Communication-Enhancing Vagueness

Abstract

I experimentally investigate how vague language changes the nature of communication in a biased strategic information transmission game. Counterintuitively, when both precise and imprecise messages can be sent, in aggregate, senders are more accurate, and receivers trust them more than when only precise messages can be sent. I also develop and structurally estimate a model showing that vague messages increase communication between boundedly rational players, especially if some senders are moderately honest. Moderately honest senders avoid stating an outright lie by using vague messages to hedge them. Then, precise messages are more informative because there are fewer precise lies.

1. Introduction

Often, people like to shade the truth by being imprecise. A salesperson might say that he or she receives “few complaints” about an appliance he or she is trying to sell, even if three of the eight customers who bought that model in the past year had complained. A friend might explain his or her absence from a Sunday afternoon gathering he or she did not wish to attend by saying he or she had been “sick all weekend”, even if his or her temperature never went above 99.5 degrees Farenheit.

One characteristic of these vague utterances is that they are explicitly imprecise. This paper experimentally investigates how allowing players the option of sending explicitly imprecise messages instead of precise messages affects strategic information transmission in a standard cheap talk setting, similar to Crawford and Sobel [1].1 Counterintuitively, allowing imprecise messages improves communication—senders on average choose more informative messages, and receivers on average treat the messages as more informative.

An additional contribution of my paper is to explain why communication becomes more effective. In an experimental setting with rational, self-interested players, the kinds of messages available do not matter because in equilibrium, no messages convey any information [1]. However, bounded rationality or honesty might cause players to communicate some information and to treat imprecise messages differently than equivalent precise messages. I find that both characteristics explain subject behavior but that bounded rationality is relatively more important.

In the experiment, the subjects communicate about a state s, which is an integer between 1 and 5. The sender observes s and chooses freely from a menu of English-language messages to send about it. In the base treatment, the menu consists of every message that specifies the state precisely, such as “The state is 3”. The rich language treatment adds three additional messages that specify an interval the state is within, either from 1 to 3, 2 to 4, or 3 to 5 (e.g., “The state is 1, 2, or 3”). After learning the message, the receiver then chooses an action a. The receiver would like to choose $a=s$, while the sender would like him or her to choose $a=s+2$, so they play a partial common interest game [4].

With either message space, with rational payoff-maximizing players, no Bayesian Nash equilibrium equilibrium exists in which m conveys any information about s, and as a result, a choices are independent of m. However, subjects in both treatments send messages that convey more information than optimal and are more trusting of messages than they should be.

This “overcommunication” finding is typical (e.g., Cai and Wang [5] or Duffy et al. [6]; see Blume et al. [7] for a general overview). The prior experimental literature has alternately explained excess communication through preferences for truth-telling [2,8,9,10,11,12,13,14] or bounded rationality [5,15,16,17].2

Comparing their relative importance can be difficult, because from a money-maximizing perspective, honest behavior will seem unsophisticated. To compare them, I develop a model which teases out distinct implications of honesty and bounded rationality across my treatments. My model adapts the one-parameter Poisson cognitive hierarchy model in Camerer et al. [18] to cheap talk and combines it with some players receiving a non-pecuniary benefit from truth-telling.

In structural non-equilibrium models such as cognitive hierarchy or level-k models, players have a (typically incorrect) belief about how other players are acting and optimize given their beliefs.3 Players vary in the sophistication of their beliefs. Level 0 ($L0$) players act non-strategically—in this context, by either being truthful or by being completely credulous—while $Lk$ players are defined inductively and believe all players are of lower levels. Due to the upward bias, for instance, $L1$ senders exaggerate s by choosing $m=s+2$. If the $L2$ receivers think a few senders are $L0$ but most are $L1$, they discount messages $m\ge 3$ by choosing $a=m-2$, but they think $m<3$ are truthful and do not discount them. Following Camerer et al. [18], I assume that the players’ levels have the Poisson distribution. Many qualitative aspects of my subjects’ behavior match this sort of reasoning.

Players vary in their honesty types in addition to their cognitive types. A player’s honesty type indicates how strong a preference for truth-telling he or she has and can be thought of as reduced-form behavior combining a variety of factors. Some subjects behave as if they dislike sending literal false messages (i.e., “lies”, following Sobel [24]’s definition). This form of honesty is the most straightforward explanation for why senders’ message choices would be qualitatively different in the two treatments.

I am agnostic about the underlying reason my subjects behave in this way, though. Recent experiments show that in non-strategic settings, subjects seem to be motivated by honesty preferences that combine a direct cost of lying and a cost for being identified as a liar [13,14]. The dislike of sending literal false messages embodied in my model’s honesty types could stem from either of these explanations.

Players who care little about being honest, players who care moderately about being honest, and players who care strongly about being honest all should exhibit distinct behaviors in my experiment. The rich message treatment enables players to truthfully exaggerate, such as by sending the message “The state is 1, 2, or 3” when the state is really 1. All players believe that precise lies are more effective deceptions than truthful exaggerations, especially when the state is high. Like weakly honest players, moderately honest players in the base treatment should not send honest messages because the monetary loss from forgoing precise lies is too high. However, moderately honest players should engage extensively in truthful exaggeration in rich treatment. Finally, strongly honest players should be honest even when truthful exaggerations are not available in the base treatment.

Subject behavior across the treatments was more consistent with moderate honesty than strong honesty. Overall, 45% of the messages sent in the rich treatment were imprecise, and 57% of the imprecise messages were truthful. On the other hand, in the base treatment $s<5$, only 15% of the messages were truthful.

I then estimate a simple structural model using maximum likelihood with two parameters of interest: $\lambda$, the average cognitive level of the subjects, and $\tau$, the fraction (and subjects’ beliefs about the fraction) of moderately honest players. Most of the subjects in my experiment employed one or two levels of exaggeration or discounting (i.e., rounds of best-response reasoning). About a quarter of the subjects showed moderate preferences for honesty.

Despite the presence of honest players, when these estimates are used to calibrate my model, they imply that honesty plays a limited role in explaining why communication increases in rich language treatment. Just as important as honesty are two other factors. First, the model implies that precise messages are more informative in rich message treatment independent of honesty. In that treatment, senders of all honesty types often choose imprecise messages instead of precise ones if they observe a low state. Removing the most misleading lies from the pool of precise messages improves the accuracy of the precise messages. Second, I estimate that players are somewhat less sophisticated in their reasoning in the rich language treatment. The reduction in the average levels of reasoning seems likely to be driven by the increased complexity of strategic calculations in the rich language treatment. In the calibrated model, this reduction in sophistication manifests largely through increased receiver trust in the messages.

Several groups of experimental papers are closely related to this paper. My experimental setup in the base treatment mirrors Cai and Wang [5] and Wang, Spezio, and Camerer [15]. They explained the overcommunication in their experiments using level-k but did not allow preferences for truth-telling in their models or investigate the impact of vagueness.4, those choices caused the computer to randomize between states in an interval. The receivers always received a point message and were not informed that it was randomly chosen.

A second pair of experimental papers investigated how senders use vague or evasive language in simpler two-state strategic information transmission settings [25,26]. These found that deception through evasive messages such as feigning ignorance is less psychologically costly than deception through direct lies.

A third set of papers investigated information unraveling in voluntary disclosure games, in which the subjects were only able to send truthful messages. Benndorf et al. [27] elicited a distribution of level-k types in a disclosure game, finding substantial numbers of $k=1$, $k=2$, and $k=3$ players. Jin et al. [28] and Deversi et al. [29] both used state spaces and payoff functions very similar to those in my experiment. Jin et al. [28] found that senders disclose favorable information but withhold less favorable information, and receivers are insufficiently skeptical when information is not disclosed. Deversi et al. [29] examined what occurs when senders can choose imprecise disclosures. They found that vagueness is profitably exploited by senders to take advantage of naive receivers, and information transmission is higher in their precise-only treatment. Hence, interestingly, my result that vagueness improves communication is reversed for disclosure games (see also Li and Schipper [30] and Hagenbach and Perez-Richet [31]). Unusually, Li and Schipper [30] found that the subjects exhibited relatively high levels of sophistication in a disclosure game with imprecise messages.

