of $U$ ensures that $s^{\ast}$ is well defined for each
$s_{\ast}\in[\underline{\theta},0)$. The desired value of $s_{\ast}$ is one
such that $s^{\ast}=\mathbb{E}[\theta|\theta\geq s_{\ast}]$. The unique
existence of such a value follows from the fact that (i) $s^{\ast}$ weakly
decreases, while $\mathbb{E}[\theta|\theta\geq s_{\ast}]$ increases, in
$s_{\ast}$ (so the two sides can be identical only at one value of
$s_{\ast}$), (ii) $s^{\ast}>\mathbb{E}[\theta]=\mathbb{E}[\theta|\theta\geq
s_{\ast}]$ if $s_{\ast}=\underline{\theta}$, and (iii)
$s^{\ast}=s_{\ast}<\mathbb{E}[\theta|\theta\geq s_{\ast}]$ if $s_{\ast}$ is
sufficiently close to $0$. ∎

###### Proof of Proposition 2.

Recall that in Proposition 1, either no information is optimal, or any optimal
signal reveals (effectively) only whether $\theta<s_{\ast}$ or not for some
$s_{\ast}\in(\underline{\theta},0)$. No information (with proposal
$p=2\mathbb{E}[\theta]$) can be obviously implemented in the proposal-first
model.

Consider any signal that falls into the second category (revealing only
whether $\theta<s_{\ast}$ or not). Under such a signal, the implemented policy
is $2\mathbb{E}[\theta|\theta\geq s_{\ast}]$ if $\theta\geq s_{\ast}$ and the
status quo policy $0$ if $s<s_{\ast}$. Suppose Proposer chooses the same
experiment and offers $p=2\mathbb{E}[\theta|\theta\geq s_{\ast}]$. Clearly,
Vetoer will accept $p$ if $\theta\geq s_{\ast}$ but vetoes if
$\theta<s_{\ast}$, which is the same outcome as in the persuasion-first model.
∎

###### Proof of Proposition 3.

Part 1: Inequality (4) holds with equality when $s=s_{*}$, and so to hold for
all $s<s_{*}$ and $s>s_{*}$ it must be that

$U^{\prime}(s_{*})=\frac{U(\mathbb{E}[\theta|\theta\geq
s_{\ast}])-U(s_{\ast})}{\mathbb{E}[\theta|\theta\geq s_{\ast}]-s_{\ast}},$

or equivalently,

$-s_{*}=\frac{u(2E[\theta|\theta\geq
s_{*}])-u(0)}{2u^{\prime}(2E[\theta|\theta\geq s_{*}])}-E[\theta|\theta\geq
s_{*}].$ (7)

As the LHS is strictly decreasing in $s_{*}$ a change in $u$ which increases
the RHS for all values of $s_{*}$ must decrease the solution to (7). So it is
sufficient to show that for any $a=2E[\theta|\theta\geq s_{*}]\in[0,1]$,

$\frac{u_{1}(a)-u_{1}(0)}{u_{1}^{\prime}(a)}\leq\frac{u_{2}(a)-u_{2}(0)}{u_{2}^{\prime}(a)}.$
(8)

Now note that

$u_{j}(a)-u_{j}(0)=\int_{0}^{a}u_{j}^{\prime}(t)d(t),$

and so

$\frac{u_{j}(a)-u_{j}(0)}{u_{j}^{\prime}(a)}=\int_{0}^{a}\frac{u_{j}^{\prime}(t)}{u_{j}^{\prime}(a)}dt$

and

$\log\left(\frac{u_{j}^{\prime}(t)}{u_{j}^{\prime}(a)}\right)=\log(u_{j}^{\prime}(t))-\log(u_{j}^{\prime}(a))=\int_{t}^{a}-\frac{u_{j}^{\prime\prime}(s)}{u_{j}^{\prime}(s)}ds.$

Hence if

$-\frac{u_{1}^{\prime\prime}(a)}{u_{1}^{\prime}(a)}\leq-\frac{u_{2}^{\prime\prime}(a)}{u_{2}^{\prime}(a)}$

for all $a$ then (8) holds. Thus $s_{*}$ and $s^{*}=E[\theta|\theta\geq
s_{*}]$ are lower under $u_{2}$ than $u_{1}$.

