# Order of Commitments in Bayesian Persuasion with Partial-informed Senders

Shih-Tang Su, Vijay G. Subramanian

###### Abstract.

The commitment power of senders distinguishes Bayesian persuasion problems
from other games with (strategic) communication. Persuasion games with
multiple senders have largely studied simultaneous commitment and signalling
settings. However, many real-world instances with multiple senders have
sequential signalling. In such contexts, commitments can also be made
sequentially, and then the order of commitment by the senders – the sender
signalling last committing first or last – could significantly impact the
equilibrium payoffs and strategies. For a two-sender persuasion game where the
senders are partially aware of the state of the world, we find necessary and
sufficient conditions to determine when different commitment orders yield
different payoff profiles. In particular, for the two-sender setting, we show
that different payoff profiles arise if two properties hold: 1) the two
senders are willing to collaborate in persuading the receiver in some
state(s); and 2) the sender signalling second can carry out a credible threat
when committing first such that the other sender’s room to design signals gets
constrained.

††conference: ; ;

##  1\. Introduction

The cornerstone feature of Bayesian persuasion (Kamenica and Gentzkow, 2011;
Rayo and Segal, 2010) is the commitment power of the sender(s). Generalized
from the seminal prosecutor-judge example in (Kamenica and Gentzkow, 2011)
with one sender and one receiver, Gentzkow and Kamanica in (Gentzkow and
Kamenica, 2016) introduced a more realistic scenario with a second sender, the
defense attorney, who can also choose a signal to persuade the judge in
addition to the prosecutor; this extended Bayesian persuasion models to
multiple senders. A body of literature has since emerged along with two
different directions within the context of fully-informed senders: competition
and equilibrium under simultaneous signal revelation (Au and Kawai, 2020); and
(iterative) commitment derivation in sequential persuasion (Li and Norman,
2018b). A general result in these works states that the competition (between
senders) increases the extent of information revelation.

In a variety of real-world problems with multiple participants but only one
decision maker – such as applying for a grant/scholarship, co-branding
marketing, lobbying, and manufacturing process selection – the senders are not
fully informed about the state of the world. However, each sender holds a
piece of information about the state of the world, and learning other senders’
private information (from their signals) may lead to a more tailored
signalling scheme. Hence, truthfully reporting her private information is
usually not an optimal strategy for a Bayes-rational sender. The above
scenarios can be modeled as a Bayesian persuasion problem with multiple
partially informed senders. When senders are partially informed, the
interactions of senders under simultaneous revelation and interactions under
sequential revelation are essentially different. With sequential signal
revelation and partially-informed senders, a sender who sends signals later on
in the sequence can learn more about the state of the world by updating her
belief using the earlier senders’ signals. Since these later on in the
sequence senders can anticipate some revealed signals while sending their own
signals, they can design signalling strategies conditional on the realization
of early senders’ signals. On the other hand, when early senders foresee that
their signals will be exploited to act against them, they will modify their
commitment to preclude or minimize such behaviors. This iterative belief-
updating procedure on commitments leads to the following questions in Bayesian
persuasion with multiple partially informed senders context: will the
commitment order of senders matter? If it matters, which sender should make
her commitment first? In this paper, we study these questions, and start with
a motivating example to help the reader better understand real-world scenarios
that fit the Bayesian persuasion with multiple partially informed senders
paradigm.

###  1.1. Motivating Example

###### Example 0.

A company $X$ searches for the most efficient way to produce its new product.
For this purpose, it needs to decide between the following combinations:
whether to apply a particular patent or not, and which manufacturing process
(advanced or mature) it should adopt. The company sponsors a university lab
$Y$ to do some simulations for its decision on the patent application. Based
on the simulation results, the company then ask its partner factory $Z$ to do
some trial runs to determine the manufacturing process. However, lab $Y$
always prefers the patent to be applied, and the factory $Z$ desires the
advanced process to be adopted. Thus, $Y$ may choose simulation environments
in favor of the patent being applied, and $Z$ may choose trial run parameters
in favor of the advanced process. However, all the simulation and trial-run
setups have to be submitted to the company beforehand. Then two questions
arise: First, does the order of submission of proposals matter? Second, if it
does, from company $X$’s perspective, which party should submit their proposal
first, $Y$ or $Z$?

In the above example, $Y$ and $Z$ play the role of senders with commitments,
and $X$ is the receiver. Since both senders only have partial (maybe
correlated) information of the state of the world, the sender who sends the
signal later, $Z$ in the example, may be able to commit to a signalling scheme
that depends on the state of the world and the realized signal of the other
sender, $Y$ in the example. Note that when $Z$ makes her commitment, she has
not yet observed the signal sent from $Y$. Hence, her goal is to make a
commitment on top of $Y$’s commitment (if $Y$ commits first) to exploit the
signal revealed from $Y$. However, whether the (potential) exploit of $Y$’s
signal can hurt $Y$ depends on the correlation of $Y$ and $Z$’s private
information and the relevant payoff matrices. Not only is our example
plausible, similar scenarios with multiple partial-informed senders can be
found in data markets (Acemoglu et al., 2019; Fallah et al., 2022), corporate
political actions (Chiroleu-Assouline and Lyon, 2020), and stress testing
(Inostroza and Pavan, 2021). Hence, our goal is to characterize broadly
applicable conditions that determine when the commitment order matters. With
this goal in mind, the two main contributions of this work are:

Contributions:

  1. (1)

To the best of our knowledge, we are the first to identify the importance of
the commitment order in multi-sender Bayesian persuasion problems. We show
that when senders are partially informed and signals are realized
sequentially, the commitment order may play a significant role in the
equilibrium strategies (commitments), and hence, the payoffs of the agents.

  2. (2)

In a two-sender setting, we present a set of sufficient conditions and a set
of necessary conditions that can identify when the commitment order matters.
An intuitive characterization of our conditions is the following – when both
senders have a conflict of interest, and the sender signalling second can post
a credible threat, the commitment order will matter.

###  1.2. Literature Review

The major distinction between Bayesian persuasion (Kamenica and Gentzkow,
2011; Rayo and Segal, 2010) and other paradigms that study games with
communication, e.g., cheap talk (Crawford and Sobel, 1982), signalling games
(Spence, 1978), and mechanism design (Myerson, 1981; Hurwicz and Reiter,
2006), is the commitment power of the sender(s). With the commitment power in
hand, each sender can influence the receiver’s (payoff-relevant) action by
committing to her (verifiable) signalling scheme, which is typically a
randomized mapping from states of the world to signals. Subsequently, the
sender(s) receives some private information about the state of the world and
applies the signalling scheme. After that, the receiver(s) choose actions.
Seminal works of Kamenica and Gentzkow (Kamenica and Gentzkow, 2011), Rayo and
Segal (Rayo and Segal, 2010), and Bergemann and Morris (Bergemann and Morris,
2016) established the research area of Bayesian persuasion, and since then
there has been a large literature on both theoretical aspects and applications
of Bayesian persuasion. To keep our discussion focused, we only discuss
closely related work and refer the reader to survey articles (Kamenica, 2019;
Bergemann and Morris, 2019) for the broader literature.

The literature closest to our work studies Bayesian persuasion with multiple
senders. Gentzkow and Kamenica (Gentzkow and Kamenica, 2016) studied the model
of fully-informed senders (symmetric information games) with simultaneous
signal revelation. They identified conditions on the information environment
such that the competition outcome is no less (Blackwell) informative than the
collusive outcome, and those conditions were, later on, proved to be
sufficient and necessary by Li and Norman (Li and Norman, 2018a). Thereafter,
Gentzkow and Kamenica (Gentzkow and Kamenica, 2017) and Au and Kawai (Au and
Kawai, 2020) both considered fully-informed senders with simultaneous signal
revelation. The former proved that the concavification approach could be
extended to multiple senders, and greater competition among senders tends to
increase information revelation for the receiver. The latter further showed
the existence of a unique symmetric equilibrium, and also that full disclosure
(of every sender) occurs when the number of senders goes to infinity. In
simultaneous commitment plus signal revelation models, the competition among
senders becomes more sophisticated when senders are partially informed. In
(Boleslavsky and Cotton, 2018), Boleslavsky and Cotton studied two senders
with independent partial information. They construct the equilibrium based on
the result that each sender’s incentive is similar to that in an all-pay
auction under complete information. Au and Kawai (Au and Kawai, 2021)
considered multiple senders with correlated information, and presented two
effects, namely the underdog-handicap effect and the good-news curse, that
drive the information disclosure. The former encourages more aggressive
disclosure because of the receiver’s bias towards a (ex-ante) stronger sender.
But the latter disincentivizes the disclosure because each sender using her
favorable signal implies that their rival is more likely to generate a strong
competing signal in response. However, when senders are partially informed,
interactions among senders can incorporate not only competitions but also
(partial) collaboration. The simultaneous commitment plus signal revelation
models essentially preclude111Except for senders with fully-aligned utilities,
which will transform the multi-sender Bayesian persuasion problem to a
coordination game. collaboration among partially informed senders since
senders cannot learn (and tailor) their commitments using other senders’
possible realized signals. Hence, to fully capture the interactions among
partially-informed senders, one should target settings beyond the simultaneous
signal revelation paradigm.

Besides the simultaneous signal revelation setting, Li and Norman (Li and
Norman, 2018b) consider a Bayesian persuasion model with sequential signal
revelation. In the Li-Norman model, senders commit sequentially to post
experiments, and every experiment is conditionally independent of each other
given the state. With this conditional independence, Li and Norman analyze
optimal signalling strategies for heterogeneous222Senders with different
utility functions. senders with same (and full333The sender can access all
possible experiments under the “reduced” state space obtained by merging each
state set not distinguishable by the sender to a single state.) information
about the state. Interestingly, their result shows that sequential persuasion
cannot yield a more informative equilibrium in comparison to simultaneous
persuasion (Gentzkow and Kamenica, 2016) and is strictly less informative in
binary state models. Their result stands on the conditional independence of
the experiments and the assumption that the order of (senders’) commitments
perfectly matches the order of signals/experiments. However, in scenarios such
as the motivating example above, this alignment may not hold, and the receiver
may have a choice about who should commit first. Besides, analogous to
simultaneous Bayesian persuasion, senders in sequential Bayesian persuasion
may be partially informed. The sequential revelation and partially informed
senders assumptions distinguish our model from the existing literature, and
leads to interesting follow-up research questions on the optimal commitment
order.

##  2\. Problem Formulation

There are two senders $S_{1},S_{2}$ and one receiver $R$ in the game. The
state space $\Theta$ is finite, and the receiver’s action space $A$ is also
finite. For simplicity of analysis, we assume that the state and action space
have the same size, i.e., $|\Theta|=|A|$, and action $a_{\theta}$ is the
receiver’s unique best response of state $\theta\in\Theta$. To further
simplify the analysis, we assume that the receiver only obtains (positive)
utility upon matching the state correctly, i.e.,

(1) $\displaystyle U_{R}(\theta_{i},a_{j})\begin{cases}1&\text{if~{}}i=j,\\\
0&\text{otherwise}.\end{cases}$

Before the true state is realized (by nature), the senders and the receiver
have a common prior belief of the distribution of the state
$p\in\Delta(\Theta)$. However, both senders obtain some private information
about the state of the world when it is realized. To avoid some trivial cases
and also to avoid overlap with the existing literature, we make four
assumptions on the senders’ private information:

###### Assumption 1.

Each sender only has partial information of the true state. To avoid
redundancy, we assume each sender $S_{k}$’s information space
$\mathcal{I}_{k}$ has the size $|\mathcal{I}_{k}|<|\Theta|$, $k\in\\{1,2\\}$.

###### Assumption 2.

Senders $S_{1}$ and $S_{2}$ have different private information, i.e.,
$\mathcal{I}_{1}\neq\mathcal{I}_{2}$.

###### Assumption 3.

Neither sender is more knowledgeable444In terms of Blackwell informativeness
(Blackwell, 1953). than the other, and the private information of each sender
is generated via an onto function $F_{k}:\Theta\rightarrow\mathcal{I}_{k}$. In
other words, $F_{k}$ is a partition of $\Theta$, and neither $F_{1}$ is more
Blackwell informative than $F_{2}$ nor $F_{2}$ is more Blackwell informative
than $F_{1}$.

###### Assumption 4.

The true state can be revealed under both senders’ truth-telling strategies.
In other words, for every $\theta_{i}\in\Theta$, there exists an
$I_{1}^{x}\in\mathcal{I}_{1}$ and an $I_{2}^{y}\in\mathcal{I}_{2}$ such that
$\mathbb{P}(\theta_{i}|I_{1}^{x},I_{2}^{y})=1$.

After both senders get their private information, $S_{1}$ sends a signal
$\omega_{1}\in\Omega_{1}$ to influence the receiver’s belief. After observing
the signal $\omega_{1}$, $S_{2}$ sends a signal $\omega_{2}\in\Omega_{2}$ to
influence the receiver’s belief. After receiving signals $\omega_{1}$ and
$\omega_{2}$, the receiver has to take action $a\in A$. Extending the sender-
preferred tie-breaking rules in (Kamenica and Gentzkow, 2011), we assume that
the receiver breaks a tie in favor of $S_{1}$ first. When the action set
maximizing both $S_{1}$ and receiver’s utilities is not a singleton, the
receiver takes an (arbitrary) action maximizing $S_{2}$’s utility from the
set.

In this model, both senders commit to their respective signalling strategies
before receiving their private information. Moreover, $S_{1}$ and $S_{2}$ make
their commitment sequentially under a pre-determined order. That is to say,
the sender who commits later can exploit another sender’s commitment to design
her signalling strategy. We will first detail the game flow (when the order of
commitment in steps 2 and 3 is determined), and then discuss how the sender’s
commitment can depend on the other sender’s signalling strategy based on the
commitment order.

###  2.1. Procedure of the game

The game in our model evolves as follows:

  1. (1)

Prior belief of the state $p\in\Delta(\Theta)$ is disclosed to all
participants.

  2. (2)

$S_{k}$ commits her signalling strategy, where the index $k\in\\{1,2\\}$ is
determined beforehand and becomes common knowledge.

  3. (3)

The other sender, $S_{-k}$, commits to her signalling strategy.

  4. (4)

Nature chooses a realized state $\theta\in\Theta$.

  5. (5)

$S_{1}$ gets her private information and sends a (randomized) signal
$\omega_{1}$ based on her commitment.

  6. (6)

$S_{2}$ gets her private information and observes $S_{1}$’s realized signal
$\omega_{1}$. Then, she sends a (randomized) signal $\omega_{2}$ according to
her commitment.

  7. (7)

The receiver observes the realized signals $\omega_{1},\omega_{2}$ and takes
action $a\in A$ to maximize the probability of matching the state.

