Vague Language and Context-Dependence:An Experimental Study.∗

Wooyoung Lim† and Qinggong Wu‡

May 22, 2017

Abstract

In this paper we broaden the existing notion of vagueness to account for linguisticambiguity due to language used in a context-dependent way. This broadened notion,termed as literal vagueness, necessarily arises in any Pareto-optimal equilibrium inmany standard conversational situations. With controlled laboratory experiments wefind that people can make use of literally vague language to effectively transmit infor-mation.

Keywords: Communication Games, Context-Dependence, Vagueness, Laboratory Ex-perimentsJEL classification numbers: C91, D03, D83

∗This study is supported by a grant from the Research Grants Council of Hong Kong (Grant No. GRF-16502015).

†Department of Economics, The Hong Kong University of Science and Technology. Email:wooyoung@ust.hk

‡Department of Decision Sciences and Managerial Economics, Chinese University of Hong Kong. Email:wqg@baf.cuhk.edu.hk

1 Introduction

Language is vague, as this very sentence exemplifies —– what does “vague” mean here,after all? Despite economists’ persisting study of language and communication, we stilllack a good explanation to account for vagueness. On the contrary, as Lipman (2009)forcefully argues, there is no benefit to vague language in a typical conversation. To ex-plain vagueness, as Lipman (2009) notes, we need to relax the standard “full rationality”assumption.

In this paper, we offer our explanation of why language is vague. Instead of deviatingfrom the full-rationality paradigm, we take an alternative perspective. In particular, webroaden the notion of vagueness and show that vagueness in this broader sense has anadvantage in many standard conversational situations.

Vagueness language admits a borderline case. For example, no one can claim that thereis a clear cutoff between red and orange that allows him/her to separate the color spectruminto two disjoint sets. Thus, if a speaker uses a vague language, listener’s posterior beliefscannot be concentrated on a precise subset of a state space. Such vague languages cannotcreate an efficiency improvement from a non-vague language.

We argue that in many cases, there is an exogenous relation between messages anda particular sub-dimension of states they conventionally signify. The other dimensionwith which the literal meaning of messages does not have any exogenous relation can beregarded as context. One important observation from our daily life conversations is thatthe presence of context and the use of a language dependent upon the context may impressa listener that the language is vague. We call such context-dependent use of languageand its consequential vagueness literal vagueness. We show that literal vagueness canindeed be a fundamental property of the optimal language in many standard conversationsin which the context — either directly payoff-relevant or not — matters. Thus literalvagueness has a solid efficiency foundation, which makes it a plausible explanation forwhy language is often vague.

We then experimentally investigate whether people could make use of literally vaguelanguages to efficiently communicate. We consider a conversational environment in whichtwo speakers speak sequentially to a listener, and the way the later speaker talks may relyon what the earlier speaker has said. In this simple environment in which the optimallanguage is necessarily literally vague, we observed that subjects indeed tended to do so.Several variations of the environment with varying degrees of complexity in coordinatingon the optimal, literally vague language provide further supporting evidence. Althoughour experimental result alone does not suffice to identify the efficiency advantage of literal

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vagueness as the explanation for the omnipresence of vague languages, we view them asan important first step to understanding vagueness of language.

In the rest of the introduction, we shall discuss the theoretical aspects of vaguenessand introduce our notion of literal vagueness.

Vagueness

Take the simplest case of communication: A speaker wishes to tell the state of the worldω ∈ Ω to a listener by sending the latter a message m ∈M , where Ω is the set of all statesand M the set of all available messages. For simplicity assume Ω and M are finite.

The language of the speaker is formally his message-sending strategy λ ∶ Ω → ∆(M)where ∆(M) is the set of all lotteries over M . Thus, if the speaker knows ω is the state,he uses the lottery λ(ω) to draw a message.

