Commitment and Randomization in Communication

Commitment is often valuable. In the context of communication, this fact is brought out by the contrast of Sender’s payoff in Bayesian persuasion versus cheap talk. For any prior, and any profile of Sender and Receiver’s preferences, Sender’s payoff is always weakly higher under Bayesian persuasion than in any cheap-talk equilibrium.¹¹1In fact, Bayesian persuasion provides the tight upper bound on Sender’s equilibrium payoff under any communication protocol, such as disclosure or signaling. In this paper, we ask: when does commitment make Sender strictly better off?

Answering this question would contribute to our understanding of circumstances that incentivize building strong institutions immune to influence (North 1993; Lipnowski, Ravid, and Shishkin 2022) or building a reputation for a degree of honesty (Best and Quigley 2024; Mathevet, Pearce, and Stacchetti 2024).

We focus exclusively on environments with finitely many states and actions. We show that, generically, Sender with commitment values that commitment if and only if he values randomization (Theorem 1). In other words, the Bayesian persuasion payoff is achievable in a cheap-talk equilibrium if and only if a partitional experiment is a solution to the Bayesian persuasion problem. Moreover, if Sender’s preferred equilibrium in a cheap-talk game necessarily involves randomization, then Sender values commitment (Theorem 2).

Theorems 1 and 2 respectively consider willingness-to-accept (WTA) and willingness-to-pay (WTP) for commitment. Theorem 1 considers a Sender endowed with commitment and establishes that his WTA for commitment is strictly positive if and only if his WTA for use of randomization is strictly positive. Theorem 2 considers a Sender without commitment (one engaged in cheap talk) and establishes that his WTP for commitment is strictly positive if his WTA for use of randomization is strictly positive.

The link between commitment and randomization does not, on its own, address the question of “how often” (i.e., for what share of preferences), Sender finds commitment (or, equivalently, randomization) valuable. Theorems 1 and 2 would be of substantially less interest if it were the case that (in the finite worlds we consider), commitment and randomization are almost always valuable, with only exceptions being knife-edge cases such as completely aligned or completely opposed preferences.²²2Denoting Sender’s utility by $u_{S}$ and Receiver’s utility by $u_{R}$, it is easy to see that when $u_{S}=u_{R}$, neither commitment nor randomization is valuable (because full information is optimal and achievable via a cheap-talk equilibrium). Similarly, when $u_{S}=-u_{R}$, neither commitment nor randomization are valuable (because no information is optimal and achievable via a cheap-talk equilibrium). We show, however, that this is not the case. In fact, we uncover a potentially surprising connection between the likelihood that commitment has no value and the cardinality of the action set.

Formally, let $\left|A\right|$ denote the cardinality of the action set. Suppose that for each action-state pair, we draw Sender’s utility i.i.d. from some distribution $F$ and we draw Receiver’s utility i.i.d. from some distribution $G$. We assume that Sender’s utility draw is independent of Receiver’s. For any number of states, for any interior prior, for any atomless distributions of preferences ($F$ and $G$), the likelihood that commitment has no value is bounded below by $\frac{1}{\left|A\right|^{\left|A\right|}}$; moreover, as the number of states grows large, the likelihood that commitment has no value converges precisely to $\frac{1}{\left|A\right|^{\left|A\right|}}$ (Theorem 3). So, if the action set is binary and there are many states, the share of preference profiles for which commitment has no value is approximately $\frac{1}{4}$.

In this section, we consider a Sender with commitment power, who can choose his messaging strategy prior to being informed of the state. We ask whether this commitment power makes Sender strictly better off. We link the value of commitment to Sender’s behavior under commitment, in particular to whether Sender has a strict preference for randomization.