Part 2: We first show that the RHS of (7) is increasing in
$s^{*}=E[\theta|\theta\geq s_{*}]$. For this we note the the RHS is equal to

$\frac{u(2s^{*})-u(0)-2u^{\prime}(2s^{*})s^{*}}{2u^{\prime}(2s^{*})}.$

Differentiating with respect to $s^{*}$, the derivative is proportional to

$\displaystyle
2u^{\prime}(2s^{*})[2u^{\prime}(2s*)-2u^{\prime}(2s^{*})-4s^{*}u^{\prime\prime}(2s^{*})]-4u^{\prime\prime}(2s^{*})[u(2s^{*})-u(0)-2u^{\prime}(2s^{*})s^{*}]$
$\displaystyle=-4u^{\prime\prime}(2s^{*})[u(2s^{*})-u(0)]$ $\displaystyle\geq
0,$

and so the RHS is increasing.

A rightward LR shift increases $s^{*}=E[\theta|\theta\geq s_{*}]$ for any
$s_{*}$. And so given the that LHS of (7) is strictly decreasing in $s_{*}$,
and the RHS is increasing, $s_{*}$ must decrease and
$s^{*}=E[\theta|\theta\geq s_{*}]$ must increase for (7) to continue to hold.

∎

###### Proof of Proposition 4.

We first prove that no information is optimal if
$\mu_{0}\geq\max\\{0,\phi(h)\\}$. If $\mu_{0}\geq\phi(1)$ (equivalently,
$\psi(\mu_{0})=1$) then the result is trivial because Vetoer will accept 1
with no information. In what follows, suppose $\mu_{0}<\phi(1)$. Let
$\overline{U}$ denote the _smallest_ concave function uniformly above $U$.
Since $\psi(\mu)\leq 2((1-\mu)\ell+\mu h)$ for all $\mu$ and
$-c(1-2((1-\mu)\ell+\mu h))$ is concave in $\mu$, we have

$-c(1-2((1-\mu)\ell+\mu h))\geq\overline{U}(\mu)\geq U(\mu)=-c(1-\psi(\mu)).$

For $\mu\geq\max\\{0,\phi(h)\\}$, $\psi(\mu)=2((1-\mu)\ell+\mu h)$, and thus
$\overline{U}(\mu)=U(\mu)$. Since $\overline{U}$ represents the maximal
attainable payoff, it follows that no information is optimal whenever
$\mu_{0}\geq\max\\{0,\phi(h)\\}$.

For the second result, notice that the straight line that connects between
$(0,U(0))$ and $(\phi(h),U(\phi(h)))$ necessarily lies weakly below
$\overline{U}(\mu)$. Therefore, a _sufficient_ condition for no information to
be suboptimal at $\mu$ close to $0$ is

$U^{\prime}(0)<\frac{U(\phi(h))-U(0)}{\phi(h)}\leq\overline{U}^{\prime}(0).$

Applying the above derivation, we can show that this inequality is equivalent
to

$c^{\prime}(1-\psi(0))\psi^{\prime}(0)=c^{\prime}(1-2\ell)2\ell<(c(1)-c(1-h))\frac{2(h-\ell)}{h-2\ell},$

which necessarily holds when $\ell$ is close to $0$. ∎

##  Appendix B Example: More Than Two Messages

We consider the persuasion-first model in which both Proposer and Vetoer have
linear loss functions; $\theta$ can be either $0$, $\ell=0.1$, or $h=0.5$; and
the prior probabilities of Vetoer’s ideal point are given by $(0.7,0.2,0.1)$.
In this case, Proposer obtains the following expected payoff by providing full
information (and optimally proposing $0$, $0.2$, or $1$, depending on
$\theta$):

$U_{FI}=0.7(-1)+0.2(-0.8)+0.1(0)=-0.86.$

We now show that no binary experiment can achieve the same payoff.