###  2.2. Receiver’s best response

Given the procedure of the game, the receiver takes action after observing the
realization of both senders’ signals, $\omega_{1},\omega_{2}$. Given that her
objective is to match the state correctly, the receiver’s best response is
taking action $a^{*}\in A$ such that

$\displaystyle a^{*}\in\arg\max_{a\in
A}\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{R}(\theta,a)|\omega_{1},\omega_{2}].$

However, the tie-breaking rule, although indifferent to the receiver, plays a
significant role on senders’ signalling strategy, especially when multiple
senders are considered in the model. As mentioned above, we assume the
receiver breaks a tie in favor of $S_{1}$ first and then $S_{2}$. The formal
expression of the tie-breaking rule is presented below:

$\displaystyle\text{Let}~{}A_{R}\coloneqq\arg\max_{a\in
A}\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{R}(\theta,a)|\omega_{1},\omega_{2}]\text{,~{}}A_{S_{1}}\coloneqq\arg\max_{a^{\prime}\in
A_{R}}\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{S_{1}}(\theta,a^{\prime})|\omega_{1},\omega_{2}],$
(2)
$\displaystyle\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{S_{1}}(\theta,a^{*})|\omega_{1},\omega_{2}]=\max_{a^{\prime}\in
A_{R}}\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{S_{1}}(\theta,a^{\prime})|\omega_{1},\omega_{2}],$
(3)
$\displaystyle\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{S_{2}}(\theta,a^{*})|\omega_{1},\omega_{2}]=\max_{a^{\prime\prime}\in
A_{S_{1}}}\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U_{S_{2}}(\theta,a^{\prime})|\omega_{1},\omega_{2}].$

To avoid ambiguity,
$\mathbb{E}_{p,\Gamma_{1},\Gamma_{2}}[U(\theta,a)|\omega_{1},\omega_{2}]$
represents the expected utility conditional on the realized signals
$\omega_{1}$ and $\omega_{2}$ under the prior $p$, $S_{1}$’s commitment
$\Gamma_{1}$, and $S_{2}$’s commitment $\Gamma_{2}$, given action $a$ is
taken. In short, the expectation is taken only on the prior. Equation (2)
states that the receiver chooses an (arbitrary) action in the set which
maximizes $S_{1}$’s expected utility while she is indifferent. Equation (3)
states if there is still a tie after maximizing both $S_{1}$ and receiver’s
expected utility, the receiver chooses an (arbitrary) action in the set which
maximizes $S_{2}$’s expected utility. For simplicity of representation, we
abuse notation and let $a^{*}(\omega_{1},\omega_{2})$ represents the
receiver’s best response under $\omega_{1}$ and $S_{2}$’s signal $\omega_{2}$
hereafter.

###  2.3. Assumption on commitments

After defining the receiver’s best response, we detail why a sender $S_{k}$
may be able to utilize another sender $S_{-k}$’s commitment when she commits
later. The key is the pre-determined order of the sequential signalling. Since
$S_{1}$ always sends her signal before $S_{2}$, the signalling schemes of the
senders are asymmetric. In other words, $S_{1}$ cannot commit to a signalling
scheme that depends on the realization of $\omega_{2}$, but $S_{2}$ can commit
to a signalling scheme based on the realization of $\omega_{1}$. Next we
clarify the information that $S_{2}$ can exploit while making her commitment.
To avoid ambiguity, we discuss both commitment orders and will introduce our
cornerstone assumption, namely permutation-free commitments. Before the
discussion, we note that the superscript $f,s$ of the commitment notation
$\Gamma^{f},\Gamma^{s}$ denote the commitment order, and the subscript denote
the sender and another sender’s commitment (if available), e.g.,
$\Gamma^{s}_{2,\Gamma^{f}_{1}}$ denotes $S_{2}$’s commitment when she commits
after $S_{1}$ under $S_{1}$’s commitment $\Gamma^{f}_{1}$.
Case 1: $S_{1}$ commits to her signalling strategy first

  * •

Since sender $S_{1}$ has no observation of sender $S_{2}$’s signal while
sending her signal and has to commit first, her commitment $\Gamma^{f}_{1}$
can be represented as a function
$\Gamma^{f}_{1}:\mathcal{I}_{1}\rightarrow\Delta(\Omega_{1})$, where
$|\Omega_{1}|\leq|\mathcal{I}_{1}|$.

  * •

Sender $S_{2}$ can utilize sender $S_{1}$’s commitment and realized signal to
make her commitment. Hence, sender $S_{2}$’s commitment
$\Gamma^{s}_{2,\Gamma^{f}_{1}}$ can be represented as a function
$\Gamma^{s}_{2,\Gamma^{f}_{1}}:\mathcal{I}_{2}\times\Omega_{1}\rightarrow\Delta(\Omega_{2})$.

Case 2: $S_{2}$ commits to her signalling strategy first

  * •

Even though sender $S_{1}$ knows $S_{2}$’s commitment $\Gamma^{f}_{2}$, she
still has no observation of sender $S_{2}$’s signal while sending her signal.
Hence, the commitment $\Gamma^{s}_{1,\Gamma^{f}_{2}}$ is a function
$\Gamma^{s}_{1,\Gamma^{f}_{2}}:\mathcal{I}_{1}\rightarrow\Delta(\Omega_{1})$,
where $|\Omega_{1}|\leq|\mathcal{I}_{1}|$. In other words, sender $S_{1}$ can
adjust her commitment based on $S_{2}$’s commitment, but cannot tailor her
commitments by utilizing $S_{2}$’s realized signals.

  * •

Sender $S_{2}$ wants to utilize $S_{1}$’s realized signal to make her
(optimal) commitment. However, $S_{2}$ has to commit before sender $S_{1}$ in
this case. Since we don’t want each signal token to possess an implied
meaning, thereby allowing $S_{1}$ to act against $S_{2}$ by just reordering
the signal tokens, we assume that the $S_{2}$ commits to a permutation-free
commitment.

###### Definition 1.

A commitment
$\Gamma^{f}_{2}:\mathcal{I}_{2}\times\Omega_{1}\rightarrow\Delta(\Omega_{2})$
is permutation-free if for every commitment $\Gamma^{s}_{1}$ of $S_{1}$, there
is no permutation matrix $M$ such that

$\displaystyle\mathbb{E}_{p,\Gamma^{s}_{1}}[U_{S_{2}}|\Gamma^{f}_{2}(I_{2},M(\omega_{1}))]>\mathbb{E}_{p,\Gamma^{s}_{1}}[U_{S_{2}}|\Gamma^{f}_{2}(I_{2},\omega_{1})].$

Next, we justify why we believe that the permutation-free assumption on
commitments is appropriate when $S_{2}$ has to commit first. When we exploit
Bayesian persuasion in real-world problems, commitments are usually
implemented via experiments (which can be operated/supervised by a third
party). Hence, when $S_{2}$ has to commit first and wants to exploit $S_{1}$’s
realized signal, she can propose different experiments depending on $S_{1}$’s
signal realization. Unless we assume some signal tokens have implied meaning
or $S_{2}$ has some prior knowledge about $S_{1}$’s signalling strategy,
$S_{2}$ cannot know the mapping from signal $\omega_{1}$ to the interim
distribution of the states while making her commitments. However, when an
experiment of $S_{2}$ has to be executed, $S_{1}$’s signal realization and her
commitments are both common knowledge. Hence, $S_{2}$ can commit to
experiments with her private signal and the mock signal realizations of
$S_{1}$, and ask the operator of her experiments to use a
permutation/reordering of $S_{1}$’s signal tokens that works best for her.
These experiments can be appropriately executed, and this permutation-free
commitment can be fulfilled. Moreover, not embracing the permutation-free
property implies that $S_{1}$ and $S_{2}$ have an ex-ante consensus on the
meaning of $S_{1}$’s signal tokens, but $S_{1}$ can violate their ex-ante
consensus to act against $S_{2}$. If $S_{2}$ can lose by sender $S_{1}$
permuting her signal tokens, $S_{2}$’s experiments will reveal her (partial)
private information only if $S_{1}$ cannot exploit this revelation against her
via permutations. This will significantly reduce the utility of being able to
choose the commitment order. To further illustrate this, a numerical example
is presented in Appendix 6 where $S_{2}$’s optimal commitment ends up being
independent of $S_{1}$’s signal realization when the permutation-free property
is violated.

###  2.4. Objectives of senders

Before we proceed, we list the objective functions of senders for both
commitment orders. For simplicity of representation, we abuse notations and
let $\Gamma_{k}^{f*}$ denotes the optimal commitment when $S_{k}$ commits
first, and $\Gamma_{k}^{s*}$ denotes the optimal commitment when $S_{k}$
commits last.

When $S_{1}$ commits first, the objective functions of $S_{1}$ and $S_{2}$ are
the following:

$\displaystyle\Gamma_{1}^{f*}$
$\displaystyle\in\arg\max_{\Gamma_{1}^{f}\in\mathbf{\Gamma_{1}}}\mathbb{E}_{p,\Gamma_{1}^{f},\Gamma_{2}^{s}(\Gamma_{1}^{f})}[U_{S_{1}}]$
(4)
$\displaystyle\text{~{}s.t.~{}}\mathbb{E}_{p,\Gamma_{1}^{f},\Gamma_{2}^{s}(\Gamma_{1}^{f})}[U_{S_{2}}]=\max_{\Gamma_{2}^{s^{\prime}}\in\mathbf{\Gamma_{2}}}\mathbb{E}_{p,\Gamma_{1}^{f},\Gamma_{2}^{s^{\prime}}}[U_{S_{2}}],$
(5) $\displaystyle\Gamma_{2}^{s*}$
$\displaystyle\in\arg\max_{\Gamma_{2}^{s}\in\mathbf{\Gamma_{2}}}\mathbb{E}_{p,\Gamma_{1}^{f*},\Gamma_{2}^{s}}[U_{S_{2}}].$

When $S_{2}$ commits first with permutation-free commitments, the objective
functions of $S_{1}$ and $S_{2}$ are the following:

$\displaystyle\Gamma_{1}^{s*}$
$\displaystyle\in\arg\max_{\Gamma_{1}^{s}\in\mathbf{\Gamma_{1}}}\mathbb{E}_{p,\Gamma_{1}^{s},\Gamma_{2}^{f*}}[U_{S_{1}}]\text{~{}s.t.~{}}\mathbb{E}_{p,\Gamma_{1}^{s},\Gamma_{2}^{f*}}[U_{S_{2}}]=\max_{\Gamma_{1}^{s^{\prime}}\in\mathbf{\Gamma_{1}^{M}}(\Gamma_{1}^{s})}\mathbb{E}_{p,\Gamma_{1}^{s^{\prime}},\Gamma_{2}^{f*}}[U_{S_{2}}],$
$\displaystyle\text{~{}where~{}}\mathbf{\Gamma_{1}^{M}}(\Gamma_{1}^{s})\text{~{}is
the set of commitments where the signals are permuted}$ (6)
$\displaystyle~{}\text{when compared to~{}}\Gamma_{1}^{s}.$ (7)
$\displaystyle\Gamma_{2}^{f*}$
$\displaystyle\in\arg\max_{\Gamma_{2}^{f}\in\mathbf{\Gamma_{2}}}\mathbb{E}_{p,\Gamma_{1}^{s},\Gamma_{2}^{f}}[U_{S_{2}}]\text{~{}s.t.~{}}\mathbb{E}_{p,\Gamma_{1}^{s},\Gamma_{2}^{f}}[U_{S_{1}}]=\max_{\Gamma_{1}^{s^{\prime}}\in\mathbf{\Gamma_{1}}}\mathbb{E}_{p,\Gamma_{1}^{s^{\prime}},\Gamma_{2}^{f}}[U_{S_{1}}].$

##  3\. When the Commitment Order Matters – Simple conditions

In Section 2.3, we discussed the rationale for why the permutation-free
property (on commitments) is an appropriate assumption in sequential
commitment problems. In this section, we will present an intuitive set of
conditions when the commitment order matters, i.e., when different expected
utilities are obtained for different commitment orders so that senders or the
receiver would have a preference between them. We start by presenting a
numerical example that shows that the commitment order can result in a
credible threat even though the order of signalling stays unchanged.

###  3.1. Credible threats

Let’s consider a game with two senders and one receiver. There are 4 possible
states of the world $\Omega=\\{TL,TR,BL,BR\\}$ (top-bottom and left-right).
$S_{1}$ knows whether the state of the world is top or bottom, i.e.,
$\mathcal{I}_{1}=\\{T,B\\}$, and $S_{2}$ knows whether the state of the world
is left or right, i.e., $\mathcal{I}_{2}=\\{L,R\\}$. $S_{1}$ can send signals
$\omega_{1}\in\\{\omega_{1,1},\omega_{1,2}\\}$ and $S_{2}$ can send signals
$\omega_{2}\in\\{\omega_{2,1},\omega_{2,2}\\}$ to influence the receiver’s
action. The distribution of the state of the world (prior) is common
knowledge, listed in Table 1(a). The senders’ utilities are provided in Table
1(b). The receiver’s goal is to guess the state of the world accurately: she
receives payoff 1 when she guesses correctly and 0 otherwise.

Table 1. State distribution and senders’ utilities

| L | R
---|---|---
T | $0.1$ | $0.2$
B | $0.4$ | $0.3$
(a) Distribution of the states

| $a_{TL}$ | $a_{TR}$ | $a_{BL}$ | $a_{BR}$
---|---|---|---|---
$(U_{S_{1}},U_{S_{2}})$ | 1,0 | 2,2 | 0,0 | 0,3
(b) Utilities of senders

In this game, $S_{1}$ prefers top over bottom and $S_{2}$ prefers right over
left. Now, we study the optimal commitments of $S_{1}$ and $S_{2}$ under
different commitment orders. First, let’s start with the scenario where
$S_{1}$ make her commitment first.

####  3.1.1. $S_{1}$ commits first

If $S_{2}$ didn’t appear in the game, $S_{1}$ would act as if in a classical
1-sender Bayesian persuasion model: sending a signal which suggests action
$a_{TL}$ or $a_{TR}$ when she knows the true state is in the top half and
perform a mixed signalling strategy when the true state is in the bottom half.
However, in the presence of $S_{2}$, $S_{1}$ knows that as long as she can
guarantee that the posterior probability of $BR$ is smaller than or equal to
$TR$, $S_{2}$, for the sake of maximizing her expected payoff, will commit to
a signalling strategy which helps $S_{1}$ since $BL$ is inferior to $TR$ for
$S_{2}$. Bearing this in mind, $S_{1}$ will make the following commitment by
anticipating $S_{2}$’s commitment.
$S_{1}$’s commitment: Send a suggestion of $T$ with probability 1 when her
private information is $T$ and probability $\frac{2}{3}$ when her private
information is $B$.

Now, since $S_{1}$ has made her commitment, $S_{2}$ makes her commitments on
top of it.
$S_{2}$’s commitment:

  * •

While observing $S_{1}$’s signal suggesting $T$, send a suggestion $R$ with
probability 1 when her private information is $R$ and probability
$\frac{3}{4}-\epsilon$ when her private information is $L$. And send a
suggestion of $L$ with probability $\frac{1}{4}+\epsilon$ while observing $L$,
where $\epsilon$ is infinitesimal, i.e., $|\epsilon|\ll 1$.

  * •

While observing $S_{1}$’s signal suggesting $B$, send a suggestion $R$ with
probability 1 when her private signal is $R$ and probability
$\frac{3}{4}-\epsilon$ when her private signal $L$. And send a suggestion of
$L$ with probability $\frac{1}{4}+\epsilon$ while observing $L$, where
$\epsilon$ is infinitesimal, i.e., $|\epsilon|\ll 1$.