For an intuition about the only-if direction, suppose that there is a partitional optimal persuasion profile $\left(\sigma,\rho\right)$. Let $M_{\sigma}$ be the set of messages that are sent in equilibrium. For each $m\in M_{\sigma},$ let $\Omega_{m}$ be the set of states that lead to message $m$, and let $\mu_{m}$ be the belief induced by $m$. For a generic set of environments, Receiver’s optimal action given belief $\mu_{m}$ (call it $a_{m}$) is unique. Since $A$ is finite, $a_{m}$ must be the uniquely optimal action in a neighborhood of beliefs around $\mu_{m}$. Subtly, this implies that every action $a_{m}$ taken in equilibrium must be Sender’s preferred action, among the actions taken in equilibrium, in all states where action $a_{m}$ is taken. That is a mouthful, so in other words: let $A^{*}=\left\{a_{m}|m\in M_{\sigma}\right\}$; for each $a_{m}\in A^{*}$, we must have $u_{S}\left(a_{m},\omega\right)\geq u_{S}\left(a_{m^{\prime}},\omega\right)$ for all $a_{m^{\prime}}\in A^{*}$ and all $\omega\in\Omega_{m}$. Why does this hold? If it were not the case, Sender could attain a higher payoff with an alternative strategy. Suppose $u_{S}\left(a_{m},\omega\right)<u_{S}\left(a_{m^{\prime}},\omega\right)$ for some $a_{m^{\prime}}\in A^{*}$, $\omega\in\Omega_{m}$. Sender could send $m^{\prime}$ in $\omega$ with a small probability and still keep $a_{m}$ optimal given $m$. Finally, the fact that $u_{S}\left(a_{m},\omega\right)\geq u_{S}\left(a_{m^{\prime}},\omega\right)$ for all $a_{m^{\prime}}\in A^{*}$ and all $\omega\in\Omega_{m}$ implies that strategy under consideration also constitutes a cheap-talk equilibrium.¹³¹³13Deviating to an on-path message $m\in M^{*}$ obviously cannot be profitable; for any off-path message $m\neq M^{*}$, we can just set $\text{$\sigma_{R}$$\left(\cdot|m\right)$}=\sigma_{R}\left(\cdot|m^{*}\right)$ for some $m^{*}\in M^{*},$ thus ensuring that this deviation is also not profitable. Hence, commitment is not valuable.

As the intuition above suggests, Theorem 1 can easily be extended to establish a threefold equivalence. Generically, the following imply each other: (i) commitment is valuable, (ii) committed Sender values randomization, and (iii) any optimal persuasion profile induces a belief under which Receiver has multiple optimal actions (see Theorem 1^′ in the Appendix).

We postpone the discussion of the converse direction until the next section, as the intuition for it is related to the intuition for Theorem 2. Formal proofs are in the Appendix.

In this section, we consider a Sender without commitment power who engages in a cheap-talk game. We ask whether he would be strictly better off if he had commitment power. We link the value of such commitment to Sender’s behavior in Sender-preferred cheap-talk equilibria, in particular to whether Sender necessarily randomizes in such equilibria.

Theorem 2 and the if-direction of Theorem 1 both derive from the following result. Generically, if a cheap-talk equilibrium yields the persuasion payoff, then there is a partitional $\sigma$ and a (pure strategy) $\rho$ such that $(\sigma,\rho)$ is a cheap-talk equilibrium and yields the persuasion payoff. We build this result (Lemma 4 in the Appendix) in two steps.

The first step (Lemma 5 in the Appendix) shows that, generically, if $\left(\sigma,\rho\right)$ is R-BR and yields the persuasion payoff, then $\rho$ must be a pure strategy on-path. Consider toward contradiction that there is an $m$ sent with positive probability under $\sigma$, and there are two distinct actions, say $a$ and $a^{\prime}$, in the support of $\rho\left(\cdot|m\right)$. It must be that both Sender and Receiver are indifferent between $a$ and $a^{\prime}$ under belief $\mu_{m}$: Receiver has to be indifferent because $\left(\sigma,\rho\right)$ is R-BR; Sender has to be indifferent because $\left(\sigma,\rho\right)$ yields the persuasion payoff, which maximizes $U_{S}$ over all persuasion profiles.¹⁴¹⁴14If Sender strictly prefers one action over the other, say $a$ over $a^{\prime}$, at $\mu_{m}$, then Sender would obtain a higher payoff if Receiver always takes $a$ following $m$ (which would remain R-BR given Receiver’s indifference). The result then follows from establishing that such a coincidence of indifferences generically cannot arise when Sender is optimizing. For some intuition for why this is the case, consider Figure 1 which illustrates this result when there are three states. Suppose $a_{1}$ and $a_{2}$ are in the support of $\rho\left(\cdot|m\right)$. Region $R_{i}$ denotes beliefs where Receiver prefers $a_{i}$. Region $S_{i}$ denotes beliefs where Sender prefers $a_{i}$. Generically, the border between $R_{1}$ and $R_{2}$ is distinct from the border between $S_{1}$ and $S_{2}$ and thus the two borders have at most one intersection, $\mu_{m}$. Moreover, generically $\mu_{m}$ (if it exists) is an interior belief. But now, Sender could deviate to an alternate strategy that induces beliefs $\mu_{1}$ and $\mu_{2}$ instead of $\mu_{m}$, with Receiver still indifferent between $a_{1}$ and $a_{2}$ at both $\mu_{1}$ and $\mu_{2}$. Suppose that Receiver takes action $a_{i}$ following belief $\mu_{i}$. This strategy is still R-BR for Receiver and gives Sender a strictly higher payoff. Thus, we have reached a contradiction. With more than three states and more than two actions, the proof that the coincidence of indifferences generically cannot arise is conceptually similar but notationally more involved. It is presented in the Appendix as Lemma 2.