Multiple states can be grouped together if the same lottery is used in them. In otherwords, language λ induces a partition Π on Ω such that for any two states ω and ω′, Πλ(ω) =Πλ(ω′) if and only if λ(ω) = λ(ω′). Language λ is not vague if the following is satisfied:

V. For any two states ω and ω′, if Πλ(ω) ≠ Πλ(ω′) then λ(ω) and λ(ω′) have disjointsupports.

Otherwise λ is vague. What is vagueness meant to capture? Note that if a languageλ is not vague then for each message m ever used there is a unique block π(m) ∈ Πλ suchthat π(m) contains the states in which m is drawn with positive probability. Thereforethe message m helps the listener precisely narrow down the possible states to π(m) in thesense that the listener’s posterior is simply his prior concentrated on π(m). If λ is vaguethen it does not induce such sharp demarcation of Πλ: Upon receiving some message mthat is used with positive probability in distinct blocks π ∈ Πλ and π′ ∈ Πλ the listenerremains uncertain which block obtains. Theorem 1 of Lipman (2009) establishes thatunder common interest there is always a Pareto-optimal language that is not vague, thusnegating the necessity of vague language.1

1The definition we give here is weaker than that given in Lipman (2009), as the latter rules out anyrandomization. The difference is minimal, though, because Condition V implies the weaker version onlyadmits randomization among messages that are entirely synonymous. It is straightforward to verify thatall results in Lipman (2009) hold under the weaker definition as well.

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Literal Vagueness

We would like to use an example to demonstrate that vagueness as defined above mayexclude languages that typically impress us as vague.

Example 1. In a conference, Bob, a graduate student, asks Alice, a distinguished scholar,“How do you like my talk”. Alice’s reply depends on: 1) whether she likes Bob’s talk (L) ornot (NL), and 2) whether she has time for further conversation (T ) or is in a hurry to thenext session (NT ). If Alice does not have time she says “It’s interesting.” (l). If she has timeshe says “It’s interesting” if she likes the talk, or “The research can benefit from some furtherdevelopment” (nl) if she does not like the talk.

Is Alice’s language vague? We can formalize the example with state space Ω = L,NL×T,NT andM = l, nl. Alice’s language is technically not vague as defined above, becauseshe uses only nl in state (NL,T ) and only l in other states. However, Alice’s language maystrike Bob as vague. For instance, if Alice says “It’s interesting”, Bob is uncertain whetherAlice indeed likes the paper, or is just in a hurry to end the conversation.

Taking a closer look, we see that Bob’s confusion is rooted in the relation of the mes-sages “It’s interesting” and “The research can benefit from some further development” tothe states. These messages have a literal interpretation that only concerns Alice’s opinionabout Bob’s talk and is irrelevant whether Alice has time. Literal interpretation repre-sents an exogenous relation between messages and the states they conventionally signify.Thus, an impression of vagueness would result if the language does not precisely demar-cate the “literal state space” L,NL as a non-vague language does for the true state space.

Languages like Alice’s are deemed as non-vague according to the definition above be-cause in the standard model of conversation, Ω and M are taken as abstract sets lackingstructure and relation. No exogenous literal interpretation exists outside the speaker’sidiosyncratic use of the messages, and thus interpretation is purely endogenous. In thebackground of everyday conversation, on the other hand, there is a focal language whichendows messages with exogenous literal interpretation, and the sense of vagueness of-ten occurs because of the inconsistency between the common literal interpretation andan individual’s idiosyncratic language use, as is the case in Example 1. Thus it would bebeneficial to study vagueness in a model which literal interpretation can be built in.

For this purpose we alter the standard model as follows: Suppose the state space hasa two-dimensional product structure such that Ω = F × C, where a typical f ∈ F is calleda feature and c ∈ C a context. The message space M has an exogenous literal relationwith F , but not C, that is, messages in M are conventionally used to describe features but

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has no literal relation with the context.2 For instance “I’m well” is conventionally usedto describe wellness but not hurriedness. Given this setup we propose a new notion ofvagueness. Specifically, a language λ is said to be not literally vague if Πλ satisfies thefollowing conditions:

L1. For any two states ω and ω′, if Πλ(ω) ≠ Πλ(ω′) then λ(ω) and λ(ω′) have disjointsupports.

L2. For any π ∈ Πλ there exist Fπ ⊂ F and Cπ ⊂ C such that π = Fπ ×Cπ.

Otherwise λ is said to be literally vague.

If a language is not literally vague then it is not vague, because Condition L1 is iden-tical to Condition V. Thus literal vagueness is broader than vagueness. The additionalCondition L2 further demands that, for a language to be not literally vague, every mes-sage m ever sent precisely narrows down the possible features for the listener in the sensethat upon receiving m the listener’s posterior about F is his prior concentrated on Fπ(w)where π(w) is the unique block of Πλ with m being used with positive probability.

Alice’s language in Example 1 is literally vague, because, given the message “It’s inter-esting”, Bob’s prior does not become concentrated on any subset of F = L,NL.