#### Optimal proposal depending on Vetoer’s posterior belief.

If Vetoer’s belief over $\theta$ is given by
$(\mu_{0},\mu_{\ell},1-\mu_{0}-\mu_{\ell})$ then he is willing to accept $p$
if and only if

$\displaystyle-\mu_{0}p-\mu_{\ell}|p-0.1|-(1-\mu_{0}-\mu_{\ell})|p-0.5|\geq-\mu_{\ell}0.1-(1-\mu_{0}-\mu_{\ell})0.5.$

The highest $p$ that Vetoer would accept (equivalently, Proposer’s optimal
proposal) is given by

$\displaystyle p(\mu_{0},\mu_{l}):=\left\\{\begin{array}[]{ll}0&\text{if
}\mu_{0}>1/2\\\ \frac{0.2\mu_{\ell}}{2(\mu_{0}+\mu_{\ell})-1}&\text{if
}\mu_{0}\in(1/2-0.8\mu_{\ell},1/2]\\\ 1-\mu_{0}-0.8\mu_{\ell}&\text{if
}2\mu_{0}+1.6\mu_{\ell}\leq 1.\end{array}\right.$ (12)

Notice that if Proposer provides no information then, since the prior
probability of $\theta=0$ is $0.7$, Vetoer would never accept any positive
proposal, so Proposer’s expected payoff is

$U_{NO}=-1<U_{FI}.$

#### Optimal binary experiment.

Consider the following binary experiment: If $\theta=0$ then $s=0$ with
probability $1$. If $\theta=0.1$ then $s=0$ with probability $1/6$ and $s=1$
with probability $5/6$. Finally, if $\theta=0.5$ then $s=1$ with probability
$1$. Together with the prior $(0.7,0.2,0.1)$, this leads to posterior
$(21/22,1/22,0)$ with probability $11/15$ (when $s=0$) and $(0,5/8,3/8)$ with
probability $4/15$ (when $s=1$). The derivation above implies that Proposer
will propose $0$ in the former case and $1/2$ in the latter case. The
resulting expected payoff is

$\displaystyle
U_{BI}=\frac{11}{15}(-1)+\frac{4}{15}(-0.5)=-\frac{13}{15}\approx-0.8667<U_{FI}=-0.86.$

To understand this result, notice that analogous to the binary analysis
(Example 1), Proposer optimally induces beliefs for which proposal $p=h=0.5$
is made and accepted; this is the reason why this experiment is an optimal
binary experiment, as shown shortly. With only two messages, however, the
remaining $\ell=0.1$ vetoers must be pooled with $0$, at the cost of
maintaining the status quo more often.

We conclude this section by proving that the above binary experiment is an
optimal experiment. To begin with, notice that at least one message (signal
realization) should have $\mu_{0}\geq 0.7>0.5$, in which case the induced
policy is $0$. Hence, without loss, we can assume that $s\in\\{0,p\\}$ where
$s$ denotes the policy induced after each realization.

Let $\sigma(\theta)\in[0,1]$ denote the probability of $s=p$ given $\theta$.
Since $p$ is at least weakly higher with more weight on $0.5$ than $0.1$, and
at least weakly higher with more weight on $0.1$ than $0$, it incurs no loss
of generality to restrict attention to $\sigma(0.5)=1$ and either
$\sigma(0)=0$ or $\sigma(0.1)=1$. In other words, $p$ (as opposed to $0$)
should be induced with probability $1$ if $\theta=0.5$. In addition, $p$
should be induced first when $\theta=0.1$ and then (possibly) when $\theta=0$;
if $\sigma(0.1)<1$ while $\sigma(0)>0$ then, for example, $p$ can be raised
without reducing the probability of inducing it by raising $\sigma(0.1)$ while
lowering $\sigma(0)$.

(i) When $\sigma(0.1)=1$: Consider the case in which $\sigma(0.1)=1$ and
$\sigma(0)=\sigma\in[0,1]$. Then $s=p$ is chosen with probability
$0.3+0.7\sigma$ and $0$ is chosen with probability $0.7(1-\sigma(0))$. The
posterior following $s=p$ is then

$\mu_{0}=\frac{7\sigma}{3+7\sigma}\text{ and }\mu_{\ell}=\frac{2}{3+7\sigma}.$

Obviously any $\sigma$ for which $\mu_{0}>1/2$ cannot be optimal, and there is
no $\sigma$ for which
$2\mu_{0}+1.6\mu_{\ell}=\frac{3.2+14\sigma}{3+7\sigma}\leq 1$. This implies
that the resulting action is determined by the second case in (12):