We notice that $S_{2}$’s commitment is independent of $S_{1}$’s realized
signal. This is merely a happenstance in this example; this makes the analysis
simpler too. Given the commitments above, the expected utilities of
$S_{1}$,$S_{2}$ and the receiver are
$[\mathbb{E}_{p}[U_{S_{1}}],\mathbb{E}_{p}[U_{S_{2}}],\mathbb{E}_{p}[U_{R}]]=[1.2,1.8,\tfrac{1}{3}]$555See
Appendix 7.5 for detailed calculation..

####  3.1.2. $S_{2}$ commits first

Now, consider the game where $S_{2}$ commits first. Although $S_{2}$ does not
know $S_{1}$’s signalling strategy while making her commitment, her commitment
can still use $S_{1}$’s signal realization. This is because, when her
experiment is conducted, $S_{1}$’s signal realization is common knowledge.
Therefore, she can commit to the following signal and use this to threaten
$S_{1}$ credibly.
$S_{2}$’s commitment

  * •

While observing $S_{1}$’s suggestion of $T$, send a suggestion $R$ with
probability 1, whatever her private information is.

  * •

While observing $S_{1}$’s suggestion of $B$, send a suggestion $R$ with
probability 1 when her private information is $R$ and probability
$\frac{3}{4}$ when her private information is $L$. And send a suggestion $L$
otherwise.

Given this commitment of $S_{2}$, $S_{1}$, now, cannot hope that $S_{2}$ will
help her when $S_{1}$ signals her private information is $T$. Hence, $S_{1}$
has to make a more conservative commitment.
$S_{1}$’s commitment

  * •

Send a suggestion $T$ with probability 1 when her private information is $T$
and probability $\frac{1}{2}$ when $S_{1}$’s private information is $B$.

  * •

Send a suggestion $B$ otherwise.

Now, we have to check whether $S_{2}$ will deviate from her original
commitment to verify optimality from $S_{2}$’s perspective. Since the receiver
breaks a tie in favor of $S_{1}$ and then $S_{2}$, using any signal $L$ with
probability $>0$ when $S_{1}$ sends a suggestion $T$ will strictly lower her
expected utility. When $S_{1}$ suggests $B$, the true state is either $BL$ or
$BR$, and her commitment already maximizes her expected utility by making the
receiver weakly prefer $a_{BR}$ (given the receiver’s tie-breaking rule and
the indifference of $S_{1}$’s utility between $a_{BL}$ and $a_{BR}$). Hence,
her original commitment is the best response to the $S_{1}$’s commitment. By
calculating666See Appendix 7.6 for detailed calculation. the expected utility
of agents, we obtain:
$[\mathbb{E}_{p}[U_{S_{1}}],\mathbb{E}_{p}[U_{S_{2}}],\mathbb{E}_{p}[U_{R}]]=[1.1,3,0.4]$.

####  3.1.3. Comparison of utilities

Next, we compare the utilities for senders and the receiver between the two
commitment orders and assess their value for each agent. Both senders want to
commit first, and the receiver has a preference on letting $S_{2}$ commit
first. Moreover, if senders can trade their commitment order and $S_{1}$ is
initially endowed the first commitment slot, every transfer $x\in(0.1,0.2)$
from $S_{2}$ to $S_{1}$ makes both senders better off by trading the first
commitment slot from $S_{1}$ to $S_{2}$. The example shows an interesting
phenomenon: when senders only have partial information and have to send
signals sequentially, they may collaborate on the signals they commit and
collude on the order of commitment if there is a market for senders to trade
their commitment position. For readers curious about whether there is a
scenario that both senders and the receiver are better off after a transfer of
the commitment slot, an example is provided in Section 4.1.

###  3.2. Collaborative States

Motivated by the above example, we first present an easily rationalized
proposition capturing a set of scenarios where the commitment order matters
and prove a generalized theorem afterward. The proposition captures scenarios
similar to the example discussed above, wherein $S_{1}$’s and $S_{2}$’s
interests are aligned if $S_{1}$ reveals her private information truthfully.
In these scenarios, both senders want to persuade the receiver to take the
same action. This implies that senders can collaborate on their signalling
strategies in some states. We call this type of states collaborative states
and define them formally below.

###### Definition 2.

A state $\hat{\theta}\in\Theta$ is a collaborative state if the following two
conditions hold:

  1. (1)

$\hat{\theta}\in\arg\max_{\theta\in\hat{I}_{1}}U_{S_{1}}(\theta,a_{\theta})$,

  2. (2)

$\hat{\theta}\in\arg\max_{\theta\in\hat{I}_{1}}U_{S_{2}}(\theta,a_{\theta})$,

where $\hat{I}_{1}$ is the information set containing state $\hat{\theta}$,
i.e., $\hat{I}_{1}=F_{1}(\hat{\theta})$.

Definition 2 states that when the $S_{1}$ reveals her private information
truthfully while $\theta=\hat{\theta}$, $S_{2}$ is willing to collaborate with
$S_{1}$ to persuade the receiver towards taking their most preferred action.
Note the asymmetry of the definition vis-a-vis the sender identities. This
occurs because $S_{1}$ signals before $S_{2}$. With the definition of
collaborative states in place, we present a preliminary result about
conditions when the commitment order matters.

###### Proposition 1.

Given a prior $p$ and both senders’ utility functions $U_{S_{1}},U_{S_{2}}$,
the commitment order matters if there exists a private information set
$I_{1}\in\mathcal{I}_{1}$ satisfying the following conditions:

  1. (1)

(Existence of collaborative state) There exists a collaborative state
$\hat{\theta}\notin I_{1}$ satisfying the following two inequalities:

(8) $\displaystyle
U_{S_{2}}(\hat{\theta},a_{\hat{\theta}})<\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{2}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}]$
(9) $\displaystyle
U_{S_{1}}(\hat{\theta},a_{\hat{\theta}})>\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{1}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}],$

where $\Gamma_{2}^{*}(I_{1})$ denotes $S_{2}$’s optimal signalling strategy
while $S_{1}$ truthfully reveals her private signal $I_{1}$, and
$a^{*}(I_{1},\omega_{2})$ denotes the receiver’s best response under $I_{1}$
and $S_{2}$’s signal $\omega_{2}$.

  2. (2)

(A demand of collaboration) Let $\mathbf{\hat{\Gamma}}_{1}$ be the set of
commitments with a signal $\hat{\omega}_{1}\in\Omega_{1}$ satisfying
$\mathbb{P}(I_{1}|\hat{\omega}_{1})=1$. For any commitment order, the optimal
commitment(s) of $S_{1}$ must belong to this set, i.e.,
$\Gamma_{1}^{f*},\Gamma_{1}^{s*}\subseteq\mathbf{\hat{\Gamma}}_{1}$.

  3. (3)

(Existence of a credible threat) The preference orders of $S_{1}$ and $S_{2}$
on $\Phi(I_{1})$ are polar opposites, where $\Phi(I_{1})$ is the receiver’s
possible optimal action set given $S_{1}$’s private information $I_{1}$, i.e.,
$\Phi(I_{1})=\\{a_{\theta}|\theta\in I_{1}\\}$.

We provide a detailed proof in Appendix 7.1. The core of the proof is to
construct a credible threat using a collaborative state. To make the
construction possible, we need the existence of another signal which gives
$S_{1}$ a lower expected utility (based on the receiver’s best response). This
other signal gives $S_{1}$ an incentive to commit to a mixed signalling
strategy, which mixes this signal with the signal corresponding to the
collaborative state. Otherwise, $S_{1}$’s private information can be elicited
by $S_{2}$ when $S_{1}$’s private information contains the collaborative
state, and credible threats are no longer needed since $S_{1}$’s and $S_{2}$’s
interests are aligned. The existence of this other signal is implied by
condition 2 in non-trivial scenarios, where the details are stated in Claim 1.
Moreover, conditions 1 and 3 guarantee a conflict of interest between $S_{1}$
and $S_{2}$’s on mixing other states with the collaborative state. Hence, if
$S_{1}$ gets to commit first, she can propose a more aggressive mixed strategy
because the best commitment of $S_{2}$, after $S_{1}$’s commitment, is to
collaborate with $S_{1}$’s mixed strategy. However, if $S_{2}$ can commit
first, she is willing to collaborate, but she does not want $S_{1}$ to mix
other states with this collaborative state. Hence, she can create a credible
threat to reduce (but not eliminate) $S_{1}$’s strategy of mixing other states
with the collaborative one. Hence, the commitment order matters.

###### Claim 1.

When the inequality (9) holds, condition 2 in Proposition 1 implies that one
of the following two conditions is true:

  1. (1)

The signal $\hat{\omega}_{1}$ of condition 2 in Proposition 4 is not the
signal realization which gives $S_{1}$ the highest expected utility in
$S_{1}$’s optimal commitments.

  2. (2)

Every signal realization gives $S_{1}$ the same expected utility.

##  4\. General Results On When The Commitment Order Matters

Note that while Proposition 1 provides a set of sufficient conditions when the
commitment order matters in a constructive manner, each condition in
Proposition 1 has its deficiency. First, the requirement of a collaborative
state is a strong condition. $S_{2}$ may be willing to partially collaborate
with $S_{1}$ after she has maximized the probability of her most-preferred
action under a given $I_{1}$. In other words, a collaboration may happen in a
state where its corresponding action is $S_{2}$’s second-best action (under
the given information set of $S_{1}$). Second, the verification of demand of
collaboration requires knowledge about the commitment space’s structure777When
the utilities can be written in a matrix form, it is usually straightforward
to see whether there is an information set of $S_{1}$ that gives her a lower
utility compared to the collaborative state.. When the state space grows, this
can be a tedious task without additional constraints. Third, although the
credible-threat condition guarantees the uniqueness of $S_{2}$’s optimal
signalling strategy and a conflict of interest between $S_{1}$ and $S_{2}$ on
$I_{1}$, a polar opposite preference ordering is only one type where a
conflict of interest between $S_{1}$ and $S_{2}$ happens. As long as $S_{1}$
and $S_{2}$ have an opposite preference on a mixture of two information sets
of $S_{1}$, $S_{2}$ may be able to construct a credible threat using this
opposite preference. Before stating the generalized result, we start with an
example where the commitment order matters but the conditions in Proposition 1
are violated in Section 4.1. Hence, to remove the need for the knowledge of
the commitment space, the requirement of collaborative states, and the
assumption of polar opposite preference ordering, we will present a set of
sufficient conditions generalized from Proposition 1.

###  4.1. An example highlighting the lack of generality of Proposition 1

As mentioned above, we will present an example where the commitment order
matters, but conditions in Proposition 1 are either violated or not needed.
Besides, we utilize this example to demonstrates a scenario where transferring
the commitment slot increases the utilities of both senders and also the
receiver, i.e., the social welfare increases. In this example we highlight
that a collaborative state is not necessary for $S_{1}$ to mix her private
information sets. Besides, a conflict of interest between $S_{1}$ and $S_{2}$
on a mixture of a pair of $S_{1}$’s information sets does not demand (polar)
opposite preference ordering (on corresponding actions) in any of $S_{1}$’s
private information sets.

Although the smallest example that meets our need requires 7 states, to
simplify the description of $S_{1}$’s and $S_{2}$’s private information
structures, we add two dummy states with prior probability $0$ and expand the
example to a $9$-state model, where each sender has three possible types. To
avoid confusion and to simplify the discussion on sender’s preference
ordering, we skip the definition of $S_{1}$’s and $S_{2}$’s utilities on dummy
states.

###### Example 0.

There are $3\times 3=9$ states of the world $\theta_{XY}\in\Theta$, where
$X\in\\{T,M,B\\}$ and $Y\in\\{L,C,R\\}$. $S_{1}$ knows the information of $X$
and $S_{2}$ knows the information of $Y$, i.e., $\mathcal{I}_{1}=\\{T,M,B\\}$
and $\mathcal{I}_{2}=\\{L,C,R\\}$. The prior on the state is provided in Table
2(a), and the utilities of $S_{1}$ and $S_{2}$ are provided in Table 2(b). In
this example, $S_{1}$ sends signals prior to $S_{2}$, and the receiver’s
objective is to maximize the probability of guessing the true state. To make
notation less cumbersome, we use $a_{XY}$ to represent $a_{\theta_{XY}}$.

Table 2. State distribution and senders’ utilities

| L | C | R
---|---|---|---
T | $0.05$ | $0$ | $0.05$
M | $0.12$ | $0.2$ | $0.02$
B | $0.35$ | $0.21$ | $0$
(a) Distribution of the states

| $a_{TL}$ | $a_{TR}$ | $a_{ML}$ | $a_{MC}$ | $a_{MR}$ | $a_{BL}$ | $a_{BC}$
---|---|---|---|---|---|---|---
$(U_{S_{1}},U_{S_{2}})$ | 0,0 | 5,5 | 1,0 | 2,2 | 0,3 | 0,1 | 1,4
(b) Utilities of senders

In this example, we demonstrate that all the conditions in Proposition 1 are
either violated or not needed. First, although there exists a (unique)
collaborative state $\theta_{TR}$ satisfying condition 1, $S_{1}$’s and
$S_{2}$’s preference order on states $\\{\theta_{BL},\theta_{BC}\\}$ are fully
aligned, instead of a polar opposite preference required by Proposition 1.
Actually, persuading receiver to take action $a_{TR}$ is best not only for
$S_{1}$ but also for $S_{2}$. Hence, $S_{2}$ will not threaten $S_{1}$ on her
signals suggesting $a_{TR}$, and the signalling strategy on persuading
$a_{TR}$ will be independent of the commitment order. Second, although the
second condition (a demand of collaboration) holds on $S_{1}$’s private
information $I_{1}=B$, Proposition 1 provides no guidance on how senders will
collaborate without an effective collaborative state for the construction of
collaborative signalling strategies. Thus, the second condition is not helpful
without the existence of collaborative state. Third, none of the preference
orders of $S_{1}$ and $S_{2}$ are polar-opposite under $S_{1}$’s truthful
report. Since all conditions are either defunct or violated, Proposition 1
cannot help determine whether the commitment order matters or not in this
example.

In actuality though, the commitment order does matter here. Note that $S_{1}$
prefers a mixture of her private information $M,B$ to persuade the receiver in
taking action $a_{MC}$, but $S_{2}$ prefers a separation of $S_{1}$’s private
information $M$ and $B$. This gives $S_{2}$ a chance to pose a credible
threat. Note that $\theta_{MC}$ is not a collaborative state because $S_{2}$
will first persuade the receiver to take $a_{MR}$, and then help $S_{1}$
persuade the receiver to take $a_{MC}$ when she cannot increase the value of
$\mathbb{P}(a_{MR})$. To verify that the commitment order matters in this
example, we calculate the utilities of the senders $S_{1}$, $S_{2}$ and the
receiver under both commitment orders. For simplicity of representation, let
$\omega_{1}^{X}$ denote $S_{1}$’s signal suggesting the action set
$\\{a_{XL},a_{XC},a_{XR}\\}$, $X\in\\{T,M,B\\}$ and $\omega_{2}^{Y}$ denote
$S_{2}$’s signal suggesting the action set $\\{a_{TY},a_{MY},a_{BY}\\}$,
$Y\in\\{L,C,R\\}$. In other words, $\omega_{1}^{X}$ implies
$\mathbb{P}(\theta_{XY})=\max\\{\mathbb{P}(\theta_{TY}),\mathbb{P}(\theta_{MY}),\mathbb{P}(\theta_{BY})\\}$
for every $Y\in\\{L,C,R\\}$ and $\omega_{2}^{Y}$ implies
$\mathbb{P}(\theta_{XY})=\max\\{\mathbb{P}(\theta_{XL}),\mathbb{P}(\theta_{XC}),\mathbb{P}(\theta_{XR})\\}$
for every $X\in\\{T,M,B\\}$. We will use $\omega_{1}^{X}$ and $\omega_{2}^{Y}$
to state the commitments in this example.