The Efficiency Advantage of Literal Vagueness

We wish to demonstrate that, unlike strictly vague languages, literally vague languagesmay have an efficiency advantage. To show that we adopt the standard cheap talk modelwith common interest as Lipman (2009) does:3 The speaker is informed of the state ω ∈ Ω

and the listener is not. The listener’s job is to choose an action a from a set of availableactionsA, and consequently both players receive a payoff of u(ω, a). To help the listener, thespeaker sends a message m ∈M . To incorporate literal vagueness we impose the productstructure Ω = F × C on the state space as discussed above. We do not require a commonprior. However, we do assume that each state is believed by at least one of the players aspossible i.e. its probability is positive.

The following proposition establishes the potential efficiency advantage of literallyvague languages by showing that if the choice problem is not trivial and if messages arelimited then there are payoff functions u given which the optimal language must be liter-ally vague.

2The notion of context in our framework resembles the “prior collateral information” in Quine (1960).3An earlier, non-formal study on language use in this situation is due to Lewis (1969).

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Proposition 1. If ∣F ∣ ≥ 2, ∣C ∣ ≥ 2, ∣A∣ ≥ 2 and min(∣F ∣∣C ∣ − 1, ∣A∣) ≥ ∣M ∣ ≥ 2 then there isa payoff function u ∶ Ω × A ↦ R given which the sender’s language in any Pareto-optimalperfect Bayesian equilibrium must be literally vague.

Proof. Pick distinct f, f ′ ∈ F , c, c′ ∈ C and a, a′ ∈ A. Let A∗ be an arbitrary subset of A/a, a′such that ∣A∗∣ = ∣M ∣ − 2. Note that A∗ can be the empty set. Such A∗ exists because ∣A∣ ≥∣M ∣ ≥ 2 implies ∣A/a, a′∣ = ∣A∣ − 2 ≥ ∣M ∣ − 2 ≥ 0.

Consider the utility function u such that:

1. a is the unique optimal action in states (f, c) and (f ′, c′).

2. For any action a∗ ∈ A∗ there is a unique state (f∗, c∗) ∉ (f, c), (f ′, c′), (f, c′) such thata∗ is the unique optimal action in (f∗, c∗).

3. a′ is the unique optimal action in the rest of the states, including (f, c′).

Such u exists because ∣A∗∣ ≤ ∣F ∣∣C ∣−3. By construction of u there is a partition P on Ω suchthat:

1. For any p ∈ P there is some α(p) ∈ A∗ ∪a, a′ where α(p) is the unique optimal actionin any state ω ∈ p.

2. α(p) ≠ α(p′) if p ≠ p′.

Let λ be the speaker’s language in a Pareto-optimal perfect Bayesian equilibrium.Clearly λ can be described by the bijection µ ∶ P ↦ M such that λ(ω) = µ(P (ω)). Mes-sage µ(P (ω)) can be interpreted as the recommendation “the optimal action is α(P (ω)) inthe current state”. Therefore Πλ = P . Thus (f, c), (f ′, c′) ∈ Πλ since only for these twostates the optimal action is a. λ is literally vague because (f, c), (f ′, c′) is not a productof any subsets of F and C.

Remark 1. The conditions ∣F ∣ ≥ 2, ∣C ∣ ≥ 2 and ∣A∣ ≥ 2 reflect that the state space andthe decision problem are not trivial. ∣M ∣ ≥ 2 reflects that communication is not trivial.∣F ∣∣C ∣ − 1 ≥ ∣M ∣ implies that available messages are not adequate to fully describe thestate space, that is, it is not possible to use a distinct message to denote each state. Itis noteworthy that the whole issue of vagueness would be less of a concern if messageswere not limited, for obviously full description is best if feasible.

Remark 2. That ∣A∣ ≥ ∣M ∣ is not a necessary condition. The assumption is made tosimplify the proof, which is based on a particular construction. Although the construction

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may strike us as contrived and peculiar, the spirit of the idea underlying optimality ofa literally vague language does not crucially rely on that ∣A∣ ≥ ∣M ∣, and is quite general.The spirit is the following: if the context-dimension of the state is also payoff-relevant,and the messages are not abundant enough, then in the optimal language a message maybe used in a context-dependent way, because the messages, albeit literally descriptiveonly of features, need to contain information about the context as well. Therefore, tocorrectly interpret which subset of features a message points to, it is necessary to knowthe corresponding context; in contrast, without knowledge of the context, the message isvague in terms of which subset of features it points to.