Many other papers investigate the role of vague language in other forms of cheap talk games, including asymmetric coordination games with private information [32], leader-follower public goods games [33], delegation games [34], cheating games [35], three-player common interest context-dependent communication games [36], and real-world communication about intentions on a TV game show [37]. Like this paper, these papers found that subjects, given the choice, prefer imprecise messages to outright lies, and followers are too trusting of these messages. In some, vagueness is beneficial either because it can mask incentives that, if known, would make an attractive equilibrium strategically unsustainable [32,33], because it makes information more credible [34], or because it enables more efficient communication of context-dependent information [36].

Relative to these sets of papers, my contribution is to show that vagueness can have positive effects on communication in strategic information transmission contexts and offer an explanation for why it has positive effects.5

Section 2 describes my experiment, and Section 3 provides descriptions and reduced-form analyses of subject behavior, focused on motivating my model. Section 4 presents my model and its predictions, after which Section 5 details my structural estimates. Finally, Section 6 concludes the paper.

2. Experimental Design

My experimental design builds on those of Cai and Wang [5] and Wang, Spezio, and Camerer [15], which are discrete versions of the Crawford and Sobel [1] cheap talk game. In each period, subjects were paired up, with one member of each pair being the sender and one being the receiver. A state of the world $s\in S\equiv \{1,2,\dots ,5\}$ was then randomly determined for each pair. Each state was equally likely. The sender then learned s and chose $m\in M$ to send to the receiver. The message space M—the set of possible messages—varied across treatments. Finally, the receiver learned m and chose $a\in A=S$.

The sender and receiver payoffs (in experimental units) were

$$\begin{array}{cc}\hfill u_s(s,a)& =100-25\times {(s+2-a)}^{1.2}\hfill \\ \hfill u_r(s,a)& =100-25\times {(s-a)}^{1.2}\hfill \end{array}$$

The subjects were presented with payoff tables similar in format to

Table 1. Payoff tables.

	Sender Payoffs					Receiver Payoffs
Action Choice	State					State
	$\mathit{s}=\mathbf{1}$	$\mathit{s}=\mathbf{2}$	$\mathit{s}=\mathbf{3}$	$\mathit{s}=\mathbf{4}$	$\mathit{s}=\mathbf{5}$	$\mathit{s}=\mathit{1}$	$\mathit{s}=\mathbf{2}$	$\mathit{s}=\mathbf{3}$	$\mathit{s}=\mathbf{4}$	$\mathit{s}=\mathbf{5}$
$a=1$	43	7	−32	−72	−115	100	75	43	7	−32
$a=2$	75	43	7	−32	−72	75	100	75	43	7
$a=3$	100	75	43	7	−32	43	75	100	75	43
$a=4$	75	100	75	43	7	7	43	75	100	75
$a=5$	43	75	100	75	43	−32	7	43	75	100

below, although with neutral labels instead of “sender”, “receiver”, “action”, and “state”.

There were three treatments. In the base treatment, the sender could send only precise messages of the form “s is x” (i.e., the message space was $M_B=\{1,2,3,4,5\}$). In the rich language treatment, the sender could choose to send either precise messages or imprecise messages. Imprecise messages specified an interval of three numbers that s could be within, such as “s is 2, 3, or 4”, so the message space was $M_R=\{1,2,3,4,5,\{1,2,3\},\{2,3,4\},\{3,4,5\}\}$.

Finally in a third treatment (the noise treatment), the sender observed s imperfectly and could send precise messages or imprecise messages ($M_N=M_R$). Instead of observing s, the sender observed $\sigma \in \Sigma =\left\{\right\{1\},\{2\},\{3\},\{4\},\{5\},\{1,2,3\},\{2,3,4\},\{3,4,5\left\}\right\},$ where $\sigma$ was equally likely to be any element of $\Sigma$ and s was equally likely to be any element of $\sigma$. In practice, choices in the noise and rich treatments were indistinguishable, so I added the noise data to the rich data, setting $s=E[s\mid \sigma ]$ for noise observations.6

Three principles guided the construction of $M_R$. First, having only a few additional messages made the choices easier for the subjects and simpler to analyze subsequently. Second, contiguous messages with intermediate precision of meaning suggest a natural interpretation of low, medium, and high. Third, the precise messages were uniformly distributed over the state space, and I chose to maintain that property for the imprecise messages.

Instructions for each treatment were handed out and read aloud at the start of that treatment. (Appendix B contains the experimental instructions.) The experiment was conducted using z-Tree software [42] at Clemson University in the fall of 2011. There were 42 subjects, 18 (42%) of whom were female. 24 (57%) subjects were first- or second-year college students, while 18 (43%) were third-year, fourth-year, or graduate students. The subjects were mostly College of Business and Behavioral Sciences undergraduates: 32 (76%) had majors in the business school, 4 (10%) had humanities majors, and 6 (14%) had science or engineering majors.

The subjects alternated roles and were randomly rematched each period in order to avoid repeated game effects. All subjects participated in all treatments, and the order of the treatments was randomized for each session.7 In total, there were 268 rounds of the base treatment, 204 rounds of the vague treatment, and 220 rounds of the noise treatment (there were 4–6 rounds for each treatment). Note that a round yields one message choice observation and one action choice observation. Each experimental unit was worth USD 0.01. The average earnings were around USD 24, which included a USD 6 payment for showing up.

3. Results

Table 2. Informational content and distribution of messages across treatments.

Precise Messages	State from which Sent ($E\left[\mathit{s}\right|\mathit{m}\left]\right)$			Frequency
	Base Treat	Rich Treat	$\mathit{p}$	% Base	% Rich	p
$m=1$	1.81	2.50	0.278 a	0.06	0.02 **	0.022 b
$m=2$	2.33	2.50	0.618 a	0.11	0.03 ***	<0.001 b
$m=3$	2.43	2.29	0.771 a	0.14	0.08 **	0.020 b
$m=4$	2.53	3.36 **	0.009 a	0.18	0.07 ***	<0.001 b
$m=5$	3.28	3.58 *	0.052 a	0.51	0.34 ***	<0.001 b
Imprecise Messages	State from which Sent ($\mathbf{E}\left[\mathit{s}\right|\mathit{m}]$)			Frequency
		Rich Treat	p		% Rich	p
$m=\{1,2,3\}$		2.34	0.823 c		0.07	0.703 d
$m=\{2,3,4\}$		2.45	0.794 c		0.10	0.170 d
$m=\{3,4,5\}$		3.16 ***	0.002 c		0.29 ***	<0.001 d

^a and ^c: p values from two-sided Mann–Whitney U test comparing distribution of s from which m originated between treatments for precise messages ^a, or for all messages of same expected value ^b. ^b and ^d: p values from Fisher’s exact tests comparing frequency of precise m ^b or all messages of same expected value ^d. *: Significant at 0.1 level. **: Significant at 0.05 level. ***: Significant at 0.01 level.

shows the state from which each message was sent, on average, along with the frequency of each message. Over half of messages in the base treatment were $m=5$, which led to that message not being especially informative, with $E\left[s\right|m=5]=3.28$. Generally, the lower the message, the less likely it was to be chosen, and the stronger the signal it provided that s was low. In the base treatment, $m=1$ and $m=5$ conveyed different information than the other messages: the distribution of s, conditioned on $m=1$, was significantly differerent at the 10% level from that for $m=2$ (two-tailed Mann–Whitney U test $p=0.0643$), and the distribution for $m=5$ was significantly different from $m=4$ at the 1% level (MW $p<0.001$).