$p=\frac{0.2\mu_{\ell}}{2(\mu_{0}+\mu_{\ell})-1}=\frac{0.4}{1+7\sigma}.$

Then, Proposer’s payoff for any $\sigma(0)\in[0,3/7]$ is

$-0.7(1-\sigma)-(0.3+0.7\sigma)\left(1-\frac{0.4}{1+7\sigma}\right)=-1+\frac{0.4(0.3+0.7\sigma)}{1+7\sigma},$

which is decreasing in $\sigma(0)$. Hence the optimal experiment in this class
is $\sigma(0)=0$, in which case Proposer’s expected payoff is given by

$U_{BI}^{\prime}=-1+(0.4)(0.3)=-0.88$

(ii) When $\sigma(0)=0$: Now consider the other case in which $\sigma(0)=0$
and $\sigma(0.1)=\sigma\in[0,1]$. Message $p$ then occurs with probability
$0.1+0.2\sigma$. As $\mu_{0}=0$ conditional on $s=p$, it follows from (12)
that

$\displaystyle
p(0,\mu_{\ell})=\left\\{\begin{array}[]{cc}\frac{0.2\mu_{\ell}}{2\mu_{\ell}-1}&\text{if
}\mu_{\ell}>2/5\\\ 1-0.8\mu_{\ell}&\text{if }\mu_{\ell}\leq
2/5.\end{array}\right.$

Combining this with $\mu_{\ell}=\frac{2\sigma}{2\sigma+1}$ leads to

$\displaystyle
p=\left\\{\begin{array}[]{cc}\frac{0.4\sigma}{2\sigma-1}&\text{if
}\sigma>5/6\\\ \frac{1.4\sigma+1}{2\sigma+1}&\text{if }\sigma\leq
5/6.\end{array}\right.$

Consequently, Proposer’s expected utility is

$\displaystyle-(0.9-0.2\sigma)(1)-(0.1+0.2\sigma)(1-p)$
$\displaystyle=-1+(0.1+0.2\sigma)p$
$\displaystyle=\left\\{\begin{array}[]{cc}-1+(0.1+0.2\sigma)\frac{0.4\sigma}{2\sigma-1}&\text{if
}\sigma>5/6\\\ -1+(0.1+0.2\sigma)\frac{0.4\sigma+1}{2\sigma+1}&\text{if
}\sigma\leq 5/6.\end{array}\right.$

This is maximized when $\sigma=5/6$ at which point $p=0.5$ and Proposer’s
utility is $U_{BI}=-13/15$. The optimality of this binary experiment follows
from the fact that $U_{BI}>U_{BI}^{\prime}$.