####  4.1.1. $S_{1}$ commits first

First, both $S_{1}$ and $S_{2}$ wants to maximize the probability of $a_{TR}$
according to their utility functions. Hence, $S_{1}$ will first maximize the
probability of signal $\omega_{1}^{T}$.

  * •

Send $\omega_{1}^{T}$ with probability $1$ when her private information
$I_{1}=T$.

  * •

Send $\omega_{1}^{T}$ with probability $0.25$ when her private information
$I_{1}=M$.

  * •

Send $\omega_{1}^{T}$ with probability $\tfrac{1}{7}$ when her private
information $I_{1}=B$.

Now, conditional on $S_{1}$ not sending signal $\omega_{1}^{T}$, the
distribution of states reduces to Table 4, where $w=0.735$ normalizes the
total probability to be $1$.

Table 3. Distribution of the states when $\omega_{1}\neq\omega_{1}^{T}$ | L | C | R
---|---|---|---
T | $0$ | $0$ | $0$
M | $0.09/w$ | $0.15/w$ | $0.015/w$
B | $0.3/w$ | $0.18/w$ | $0$

Table 4. Distribution of the states when $\omega_{1}=\omega_{1}^{M}$ | L | C | R
---|---|---|---
T | $0$ | $0$ | $0$
M | $0.09/w$ | $0.15/w$ | $0.015/w$
B | $0.25/w$ | $0.15/w$ | $0$

According to Table 4, $S_{2}$ knows the true state is $\theta_{MR}$ when her
private information $I_{2}=R$, regardless of the signalling strategy $S_{1}$
uses to mix $M$ and $B$. Since $S_{2}$ prefers $\theta_{MR}$, then
$\theta_{MC}$, and finally $\theta_{ML}$, $S_{2}$ will first persuade the
receiver on taking action $a_{MR}$ when $\omega_{1}\neq\omega_{1}^{T}$. When
$S_{2}$ has maximized the total probability of $a_{MR}$ taken by receiver
conditional on her signalling strategies, $S_{2}$ is willing to collaborate
with $S_{1}$ on persuading the receiver to take action $a_{MC}$ instead of
$a_{ML}$. Therefore, $S_{1}$’s optimal mixture of information sets $M$ and $B$
is to maximize the conditional probability of $B$ in her signal, while
ensuring that the posterior belief of $\theta_{MC}$ is no less than
$\theta_{MB}$. Using the concavification approach (Kamenica, 2019), the
posterior beliefs of states conditional on $S_{1}$ sending signal
$\omega_{1}^{M}$ is presented in Table 4, where $w=0.655$ normalizes the total
probability to be one.

Given $S_{1}$’s signal which mixes information sets $T$, $M$, and $B$ and the
optimal mixture of $M$ and $B$ derived above, $S_{1}$ still has a positive
probability on information set $B$ for which she has no choice but to tell the
truth. By calculating the conditional probabilities of each state conditional
on $S_{1}$’s private information and her optimal signalling strategy presented
above, we can find the complete commitment of $S_{1}$. We depict it in Figure
1(a), where each number next to an arrow from $S_{1}$’s private signal
$(T,M,B)$ to a signal $(\omega_{1}^{T},\omega_{1}^{M},\omega_{1}^{B})$
represents the conditional probability of sending a particular signal under
that private information.

Given $S_{1}$’s optimal commitment, deriving $S_{2}$’s optimal commitment is
equivalent to solving a classical single-sender Bayesian persuasion problem
for each of $S_{1}$’s realized signals when $S_{2}$ commits after $S_{1}$.
Thus, $S_{2}$’s optimal signalling strategy can be calculated in
straightforward manner via the concavification approach that solves classical
Bayesian persuasion problems. Since the result of concavification approach can
be easily verified via calculating $S_{2}$’s utility change under small
deviations, we omit the numerical computations of the concavification
procedure and present $S_{2}$’s optimal commitment below:

  * •

While observing $\omega_{1}^{T}$, send $\omega_{2}^{R}$ with probability $1$,
whatever her private information is.

  * •

While observing $\omega_{1}^{M}$, send signals using the following strategy:

    * –

Send $\omega_{2}^{R}$ with probability $1$ when her private information is
$R$, with probability $0.1$ when her private information is $C$, and with
probability $0.06$ when her private information is $L$.

    * –

Send $\omega_{2}^{C}$ with probability $0.9$ when her private information is
$C$ and with probability $0.54$ when her private information is $L$.

    * –

Send $\omega_{2}^{L}$ with probability $0.4$ when her private information is
$L$.

  * •

While observing $\omega_{1}^{B}$, send signals using the following strategy:

    * –

Send $\omega_{2}^{C}$ with probability $1$ when her private information is $C$
and with probability $0.6$ when her private information is $L$.

    * –

Send $\omega_{2}^{L}$ with probability $0.4$ when her private information is
$L$.

We remark that $S_{1}$ plays an aggressive strategy which relies on $S_{2}$’s
collaboration to persuade the receiver towards taking $S_{1}$’s preferred
action. When $S_{1}$ sends $\omega_{1}^{M}$, the interim belief of the state
distribution is Table 4 presented earlier:

We notice that state $\theta_{BL}$ has the highest probability if $S_{2}$
stays silent. Without the signal of $S_{2}$, the receiver will take action
$a_{BL}$ and $S_{1}$ experiences the lowest utility. However, since $a_{MC}$
gives $S_{2}$ a higher utility than $a_{ML}$, $S_{2}$ will collaborate with
$S_{1}$ after she maximizes the total probability of $a_{MR}$. This allows
$S_{1}$ to commit to the above signalling strategy. Last, we calculate
(detailed in Appendix 7.7) the expected utility of agents and obtain
$[\mathbb{E}_{p}[U_{S_{1}}],\mathbb{E}_{p}[U_{S_{2}}],\mathbb{E}_{p}[U_{R}]]=[2.4336,2.6884,0.286]$.

(a) When $S_{1}$ commits first

(b) When $S_{2}$ commits first

Figure 1. Sender $S_{1}$’s optimal commitment

####  4.1.2. $S_{2}$ commits first

Again, both $S_{1}$ and $S_{2}$ aim to maximize the probability of $a_{TR}$
according to their utility functions. Hence, when $S_{1}$ maximizes the
probability of signal $\omega_{1}^{T}$, $S_{2}$ has no incentive to restrict
her signal $\omega_{1}^{T}$. Therefore, before $S_{2}$ makes her commitment,
she knows $S_{1}$’s signalling strategy on $\omega_{1}^{T}$ below.

  * •

Send $\omega_{1}^{T}$ with probability $1$ when her private information
$I_{1}=T$.

  * •

Send $\omega_{1}^{T}$ with probability $0.25$ when her private information
$I_{1}=M$.

  * •

Send $\omega_{1}^{T}$ with probability $\tfrac{1}{7}$ when her private
information $I_{1}=B$.

Now, conditional on $S_{1}$ not sending signal $\omega_{1}^{T}$, the
distribution of states reduces to Table 4 presented earlier. $S_{2}$’s problem
then reduces to designing an optimal commitment under Table 4. Opposite to
$S_{1}$, $S_{2}$ prefers a separation of $S_{1}$’s information sets $M$ and
$B$. Therefore $S_{2}$ will restrict $S_{1}$’s mixture of $M$ and $B$ in her
signal $\omega_{1}^{M}$ as much as she can by posing a credible threat. Here
we detail $S_{2}$’s thought process on designing her threat strategy.

Since $S_{1}$ will never use a signal $\omega_{1}$ which mixes $M$ and $B$
such that
$\mathbb{P}(MC|\omega_{1})\leq\mathbb{P}(BC|\omega_{1})$888Otherwise, the
receiver will never take action $a_{MC}$ suggested by $S_{1}$., $S_{2}$ can
only restrict $S_{1}$’s mixture by tailoring her signal on her private signal
$L$. Ideally, $S_{2}$ has two choices: (1) increase the probability of signal
$\omega_{2}^{R}$ while receiving $\omega_{1}^{M}$ with private signal $L$, or
(2) increase the probability of signal $\omega_{2}^{C}$ while receiving
$\omega_{1}^{M}$ with private signal $L$. However, the first choice cannot
threaten $S_{1}$ since $S_{1}$ has utility $0$ under $a_{MR}$. This is because
$S_{1}$ is better off when $S_{2}$ fails to persuade the receiver to take
action $a_{MR}$999In this case, the receiver will take action $a_{ML}$ instead
of $a_{MR}$, and $S_{1}$ obtains a higher utility.. Thus, $S_{2}$ can only
pick the second choice and tailor the probability of signal $\omega_{2}^{C}$
to restrict $S_{1}$. Given the inequality
$\mathbb{P}_{p}(\theta_{BL})>\mathbb{P}_{p}(\theta_{BC})$ from the prior,
$S_{2}$ can reduce the probability $\mathbb{P}(\omega_{2}^{L}|\omega_{1}^{M})$
to $0$ to restrict $S_{1}$’s mixture of $M$ and $B$. In other words, $S_{1}$
acknowledges that the receiver will take either $a_{MR}$ or $a_{MC}$ when
$\omega_{1}^{M}$ is sent. Given the interim distribution of Table 4, $S_{2}$’s
optimal signalling scheme (after using the concavification approach to derive
$\omega_{2}^{R}$) is to send signal $\omega_{2}^{R}$ with probability $0.1$
and $\omega_{2}^{C}$ with probability $0.9$ when she observes $\omega_{1}^{M}$
and her private information is $L$. Last, when $S_{2}$ observes
$\omega_{1}^{B}$, she immediately knows $S_{1}$’s private signal is $B$
(because mixing $T$ or $M$ to suggest either $a_{BL}$ or $a_{BM}$ hurts
$S_{1}$). Thus, $S_{2}$’s optimal commitment can be solved via a
concavification approach analogous to the one-sender, binary-state Bayesian
persuasion problem. To sum up, we list $S_{2}$’s optimal signalling strategy
below:

  * •

While observing $\omega_{1}^{T}$, send $\omega_{2}^{R}$ with probability $1$,
whatever her private information is.

  * •

While observing $\omega_{1}^{M}$, send signals using the following strategy:

    * –

Send $\omega_{2}^{R}$ with probability $1$ when her private information is
$R$, with probability $0.1$ when her private information is $C$, and with
probability $0.1$ when her private information is $L$.

    * –

Send $\omega_{2}^{C}$ with probability $0.9$ when her private information is
$C$ or $L$.

  * •

While observing $\omega_{1}^{B}$, send signals using the following strategy:

    * –

Send $\omega_{2}^{C}$ with probability $1$ when her private information is $C$
and with probability $0.6$ when her private information is $L$.

    * –

Send $\omega_{2}^{L}$ with probability $0.4$ when her private information is
$L$.

Based on $S_{2}$’s commitment, $S_{1}$ knows she cannot commit to an
aggressive signalling strategy and wish $S_{2}$ to help her persuading the
receiver to take action $a_{MC}$. Moreover, her maximum mixture of $M$ and $B$
is restricted by $S_{2}$ with the satisfaction of the inequality
$\mathbb{P}(\theta_{BL}|\omega_{1}^{M})\leq\mathbb{P}(\theta_{MC}|\omega_{1}^{M})$.
Thus, given the distribution presented in Table 4, $S_{1}$ can only send
signal $\omega_{1}^{M}$ with probability $\frac{1}{2}$ when her private
information is $B$ and signal $\omega_{1}^{T}$ is not sent. In short,
$S_{1}$’s optimal commitment is to first maximize $\omega_{1}^{T}$, and then
maximize $\omega_{1}^{M}$ under the restriction of $S_{2}$’s signalling
strategy. After calculating the probability of $\omega_{1}^{T}$ and the
conditional probability of $\omega_{1}^{M}$, we summarize $S_{1}$’s optimal
commitment below and a diagram in Figure 1(b). By calculating the expected
utility of agents (detailed in Appendix 7.8), we obtain
$[\mathbb{E}_{p}[U_{S_{1}}],\mathbb{E}_{p}[U_{S_{2}}],\mathbb{E}_{p}[U_{R}]]=[2.207,3.158,0.35]$.

####  4.1.3. Comparison of utilities

Similar to the example in Section 3.1, both senders want to commit first, and
the receiver has a preference on letting $S_{2}$ commit first in this example.
However, the unique collaborative state in this example violates conditions in
Proposition 1 and the signalling strategies (the probability of suggesting
$a_{TR}$) are the same in both commitment orders. However, the commitment
order matters because of the collaboration in persuading $a_{MC}$, even though
$a_{MC}$ is not $S_{2}$’s most preferred action under $I_{1}=M$. Moreover, if
senders can trade their commitment order and $S_{1}$ is initially endowed the
first commitment slot, every transfer $x\in(0.2266,0.4616)$ from $S_{2}$ to
$S_{1}$ makes both senders and the receiver better off by trading the first
commitment slot from $S_{1}$ to $S_{2}$. If the transfer from the receiver to
$S_{1}$ is also allowed (even though the receiver doesn’t participate in this
trade), the maximum compensation $S_{1}$ could receive from giving out the
first commitment slot grows to $0.5276$. This example demonstrates a case
where both senders’ and the receiver’s utilities can increase when the first
commitment slot is transferred from $S_{1}$ to $S_{2}$.

###  4.2. General result: Sufficient conditions

With the example above, we learned that the commitment order matters in a more
general scenarios than those prsented in Proposition 1. The analysis in
Example Example suggests that when both a credible threat and a collaboration
exist, the commitment order will matter. In order to know whether a (partial)
collaboration is possible, we start by defining the conditional distribution
of $S_{1}$’s information sets and use it to represent the best response
signalling strategy of $S_{2}$ under an observed signal of $S_{1}$.

###### Definition 3.

Given a prior $p$ and $S_{1}$’s commitment $\Gamma_{1}$, a signal realization
$\omega_{1}$ indicates a conditional distribution of $S_{1}$’s information
sets, $\Delta_{\Gamma_{1}}(\mathcal{I}_{1}|\omega_{1},p)$, which we denote as
$q^{\omega_{1}}$.

When two signals101010Two signals need not be from the same commitment.
$\omega_{1},\omega_{1}^{\prime}$ have the same information set distribution,
i.e., $q^{\omega_{1}}=q^{\omega_{1}^{\prime}}$, the best response of the
receiver will be identical for these two signals. Thus, to avoid duplication
and for simplicity of comparing different signals, we abuse notation and use
$a^{*}(q^{\omega_{1}},\omega_{2})$ to represent the receiver’s best response
under given $S_{1}$’s information set distribution and $\omega_{2}$, i.e.,
$a^{*}(\omega_{1},\omega_{2}),a^{*}(\omega_{1}^{\prime},\omega_{2})$. Since
the receiver’s utility is a function of $S_{1}$’s information set distribution
and $S_{2}$’s signal, we are ready to define $S_{2}$’s best response
signalling strategies.