The rich treatment produces three related effects. First, higher messages become more accurate. For messages above $m=3$, a precise message is a stronger signal that s is higher in the rich treatment than in the base treatment, while $m=4$ is 57% closer to its literal meaning (rich language $E\left[s\right|m=4]=3.36$ versus base $E\left[s\right|m=4]=2.53$, MW $p=0.009$), and $m=5$ is 17% closer (rich language $E\left[s\right|m=5]=3.58$ versus base $E\left[s\right|m=5]=3.28$, MW $p=0.052$). $m=\{3,4,5\}$ is also 43% closer to an expected $s=4$ than is the base treament $m=4$ (rich language $E\left[s\right|m=\{3,4,5\}]=3.16$ versus base $E\left[s\right|m=4]=2.53$, MW $p=0.003$).

Second, almost half of the messages (45%) sent become imprecise, and imprecise messages are twice as likely as precise $m<5$. Imprecise messages are less likely to be sent from $s=5$ but otherwise are employed about half the time in every state.8 Such a high fraction of imprecise messages is inconsistent with many explanations of my subjects’ behavior. For example, the analysis in Section 4 found that most variants of the level-k models without honesty could predict that, at most, 20% of the messages would be imprecise, and those would predominantly be sent from $s=1$ and $s=2$.

Third, the messages overall become less exaggerated. While 51% of the base treatment messages were $m=5$, only 34% of the rich treatment messages were the highest possible. The difference was statistically significant (Fisher’s exact test $p<0.001$). The fraction of messages with an expected value of 4 (i.e., $m=4$ or $m=\{3,4,5\}$) doubled from 18% to 36% in the rich treatment (Fisher’s $p<0.001$). None of the other message frequencies were significantly different between the treatments when comparing all messages of the same expected value, except for $m=1$, which went from 6% to 2% of the messages (Fisher’s $p=0.015$).

The receivers correctly treated the messages as informative, although many systematically misinterpreted them.

Table 3. Responses to messages.

Precise Messages	Action Choice ($\mathit{E}\left[\mathit{a}\right|\mathit{m}]$)			Optimal a
	Base Treat	Rich Treat	p	Base	Rich
$m=1$	1.94	1.60	0.174 a	1	2
$m=2$	2.30	2.57	0.353 a	2	2
$m=3$	2.27	2.35	0.587 a	2	2
$m=4$	2.91	3.03	0.655 a	2	3
$m=5$	3.47	3.79 ***	0.004 a	3	4
Imprecise Messages	Action Choice ($\mathit{E}\left[\mathit{a}\right|\mathit{m}]$)			Optimal a
		Rich Treat	p		Rich
$m=\{1,2,3\}$		2.07	0.440 b		2
$m=\{2,3,4\}$		2.71 **	0.043 b		2
$m=\{3,4,5\}$		3.10	0.159 b		3

^a and ^b: p values from two-sided Mann–Whitney U test comparing distribution of a for precise messages across treatments ^a or for base treatment messages to imprecise messages of the same expected value ^b. Optimal a values are a(m) that would have maximized expected payoff in the treatment. **: Significant at 0.05 level. ***: Significant at 0.01 level.

shows both the average and optimal action choice for each message. The optimal action choice is the highest-payoff action choice given the true distribution of messages. On average, the chosen a values were too high for both treatments but showed the “correct” comparative static of lower actions when receiving a lower message.

When comparing the responses in the rich treatment to the base treatment, the typical actions did not change much. With the exception of the responses to $m=5$, for which $E\left[a\right|m=5]$ increased from 3.47 to 3.79 (MW $p=0.004$), the hypothesis that the receivers’ responses to precise messages were the same across treatments could not be rejected. In addition, the responses to $m=\{2,3,4\}$ were systematically higher than the responses to the precise treatment $m=3$ (MW $p=0.043$). These patterns were consistent with the level-k behavior, as Section 4 explains. Figure 1 shows the full distribution of relative frequencies $Pr(s=x|m,\mathrm{treatment})$ for each message in the precise and rich treatments. For $m=1$ and $m=2$, the most likely state from which the message was sent was the corresponding state $s=1$ or $s=2$, while for $m>2$, with the exception of precise $m=4$ in the rich treatment, the true state was never the most likely. Consistent with the initial level-k reasoning that sends $m=s+2$, messages $m=3$ or $m=\{2,3,4\}$ were most likely to have been sent from $s=1$ in both treatments. In the base treatment, exaggerating by two units is common with $m=4$ and $m=5$ as well, but in the rich treatment, two-unit exaggerations were less typical.

Next, Figure 2 shows the frequency of each action for every message sent ($Pr(a\mid m)$) in the base and rich treatments. Consistent with the level-k predictions, there was widespread discounting of messages $m\ge 3$ by two units, such as choosing $a=1$ in response to $m=3$. The discounting was less pronounced for $m<3$, imprecise messages, and $m=5$ in the rich language treatment.

Figure 1 and Figure 2 also show that the m and a choices were quite noisy in both treatments, with the exception of $m=1$ in the base treatment, $Pr(s\mid m)<1/2$ for any s, and generally $Pr(s\mid m)>1/10$ for all s. This noisiness means that usually, the receivers would maximize utility by adjusting their action choices inward toward $a=3$ (as in Table 3).

While the number of subjects was too low to perform a detailed analysis of how different demographic characteristics affected communication choices, I did find a few differences in how the subjects behaved based on gender or college class year. The female subjects were less trusting of messages in the precise treatment than the male subjects, with a mean distance between the message they received and the action chosen of 1.39 versus 1.18 for the male subjects (MW $p=0.048$). Gender differences were not statistically significant at the 10% level in the vague treatment or for the messages sent. I also found that the first- and second-year college students sent much more accurate messages in the rich language treatment than the third-year, fourth-year, and graduate students. The average distance between the state and the message sent was 1.08 compared with 1.39 (MW $p=0.007$). Again, however, there were no other statistically significant differences in behavior either in the precise treatment or with the action choices.

Finally, there was no evidence that the players’ strategies changed during the sessions on aggregate. In particular, the fraction of imprecise messages was roughly constant in the early and late rounds, as was the share of truthful messages and the degree of discounting by receivers, ruling out sizeable learning effects.

4. Model

This section derives predictions about the subjects’ behavior using a cognitive hierarchy model, to which I add preferences for honesty. Cognitive hierarchy and level-k models are structural nonequilibrium models of how players reason strategically. There is a hierarchy of discrete cognitive types, each with gradually more sophisticated beliefs about other players’ behavior. Because higher types use more iterations of best-response reasoning, senders of higher types exaggerate more and more due to their upward bias, while receivers of higher types discount the messages more and more. My model extends the applications of level-k from Crawford [21] and Cai and Wang [5] to cheap talk by allowing heterogenous preferences for truth-telling on the part of the senders. A secondary difference is that I assume that $Lk$ players best respond to a combination of behaviors of types lower than k.9

The model has two purposes. First, it illustrates the broad differences in behavior one should expect between my treatments if the subjects are boundedly rational and some are honest. Second, the next section estimates the model structurally, taking into account that it is not likely that any subject perfectly complies with the predicted behavior of any type.

4.1. Definitions and Assumptions

Formally, players differ in their cognitive type, honesty level, and sender or receiver role. A player’s cognitive type determines his or her beliefs about how players in the other role behave. A level-k sender ($L{k}^S$) has beliefs $B_k^S(a,m)$ about the conditional probabilities $Pr(a\mid m)$ induced by receiver behavior, and a level-k receiver ($L{k}^r$) has beliefs $B_k^r(m,s)$ about the conditional probabilities $Pr(m\mid s)$ that he or she faces.

A player’s honesty type affects his or her relative utilities from sending dishonest or honest messages. All messages that are non-false according to their literal meaning are honest (i.e., m is honest if either $m=s$ or $s\in m$) and are dishonest otherwise. I discuss this assumption further in Section 4.3. An honesty type h acts as if sending a literally true message is worth giving up h cents:

$$u_s(s,a,h)=\left\{\begin{array}{cc}100-25\ast {(s+2-a)}^{1.2}+h\hfill & \mathrm{if}s=m\mathrm{or}s\in m\hfill \\ 100-25\ast {(s+2-a)}^{1.2}\hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

A receiver’s utility function is not affected by his or her honesty type. I assume that there is a finite set of honesty types, with every $h\in H$. The vector $\tau$ denotes the distribution of honest types, and I write ${\tau }_h$ to indicate the fraction of players with honesty type h. H and $\tau$ are common knowledge.