## References

  * (1)
  * Ali, Kartik and Kleiner (2023) Ali, S Nageeb, Navin Kartik and Andreas Kleiner. 2023.  “Sequential Veto Bargaining with Incomplete Information.” Econometrica _forthcoming_.
  * Anderson and Renault (2006) Anderson, Simon P and Régis Renault. 2006.  “Advertising content.” American Economic Review 96(1):93–113.
  * Aumann and Maschler (1995) Aumann, Robert J and Michael Maschler. 1995.  Repeated games with incomplete information.  MIT press.
  * Austen-Smith (1993) Austen-Smith, David. 1993.  “Information and influence: Lobbying for agendas and votes.” American Journal of Political Science 37(3):799–833.
  * Awad (2020) Awad, Emiel. 2020.  “Persuasive lobbying with allied legislators.” American Journal of Political Science 64(4):938–951.
  * Blackwell (1953) Blackwell, David. 1953.  “Equivalent Comparisons of Experiments.” Annals of Mathematical Statistics pp. 265–272.
  * Cameron (2000) Cameron, Charles M. 2000.  Veto Bargaining: Presidents and the Politics of Negative Power.  Cambridge University Press.
  * Cameron and McCarty (2004) Cameron, Charles and Nolan McCarty. 2004.  “Models of Vetoes and Veto Bargaining.” Annual Review of Political Science 7:409–435.
  * Che (1996) Che, Yeon-Koo. 1996.  “Customer return policies for experience goods.” The Journal of Industrial Economics pp. 17–24.
  * Cotton and Li (2018) Cotton, Christopher S and Cheng Li. 2018.  “Clueless politicians: On policymaker incentives for information acquisition in a model of lobbying.” Journal of Law, Economics, and Organization 34(3):425–456.
  * Crawford and Sobel (1982) Crawford, Vincent and Joel Sobel. 1982.  “Strategic Information Transmission.” Econometrica 50(6):1431–1451.
  * Dellis (2023) Dellis, Arnaud. 2023.  “Legislative informational lobbying.” Journal of Economic Theory 208:105595.
  * Dworczak and Martini (2019) Dworczak, Piotr and Giorgio Martini. 2019.  “The Simple Economics of Optimal Persuasion.” Journal of Political Economy 127(5):1993–2048.
  * Ellis and Groll (2020) Ellis, Christopher J and Thomas Groll. 2020.  “Strategic legislative subsidies: Informational lobbying and the cost of policy.” American Political Science Review 114(1):179–205.
  * Gentzkow and Kamenica (2016) Gentzkow, Matthew and Emir Kamenica. 2016.  “A Rothschild-Stiglitz approach to Bayesian persuasion.” American Economic Review 106(5):597–601.
  * Grossman (1981) Grossman, Sanford J. 1981.  “The informational role of warranties and private disclosure about product quality.” The Journal of Law and Economics 24(3):461–483.
  * Hall and Deardorff (2006) Hall, Richard L and Alan V Deardorff. 2006.  “Lobbying as legislative subsidy.” American Political Science Review 100(1):69–84.
  * Hirsch et al. (2023) Hirsch, Alexander V, Karam Kang, Pablo Montagnes and Hye Young You. 2023\.  “Lobbyists as Gatekeepers: Theory and Evidence.” Journal of Politics 85(2):731–748.
  * Hirsch and Shotts (2015) Hirsch, Alexander V. and Kenneth W. Shotts. 2015.  “Competitive Policy Development.” American Economic Review 105(4):1646–64.
  * Kamenica and Gentzkow (2011) Kamenica, Emir and Matthew Gentzkow. 2011.  “Bayesian persuasion.” American Economic Review 101(6):2590–2615.
  * Kartik, Kleiner and Van Weelden (2021) Kartik, Navin, Andreas Kleiner and Richard Van Weelden. 2021.  “Delegation in veto bargaining.” American Economic Review 111(12):4046–4087.
  * Kolotilin (2018) Kolotilin, Anton. 2018.  “Optimal information disclosure: A linear programming approach.” Theoretical Economics 13(2):607–635.
  * Lewis and Sappington (1994) Lewis, Tracy R and David EM Sappington. 1994.  “Supplying information to facilitate price discrimination.” International Economic Review pp. 309–327.
  * Matthews (1989) Matthews, Steven A. 1989.  “Veto Threats: Rhetoric in a Bargaining Game.” Quarterly Journal of Economics 104(2):347–369.
  * Milgrom (1981) Milgrom, Paul R. 1981.  “Good news and bad news: Representation theorems and applications.” The Bell Journal of Economics pp. 380–391.
  * Potters and Van Winden (1992) Potters, Jan and Frans Van Winden. 1992.  “Lobbying and asymmetric information.” Public Choice 74:269–292.
  * Prato and Turner (2023) Prato, Carlo and Ian Turner. 2023.  “Institutional Foundations of the Power to Persuade.” Working Paper .
  * Roesler and Szentes (2017) Roesler, Anne-Katrin and Balázs Szentes. 2017.  “Buyer-optimal learning and monopoly pricing.” American Economic Review 107(7):2072–2080.
  * Romer and Rosenthal (1978) Romer, Thomas and Howard Rosenthal. 1978.  “Political Resource Allocation, Controlled Agendas, and the Status Quo.” Public Choice 33(4):27–43.
  * Romer and Rosenthal (1979) Romer, Thomas and Howard Rosenthal. 1979.  “Bureaucrats Versus Voters: On the Political Economy of Resource Allocation by Direct Democracy.” Quarterly Journal of Economics 93(4):563–587.
  * Schnakenberg (2017) Schnakenberg, Keith. 2017.  “Informational lobbying and legislative voting.” American Journal of Political Science 61(1):129–145.
  * Schnakenberg and Turner (2021) Schnakenberg, Keith and Ian Turner. 2021.  “Helping friends or influencing foes: Electoral and policy effects of campaign finance contributions.” American Journal of Political Science 65(1):88–100.
  * Schnakenberg and Turner (2023) Schnakenberg, Keith and Ian Turner. 2023.  “Formal Theories of Special Interest Influence.” Working Paper .
  * Titova (2022) Titova, Maria. 2022.  “Persuasion with verifiable information.” Working Paper .