###### Definition 4.

Given a prior $p$ and a distribution of $S_{1}$’s information set
$q^{\omega_{1}}$, $G(p,q^{\omega_{1}})$ represents the set of best response
signalling strategies of $S_{2}$ while observing $S_{1}$’s information set
distribution $q^{\omega_{1}}$.

Mathematically, an element $g\in G(p,q^{\omega_{1}})$ is a function that maps
$I_{2}\in\mathcal{I}_{2}$ to a signal distribution $r\in\Delta(\Omega_{2})$
such that
$\mathbb{E}_{p,g}[U_{S_{2}}|q^{\omega_{1}},r]\geq\mathbb{E}_{p,g^{\prime}}[U_{S_{2}}|q^{\omega_{1}},r]$
for all $g^{\prime}\not\in G(p,q^{\omega_{1}})$. For simplicity of
representation, $G(p,q^{\omega_{1}}+\alpha q^{\bar{\omega}_{1}})$ denotes
$S_{2}$’s best response under $S_{1}$’s mixed signal $\hat{\omega}_{1}$ such
that
$q^{\hat{\omega}_{1}}=\frac{1}{1+\alpha}q^{\omega_{1}}+\frac{\alpha}{1+\alpha}q^{\bar{\omega}_{1}}$.
Since $S_{2}$’s expected utility is maximized in every best response strategy
$g\in G(p,q^{\omega_{1}})$, i.e.,
$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1},g]=\mathbb{E}_{p}[U_{S_{2}}|\omega_{1},g^{\prime}]$
for all $g,g^{\prime}\in G(p,q^{\omega_{1}})$, we again abuse notation and use
$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1},G(p,q^{\omega_{1}})]$ to represent
$S_{2}$’s expected utility under $S_{1}$’s information set distribution
$q^{\omega_{1}}$, prior $p$, and $S_{1}$’s signal $\omega_{1}$.

Given the above definitions, we will present a set of sufficient conditions of
when the commitment order matters that relies on two critical insights gained
from Example Example:

  1. (1)

Both senders are willing to (partially) collaborate on at least a particular
private information state of $S_{1}$.

  2. (2)

On such a “collaborative” information state, $S_{1}$ and $S_{2}$ have a
conflict of interest on the level of collaboration, which allows $S_{2}$ to
pose a credible threat.

###### Theorem 1.

Given a prior $p$, the commitment order matters if there exists a pair of
$S_{1}$’s information sets $I_{1}^{x},I_{1}^{y}$ satisfying the following
conditions:

  1. (1)

There exist two parameters $\alpha>\beta>0$ and a signalling strategy
$\hat{\Gamma}_{2}$ satisfying the following conditions:

    1. (a)

$G(p,I_{1}^{x}+\alpha I_{1}^{y})=G(p,I_{1}^{x}+\beta I_{1}^{y})$,

    2. (b)

Let $\omega_{1}^{\alpha},\omega_{1}^{\beta}$ be two mock signals of $S_{1}$
such that $\mathbb{P}(I_{1}^{x}|\omega_{1}^{\alpha})=\frac{1}{1+\alpha}$,
$\mathbb{P}(I_{1}^{y}|\omega_{1}^{\alpha})=\frac{\alpha}{1+\alpha}$,
$\mathbb{P}(I_{1}^{x}|\omega_{1}^{\beta})=\frac{1}{1+\beta}$, and
$\mathbb{P}(I_{1}^{y}|\omega_{1}^{\beta})=\frac{\beta}{1+\beta}$, then

$\displaystyle\hat{\Gamma}_{2}(p,\omega_{1}^{\beta})\in G(p,I_{1}^{x}+\beta
I_{1}^{y})~{}\text{and}~{}\hat{\Gamma}_{2}(p,\omega_{1}^{\alpha})\notin
G(p,I_{1}^{x}+\alpha I_{1}^{y}),$

    3. (c)

$\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\alpha},\hat{\Gamma}_{2}]<\frac{\mathbb{P}(I_{1}^{x})+\beta\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\beta},\hat{\Gamma}_{2}]+\frac{(\alpha-\beta)\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|I_{1}^{y},\hat{\Gamma}_{2}]$,

    4. (d)

$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\beta},G(p,I_{1}^{x}+\beta
I_{1}^{y})]<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})]$.

  2. (2)

Sender $S_{1}$ has a higher expected utility under $I_{1}^{x}$ than
$I_{1}^{y}$, regardless of $S_{2}$’s tie-breaking rule, i.e.,

$\displaystyle\min_{g\in
G(p,I_{1}^{x})}\mathbb{E}_{p,g}[U_{S_{1}}|I_{1}^{x}]>\max_{g^{\prime}\in
G(p,I_{1}^{y})}\mathbb{E}_{p,g^{\prime}}[U_{S_{1}}|I_{1}^{y}].$

  3. (3)

There is no commitment pair $(\Gamma_{1},\Gamma_{2})$ such that

$\displaystyle\sum_{\theta_{i}\in
I_{1}^{y}}\mathbb{P}_{p,\Gamma_{1},\Gamma_{2}}(a_{i})=0\text{~{}or~{}}\sum_{\theta_{i}\in
I_{1}^{x}}\mathbb{P}_{p,\Gamma_{1},\Gamma_{2}}(a_{i})=0$

Although the conditions in Theorem 1 look technical, there are intuitions
behind each condition. The first condition in Theorem 1 states that the sender
$S_{2}$ prefers a separation of sender $S_{1}$’s information sets $I_{1}^{x}$
and $I_{1}^{y}$. Moreover, if $S_{2}$ can commit first, she can make a
credible threat on her commitment to constrain the mixing of $I_{1}^{x}$ and
$I_{1}^{y}$ in sender $S_{1}$’s signalling strategies. This generalizes
condition 3 in Proposition 1. Besides, the combination of condition 1(a) and
1(d) guarantees that $S_{2}$ can collaborate with $S_{1}$, and this
generalizes the collaborative state requirement in Proposition 1. The second
condition in Theorem 1 states that the sender $S_{1}$ prefers a mixture of
$I_{1}^{x}$ and $I_{1}^{y}$ in her signalling strategy. Combining the
intuition behind the first two conditions, the two senders have opposite
preferences on the mixture of $I_{1}^{x}$ and $I_{1}^{y}$. Based on the first
two conditions, senders $S_{1},S_{2}$ are already in a scenario where the
sender $S_{2}$ is capable of constraining $S_{1}$’s mixture of $I_{1}^{x}$ and
$I_{1}^{y}$, and a collaboration between $S_{1}$ and $S_{2}$ is possible under
$I_{1}^{x}$. Hence, the proof of Theorem 1 shares the same general principle
with the proof of Proposition 1, i.e., constructing a credible threat using an
information set where both sender collaborate on persuasion. The third
condition is a regularity condition which guarantees that the prior $p$ has
put enough probability mass in states $\theta\in I_{1}^{x}$ and $\theta\in
I_{1}^{y}$. This is to avoid corner cases where the senders $S_{1}$ and
$S_{2}$ have opposite preferences on the mixture of $I_{1}^{x}$ and
$I_{1}^{y}$, but their opposite preferences do not matter because the sender
$S_{1}$ can commit to some signalling strategies which never suggests an
action $a_{i}\in\\{a_{j}|\theta_{j}\in I_{1}^{x}\cup I_{1}^{y}\\}$. The
statement of condition 3 may demand a search of the whole space of possible
commitments, which could make this theorem difficult to apply in models with a
large state space. However, in most practical problems, verifying condition 3
can be done in straightforward manner by inspecting the prior distribution.
Hence, we believe condition 3 doesn’t undermine the contribution of Theorem 1.

###  4.3. General result: Necessary conditions

The first two conditions in Theorem 1 capture the scenario when a credible
threat by $S_{2}$ can be issued, hence constraining $S_{1}$’s signalling
strategies. This leads to the question of whether there are simpler and more
basic requirements that make credible threats possible without analyzing the
best response of commitments. Moreover, results in Theorem 1 and Proposition 1
depend on the prior. Hence, we would also like to obtain more basic conditions
that are prior independent. Last, we observe that different tie-breaking rules
of the receiver may play a role in determining whether the commitment order
matters. However, it is hard to determine the criticality of the receiver’s
tie-breaking rule under sufficient conditions because each tie-breaking rule
only plays a role under a small set of priors. With these considerations in
mind, we present in the following theorem a set of necessary conditions of
scenarios where the commitment order could matter.

###### Theorem 2.

Given a prior $p$ and both senders’ utility functions $U_{S_{1}}$ and
$U_{S_{2}}$, if both conditions below are violated, then the commitment order
does not matter:

  1. (1)

There exists a set of states
$(\theta_{\alpha},\theta_{\beta},\theta_{\gamma})$ satisfying the following
properties:

    1. (a)

$\theta_{\alpha}$ and $\theta_{\beta}$ are in the same information set of
$S_{1}$ but $\theta_{\gamma}$ is not,
i.e.,$\exists!I_{1}^{x}\ni\theta_{\alpha},\theta_{\beta}$ and
$\theta_{\gamma}\in I_{1}^{y}\neq I_{1}^{x}$. Besides, $\theta_{\alpha}$ and
$\theta_{\beta}$ are not in the same information set of $S_{2}$, i.e.,
$\exists~{}I_{2}^{x}\neq I_{2}^{y}$ s.t. $\theta_{\alpha}\in I_{2}^{x}$ and
$\theta_{\beta}\in I_{2}^{y}$. But $\theta_{\gamma}$ is in the same
information set as either $\theta_{\alpha}$ or $\theta_{\beta}$, i.e.,
$\theta_{\gamma}\in I_{2}^{x}$ or $\theta_{\gamma}\in I_{2}^{y}$. An example
relationship is depicted in Figure 2.

Figure 2. An example information structure of a state tuple in condition (a)

    2. (b)

The utilities of $S_{1}$ and $S_{2}$ satisfy

(10) $\displaystyle
U_{S_{1}}(a_{\alpha})>\max\\{U_{S_{1}}(a_{\beta}),U_{S_{1}}(a_{\gamma})\\},$
(11) $\displaystyle
U_{S_{2}}(a_{\gamma})>U_{S_{2}}(a_{\alpha})>U_{S_{2}}(a_{\beta}).$

  2. (2)

The receiver’s tie-breaking rule is belief dependent.

The first condition in Theorem 2 captures a requirement of an alignment
between $S_{1}$’s and $S_{2}$’s utilities. The requirement is considerably
weaker than the definition of collaborative states and condition 1 of Theorem
1. The commitment order may matter when both senders want to collaborate in a
pair of states, but they have a conflict of interest on mixing a third
state111111This third state is in a different information set of $S_{1}$. with
this pair. When both senders have no conflicts of interest or never
collaborate, then the commitment order will not matter. To avoid confusion, we
note here that the violation of Theorem 2 does not demand fully-aligned
preference orderings between $S_{1}$ and $S_{2}$, or polar opposite preference
orderings between $S_{1}$ and $S_{2}$. Senders can still collaborate in a set
of states and have a conflict of interest in another set of states, as long as
these two sets do not intersect. The second condition states that if the
receiver’s tie-breaking rule depends on her beliefs, then a sender may tailor
her signalling strategies to make the tie-breaking rule favor her instead of
the other sender. When this occurs, the sender committing first suffers due to
the tie-breaking rule. The sender committing later can always tailor the
receiver’s beliefs on top of the earlier sender’s commitment and make the tie-
breaking rule favor her instead.

##  5\. Discussion

We present two examples in this discussion section to demonstrate interesting
phenomena. The first example demonstrates an interesting scenario where the
commitment order matters. In this scenario, the only reason that drives
$S_{1}$ to get the first commitment slot is because she can stay silent while
committing first. The second example compares sequential commitments with
simultaneous commitments. In the second example, the sequential commitment
setting gives the receiver higher expected utility and more informative
signals than simultaneous commitments. Beside these two examples, we remark on
the difficulty of developing general algorithms to derive the optimal
commitments under incomplete information of senders to close the discussion
section.

###  5.1. Silence is golden: commit to send a non-informative signal

We present an interesting example where commitment order matters, but senders
commit completely opposite signalling strategies while committing first. When
$S_{1}$ commits first, her optimal signalling strategy is either to stay
silent or to send non-informative signals. However, when $S_{2}$ commits
first, her optimal commitment is a truth-telling strategy, and her truth-
telling strategy forces $S_{1}$’s optimal commitment to become truth-telling
as well. Hence, in the example detailed below, if the receiver can choose the
commitment order, she prefers $S_{2}$ committing first.

###### Example 0.

The state space is ternary $\Theta=\\{\theta_{1},\theta_{2},\theta_{3}\\}$
with prior distribution
$(\mathbb{P}(\theta_{1}),\mathbb{P}(\theta_{2}),\mathbb{P}(\theta_{3}))=\big{(}0.2,0.5,0.3\big{)}$.
$S_{1}$ knows the state is $\theta_{3}$ or not, but cannot distinguish
$\theta_{1}$ and $\theta_{2}$,
$\mathcal{I}_{1}=\\{\\{\theta_{1},\theta_{2}\\},~{}\\{\theta_{3}\\}\\}$.
$S_{2}$ knows the state is $\theta_{1}$ or not, but cannot distinguish
$\theta_{2}$ and $\theta_{3}$,
$\mathcal{I}_{2}=\\{\\{\theta_{1}\\}~{},\\{\theta_{2},\theta_{3}\\}\\}$.
$S_{1}$ and $S_{2}$’s utilities only depend on the receiver’s action and are
given by

$\displaystyle[U_{S_{1}}(a_{1}),U_{S_{1}}(a_{2}),U_{S_{1}}(a_{3})]=[3,0,1],\quad[U_{S_{2}}(a_{1}),U_{S_{2}}(a_{2}),U_{S_{2}}(a_{3})]=[1,0,3].$

The receiver gets utility $1$ if the index of the state matches the action,
and $0$ otherwise.

When $S_{1}$ commits first, she knows that $S_{2}$ cannot distinguish
$\theta_{2}$ and $\theta_{3}$. Hence, as long as the receiver will guess on
$a_{2}$ as opposed to $a_{3}$ given the posterior belief of the states,
$S_{2}$ is willing to collaborate with $S_{1}$ to suggest $a_{1}$. Hence, to
maximize $S_{1}$’s expected utility, her optimal commitment is to stay silent,
i.e., commit to a non-informative signal because
$\mathbb{P}(\theta_{2})>\mathbb{P}(\theta_{3})$ in the prior. Given $S_{1}$’s
non-informative commitment, $S_{2}$’s optimal commitment is to suggest $a_{1}$
while she knows the state is $\theta_{1}$, and suggest $a_{1}$ with
probability $0.4$ when the state is either $\theta_{2}$ or $\theta_{3}$ (since
$S_{2}$ cannot distinguish between $\theta_{2}$ or $\theta_{3}$).