Remark 3. That the result holds without the common prior assumption is becausethe constructed payoff function u implies that ex post optimality is achieved in a Pareto-optimal equilibrium. Thus the potential conflict of interest due to heterogeneous priorsbecomes irrelevant concerning the Pareto-optimal equilibrium because it is the most pre-ferred strategy profile by both players regardless of prior beliefs.

Context-dependence

As mentioned in Remark 2 above, the optimality of literal vagueness is due to the na-ture of the decision problem being context-dependent, that is, the payoff depends onthe context-dimension of the state of which the focal language is not literally descriptive.Consequently, the language may also be used in context-dependent ways. The same mes-sage refers to different subsets of features depending on the context that obtains, becauseinformation about the context is also worth communicating. It is not unusual that themeaning of a word depends on the context. Indeed, the whole linguistic field of pragmat-ics is dedicated to the study of context-dependence.

It is important to distinguish between two types of context-dependence embedded in alanguage. The first type refers to the case that the sender’s choice of message depends onthe context. For instance, there are two common Cantonese expressions of “thank you”,“m-goi” and “do-ze”. People would say “m-goi” to thank for something non-materialistic,e.g., “m-goi” is used when people ask for help or express graditude for a favor. In con-trast, “do-ze” is mostly used to thank for something materialistic, e.g., a gift.4 This type ofcontext-dependent use of language needs not be literally vague, because distinct messagesmay simply convey distinct information about the context without differentiating betweeninformation about the feature. The second type of context-dependence in a language refersto the case that the correct interpretation about which subset of features is signified by

4However, no such distinction in expressing gratitude exists in Mandarin Chinese.

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a message depends on the context. In this case, not only the speaker’s choice of messagedepends on the context, but he also chooses the same message to signify different subsetsof features depending on the context. In Example 1, Alice’s use of “It’s interesting” corre-sponds to this case. Thus the second type is stronger than the first type, and it necessarilyimplies literal vagueness. When thereafter we discuss context-dependence in a languagewe mean the second type.

A crucial source of context-dependence and thus literal vagueness is that the focallanguage is essentially one-dimensional: It only has terms literally relating to the feature-dimension of the state. Why would people use such an overly parsimonious language? Oneimportant reason is its simplicity. After all, using a richer language is more costly, andoften the richer language may not be feasible at all — for instance, if the common languageshared by people from different linguistic backgrounds is only descriptive of one aspect ofthe state. People with different native tongues can use facial expression in person, oremoji , and emoticon :-) on the internet, as a common language to describe emotion, butthis common language lacks terms descriptive of any other aspect of the world.

In Example 1 the context is payoff-relevant. It is possible that even if the context isnot directly payoff-relevant, people still prefer to use a literally vague language. This isparticularly the case if the context is meta-linguistic. Here we give two examples, bothvariants of the standard two-person cheap talk model with common interest.

Example 2. Alice wishes to describe to Bob, a tailor, the color she has in mind for her nextdress. Alice may have a small vocabulary for colors, only with typical terms like “blue”,“red”, or she may have a large vocabulary for colors, which in addition also includes termsdenoting subtle colors like “maroon” and “turquoise”. Alice’s vocabulary is unknown to Bob,and can be interpreted as the context-dimension of the state. Alice’s optimal language maybe literally vague.

This is an example of the more general model studied in Blume and Board (2013). Intheir model, the available messages for the speaker may vary depending on the speaker’slanguage competence. Consequently, in optimal communication speakers of different lan-guage competence may use the same message to indicate different sets of payoff-relevantstates, implying that the optimal language is vague in their model. Example 2 shows thatlanguage competence can be incorporated as the context-dimension of the state, and inthe model with the enriched state space the optimal language is no longer vague, but isinstead literally vague.

Example 3. Alice wishes to describe the height of Charlie to Bob, so that Bob can recognizeCharlie at the airport and pick him up. Moreover,

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• Alice knows whether Charlie is a professional basketball player or not.

• Bob may or may not know whether Charlie is a professional basketball player or not.

• Alice may or may not know whether Bob knows whether Charlie is a professionalbasketball player or not.