Level 0 cognitive types are non-strategic and primarily serve as conjectures about behavior by $L1$ players. $L{0}^S$ messages are truthful, and $L{0}^R$ messages are credulous.10 Strategic $Lk$ behavior is then defined inductively as responding optimally to beliefs about the other role’s behavior based on $L0,\dots ,L(k-1)$ behavior.11 A level-$k\ge 1$ sender chooses for each state a (possibly) mixed strategy over messages ${\sigma }_{s,k}(m\mid s)$ as a best response to that type’s beliefs $B_k^S(a,m)$ about receiver behavior. Likewise, a level-$k\ge 1$ receiver chooses ${\sigma }_{a,k}(a\mid m)$, which is a best response to his or her beliefs $B_k^r$ about the sender’s behavior.

Definition 1.

A type $L{0}^S$ player sends every available honest message with equal probability. A type $L{0}^R$ player chooses action $a=m$ if m is precise and $a=x$ if $m=\{x-1,x,x+1\})$.

In the Poisson cognitive hierarchy model of Camerer, Ho, and Chong [18], the frequency of level-k behavior $f\left(k\right)$ follows the Poisson distribution, and the level-k beliefs about the frequencies of type $l<k$ players ${\widehat{f}}_k\left(l\right)$ are formed by normalizing $f\left(0\right),\dots ,f(k-1)$ to sum up to one:

$$f\left(k\right)=e^{-\lambda }{\lambda }^k/k!,\mathrm{and}{\widehat{f}}_k\left(l\right)=\frac{f\left(l\right)}{\sum _{i=0}^{k-1}f\left(i\right)}.$$

By definition, $\lambda$ is the mean cognitive level in the population as well as the variance of the cognitive levels in the population.

Definition 2.

For $k\ge 1$, a type $L{k}^S$ player with honesty h has beliefs $B_k^S$ and chooses ${\sigma }_{s,k}$ that follow

$$\begin{array}{cc}\hfill B_k^S(a,m)& =\sum _{i=0}^{k-1}{\widehat{f}}_k\left(i\right){\sigma }_{r,i}(a\mid m)\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\sigma }_{s,k}(m\mid s,h)>0& \iff m\in \underset{m}{arg\; max}E[u_s(s,a,h)\mid B_k^S,m]\hfill \end{array}$$

(2)

and a type $L{k}^R$ player has beliefs $B_k^R$ and chooses ${\sigma }_{r,k}$ that follow

$$\begin{array}{cc}\hfill B_k^R(m,s)& =\sum _{i=0}^{k-1}{\widehat{f}}_k\left(i\right)\left(\sum _{h\in H}{\tau }_h{\sigma }_{s,k-1}(m\mid s,h)\right)\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\sigma }_{r,k}(a\mid m)>0& \iff a\in \underset{a}{arg\; max}E[u_r(s,a)\mid m,B_k^R]\hfill \end{array}$$

(4)

where $E[u_r(s,a)\mid B_k^R]$ is calculated via Bayes’ rule.

To operationalize Equation (2), I assume that for cases in which the senders are indifferent between multiple m, they randomize uniformly over their best messages.

It is helpful to think of honesty as falling into three levels: “weak” ($h\in [0,25)$), “moderate” ($h\in (25,57)$), and “strong” ($h>57$). These categories are based on the opportunity cost of being honest in different situations (i.e, the loss in pecuniary payoff for sending an $s=m$ or $s\in m$ message versus the best non-honest message). In the base treatment, $L{1}^S$ and $L{2}^S$ believe that they can achieve $a=s+2$ with $m=s+2$ or remain honest and achieve $a=s$ at a loss of $25{(s+2-s)}^{1.2}=25\ast 2^{1.2}\approx 57$.12 On the other hand, in the rich treatment, low-level senders perceive the opportunity cost of honesty to be either 25 or 32 for sending an imprecise message which is a truthful exaggeration. For instance, for $L{1}^S$, when $s=1$, $m=\{1,2,3\}$ has an opportunity cost of $25{(s+2-(s+1))}^{1.2}=25$.

In presenting the model’s predictions but not estimating the model, I restrict attention to a set of parameters that encompasses most reasonable distributions of the types and for which the strategies of each type of player are largely constant. The restriction also limits the number of special cases that need to be described. Let the constant $c\equiv 2^{1.2}-1\approx 1.3$. c is proportional to the difference in payoff cost to the sender or receiver having an action choice 2 away from the player’s bliss point instead of 1 (i.e., a loss of $25\ast 2^{1.2}$ instead of a loss of 25). Let ${\tau }_{32+}=\sum _{h\in H,h\ge 32}{\tau }_h$ be the share of players with honesty $h\ge 25c\approx 32$.13

Definition 3.

A standard type distribution of players has frequencies of cognitive and honesty types that are distributed with

$$\lambda \in [1.55,5]\mathit{and}{\tau }_{32+}\in \left[\frac{100+16\lambda }{1000},\frac{\lambda -c}{\lambda c}\right].$$

This $\lambda$ range is consistent with the estimates from other papers and with my preferred estimates. The restriction on $\tau$ asserts that roughly between 15% and 50% of my subjects are at least moderately honest which, again, is consistent with my estimates.

4.2. Predictions

Particular patterns of behavior for a role are usually common to pairs of cognitive types. $L{1}^R$ best responds to $L{0}^S$, implying ${\sigma }_{r,1}={\sigma }_{r,0}$. That in turn implies $L{1}^S$ and $L{2}^S$ both effectively use one round of best responses, since both types believe all receivers are credulous. If $\lambda$ is high enough that the $L2$ players believe the $L0$ players are uncommon, the $L2$ and $L3$ receivers act similarly as well. $L{2}^R$’s best respond to a fraction $f\left(0\right)/\left(f\right(0)+f(1\left)\right)=1/(\lambda +1)$ of truthful messages and a fraction $\lambda /(\lambda +1)$ of single-level exaggerated messages, which are quite close to $L{3}^R$’s beliefs for reasonable $\lambda$ choices. Likewise, the $L3$ and $L4$ senders typically engage in two rounds of non-trival best-response reasoning, and so on.

Propositions 1–4, summarized in

Table 4. Base treatment message and action predictions (Proposition 1).

State	$\mathit{h}<32{\mathit{Sender}}^{\prime }\mathit{s}\mathit{Message}$			Message	${\mathit{Receiver}}^{\prime }\mathit{s}\mathit{Action}$
	$\mathit{L}{\mathbf{0}}^{\mathit{S}}$	$\mathit{L}{\mathbf{1}}^{\mathit{S}}/\mathit{L}{\mathbf{2}}^{\mathit{S}}$	$\mathit{L}{\mathbf{3}}^{\mathit{S}}+$		$\mathit{L}{\mathbf{0}}^{\mathit{R}}/\mathit{L}{\mathbf{1}}^{\mathit{R}}$	$\mathit{L}{\mathbf{2}}^{\mathit{R}}/\mathit{L}{\mathbf{3}}^{\mathit{R}}$	$\mathit{L}{\mathbf{4}}^{\mathit{R}}+$
$s=1$	1	3	2 ${}^{†}$	$m=1$	1	1	1
$s=2$	2	4	5 ${}^{††}$	$m=2$	2	2	1 ${}^{\star }$
$s=3$	3	5	5	$m=3$	3	1	1
$s=4$	4	5	5	$m=4$	4	2	2
$s=5$	5	5	5	$m=5$	5	4	4

^†: For L5^S and higher senders, there is a $\overline{\lambda }$ cutoff such that for λ > $\overline{\lambda }$, m(1) = 5. ^††: If λ < 1.46, then for L3^S, m(2) = 4. ^⋆: If λ < $\sqrt[3]{6}$, then for L4^R, a(2) = 2.