When $S_{2}$ commits first, she knows $S_{1}$ is willing to help her to
persuade the receiver to take $a_{2}$ when the information set contains only
$\theta_{2}$ and $\theta_{3}$. Hence, her optimal commitment is, to tell the
truth, sending a signal to reveal $\theta_{1}$ when the true state is
$\theta_{1}$, and sending a signal to tell she knows the state is $\theta_{2}$
or $\theta_{3}$ otherwise. Given $S_{2}$’s truth-telling commitment and
underlying prior, $S_{1}$ knows if she doesn’t reveal $\theta_{2}$ when the
true state is $\theta_{2}$, the receiver will take $a_{3}$ when the state is
$\theta_{2}$. Hence, her optimal signalling strategy becomes a truth-telling
strategy as well.

###  5.2. Comparison to simultaneous commitments

When senders only have partial information (of the state of the world),
sequential commitments may bring more informative signals jointly for the
receiver. This is different from the result in Li-Norman (Li and Norman,
2018b), where senders have complete information. Hence, we want to compare our
example in Section 3.1 with simultaneous commitments (and simultaneous
signalling) from the receiver’s perspective. In order to make a fair
comparison, the prior and sender’s utilities are the same as the example in
Section 3.1.

Table 5. State distribution and senders’ utilities

| L | R
---|---|---
T | $0.1$ | $0.2$
B | $0.4$ | $0.3$
(a) Distribution of the states

| $a_{TL}$ | $a_{TR}$ | $a_{BL}$ | $a_{BR}$
---|---|---|---|---
$(U_{S_{1}},U_{S_{2}})$ | 1,0 | 2,2 | 0,0 | 0,3
(b) Utilities of senders

From $S_{2}$’s perspective, since $TR$ is better than $TL$ and $BR$ is better
than $BL$ based on her utility function, her commitment will try to persuade
the receiver in the best possible manner for taking action $TR$ or $BR$,
regardless of what $S_{1}$ sends. Hence, $S_{2}$’s optimal commitment is the
following:

  * •

When her private information is $R$, send a signal to suggest $R$ w.p. 1.

  * •

When her private information is $L$, send a signal to suggest $R$ with
probability $\frac{3}{4}$ and send a suggestion $L$ with probability
$\frac{1}{4}$.

From $S_{1}$’s perspective, even though she cannot observe $S_{2}$’s
commitment and signals, she can anticipate $S_{2}$’s optimal commitment via
$S_{2}$’s utility function and the prior. Hence, although signals are sent and
commitments are made simultaneously, $S_{1}$ can use her inference on
$S_{2}$’s optimal commitment to make her commitment. Thus, $S_{1}$’s
commitment is the following:

  * •

When her private information is $T$, send a signal to suggest $T$ w.p. 1.

  * •

When her private information is $B$, send a signal to suggest $T$ with
probability $\frac{2}{3}$ and send a suggestion $B$ with probability
$\frac{1}{3}$.

Given the above commitments, the receiver’s utility is calculated below:

$\mathbb{E}_{p}[U_{R}]=0.1\times 0+0.2\times
1+0.4\times\tfrac{1}{3}\times\tfrac{1}{4}+0.3\times\tfrac{1}{3}=\frac{1}{3}.$

In Section 3.1, the receiver’s utility is $\frac{1}{3}$ while $S_{1}$ commits
first and $0.4$ while $S_{2}$ commits first. In this example, simultaneous
commitment (and signalling) grants the receiver the same utility as the case
where $S_{1}$ commits first. However, it is still a lower utility compared to
the case where $S_{2}$ commits first. We note that even though the commitments
are made simultaneously, and signals are sent simultaneously, the equilibrium
commitment pairs may not be unique when senders only have partial information
about the state of the world. The non-uniqueness of equilibrium commitment
pairs makes it difficult to make a fair comparison since equilibrium selection
is also needed.

###  5.3. Algorithmic challenges for deriving the optimal commitments

Finally, we conclude by remarking that a general algorithm that solves the
optimal commitments in information design problems with multiple senders where
each sender obtains incomplete private information is challenging to develop.
The reason is that the (required) level of inference depends on the
information structure of the problem. Even when the commitment order is
aligned with the signalling order, the backward-iteration algorithm presented
in (Li and Norman, 2018b) cannot solve for the optimal commitments with
partially informed senders. In general, deriving the optimal commitments
demands iterative reasoning of the other sender’s optimal signalling strategy
under different interim beliefs of states. This process is straightforward
when senders’ information spaces are binary, but becomes cumbersome when
senders’ information spaces grow, even from binary to ternary as in Example
Example. We believe deriving a general algorithm that iteratively reasons
senders’ optimal signalling strategies under incomplete information, will lead
to further significant contributions to the field of information design, but
this is beyond the scope of this work.

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##  6\. Appendix: An example justifying permutation-free commitments

There are four possible states of the world $\Theta=\\{TL,TR,BL,BR\\}$ (top-
bottom and left-right) with prior distribution depicted in Table 1(a). Sender
$S_{1}$ knows whether the state of the world is top or bottom
$\mathcal{I}_{1}=\\{T,B\\}$ and sender $S_{2}$ knows whether the state of the
world is left or right $\mathcal{I}_{2}=\\{L,R\\}$. $S_{1}$ can send signals
$\omega_{1}\in\\{\omega_{1,1},\omega_{1,2}\\}$ and $S_{2}$ can send signals
$\omega_{2}\in\\{\omega_{2,1},\omega_{2,2}\\}$. We assume that both senders’
utilities are as given in Table 6(b), and this example requires sender $S_{2}$
to commit before $S_{1}$. All participants’ utility functions and the (prior)
state distribution are common knowledge.

Table 6. State distribution and senders’ utilities

| L | R
---|---|---
T | $0.1$ | $0.2$
B | $0.5$ | $0.2$
(a) Distribution of the states

| $a_{TL}$ | $a_{TR}$ | $a_{BL}$ | $a_{BR}$
---|---|---|---|---
$(U_{S_{1}},U_{S_{2}})$ | 2,0 | 4,2 | 3,0 | 0,3
(b) Utilities of senders

Given the prior and the utility functions, sender $S_{1}$ and $S_{2}$’s
objective functions are the following:

$\displaystyle\Gamma_{1}^{s*}$
$\displaystyle\in\arg\max_{\Gamma_{1}^{s}\in\mathbf{\Gamma_{1}}}\mathbb{E}_{p}[U_{S_{1}}|\Gamma_{2}^{s}]$
$\displaystyle\text{~{}s.t.~{}}\mathbb{E}_{p}[U_{S_{2}}|\Gamma_{1}^{s}(\Gamma_{2}^{s*})]\geq\mathbb{E}_{p}[U_{S_{2}}|\Gamma_{1}^{s^{\prime}}(\Gamma_{2}^{s^{*}})],$
$\displaystyle\text{~{}where~{}}\Gamma_{1}^{s^{\prime}}\text{~{}is a
commitment that swaps the signals of~{}}\Gamma_{1}^{s},i.e.,$
$\displaystyle\mathbb{P}_{\Gamma_{1}^{s}}(\omega_{1,1}|T)=\mathbb{P}_{\Gamma_{1}^{s^{\prime}}}(\omega_{1,2}|T)\text{~{}and~{}}\mathbb{P}_{\Gamma_{1}^{s}}(\omega_{1,1}|B)=\mathbb{P}_{\Gamma_{1}^{s^{\prime}}}(\omega_{1,2}|B).$
$\displaystyle\Gamma_{2}^{f*}$
$\displaystyle\in\arg\max_{\Gamma_{2}^{f}\in\mathbf{\Gamma_{2}}}\mathbb{E}_{p}[U_{S_{2}}|\Gamma_{1}^{s*}(\Gamma_{2}^{s})]$

Since $S_{1}$ cannot threaten $S_{2}$ via signal reordering, $S_{2}$ can
commit to signalling strategies depending on the realization of $S_{1}$’s
signal. For simplicity of the representation, let’s assume $\omega_{1,1}$ has
higher posterior belief on $S_{1}$’s private information $T$ than
$\omega_{1,2}$. $S_{2}$’s optimal commitment is the following:

  * •

While observing $\omega_{1,1}$, send $\omega_{2,1}$ with probability 1 no
matter what private information she knows.

  * •

While observing $\omega_{1,2}$ and given her private information is $R$, send
$\omega_{2,1}$ with probability 1.

  * •

While observing $\omega_{1,2}$ and given her private information is $L$, send
signal $\omega_{2,1}$ with probability $0.4-\epsilon$ and $\omega_{2,2}$ with
probability $0.6+\epsilon$, where $\epsilon$ is infinitesimal, i.e.,
$|\epsilon|\ll 1$.

Given $S_{2}$’s commitment above, the optimal commitment of $S_{1}$ is the
following:

  * •

Given her private information $T$, send $\omega_{1,1}$.

  * •

Given her private information $B$, send $\omega_{1,1}$ $40\%$ of the time and
and send $\omega_{1,2}$ $60\%$ of the time.

Given this pair of commitments, the expected utilities of all the agents are
given by $(U_{S_{1}},U_{S_{2}},U_{R})=(2.86,1.88,0.5)$.

However, when $S_{2}$ cannot commit to permutation-free commitments, $S_{1}$’s
objective function changes to the following

$\displaystyle\Gamma_{1}^{s*}\in\arg\max_{\Gamma_{1}^{s}\in\mathbf{\Gamma_{1}}}\mathbb{E}_{p}[U_{S_{1}}|\Gamma_{2}^{s}]$

If $S_{2}$ commits to the same signalling strategy, $S_{1}$ can simply swap
$\omega_{1,1}$ with $\omega_{1,2}$ to get a higher expected utility. Under the
threat of $S_{1}$’s reordering of signals, $S_{2}$’s commitment reduces to the
following form. No matter the signal sent by $S_{1}$, $S_{2}$ commits to the
following signalling scheme:

  * •

When $S_{2}$’s private information is $R$, send $\omega_{2,1}$ with
probability 1.

  * •

When $S_{2}$’s private information is $L$, send signal $\omega_{2,1}$ with
probability $40\%-\epsilon$ of the time and and send $\omega_{1,2}$
$60\%+\epsilon$ of the time, where $\epsilon$ is infinitesimal, i.e.,
$|\epsilon|\ll 1$.

Under $S_{2}$’s commitment above, $S_{1}$’s optimal commitment stays the same.
However, the expected utility of agents changes to
$(U_{S_{1}},U_{S_{2}},U_{R})=(2.74,1.76,0.56)$.

In this example, when permutation-free commitments are not enforced, $S_{2}$
is under the threat of $S_{1}$’s reordering of signals, so her optimal
strategy is to make her commitment independent of $S_{1}$’s signal
realizations. Here, both senders suffer (utility losses) from the lack of
permutation-free commitments.

##  7\. Appendix: Proofs and Calculations

###  7.1. Proof of Proposition 1

###### Proof.

According to condition 2, we have a signal $\hat{\omega}_{1}$ which tells
$S_{1}$’s private information is $I_{1}$, i.e.,
$\mathbb{P}(I_{1}|\hat{\omega_{1}})=1$. First, we prove $S_{1}$ does not
prefer sending the signal $\hat{\omega}_{1}$. When $S_{1}$ sends signal
$\hat{\omega}_{1}$, $S_{2}$ knows $S_{1}$’s private information is $I_{1}$.
Let $\Gamma_{2}^{*}(I_{1})$ denote $S_{2}$’s optimal signalling strategy while
receiving $\hat{\omega}_{1}$, inequality (9) states $S_{1}$’s utility under
action $a_{\hat{\theta}}$ is higher than her expected utility conditional on
her signal $\hat{\omega}_{1}$. Thus, $S_{1}$ prefers a mixture of her
information set $I_{1}$ and the information set which contains the
collaborative state $\hat{\theta}$, called $\hat{I}_{1}$.

We notice that if $S_{2}$’s optimal signalling strategy while receiving
$\hat{\omega}_{1}$ is not unique, the conditional expectation
$\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{1}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}]$
may not be well-defined since different $S_{2}$’s tie-breaking strategies may
vary $S_{1}$’s expected utility. To deal with the non-uniqueness of
$\Gamma_{2}^{*}(I_{1})$, we assume the polar opposite preference ordering on
the receiver’s possible optimal action set given $S_{1}$’s private information
$I_{1}$, called $\Phi(I_{1})$ in condition 3. Therefore, when $S_{2}$ is
indifferent between two different actions $a_{m}$ and $a_{n}$ where
$a_{m},a_{n}\in\Phi(I_{1})$, $S_{1}$ is indifferent between $a_{m}$ and
$a_{n}$ too. Thus, the conditional expectation
$\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{1}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}]$
in inequality (9) is well-defined, and $S_{1}$ prefers a mixture of $I_{1}$
and $\hat{I}_{1}$ when the inequality (9) holds.

Next, we prove $S_{2}$ prefers a separation of $I_{1}$ and $\hat{I}_{1}$.
Given assumption 4 in the problem formulation, where the true state can be
revealed under both senders’ truth-telling strategies, $S_{2}$ knows the true
state while observing the signal realization $\hat{\omega}_{1}$ and her
private information $I_{2}$, and her optimal signalling strategy under
$\hat{\omega}_{1}$ can be calculated directly via the concavification approach
— this is equivalent to solving classic Bayesian persuasion problem in
(Kamenica and Gentzkow, 2011) with finite states. According to the inequality
8 stated in condition 1 and the definition of collaborative state, for every
$\theta_{k}\in\hat{I}_{1}$,the following inequality holds:

$\displaystyle
U_{S_{2}}(\theta_{k},a_{\theta_{k}})<\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{2}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}].$

Thus, $S_{2}$ enjoys a higher expected utility while observing
$\hat{\omega}_{1}$ than observing the signal telling her $S_{1}$’s private
information is $\hat{I}_{1}$. Thus, $S_{2}$ prefers a separation of $I_{1}$
and $\hat{I}_{1}$.

When condition 3 in Proposition 1 holds, i.e., preference orders of senders on
$\Phi(I_{1})$ are polar opposite, $S_{2}$’s expected-utility maximization
strategy (via the concavification approach) will directly minimize $S_{1}$
expected utility when $\hat{\omega_{1}}$ is realized. Therefore, from
$S_{1}$’s objective, $S_{1}$ should minimize the probability
$\mathbb{P}(\hat{\omega_{1}})$ to reduce the probability where $S_{2}$
observes $\hat{\omega_{1}}$. Hence, $S_{1}$ will increase
$\mathbb{P}(\omega_{1}|I_{1})$ for every $\omega_{1}$ such that
$\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}]>\mathbb{E}_{p}[U_{S_{1}}|\hat{\omega_{1}}]$.
This includes the signal which suggests the receiver to take the collaborative
action $a_{\hat{\theta}}$. Hence, when $S_{1}$ commits first, she will
maximize the probability of suggesting the collaborative state $\hat{\theta}$
via mixing $I_{1}$ with $\hat{I}_{1}$. Since $\hat{\theta}$ is a collaborative
state, $S_{2}$ will collaborate with $S_{1}$ while observing $S_{1}$’s signal
which suggests $a_{\hat{\theta}}$.