The second step (Lemma 6 in the Appendix) shows that, generically, if $\left(\sigma,\rho\right)$ is a cheap-talk equilibrium that yields the persuasion payoff, and $\rho$ is a pure strategy on-path, then there is a partitional cheap-talk equilibrium that yields the persuasion payoff. This is easy to see. Generically, for any $\omega$ and any $a\neq a^{\prime}$, we have $u_{S}\left(a,\omega\right)\neq u_{S}\left(a^{\prime},\omega\right)$. Now, consider some cheap-talk equilibrium $\left(\sigma,\rho\right)$ that yields the persuasion payoff with $\rho$ is a pure strategy on-path. If $\sigma$ is partitional, our result is immediate. Suppose to the contrary that in some $\omega$, both $m$ and $m^{\prime}$ are sent with positive probability. Then, $m$ and $m^{\prime}$ must induce the same action: if $m$ induces some $a$ and $m^{\prime}$ induces a distinct $a^{\prime}$, the fact that $u_{S}\left(a,\omega\right)\neq u_{S}\left(a^{\prime},\omega\right)$ would mean that $\sigma$ cannot be S-BR. Given that any two messages sent in $\omega$ induce the same action, we can define $\rho\left(\sigma\left(\omega\right)\right)$ as the action that Receiver takes in state $\omega$ given $\left(\sigma,\rho\right)$.

Now, we can consider an alternative, partitional profile $\left(\hat{\sigma},\hat{\rho}\right)$. Let $f$ be any injective function from $A$ to $M$. Let $\hat{\sigma}\left(\omega\right)=f\left(\rho\left(\sigma\left(\omega\right)\right)\right)$ and $\rho\left(f\left(a\right)\right)=a$. It is immediate that $\left(\hat{\sigma}_{S},\hat{\sigma}_{R}\right)$ is also a cheap-talk equilibrium and yields the persuasion payoff.

It is perhaps worth noting that 1 and 2 jointly imply the following:

Theorems 1 and 2 would not be particularly interesting if it turned out that both commitment and randomization are almost always valuable.

When $u_{S}=u_{R}$ or $u_{S}=-u_{R}$, it is easy to see that neither commitment nor randomization are valuable, but those are knife-edge cases and it is important to show that commitment has no value in a broader class of environments. We do so in this section.

To formalize our result, we fix an arbitrary prior $\mu_{0}$ and generate random environments by drawing Sender’s utility for each action-state i.i.d. from some atomless distribution $F$ and Receiver’s utility for each action-state i.i.d from some atomless distribution $G$. We further assume that for each $\left(a,\omega\right)$, the random variables $u_{S}\left(a,\omega\right)$ and $u_{R}\left(a,\omega\right)$ are independent from one another.

We should note that this structure does not preclude any particular configuration of preferences. For any $F$ and $G$, with some probability the environment will be such that Sender’s and Receiver’s preferences are perfectly aligned, with some probability they will be completely opposed, with some probability they will be aligned in some states but not others, etc.

Fixing $A$ and $\Omega$, we thus generate stochastic environments and can ask: what is the probability that commitment (or equivalently randomization) has no value. Our main theorem in this section establishes results about $\Pr\left(\text{commitment has no value}\right)$ that turn out to be independent of $F$, $G$, and $\mu_{0}$.