,

Table 5. Rich treatment message predictions (Propositions 2 and 3).

State	$\mathit{h}<25{\mathit{Sender}}^{\prime }\mathit{s}\mathit{Action}$			$\mathit{h}>32{\mathit{Sender}}^{\prime }\mathit{s}\mathit{Action}$
	$\mathit{L}{\mathbf{0}}^{\mathit{S}}$	$\mathit{L}{\mathbf{1}}^{\mathit{S}}/\mathit{L}{\mathbf{2}}^{\mathit{S}}$	$\mathit{L}{\mathbf{3}}^{\mathit{S}}+$	$\mathit{L}{\mathbf{0}}^{\mathit{S}}$	$\mathit{L}{\mathbf{1}}^{\mathit{S}}/\mathit{L}{\mathbf{2}}^{\mathit{S}}$	$\mathit{L}{\mathbf{3}}^{\mathit{S}}+$
$s=1$	1, $\{1,2,3\}$	3, $\{2,3,4\}$	$\{3,4,5\}{}^{†}$	1, $\{1,2,3\}$	$\{1,2,3\}$	$\{1,2,3\}{}^{††}$
$s=2$	2, $\{1,2,3\}$, $\{2,3,4\}$	4, $\{3,4,5\}$	5 ${}^{†}$	2, $\{1,2,3\}$, $\{2,3,4\}$	$\{2,3,4\}$	$\{2,3,4\}{}^{††}$
$s=3$	3, $\{1,2,3\}$, $\{2,3,4\}$, $\{3,4,5\}$	5	5	3, $\{1,2,3\}$, $\{2,3,4\}$, $\{3,4,5\}$	$\{3,4,5\}$	$\{3,4,5\}$
$s=4$	4, $\{2,3,4\}$, $\{3,4,5\}$	5	5	4, $\{2,3,4\}$, $\{3,4,5\}$	4 $\{3,4,5\}$	$\{3,4,5\}{}^{\star }$
$s=5$	5, $\{3,4,5\}$	5	5	5, $\{3,4,5\}$	5	5

^†: If λ < 2.73 (respective of λ < 1.88), then L3^S (L3^S + L4^S) send m(1) = {2,3,4} and m(2) = {3,4,5}. ^††: If h is beneath an λ-dependent cutoff, L4^S and higher send m(1) = {3,4,5} and m(2) = 5. ^⋆: If h < 36, then if h is beneath a λ-dependent cutoff, L4^S and higher send m(4) = 5.

and

Table 6. Rich treatment action predictions (Proposition 4).

Precise Message	${\mathit{Receiver}}^{\prime }\mathit{s}\mathit{Action}$			Imprecise Message	${\mathit{Receiver}}^{\prime }\mathit{s}\mathit{Action}$
	$\mathit{L}{\mathbf{0}}^{\mathit{R}}/\mathit{L}{\mathbf{1}}^{\mathit{R}}$	$\mathit{L}{\mathbf{2}}^{\mathit{R}}/\mathit{L}{\mathbf{3}}^{\mathit{R}}$	$\mathit{L}{\mathbf{4}}^{\mathit{R}}+$		$\mathit{L}{\mathbf{0}}^{\mathit{R}}/\mathit{L}{\mathbf{1}}^{\mathit{R}}$	$\mathit{L}{\mathbf{2}}^{\mathit{R}}/\mathit{L}{\mathbf{3}}^{\mathit{R}}$	$\mathit{L}{\mathbf{4}}^{\mathit{R}}+$
$m=1$	1	1	1	$m=\{1,2,3\}$	2	1	1
$m=2$	2	2	2	$m=\{2,3,4\}$	3	2 ${}^{\circ }$	2 ${}^{\circ }$
$m=3$	3	1	1	$m=\{3,4,5\}$	4	3	3 ${}^{▴}$
$m=4$	4	2	2
$m=5$	5	4	4

^∘: If ${\tau }_{32+}$ < 1/3, then if λ is greater than a ${\tau }_{32+}$-dependent cutoff, for L3^R and higher, a({2,3,4}) = 1. ^▴: If λ is greater than a ${\tau }_{32+}$-dependent cutoff, for L5^R and higher, a({3, 4, 5}) = 2.

, characterize the behavior of senders and receivers drawn from a standard type distribution in my experiment. See Appendix A for proofs.

Proposition 1.

If the players have a standard type distribution, and no players are strongly honest, then the following is true for weakly honest senders and all receivers in the base treatment:

(i) 

$L{1}^S$ and $L{2}^S$ exaggerate one level if possible ($m=min\{s+2,5\}$);

(ii) 

$L{2}^R$ and $L{3}^R$ discount $m=3$ and $m=4$ by two units ($a=m-2$), $m=5$ by one unit ($a\left(5\right)=4$), and $m=1$ or $m=2$ by zero units ($a=m$);

(iii) 

$L{3}^S$ and higher-level senders send the highest message $m=5$ except possibly when $s=1$ or $s=2$ (in which case $m\left(1\right)=2$ and $m\left(2\right)=4$); and

(iv) 

$L{4}^R$ and higher-level receivers discount $m=5$ one unit ($a\left(5\right)=4$), $m=2$ by zero or one unit ($a\left(2\right)=1$ or $a\left(2\right)=2$), and all other $m<5$ by two units ($m=max\{m-2,1\}$.

Moderately honest senders make identical choices to weakly honest senders, except that $L{1}^S$ and $L{2}^S$ choose $m\left(4\right)=4$.

Table 4 summarizes the types’ predicted behaviors in the base treatment. Proposition 1 predicts that $m<3$ messages should be relatively rare, relatively truthful, and the receivers should trust them more. The truthfulness of the low messages does not stem from honesty. For $m\ge 3$, many messages are exaggerations by two units, and the receivers discount them, often by two units. Finally, most of the messages are $m=5$.

In the base treatment, for players with $h\ge 57$, $L{1}^S$ and $L{2}^S$ would send $m=s$. Then, if ${\tau }_{57+}$ was low, $L{2}^R$ and higher would discount it by the same amount as in Proposition 1 or by a smaller amount if ${\tau }_{57+}$ was higher. That discounting makes honesty more costly to $L{3}^S$ and higher, so unless there is a significant share of players with h significantly higher than 57, the behavior for the higher types becomes similar to that in Proposition 1. Thus, strong honesty would cause a sizeable share of honest messages and trusting action choices for $m\ge 3$ as well as for for $m<3$.

In the rich treatment, $L{0}^S$ randomize between truthful precise and truthful imprecise messages. This leads to $L{1}^S$ and $L{2}^S$ with $h>32$ choosing imprecise messages $m\left(s\right)=\{s,s+1,s+2\}$, which exaggerate the state in a truthful way. In addition, weakly honest $L{1}^S$ and $L{2}^S$ send imprecise m deceptively ($m\left(s\right)=\{s+1,s+2,s+3\}$) half the time in states $s=1$ and $s=2$.

Both honest and non-honest level-$k\ge 3$ senders employ imprecise messages, but exactly how they do so on depends whether they believe imprecise messages will be treated skeptically or credulously. Weakly honest level-$k\ge 3$ senders often choose to masquerade as honest or naive types by sending imprecise messages but only when the state is low ($s<3$). For example, for these senders, the message $\{3,4,5\}$ is an attractive deception in $s=1$ because they believe there is a good chance that $a\left(\right\{3,4,5\left\}\right)=3$, regardless of exactly how low-level senders use imprecise m.

Whether there is truthful exaggeration by moderately honest level-$k\ge 3$ senders depends on what they believe receivers believe. If relatively few of the low-level imprecise messages are deceptive—which occurs either if $\lambda$ is low enough or ${\tau }_{32+}$ is high enough—then the $L2$ and $L3$ receivers are fairly trusting of imprecise messages and choose $a\left(\right\{x,x+1,x+2\left\}\right)=x$ when they usually choose $a(x+1)=x-1$. Because of these a choices, moderately honest level-$k\ge 3$ senders also send imprecise messages in states $s<5$. In contrast, if a critical mass of non-deceptive messages is not available, low-level receivers are less trusting of imprecise messages, and these senders make use of imprecise messages in the same way as weakly honest senders because honesty is too costly.