However, when sender $S_{2}$ commits first, the inequality (8) stated in
condition 1, i.e.,
$U_{S_{2}}(\hat{\theta},a_{\hat{\theta}})<\mathbb{E}_{p,\Gamma_{2}^{*}(I_{1})}[U_{S_{2}}(\theta,a^{*}(I_{1},\omega_{2}))|I_{1}]$,
guarantees that $S_{2}$ prefers the realized signal $\hat{\omega_{1}}$ instead
of the realized signal which suggests her to collaborate toward $\hat{\theta}$
via a mixture of $I_{1}$ and $\hat{I}_{1}$. Therefore, when sender $S_{2}$
commits first, she will commit to a signalling strategy which minimizes the
probability of $I_{1}$ mixed with $\hat{I}_{1}$ (in order to increase the
total probability of observing $\hat{\omega_{1}}$). Since condition 2
guarantees that $\mathbb{P}(\hat{\omega}_{1})>0$ for every optimal
commitments, we will never fall into the corner case where $S_{1}$ and $S_{2}$
have an opposite preference on the mixture of $I_{1}$ and $\hat{I}_{1}$, but
one of the sender’s strategy of minimizing/maximizing the mixture of $I_{1}$
and $\hat{I}_{1}$ cannot be achieved owing to the low value of
$\mathbb{P}(I_{1})$ under the given prior. Hence, according to the opposite
strategies of $S_{1}$ and $S_{2}$ on the mixture of $I_{1}$ and $\hat{I}_{1}$,
the commitment order matters. ∎

###  7.2. Proof of Claim 1

###### Proof.

We prove this claim by contradiction. First, assume that both conditions are
violated and the signal $\hat{\omega}_{1}$ revealing $I_{1}$ is the signal
realization which gives $S_{1}$ the highest expected utility under an
equilibrium derived by solving the objectives stated in Section 2.4, where
each sender’s commitment is an optimal one under the given commitment order.
Then, let signal $\bar{\omega}_{1}$ represents the signal that $S_{1}$ sends
when the true state is $\hat{\theta}$, the violation of condition 2 guarantees
that the following inequality holds

$\displaystyle\mathbb{E}_{p}[U_{S_{1}}(a^{*}_{\Gamma_{1}^{*},\Gamma_{2}^{*}}(\hat{\omega}_{1},\omega_{2}))]>\mathbb{E}_{p}[U_{S_{1}}(a^{*}_{\Gamma_{1}^{*},\Gamma_{2}^{*}}(\bar{\omega}_{1},\omega_{2}))],$

where $a^{*}_{\Gamma_{1}^{*},\Gamma_{2}^{*}}(\omega_{1},\omega_{2})$ is the
best response of the receiver under a realized pair of signal
$\omega_{1},\omega_{2}$ and both senders’ commitments $\Gamma_{1}^{*}$ and
$\Gamma_{2}^{*}$. Since the signal $\hat{\omega}_{1}$ reveals the private
signal $I_{1}$, we can construct another commitment $\Gamma_{1}^{\prime}$
below, where $\epsilon\ll 1$:

  1. (1)

$\mathbb{P}(\hat{\omega}_{1}^{\prime})=\mathbb{P}(\hat{\omega}_{1})+\epsilon\mathbb{P}(\bar{\omega}_{1})$
and
$\mathbb{P}(\theta|\hat{\omega}_{1}^{\prime})=\frac{1}{1+\epsilon}\mathbb{P}(\theta|\hat{\omega}_{1})+\frac{\epsilon}{1+\epsilon}\mathbb{P}(\theta|\bar{\omega}_{1})$
for all $\theta\in\Theta$;

  2. (2)

$\mathbb{P}(\bar{\omega}_{1}^{\prime})=(1-\epsilon)\mathbb{P}(\bar{\omega}_{1})$,
and the interim beliefs of $\bar{\omega}_{1}^{\prime}$ and $\bar{\omega}_{1}$
are the same, i.e.,
$\mathbb{P}_{p,\Gamma_{1}^{\prime}}\big{(}\theta|\bar{\omega}_{1}^{\prime}\big{)}=\mathbb{P}_{p,\Gamma_{1}}\big{(}\theta|\bar{\omega}_{1}\big{)}$
for all $\theta\in\Theta$;

  3. (3)

For every $\omega_{1}$ under the commitment $\Gamma_{1}^{\prime}$ such that
$\omega_{1}\neq\bar{\omega}_{1}$ and $\omega_{1}\neq\hat{\omega}_{1}$,
$\mathbb{P}_{p,\Gamma_{1}^{\prime}}(\omega_{1})=\mathbb{P}_{p,\Gamma_{1}^{*}}(\omega_{1})$
and
$\mathbb{P}_{p,\Gamma_{1}^{\prime}}\big{(}\theta|\omega_{1}\big{)}=\mathbb{P}_{p,\Gamma_{1}^{*}}\big{(}\theta|\omega_{1}\big{)}$
for all $\theta\in\Theta$;

As assumed earlier, $\hat{\omega}_{1}$ is the signal which gives $S_{1}$ the
highest expected utility with $\mathbb{P}(I_{1}|\hat{\omega}_{1})=1$, and
$\bar{\omega}_{1}$ is the signal $S_{1}$ sends to collaborate with $S_{2}$ on
persuading the receiver to take $a_{\hat{\theta}}$. Given the inequality (9),
$S_{2}$’s optimal signalling strategy belongs to one of the following two
cases:

Case 1: $S_{2}$ cannot elicit the collaborative state $\hat{\theta}$ under the
signal $\hat{\omega}_{1}^{\prime}$.
In this case, because $\epsilon\ll 1$, the optimal commitment of sender
$S_{2}$ stays unchanged between $\Gamma_{1}^{*}$ and $\Gamma_{1}^{\prime}$
since $S_{2}$ cannot successfully persuade the receiver to take
$a_{\hat{\theta}}$ and she has already chosen her optimal partition of $I_{1}$
in her commitment. Therefore, when $\epsilon\ll 1$, changing the commitment
from $\Gamma_{1}^{*}$ to $\Gamma_{1}^{\prime}$ in this case will not affect
$S_{2}$’s optimal signalling strategy. However, the revised commitment
$\Gamma_{1}^{\prime}$ has a higher total probability of
$\hat{\omega}_{1}^{\prime}$ comparing to the signal $\hat{\omega}_{1}$ under
$\Gamma_{1}^{*}$. Given the inequality
$\mathbb{E}_{p}[U_{S_{1}}(a^{*}_{\Gamma_{1}^{*},\Gamma_{2}^{*}}(\hat{\omega}_{1},\omega_{2}))]>\mathbb{E}_{p}[U_{S_{1}}(a^{*}_{\Gamma_{1}^{*},\Gamma_{2}^{*}}(\bar{\omega}_{1},\omega_{2}))]$
and the knowledge that both the optimal commitments of $S_{2}$ and the
receiver’s best response stay unchanged, $S_{1}$ will enjoy a higher expected
utility under the the commitment $\Gamma_{1}^{\prime}$, which contradicts the
assumption that $\Gamma_{1}^{*}$ is the optimal commitment of $S_{1}$. Hence,
one of conditions in Claim 1 must hold in this case.

Case 2: $S_{2}$ can elicit the collaborative state $\hat{\theta}$ under the
signal $\hat{\omega}_{1}^{\prime}$.
This case occurs when $S_{2}$’s private information set $I_{2}\ni\hat{\theta}$
contains no state that belongs to $S_{1}$’s information set $I_{1}$, i.e.,
$\theta_{i}\notin I_{2},\forall\theta_{i}\in I_{1},\hat{\theta}\in I_{2}$.
Therefore, even under the signal $\hat{\omega}_{1}^{\prime}$ which mixes
$\hat{\omega}_{1}$ with the signal containing the collaborative states,
$S_{2}$ can still persuade the receiver to take $a_{\hat{\theta}}$ with some
probability (according to the prior). Besides, persuading the receiver to take
$a_{\hat{\theta}}$ benefits both senders. However, after maximizing the signal
$S_{2}$ used to persuade the action $a_{\hat{\theta}}$, $S_{2}$’s optimal
signalling strategy conditional on $a_{\hat{\theta}}$ not sent is the same as
her optimal signalling strategy while receiving $\hat{\omega}_{1}$. Therefore,
$S_{2}$’s optimal signalling changes by a small amount (because $\epsilon\ll
1$). Now, since $\hat{\theta}$ is a collaborative state, $S_{2}$’s adjustment
on her optimal signalling strategy from $\hat{\omega}_{1}$ under
$\Gamma_{1}^{*}$ to $\hat{\omega}_{1}^{\prime}$ under $\Gamma_{1}^{\prime}$
not only benefits her but also benefits $S_{1}$. Moreover, after maximizing
the probability of the signal persuading $a_{\hat{\theta}}$, the $S_{2}$’s
optimal signalling strategy stays unchanged (reduced to the case 1 scenario).
Hence, $S_{1}$ enjoys a higher expected utility under the the commitment
$\Gamma_{1}^{\prime}$ in this case (because $S_{2}$’s optimal signalling
strategy gives $S_{1}$ a higher utility than the utility in case 1, and
$S_{1}$ enjoys a higher expected utility in case 1).

In both cases, $S_{1}$’s expected utility increases. This violates the
assumption that $S_{1}$ has the highest utility in $\hat{\omega}_{1}$ when
condition 2 is violated. Hence, one of the conditions in Claim 1 must hold.

∎

###  7.3. Proof of Theorem 1

First, let us review the statements in condition 1 for the readability of the
proof.

Condition 1: There exist two parameters $\alpha>\beta>0$ and a signalling
strategy $\hat{\Gamma}_{2}$ satisfying the following conditions:

  1. (a)

$G(p,I_{1}^{x}+\alpha I_{1}^{y})=G(p,I_{1}^{x}+\beta I_{1}^{y})$,

  2. (b)

Let $\omega_{1}^{\alpha},\omega_{1}^{\beta}$ be two mock signals of $S_{1}$
such that $\mathbb{P}(I_{1}^{x}|\omega_{1}^{\alpha})=\frac{1}{1+\alpha}$,
$\mathbb{P}(I_{1}^{y}|\omega_{1}^{\alpha})=\frac{\alpha}{1+\alpha}$,
$\mathbb{P}(I_{1}^{x}|\omega_{1}^{\beta})=\frac{1}{1+\beta}$, and
$\mathbb{P}(I_{1}^{y}|\omega_{1}^{\beta})=\frac{\beta}{1+\beta}$, then

$\displaystyle\hat{\Gamma}_{2}(p,\omega_{1}^{\beta})\in G(p,I_{1}^{x}+\beta
I_{1}^{y})~{}\text{and}~{}\hat{\Gamma}_{2}(p,\omega_{1}^{\alpha})\notin
G(p,I_{1}^{x}+\alpha I_{1}^{y}),$

  3. (c)

$\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\alpha},\hat{\Gamma}_{2}]<\frac{\mathbb{P}(I_{1}^{x})+\beta\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\beta},\hat{\Gamma}_{2}]+\frac{(\alpha-\beta)\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|I_{1}^{y},\hat{\Gamma}_{2}]$,

  4. (d)

$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\beta},G(p,I_{1}^{x}+\beta
I_{1}^{y})]<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})]$.

Given (a), if $S_{1}$ commits first and the two mock signals
$\omega_{1}^{\alpha},\omega_{1}^{\beta}$ described in (b) are used in
$S_{1}$’s signalling strategy, $G(p,I_{1}^{x}+\alpha
I_{1}^{y})=G(p,I_{1}^{x}+\beta I_{1}^{y})$ states that the set of $S_{2}$’s
optimal signalling strategies under $\omega_{1}^{\alpha}$ is the same as the
set of $S_{2}$’s optimal signalling strategies under $\omega_{1}^{\beta}$. In
other words, $\omega_{1}^{\alpha}$ and $\omega_{1}^{\beta}$ are two different
mixing schemes using $S_{1}$’s information sets $I_{1}^{x}$ and $I_{1}^{y}$
that elicit the same response from $S_{2}$. According to the statement of
condition 2, i.e., $\min_{g\in
G(p,I_{1}^{x})}\mathbb{E}_{p,g}[U_{S_{1}}|I_{1}^{x}]>\max_{g^{\prime}\in
G(p,I_{1}^{y})}\mathbb{E}_{p,g^{\prime}}[U_{S_{1}}|I_{1}^{y}]$, when $S_{2}$’s
action and the receiver’s action both stay unchanged, $S_{1}$ prefers signal
$\omega_{1}^{\alpha}$ over signal $\omega_{1}^{\beta}$. The logic is the
following: First, suppose the receiver takes action $a\in I_{1}^{x}$, sending
signal $\omega_{1}^{\alpha}$ mixes a higher portion of $I_{1}^{y}$ with
$I_{1}^{x}$ than sending signal $\omega_{1}^{\beta}$. This implies that the
receiver takes action $a\in I_{1}^{x}$ with a higher total probability. Based
on the inequality $\min_{g\in
G(p,I_{1}^{x})}\mathbb{E}_{p,g}[U_{S_{1}}|I_{1}^{x}]>\max_{g^{\prime}\in
G(p,I_{1}^{y})}\mathbb{E}_{p,g^{\prime}}[U_{S_{1}}|I_{1}^{y}]$, increasing the
total probability of a set of actions $a\in I_{1}^{x}$ by reducing the same
amount of probability on a set of actions $a^{\prime}\in I_{1}^{y}$ benefits
$S_{1}$; second, suppose the receiver takes action $a^{\prime}\in I_{1}^{y}$
under both signal $\omega_{1}^{\alpha}$ and signal $\omega_{1}^{\beta}$,
$S_{1}$ still prefers using the signal $\omega_{1}^{\alpha}$ when condition 2
holds, since under this circumstance signal $\omega_{1}^{\alpha}$ uses less
probability mass of states in $I_{1}^{x}$ than signal $\omega_{1}^{\beta}$. In
short, combining the statements in condition 1.(a) and condition 2, we know
$S_{1}$ prefers sending signal $\omega_{1}^{\alpha}$ over signal
$\omega_{1}^{\beta}$. Furthermore, when we consider the statements in
condition 1.(c) and 2, the inequality in 1.(c) implies $S_{1}$ can persuade
the receiver to take action $a\in I_{1}^{x}$ under signal
$\omega_{1}^{\beta}$. (Otherwise, the inequality is violated since the
receiver will take the same action $a_{a}^{\prime}\in I_{1}^{y}$ under signal
$\omega_{1}^{\alpha}$, signal $\omega_{1}^{\beta}$, and the signal which
reveals $I_{1}^{y}$.) Therefore, combining statements in condition 1.(a),
condition 1.(c), and condition 2, we know $S_{1}$ can persuade the receiver
taking actions $a\in I_{1}^{x}$ under signal $\omega_{1}^{\alpha}$ and signal
$\omega_{1}^{\beta}$, and $S_{1}$ prefers sending signal
$\omega_{1}^{\alpha}$.