Denote the action space by $A=\left\{a_{1},a_{2},...,a_{\left|A\right|}\right\}$ and denote $\left|A\right|$ elements of $M$ by $m_{1}$ through $m_{\left|A\right|}$. Let $\Omega_{i}$ be the set of states where $a_{i}$ is Sender’s ideal action. The requesting messaging strategy sets $\sigma\left(\omega\right)=m_{i}$ for $\omega\in\Omega_{i}$.¹⁵¹⁵15Generically, $\Omega_{i}$ and $\Omega_{j}$ do not intersect. An obliging action strategy sets $\rho\left(m_{i}\right)=a_{i}$. A profile that consists of the requesting and an obliging strategy yields Sender’s ideal payoff.

Say that an environment is obedient if for each $\Omega_{i}$ and each $a_{j}\in A$, we have

If the environment is obedient, a profile that consists of the requesting and an obliging strategy clearly constitutes a cheap-talk equilibrium. Since such a profile yields Sender’s ideal payoff, commitment clearly has no value if the environment is obedient.¹⁶¹⁶16The obedience condition also appears in Antic, Chakraborty, and Harbaugh (2022) and Aybas and Callander (2024). In Antic, Chakraborty, and Harbaugh (2022), it is a necessary condition for the possibility of subversive conversations: without it, a third-party (Receiver) with veto power would prevent a committee (Sender) from implementing a project solely based on the information that the committee wants to do so. Aybas and Callander (2024) consider preferences of the form $u_{R}\left(a,\omega\left(\cdot\right)\right)=\omega\left(a\right)^{2}$ and $u_{S}\left(a,\omega\left(\cdot\right)\right)=\left(\omega\left(a\right)-b\right)^{2}$ for some $b>0$ where $\omega:A\rightarrow\mathbb{R}$ is the realized path of a Brownian motion. They identify features of $b$ and $A$ that make the environment obedient.

Now, for any $\Omega_{i}$ that is not empty, the probability that inequality $\left(\ref{eq:obedient}\right)$ is satisfied is $\frac{1}{\left|A\right|}$, since for each $a_{j}$, $u_{R}\left(a_{i},\omega\right)$ and $u_{R}\left(a_{j},\omega\right)$ are i.i.d. with an atomless distribution. Moreover, given two non-empty $\Omega_{i}$ and $\Omega_{j}$, the probability that inequality $\left(\ref{eq:obedient}\right)$ is satisfied for $\Omega_{j}$ is independent of the probability that it is satisfied for $\Omega_{i}$. Thus if every $\Omega_{i}$ is non-empty, the probability that the environment is obedient (i.e., inequality $\left(\ref{eq:obedient}\right)$ is satisfied for each of the $\Omega_{i}$ sets) is $\left(\frac{1}{\left|A\right|}\right)^{\left|A\right|}$, or $\frac{1}{\left|A\right|^{\left|A\right|}}$.

If an $\Omega_{i}$ is empty, inequality $\left(\ref{eq:obedient}\right)$ is satisfied vacuously for that $\Omega_{i}$. Thus, for any $\mu_{0}$ and any atomless $F$ and $G$ the overall probability that the environment is obedient must be weakly greater than $\frac{1}{\left|A\right|^{\left|A\right|}}$. Since commitment has no value in obedient environments, we conclude that $\Pr\left(\text{commitment has no value}\right)\geq$ $\frac{1}{\left|A\right|^{\left|A\right|}}.$

We establish the second part of the theorem by showing that as $\left|\Omega\right|$ grows large: (i) the likelihood that an $\Omega_{i}$ is empty converges to zero so $\Pr\left(\text{obedience}\right)$ converges to $\frac{1}{\left|A\right|^{\left|A\right|}}$, and (ii) $\Pr\left(\text{commitment has no value}\right)$ converges to $\Pr\left(\text{obedience}\right)$.

Part (i) is easy to see. For any $a\in A$, as $\Omega$ grows large, the chance that there is no state where $a$ is Sender’s ideal action vanishes.