Therefore, if imprecise messages are sent frequently from $s=3$ and $s=4$, and $m=\{2,3,4\}$ and $m=\{3,4,5\}$ are trusted more than the corresponding precise messages, it cannot be the case that $\lambda$ is high and ${\tau }_{32+}$ is low. In order to generate the pattern of imprecise messages observed in the data, there must be higher-honesty senders sending imprecise messages from $s=3$ and $s=4$ as well.

Table 5 and Table 6 organize the predictions of the following propositions describing behavior in the rich treatment.

Proposition 2.

If players have a standard type distribution, then weakly honest players in the rich treatment choose identical strategies to those in Proposition 1 above, except for the following:

(i) 

$L{1}^S$ and $L{2}^S$ in states $s=1$ and $s=2$ send $m\left(s\right)=\{s+1,s+2,s+3\}$ with probability $1/2$;

(ii) 

$L{3}^S$ or higher-level senders in state $s=1$ send $m\left(1\right)=\{3,4,5\}$ or $m\left(1\right)=\{2,3,4\}$;

(iii) 

$L{3}^S$ and $L{4}^S$ senders in state $s=2$ may send $m\left(2\right)=\{3,4,5\}$.

Weakly honest senders do send imprecise messages but only in low states ($s<3$). In contrast, senders who care more about honesty tend to send imprecise messages in every state except $s=5$.

Proposition 3.

If players have a standard type distribution, then moderately or strongly honest senders ($h>32$) in the rich treatment behave as in Proposition 2 above, except for the following:

(i) 

$L{1}^S$ and $L{2}^S$ truthfully exaggerate in $s<4$ and send $m\left(4\right)=4$ and $m\left(4\right)=\{3,4,5\}$ with equal probability;

(ii) 

$L{3}^S$ truthfully exaggerate in $s<5$, except possibly in $s=2$;

(iii) 

$L{4}^S$ and $L{5}^S$ truthfully exaggerate in $s=3$ and may also truthfully exaggerate in $s=1$, $s=2$, and $s=4$.

While the senders change their behavior in the rich message treatment, the receivers’ predicted responses to precise messages do not change. However, receivers generally trust imprecise messages more than precise ones (e.g., sophisticated receivers choose $a\left(\right\{3,4,5\left\}\right)=3$, while they would choose $a\left(4\right)=2$).

Proposition 4.

If players have a standard type distribution, then in the rich treatment, receivers behave as in Proposition 1 above, except for the following:

(i) 

$L{0}^R$ and $L{1}^R$ do not discount imprecise messages ($\left(a\right(\{x-1,x,x+1\})=x$);

(ii) 

$L{2}^R$ and higher discount imprecise messages by one unit ($\left(a\right(\{x-1,x,x+1\})=x-1$), except possibly discounting $m=\{2,3,4\}$ by two units ($\left(a\right(\{2,3,4\})=1$);

(iii) 

$L{4}^R$ and higher do not discount precise $m=2$ ($a\left(2\right)=2$).

4.3. Model Discussion

Propositions 1–4 provide an explanation for why there is more communication in the rich treatment. Under a standard type distribution and, indeed, under most other type distributions, many player types who would send $m=5$ from $s<5$ in the base treatment instead send a different message in the rich treatment but continue to send $m=5$ from $s=5$ at the same rate. These messages therefore become more accurate in the rich treatment. Furthermore, the imprecise messages sent with $M_R$ are more accurate than the corresponding messages that would have been sent with $M_B$. For instance, $L{1}^S$ and higher send some $m=\{3,4,5\}$ from $s=3$ and $s=4$, whereas $m=3$ and $m=4$ are usually sent from $s=1$ and $s=2$ with $M_P$. The receivers do not change their responses much, but because they discount the common $m=5$ and $m=\{3,4,5\}$ the least, on average, they are more trusting of the messages they receive.

The model’s predictions match several patterns in the data. First, the model predicts that the sender’s behavior will be relatively more responsive than the receiver’s behavior to the rich message treatment. Precise messages $m=4$ and $m=5$ are predicted to become more accurate, since they are less likely to be sent from $s\ne m$ (Propositions 2 and 3). For many parameter values, the messages sent from $s<5$ are also predicted to often be imprecise. The receivers, on the other hand, are predicted to not change how they treat precise messages across the treatments (Proposition 4 (iii)) and to be somewhat more trusting of imprecise messages $\{2,3,4\}$ and $\{3,4,5\}$ than corresponding precise messages (Proposition 4 (i) and (ii)). Note that all of the comparative statics implicitly hold $\lambda$ constant. However, $M_R$ is more complex, so $\lambda$ might fall as well.

The form of honesty preferences in the model generates (1) receivers treating $m=\{x,x+1,x+2\}$ systematically differently than $m=x+1$ and (2) half of messages sent in the rich treatment being imprecise messages. Many alternative forms of honesty preference do not predict either. For instance, theories in which the senders care about the payoff disappointment experienced by the receiver, because $L{0}^R$ and $L{1}^R$ interpret $m=\{x,x+1,x+2\}$ and $m=x+1$ in the same way, would predict no systematic difference in how the senders used those messages. Likewise, there would be no systematic differences if the senders cared only about the de facto level of deception (i.e., $E\left[s\right]-s$, where E is the receivers’ expectation) instead of caring about deviating from the literal meaning of a message.

A major difference between the predicted and observed behavior is that the model predicts more skepticism about precise $m<4$ than is observed. $L{2}^R$ senders and higher are predicted to play $a\left(1\right)=a\left(3\right)=1$. In part for this reason, I introduce noisy choice when estimating the model parameters.14

5. Structural Estimates

In this section, I estimate the parameters of my model. Many papers have structurally estimated level-k models before (e.g., Crawford and Iriberri [43]), but to my knowledge, none have jointly estimated varying cognition levels and varying preferences. One goal for these estimates is to conduct a “horse race” between honesty and bounded rationality, but by using these estimates, I can also answer additional questions, such as how the different message spaces affect communication per se.

I assume that the subjects choose a or m, influenced by logistic errors of precision $\beta$ and independent across rounds and treatments. These errors capture that level-k reasoning is subject to error and unobserved utility shocks may occur, as well as any other sources of error. Hence, a level-k sender or receiver chooses each m or a with a probability

$$\begin{array}{cc}\hfill Pr(m\mid s,k,h,\beta ,T)& =\frac{exp(\beta E[u_s(a,s,h)\mid m,B_k^S,T])}{\sum _{m^{\prime }\in M_T}exp(\beta E[u_s(a,s,h)\mid m^{\prime },B_k^S,T])}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill Pr(a\mid m,k,\beta ,T)& =\frac{exp(\beta E[u_r(a,s)\mid B_k^R,T])}{\sum _{a^{\prime }\in A}exp(\beta E[u_r(a^{\prime },s)\mid B_k^R,T])}\hfill \end{array}$$

(6)

where $B_k^R$ is a level-k sender’s beliefs, $B_k^R$ is a level-k receiver’s beliefs, T is the treatment, and h is a sender’s honesty type. $\beta$ measures the noisiness of player choices. As $\beta \to 0$, each choice becomes equally likely, while as $\beta \to \infty$, players best respond exactly according to their beliefs. For all $\beta$, the $L0$ players exactly follow the strategies specified in Definition 1.

For my preferred estimates, I restrict the honest types to either have $h=0$ (the “not honest” type) or $h=42$ (the “moderately honest” type), because it is hard to precisely identify the frequencies of more than two honesty types. I motivate this choice and discuss exactly what conclusions can be drawn about the distribution of honesty after presenting my base estimates.