Now let’s explore $S_{2}$’s preference on signals/information sets. Given
condition 1.(d), the inequality
$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\beta},G(p,I_{1}^{x}+\beta
I_{1}^{y})]<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})]$ states that
$S_{2}$ prefers a (pure) signal eliciting $S_{1}$’s information $I_{1}^{y}$
over a mixed signal $\omega_{1}^{\beta}$. Moreover, condition 1.(a) states
that the set of $S_{2}$ optimal signalling strategies under
$\omega_{1}^{\beta}$ is the same as the set of $S_{2}$ optimal signalling
strategies under $\omega_{1}^{\alpha}$. Thus, with $\alpha>\beta$ and the
inferred fact that $S_{1}$ can persuade the receiver taking $a\in I_{1}^{x}$
under signal $\omega_{1}^{\alpha}$ and signal $\omega_{1}^{\beta}$, we can
derive the inequality
$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\alpha},G(p,I_{1}^{x}+\alpha
I_{1}^{y})]<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})]$ from
$\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\beta},G(p,I_{1}^{x}+\beta
I_{1}^{y})]<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})]$. Combining the
two inequalities derived above, we can write the following inequality:

(12)
$\displaystyle\max\big{\\{}\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\alpha},G(p,I_{1}^{x}+\alpha
I_{1}^{y})],\mathbb{E}_{p}[U_{S_{2}}|\omega_{1}^{\beta},G(p,I_{1}^{x}+\beta
I_{1}^{y})]\big{\\}}<\mathbb{E}_{p}[U_{S_{2}}|I_{1}^{y},G(p,I_{1}^{y})].$

Given the inequality (12), we can conclude that $S_{2}$ prefers a separation
of $S_{1}$’s information sets $I_{1}^{x}$ and $I_{1}^{y}$.

Now, condition 1.(b) and condition 1.(c) guarantee the existence of a
signalling strategy $\hat{\Gamma}_{2}$ satisfying the following three
conditions:

  * •

$\hat{\Gamma}_{2}(p,\omega_{1}^{\beta})\in G(p,I_{1}^{x}+\beta I_{1}^{y})$,

  * •

$\hat{\Gamma}_{2}(p,\omega_{1}^{\alpha})\notin G(p,I_{1}^{x}+\alpha
I_{1}^{y})$,

  * •

$\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\alpha},\hat{\Gamma}_{2}]<\frac{\mathbb{P}(I_{1}^{x})+\beta\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|\omega_{1}^{\beta},\hat{\Gamma}_{2}]+\frac{(\alpha-\beta)\mathbb{P}(I_{1}^{y})}{\mathbb{P}(I_{1}^{x})+\alpha\mathbb{P}(I_{1}^{y})}\mathbb{E}_{p}[U_{S_{1}}|I_{1}^{y},\hat{\Gamma}_{2}]$.

Therefore, $\hat{\Gamma}_{2}$ is a signalling strategy that can threaten
$S_{1}$. Given $\hat{\Gamma}_{2}$, $S_{1}$ is better off by committing to
signalling strategies using signal $\omega_{1}^{\beta}$ instead of signal
$\omega_{1}^{\alpha}$, guaranteed by the condition 1.(c). This increases the
total probability of the signal which elicits $S_{1}$’s information
$I_{1}^{y}$, and then increases $S_{2}$’s expected utility (because $S_{2}$
prefers separation over mixture on the information sets $I_{1}^{x}$ and
$I_{1}^{y}$). In summary, given conditions (a)-(d) in condition 1 and the
inequality in condition 2, $S_{2}$ has a conflict of interest with $S_{1}$ on
the mixture of $I_{1}^{x}$ and $I_{1}^{y}$. However, $S_{2}$ is willing to
(partially) collaborate with $S_{1}$ under $I_{1}^{x}$ when $S_{2}$ has no
alternatives (because $G(p,I_{1}^{x}+\alpha I_{1}^{y})=G(p,I_{1}^{x}+\beta
I_{1}^{y})$).

At a high level, a collaboration under $I_{1}^{x}$ benefits both $S_{1}$ and
$S_{2}$ no matter who commits first. However, $S_{1}$ and $S_{2}$ hold
different opinions on the scale of collaboration, where $S_{1}$ want to
maximize the total probability of collaboration, but $S_{2}$ only wants to
collaborate when she has no alternative (signalling strategies that are
incentive-compatible to the receiver). Given the first two conditions, if the
prior distribution supports it, then both senders will collaborate on
$I_{1}^{x}$. Condition 1.(b) and 1.(c) guarantees that $S_{2}$ can construct a
credible threat to reduce the total probability of collaboration when she
commits first, and this makes the commitment order matter. Finally, condition
3 serves as a regularity condition which ensures that the prior distribution
supports the collaboration and the (potential) credible threat. Hence, when
three conditions in Theorem 1 hold, the commitment order matters.

###  7.4. Proof of Theorem 2

The idea of this proof is to prove that the commitment order may matter if one
of the conditions is not violated.

First, if the receiver’s tie-breaking rule is belief-dependent, the sender who
commits later can tailor the posterior beliefs to make the tie-breaking rule
favor her. Hence, unless all tie-breaking decisions made by the receiver are
indifferent to both senders $S_{1}$ and $S_{2}$, both prefer to commit last.
In order to verify whether the commitment order matters when condition 1
holds, we assume that condition 2 is violated.

Suppose the receiver’s tie-breaking rule is belief independent; we then prove
that satisfying condition 1 may make the commitment order matters. The idea of
proving the above statement is to construct the minimum requirements such that
(partial) collaboration and credible threats can both occur. We will
illustrate the conditions required for a (partial) collaboration and a
credible threat below, respectively.

First, we derive the minimum requirement of a (partial) collaboration. $S_{1}$
and $S_{2}$ will collaborate when their preferences (ordering) on at least a
pair of states align, called $\theta_{\alpha}$ and $\theta_{\beta}$ hereafter
for simplicity of representation. Moreover, if collaboration occurs in
$\theta_{\alpha}$ and $\theta_{\beta}$, $S_{1}$ must be unable to distinguish
$\theta_{\alpha}$ and $\theta_{\beta}$ using her private information,
i.e.,$\exists!I_{1}^{x}\ni\theta_{\alpha},\theta_{\beta}$. Otherwise, $S_{1}$
will separate these two states herself (if it is best for her) and no
collaboration occurs. Given assumption 4 in this chapter, $S_{2}$ can learn
the true state via her private information when $S_{1}$ reveals her
information truthfully. This demands $\theta_{\alpha}$ and $\theta_{\beta}$
must belong to different information sets of $S_{2}$. To make a collaboration
on $\theta_{\alpha}$ and $\theta_{\beta}$ possible, $S_{1}$ and $S_{2}$ must
have aligned preference orders on these two states. Without loss of
generality, we assume $U_{S_{1}}(a_{\alpha})>U_{S_{1}}(a_{\beta})$ and
$U_{S_{2}}(a_{\alpha})>U_{S_{2}}(a_{\beta})$. Collecting the points together,
to construct a (partial) collaboration, we need the following conditions to
hold:

  * •

$\exists!I_{1}^{x}\ni\theta_{\alpha},\theta_{\beta}$

  * •

$\exists!I_{2}^{x}\ni\theta_{\alpha}$ and $\exists!I_{2}^{y}\ni\theta_{\beta}$
such that $I_{2}^{x}\neq I_{2}^{y}$.

  * •

$U_{S_{1}}(a_{\alpha})>U_{S_{1}}(a_{\beta})$

  * •

$U_{S_{2}}(a_{\alpha})>U_{S_{2}}(a_{\beta})$

Next, we derive the minimum requirements for a (potential) conflict of
interest. Before the analysis, we pause to note that a conflict of interest is
not guaranteed under our requirements, because we do not specify the utilities
of $S_{1}$ and $S_{2}$ on the receiver’s best-response action corresponding to
other states within the same information set of $S_{1}$ (if any). Precisely,
when $I_{1}^{y}$ is the information set involved in a (potential) conflict of
interest between senders, we do not specify
$U_{S_{1}}(\theta_{\tau},a_{\tau})$ and $U_{S_{2}}(\theta_{\tau},a_{\tau})$
for every state $\theta_{\tau}\in I_{1}^{y}$ not belonging to the set of
states where we construct a (potential) conflict of interest.

A (potential) conflict of interest arises when $S_{1}$ and $S_{2}$ prefer
different mixed strategies on a set of $S_{1}$’s information sets. Since we
are interested in the minimum requirements, we search for conflict of
interests on a pair of information sets, called $I_{1}^{x}$ and $I_{1}^{y}$,
where the information set $I_{1}^{x}$ is the same set where collaboration may
occur. (Because we demand the occurrence of collaboration and a conflict of
interests. If only collaborations occur, $S_{1}$ and $S_{2}$ can reach a
consensus on their (collectively) optimal commitments. If only a conflict of
interests occurs, $S_{1}$ will commit to a maximin strategy, and $S_{2}$ will
commit to a minimax strategy. This is because $S_{1}$ always sends signals
before $S_{2}$.)

Before constructing a (potential) conflict of interest, we assume
$\theta_{\gamma}$ is a state that belongs to $S_{1}$’s information
$I_{1}^{y}$. As presented earlier, a (potential) conflict of interest requires
that $S_{1}$ and $S_{2}$ have difference preference on a mixture of
$I_{1}^{x}$ and $I_{1}^{y}$. Here we assume121212For the opposite conflict of
interest, i.e., $S_{1}$ prefers a separation over mixture and Sender2 prefers
a mixture over separation, the whole analysis is exactly analogous with a swap
of senders’ utility inequalities. The inequalities restricting $S_{1}$ will
now restrict $S_{2}$, and vice versa. In short, we just swap $S_{1}$ with
$S_{2}$ in condition 1.(b). $S_{1}$ prefers a mixture over separation and
$S_{2}$ prefers a separation over mixture.

Since $S_{1}$ prefers a mixture of $I_{1}^{x}$ and $I_{1}^{y}$, there must
exists a state $S_{\gamma}\in I_{1}^{y}$ such that
$U_{S_{1}}(a_{\alpha})>U_{S_{1}}(a_{\gamma})$, where $\theta_{\alpha}\in
I_{1}^{x}$ as used earlier. Similarly, when $S_{2}$ prefers a separation over
a mixture, there must exists a state $S_{\gamma}\in I_{1}^{y}$ such that
$U_{S_{2}}(a_{\alpha})<U_{S_{2}}(a_{\gamma})$. To avoid a violation of
assumption 1, i.e., each sender only has partial information about the state
of the world, in a ternary state space corner case, we let $\theta_{\gamma}$
belongs to either $I_{2}^{x}$ or $I_{2}^{y}$. To sum up, to construct a
minimum (potential) conflict of interest op top of a partial collaboration,
the following additional conditions hold:

  * •

$U_{S_{1}}(a_{\alpha})>U_{S_{1}}(a_{\gamma})$

  * •

$U_{S_{2}}(a_{\alpha})<U_{S_{2}}(a_{\gamma})$

  * •

Either $\theta_{\gamma}\in I_{2}^{x}$ or $\theta_{\gamma}\in I_{2}^{y}$.

Now, we have constructed the minimum requirement of collaboration and the
minimum requirement of a conflict of interest. The collection of information
structure requirements is the statement of condition 1.(a) in Theorem 2, and
the requirements on $S_{1}$’s and $S_{2}$’s utilities are summarized in the
inequalities in condition 1.(b). Therefore, when condition 1 is satisfied, the
commitment order may matter because of the co-occurrence of collaboration and
a conflict of interest. Thus, when both conditions are violated, the
commitment order does not matter.

###  7.5. Utility calculation of Example 3.1, Sender 1 commits first

(14) $\displaystyle\mathbb{E}_{p}[U_{S_{1}}]$
$\displaystyle=(0.2+0.3\times\tfrac{2}{3}+0.4\times\tfrac{2}{3}\times\tfrac{3}{4})\times
2+0=1.2$ $\displaystyle\mathbb{E}_{p}[U_{S_{2}}]$
$\displaystyle=(0.2+0.3\times\tfrac{2}{3}+0.4\times\tfrac{2}{3}\times\tfrac{3}{4})\times
2+\tfrac{1}{3}\times(0.3+0.4\times\tfrac{3}{4})\times 3$
$\displaystyle=1.2+0.6=1.8$ (15) $\displaystyle\mathbb{E}_{p}[U_{R}]$
$\displaystyle=0.2+0.3\times\tfrac{1}{3}+0.4\times\tfrac{1}{3}\times\tfrac{1}{4}=\tfrac{1}{3}$

###  7.6. Utility calculation of Example 3.1, Sender 2 commits first

(17) $\displaystyle\mathbb{E}_{p}[U_{S_{1}}]$
$\displaystyle=(0.2+0.3\times\tfrac{1}{2}+0.4\times\tfrac{1}{2})\times
2+0=1.1$ $\displaystyle\mathbb{E}_{p}[U_{S_{2}}]$
$\displaystyle=(0.2+0.3\times\tfrac{1}{2}+0.4\times\tfrac{1}{2})\times
2+\frac{1}{2}\times(0.3+0.4\times\tfrac{3}{4})\times 3$
$\displaystyle=1.1+0.9=2$ (18) $\displaystyle\mathbb{E}_{p}[U_{R}]$
$\displaystyle=0.2+0.3\times\tfrac{1}{2}+0.4\times\tfrac{1}{2}\times\tfrac{1}{4}=0.4$

###  7.7. Utility calculation of Example 4.1, Sender 1 commits first

(21) $\displaystyle\mathbb{E}_{p}[U_{S_{1}}]$
$\displaystyle=(0.05+0.05+0.03+0.05+0.005+0.05+0.03)\times 5$
$\displaystyle+(0.09\times 0.54+0.15\times 0.9+0.25\times 0.54+0.15\times
0.9)\times 2$ $\displaystyle+(0.25\times 0.4+0.09\times 0.46)+0.03\times
2=2.4336$ $\displaystyle\mathbb{E}_{p}[U_{S_{2}}]$
$\displaystyle=(0.05+0.05+0.03+0.05+0.005+0.05+0.03)\times 5$
$\displaystyle+(0.09\times 0.54+0.15\times 0.9+0.25\times 0.54+0.15\times
0.9)\times 2$ $\displaystyle+(0.015\times 4+0.09\times 0.06)\times
3+0.03\times 2\times 4+0.02$ $\displaystyle=2.6884$
$\displaystyle\mathbb{E}_{p}[U_{R}]$ $\displaystyle=0.05+0.02\times
0.75+0.2\times 0.75\times 0.9+0.12\times 0.75\times 0.36$
$\displaystyle+0.35\times\tfrac{1}{7}\times 0.4+0.21\times\tfrac{1}{7}\approx
0.286.$

###  7.8. Utility calculation of Example 4.1, Sender 2 commits first

(24) $\displaystyle\mathbb{E}_{p}[U_{S_{1}}]$
$\displaystyle=(0.05+0.05+0.03+0.05+0.005+0.05+0.03)\times 5$
$\displaystyle+(0.09\times 0.9+0.15\times 0.9+0.15\times 0.9+0.09\times
0.9)\times 2$ $\displaystyle+(0.15\times 0.6+0.09)=2.207$
$\displaystyle\mathbb{E}_{p}[U_{S_{2}}]$
$\displaystyle=(0.05+0.05+0.03+0.05+0.005+0.05+0.03)\times 5$
$\displaystyle+(0.09\times 0.9+0.15\times 0.9+0.15\times 0.9+0.09\times
0.9)\times 2$ $\displaystyle+(0.015\times 3+0.009\times 2)\times 3+0.09\times
2\times 4+0.06=3.158$ $\displaystyle\mathbb{E}_{p}[U_{R}]$
$\displaystyle=0.05+0.02\times 0.75+0.2\times 0.75\times 0.9$
$\displaystyle+0.35\times\tfrac{3}{7}\times 0.4+0.21\times\tfrac{3}{7}=0.35.$