To establish part (ii), say that an environment is jointly-inclusive if for every action $a$, there is some state $\omega$ such that $a$ is the ideal action for both Sender and Receiver in $\omega$. Analogously to part (i), it is easy to see that as $\Omega$ grows large, the probability that the environment is jointly-inclusive converges to $1$. To complete the proof, we argue that, generically, if the environment is jointly-inclusive and commitment has no value, then the environment must be obedient. First, we know from Theorem 2, that there is a partitional profile $\left(\sigma,\rho\right)$ that is a cheap-talk equilibrium and yields the persuasion payoff. Next, we note that every action $a\in A$ must be induced by $\left(\sigma,\rho\right)$: there is a state $\omega$ where $a$ is both Sender’s and Receiver’s ideal action, so if $a$ were never taken, the committed Sender could profitably deviate by sometimes¹⁷¹⁷17Sender could reveal $\omega$ with some probability $\epsilon$; Receiver’s response to all other messages would remain unchanged if $\epsilon$ is sufficiently small. revealing $\omega$ and thus inducing $a$, thus contradicting the fact that $\left(\sigma,\rho\right)$ yields the persuasion payoff. This in turn implies that, for every $\omega$, the action induced in $\omega$, $\rho\left(\sigma\left(\omega\right)\right)$, must be Sender’s ideal action in $\omega$. If Sender strictly preferred some other $a^{\prime}$ in $\omega$, $\left(\sigma,\rho\right)$ could not be S-BR as the cheap-talk Sender would profitably deviate and set $\sigma$$\left(\omega\right)$ to be whatever message induces $a^{\prime}$; since all actions are induced by $\left(\sigma,\rho\right)$, there must be such a message. Taking stock, we have established that $\left(\sigma,\rho\right)$ is a partitional profile that is R-BR (since it is a cheap-talk equilibrium) and induces Receiver to take Sender’s ideal action in every state. But this means that every message sent under $\sigma$ fully reveals what action is ideal for Sender, and Receiver obliges and takes that action. Hence, the environment is obedient.

We conclude this section with a few comments.

First, as the argument above makes clear, when the state space is large, Sender does not value commitment only if he can obtain his ideal payoff in a cheap-talk equilibrium.¹⁸¹⁸18Formally, as $\left|\Omega\right|$ goes to infinity, the probability of an environment such that Sender does not value commitment even though he does not obtain his ideal payoff converges to zero. With a smaller state space, however, cheap-talk and persuasion payoffs can coincide even if they are substantially lower than the ideal payoff.

Second, the assumption that Sender’s and Receiver’s utility are drawn from distributions that are i.i.d. across action-state pairs is more palatable if we think of $A$ and $\Omega$ as being not merely finite but also “unstructured,” without a natural metric. For example, if $A$ includes actions such as “buy one apple” and “buy two apples”, or $\Omega$ includes states such as “temperature will be 88 Fahrenheit” and “temperature will be 89 Fahrenheit,” then assuming that $u_{S}\left(a,\omega\right)$ is independent of $u_{S}\left(a^{\prime},\omega^{\prime}\right)$ as soon as $a\neq a^{\prime}$ or $\omega\neq\omega^{\prime}$ would be unreasonable.

Third, in the second part of Theorem 3, a reader might be concerned that, by keeping $F$ and $G$ fixed as $\Omega$ grows, we are “squishing” utilities together and making the difference in payoffs become vanishingly small. We could let $F$ and $G$ depend on $\left|A\right|$ and $\left|\Omega\right|$ in an arbitrary way, however, and the Theorem would still hold. We formulate the Theorem with a fixed $F$ and $G$ solely for ease of exposition.

Fourth, the obedience condition seems to have some flavor of alignment of Sender and Receiver’s preferences. While that may the case, the obedience condition does not preclude the possibility that Receiver is much worse off than she would be if Sender and Receiver’s preferences were fully aligned. For instance, consider the prosecutor-judge example and suppose that the prior is $0.7$ rather than $0.3$; then, the environment is obedient but Receiver obtains no information.

Our analysis suggests some potential directions for future research.

We establish that, generically, commitment has zero value if and only if randomization has zero value for a committed Sender. A natural question would be whether a small (or large) value of commitment implies or is implied by a small (or large) value of randomization.

Throughout, we focus on how Sender’s commitment impacts Sender’s payoff. One could also explore the impact of Sender’s commitment on Receiver’s payoff.