Given a treatment T, in which subject i made $A_i^T$ action choices $a_{i,1},\dots ,a_{i,A_i}$ and $M_i^T$ message $m_{i,1},\dots ,m_{i,M_i}$, the probability of i’s observed choices for a given parameterization is

$$L_i(\lambda ,h,\beta ,T)=\sum _{k\in K}f\left(k\right)\sum _{h\in H}{h}_h\left(\prod _{j=1}^{M_i^T}Pr(m_{i,j}\mid s,k,h,\beta ,T)\right)\left(\prod _{j=1}^{A_i^T}Pr(a_{i,j}\mid m,k,\beta ,T)\right).$$

(7)

For a set of subjects N, the log likehihood of their observed behavior in a set of treatments $\mathcal{T}$ is

$$LL(\lambda ,h,\beta )=\sum _{T\in \mathcal{T}}\sum _{i\in N}log{L}_i(\lambda ,h,\beta ,T)$$

(8)

I estimate the parameters by maximizing $LL$ and then derive confidence intervals through non-parametric bootstrapping.15

Column 1 of

Table 7. Parameter estimates.

	(1)	(2)	(3)	(4)	(5)
	Combined	Rich	Base	Senders	Receivers
	Treatments	Treatment	Treatment	Only	Only
$\lambda$	3.294	2.722	3.63	3.381	2.712
	[2.722, 3.79]	[2.419, 3.659]	[3.047, 4.5]	[2.341, 3.998]	[2.244, 3.387]
${\tau }_{42}$	0.267	0.209	0.077	0.274	0.175
	[0.132, 0.316]	[0.015, 0.298]	[0.0, 0.399]	[0.0, 0.321]	[0.0, 0.967]
$\beta$	0.027	0.029	0.025	0.026	0.03
	[0.023, 0.034]	[0.024, 0.036]	[0.02, 0.033]	[0.022, 0.032]	[0.02, 0.04]
n	1384	848	536	692	692
$LL$	−2045.688	−1379.507	−660.605	−1066.303	−977.101
$BIC$	4113.075	2779.243	1340.062	2152.226	1973.821

Maximum likelihood estimates of mean steps of thinking ($\lambda$) and share of players with utility from honesty h = 42 (${\tau }_{42}$), where 95% BCa confidence intervals are reported beneath estimates.

reports the parameter estimates for my entire dataset. The estimated mean steps of reasoning $\lambda =3.294$ was higher than that typical in the literature. However, given the definitions of $Lk$ behavior for each role, the effective amounts of sophistication were similar. These estimates imply that 32% of the senders engaged in one iteration of strategic reasoning and 40% of the senders engaged in two iterations, while 16% of the receivers did not reason strategically (they were credulous). Additionally, 42% of the receivers engaged in one iteration of strategic reasoning, and 42% engaged in two or more iterations.16 Cai and Wang [5] and Wang et al. [15] found comparable results. About 27% of the subjects had $h=42$ or were believed to have $h=42$. Finally, the choices were noisy. For example, the estimated precision $\beta =0.027$ implies that moderately honest types sent 10–20 percentage points fewer imprecise messages than they would without noise and that the weakly honest types sent 20 percentage points more imprecise messages than they would without noise.

The next columns report my model, estimated separately using the data from the rich treatment only (column 2) and the base treatment only (column 3). Reasoning was almost one level less sophisticated in the rich treatment than the base treatment, probably reflecting the greater difficulty of mental calculations involving $M_R$. The choices were approximately as noisy. Not surprisingly, the confidence interval for ${\tau }_{42}$ in the base treatment was wide—${\tau }_{42}$ could not easily be identified using choices over $M_P$—but the ${\tau }_{42}$ estimates in model 3 were similar to those in the combined model (model 1).

The final two columns in Table 7 report the model estimates using only the sender choice data (column 4) or receiver choice data (column 5). That the $\lambda$ estimates were even lower in model 5 than model 4 support the somewhat strange implication of my cognitive hierarchy model that receivers are less strategically sophisticated than senders (i.e., only $L{0}^S$ senders are naive, but $L{0}^R$ and $L{1}^R$ both act naively). If this assumption were inaccurate, one would expect the $\lambda$ estimate in model 5 to be higher than the estimate in model 4. It seems likely that the strategic reasoning that receivers must perform is more difficult than the strategic reasoning that senders must perform.17

The model pair of 2 and 3 and the pair of models 4 and 5 can be thought of as alternate models that relax one of the assumptions of my level-k model. The combination of models 2 and 3 allows $\lambda$ and $\beta$ to vary by treatment instead of forcing strategic sophistication to be constant. This treatment-varying specification (models 2 and 3) nests model 1 as a special case, and a likelihood ratio test comparing the null hypothesis (model 1) to the alternate (models 2 and 3) rejects the null at the 0.05 significance level ($D=11.152$, ${\chi }^2\left(3\right)=0.011$). This is evidence of the systematic differences in the level and noisiness of reasoning between the two treatments.18 A similar likelihood ratio test comparing the null hypothesis (model 1) to the alternate model with role-varying behavior (models 4 and 5) failed to reject the null hypothesis ($D=4.568,{\chi }^2\left(3\right)=0.206$). Comparisons using the Bayesian Information Criterion, which assigns more importance to model parsimony, support the role- and treatment-invariant model (model 1) ($BIC=4113.1$) over either the treatment-varying or role-varying model ($BIC=4123.62$ and $BIC=4130.2$, respectively, for the combined models).

5.1. Honesty

The distribution of honesty was identified in two ways. First, the strength of the players’ preferences for honesty was identified through the qualitatively different behavior of weakly, moderately, and strongly honest senders. For instance, senders with moderate honesty should reliably and truthfully exaggerate in the rich treatment but not reliably send truthful messages in the precise treatment. Second, the fraction of players of a given honesty type (i.e., ${\tau }_h$ for a given h) was identified through the magnitudes of honest behavior and through their indirect effect on how receivers and non-honest senders behave. Section 4 described some of the identifying differences in greater detail. For instance, if ${\tau }_{42}$ is low and $\lambda$ is high, the receivers should be relatively skeptical of imprecise m, and the senders should send relatively few imprecise m from states $s=3$ and $s=4$. Figure 3 shows the distribution of ${\widehat{\tau }}_{42}$ from the bootstrap for my preferred model. Most estimates are between $0.15$ and $0.35$, although there is an almost invisible left tail for the lower estimates.

Unfortunately, the log-likelihood function was relatively flat with respect to the strength of honesty preferences within each of the h intervals of not honest, moderately honest, and strongly honest. Figure 4 shows the maximum log-likelihoods from two-honesty-type models, where one type does not care about honesty and the other type is h. Intuitively, for low honesty, $\tau$ was estimated entirely from how it interacted with utility shocks in the message and action choice (Equations (5) and (6)), while within each of the other h intervals, the variation in behavior was small, so pinning down h precisely was difficult. The $LL$ may have also been flat with respect to h, because honesty has limited importance in my context. These problems also made it infeasible to estimate the frequencies of the honesty types in a richer model that contained two honest types $25<h_m<57<h_s$. Because the behavior is similar for any moderately or strongly honest type, the parameters ${\tau }_{h_m}$ and ${\tau }_{h_s}$ are colinear and cannot be estimated precisely.

Due to these difficulties, my approach is to assume that people are either fully self-interested or have a single degree of honesty h. For each h value, Figure 5 shows the fraction of players estimated to be honest at this level and the 95% BCa confidence intervals. For any moderate honesty level, at least around 15% of players can be classified as honest with 95% confidence. I focus on $h=42$ because the log-likelihood achieves its maximum at that value (see Figure 4).

Despite the underlying identification problems, there is evidence that a substantial minority of subjects had at least moderate preferences for truth-telling and believed that other subjects did as well. While I am could not estimate the exact distribution of honesty in my subjects, I am able to show both that it was not extremely weak (all subjects having $h<25$) and that only a minority of the subjects had moderate or strong honesty preferences.

5.2. Why Vague Language Matters

Counterfactual comparisons using the structural model can also answer the question of why vague language improves communication. Communication might increase in the rich treatment either because it affects subjects’ decision processes in terms of reducing their sophistication or increasing choice noisiness, because the change in the message space opens up additional ways to communicate, which improves communication overall, or because honesty increases sender accuracy or receiver trust more under the more complex message space. To gauge these explanations’ relative importance, I hold some factors as fixed and evaluate how the predicted communication change as a single factor changed. For example, if strategic sophistication is fixed at the base treatment levels, and all players are assumed to be weakly honest, the predicted improvement in communication from adding imprecise messages to the message space is a measure of how important the change in the message space is according to my model.

To measure communication, I calculate the predicted mean distance between messages and states $E\left[\right|m-s\left|\right]$ to measure how informative the messages are and the predicted mean distance between actions and messages $E\left[\right|a-m\left|\right]$ to measure how trusting the action choices are.19 Given a set of a parameters $\lambda$, $\tau$, and $\beta$ and a message space M, the mean distance between the messages and actions is the distance between each $(m,a)$ combination weighted by the predicted probability of observing that combination:

$$E\left[\right|m-a\left|\right]=\sum _{m\in M}\sum _{a\in A}\widehat{Pr}\left(m\right)\widehat{Pr}(a\mid m)·|a-m|,$$

(9)

where $\widehat{Pr}\left(m\right)$ is the predicted probability that message m is sent (obtained from Equation (5)) and $\widehat{Pr}(a\mid m)$ is the predicted conditional probability (obtained from Equation (6)). $E\left[\right|m-s\left|\right]$ is defined similarly but with weights $Pr\left(s\right)=1/5$ instead of $\widehat{Pr}\left(m\right)$.

Table 8. Predicted mean distances of messages from state ($|m-s|$) and actions from message ($|a-m|$) from calibrating model with alternate message spaces and parameter estimates from base or rich language treament behavior.

Counterfactual	$\mathit{E}\left[\right|\mathit{m}-\mathit{s}\left|\right]$	$\mathit{E}\left[\right|\mathit{a}-\mathit{m}\left|\right]$
Base treatment actual data	1.52	1.30
Base message space, base honesty, base rationality	1.46	1.28
Rich message space, base honesty, base rationality	1.30	1.33
Base message space, rich honesty, base rationality	1.42	1.27
Base message space, base honesty, rich rationality	1.37	1.08
Rich message space, rich honesty, base rationality	1.25	1.25
Base message space, rich honesty, rich rationality	1.31	1.06
Rich message space, base honesty, rich rationality	1.23	1.14
Rich message space, rich honesty, rich rationality	1.16	1.05
Rich treatment actual data	1.22	0.98

reports these measures for every combination of behavioral parameters and each message space. The rich language parameters are model 2’s: the MLE parameter estimated using only rich language treatment data. Rich honesty sets ${\tau }_{42}=0.209$, while rich rationality sets $\lambda =2.722$ and $\beta =0.029$. The base treatment parameters are those of model 3. Base honesty sets ${\tau }_{42}=0.077$, while base rationality sets $\lambda =3.63$ and $\beta =0.025$.

The first two rows compare the average distances in the base treatment data to the predicted average distances from calibrating the model with the parameter estimates from MLE using the base treatment data. The mean distance $E\left[\right|m-s\left|\right]$ in the data was 1.52 in comparison with the predicted $E\left[\right|m-s\left|\right]$ of 1.46, while for the receiver mean $E\left[\right|m-a\left|\right]$ in the data, it was 1.30 in comparison with the predicted distance of 1.28. The calibrated model closely matches senders’ average exaggeration and receivers’ average skepticism.

The last two rows provide the same comparison for the rich language treatment. For the senders, the mean distance in the data was 1.22, while the model predicts a mean distance of 1.16. For the receivers, the mean distance was 0.98, while the model predicts a mean distance of 1.05.

The intermediate rows use alternative model calibrations to show which factors are important in the model for explaining the observed increases in communication. The three rows after the base treatment model each switch one factor from its value in the base treatment model to its value in the rich language treatment model. The next three rows then switch two factors from their values in the base treatment model to their values in the rich language treatment model. Factors that cause larger reductions in sender bias or receiver skepticism are more important in the model for explaining the increase in communication. To the extent that the model accurately captures aggregate behavior, these factors are likewise more important for explaining what caused the changes in subject behavior across the treatments.

For senders, the most important factor appeared to be the message space. Switching to a rich message space in the model but holding the behavioral parameters fixed at the base model values causes the mean $|m-s|$ to fall to 1.30 from 1.46, a larger effect than for switching any other single factor. Similarly, when two factors were switched, the cases that come closest to matching the predicted mean $|m-s|$ of 1.16 in the rich language treatment are the cases in which the message space and one behavioral parameter are switched to their values in the rich language model.

In my model and in my reduced form results, a rich message space improves sender accuracy because many senders shift from $m=4$ and $m=5$ to imprecise messages, making the high-m precise messages more accurate. This shift to imprecise messages, while most pronounced in the model for honest senders, occurred with all senders in low states.

For receivers, the most important factor appears to be the bounded rationality parameters of the mean steps of thinking $\lambda$ and choice precision $\beta$. Switching to the rich language model $\lambda$ and $\beta$ parameter values but holding the message space and share of moderately honest players fixed at the base model values causes the predicted $E\left[\right|a-m\left|\right]$ to fall to 1.08 from 1.28, a larger effect than for switching any other single factor. When two factors are switched, the cases that comes closest to matching the predicted $E\left[\right|a-m\left|\right]$ of 1.05 in the rich language treatment are the cases in which the rationality parameters and one other parameter are switched to their values in the rich language model.

The increased cognitive complexity of strategic thinking in the rich message treatment made subjects worse at strategic reasoning, which manifests in the model estimates as a reduction in $\lambda$. Reducing $\lambda$ makes the receivers more trusting, because the lower-level receivers engage in fewer rounds of iterated reasoning, which tends to mechanically increase receiver skepticism.

Interestingly, for the receivers, the model predicts that switching to the rich message space would cause a slight increase in the average receiver skepticism if honesty were not present. Switching to a rich message space in the model but holding the behavioral parameters fixed at the base model values causes the predicted $E\left[\right|a-m\left|\right]$ to rise from 1.28 to 1.33, and likewise, switching to a rich message space in the model with the rich language model’s $\lambda$ and $\beta$ and base model’s ${\tau }_{42}$ causes the predicted $E\left[\right|a-m\left|\right]$ to rise from 1.08 to 1.14.

In summary, three factors explain the increase in communication produced by the rich treatment. The richer message space made the senders more accurate, while the increased difficulty of reasoning for $M_R$ made the receivers more trusting. The presence of honesty also reduced receiver skepticism when they were interpreting the $M_R$ messages.

6. Conclusions

It is natural to think that when people make use of less precise language, communication will become less effective if they do so. Surprisingly, in a standard biased information transmission setting with high sender bias, when the subjects could choose between more or less precise messages, the distribution of messages conveyed more information, and the receivers were more trusting than when the subjects could only send precise messages.

Communication was improved because the senders changed the least accurate messages they sent to imprecise ones if they were available, making both the precise messages and imprecise messages more informative as a result. It is also improved because the receivers were somewhat more trusting due to both the increased cognitive difficulty of strategic thinking and the greater prevalence of honest message choices. A larger share of honest messages was sent due to the lower opportunity costs of honesty when honest imprecise messages could be chosen.

Strategic information transmission games are a natural setting in which to expect strategic miscalculations to be important, as the incentives for exaggeration are especially salient. At the same time, what norms of honesty should operate are unclear. Honesty did play a small role in explaining why communication increases when the message space was enriched by adding explicitly imprecise messages. In a population with honesty, the receivers’ trust in the messages increased slightly more with a rich message space than it would otherwise. Nonetheless, most of the improvement in communication seemed to be due to the change in the message space or bounded rationality combined with the increased complexity of the message space. In other strategically simpler settings or those in which honesty is more salient, honesty might be relatively more important.