Alice can also describe Charlie as “tall” or “short”. Her optimal choice of the word candepend both on Charlie’s height and on whether she believes Bob knows Charlie is a pro-fessional basketball player or not. For instance, if she believes Bob knows Charlie is aprofessional basketball player then she says Charlie is “tall” only if his height is above 6foot 10, whereas if she believes Bob does not know then Charlie is “tall” only if his heightis above 6 foot 2. If we take Alice’s belief about Bob’s knowledge as the context, then herlanguage is literally vague.

This example is adapted from a more casual discussion in Lipman (2009). A more for-mal model of the example is as follows: The listener’s knowledge about the payoff-relevantstate space is not common knowledge. In particular, prior to the conversation the listenermay receive with some probability an informative private signal which narrows down thepossible set of payoff-relevant states. Moreover, the speaker may also receive a private(possibly noisy) signal which tells him whether the listener has received that informativesignal or not. It is very easy to construct a typical decision problem under which, in theoptimal equilibrium, the meaning of a message from the speaker depends on the signalthat he receives. If we do not incorporate the speaker’s signal as part of the state thenthe optimal language is vague. However, in the enriched model in which the speaker’s sig-nal is considered as the context and the payoff-relevant state as the feature, the optimallanguage is not vague but literally vague.

Finally, we show an example demonstrating that when there are multiple speakers whospeak sequentially, the way later speakers talk may rely on what earlier speakers havesaid. Earlier messages become the context on which later messages depend – the contextof the bilateral conversation between a later speaker and the listener is then endogenouslygenerated in the larger-scale multilateral conversation. The following example shows oneof such situations.

Example 4. Alice and Bob interviewed a job candidate. Alice observes the candidate’sability A and Bob observes his personality B. A and B are independently and uniformlydistributed over [0,100]. The best decision is to hire only if A +B ≥ 100.

Bob and Alice sequentially report their observations in a binary fashion to the committeechair Charlie who is responsible for the recruitment decision, with Alice speaking first. The

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best strategy is the following: Alice reports “A is high” if A ≥ 50 and “A is low” otherwise.If Alice reports “A is high” then Bob reports “B is high” if B ≥ 25 and “B is low” otherwise;if Alice reports “A is low” then Bob reports “B is high” if B ≥ 75 and “B is low” otherwise.Charlie hires the candidate if and only if Bob reports “B is high”.

Alice’s report provides the context with which Bob’s report is to be correctly interpreted.Bob’s language is literally vague because, for instance, the correct interpretation of “B ishigh” depends on Alice’s report.

This example will serve as the benchmark model for our experiments.

Related Literature

Economists have been studying strategic language use since the canonical “cheap talk”model proposed by Crawford and Sobel (1982). Despite the formidable literature sincegenerated on this subject, only until recently was linguistic vagueness given the academicattention it deserved when Lipman (2009), posing the question of “why is language vague”,argued that vague language is not optimal if all parties in the conversation a la Crawfordand Sobel (1982) have 1) common interest and 2) full rationality. Following in this quest,Blume and Board (2013) explore a situation in which the linguistic capability of someof the conversing parties are unknown and show that this uncertainty could make theoptimal language vague. The same authors further investigate the effect of higher-orderuncertainty about linguistic ability on communication in Blume and Board (2014a), andfind that, in the common interest case, vagueness persists but efficiency loss due to higher-order uncertainty is small. The relation between our paper and the above papers has beendiscussed in depth in the Introduction.

It is well known that even in the presence of conflict of interest endogenous vaguelanguage still has no efficiency advantage in the cheap talk framework. However, Blumeet al. (2007) show that exogenous noise in communication, which forces vagueness upon thelanguage, can bring Pareto improvement. Blume and Board (2014b) further confirm thatthe speaker may intentionally take advantage of the noise to introduce more vaguenessin the language. These papers differ from ours in that we focus on the common interestenvironment, and moreover we study the efficiency-foundation of endogenous vagueness.

Context-dependence is the source of literal vagueness in our paper. Given commoninterest, context-dependence arises when available messages are not sufficient to fullycommunicate the complexity of the situation. Within the cheap talk framework, Tian(2016) discusses that, when the message space is small, how the optimal language changeswith the common prior, that is, the context. In the framework of experimentation, formally

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similar to a model of sequential communication with limited messages and/or memories,Smith et al. (2016) and Wilson (2014) investigate how a participant optimally uses hislanguage conditioning on contexts, which are messages received from earlier participants.Indeed, an implication of the characterization of the Pareto-ranking of communicationmechanisms due to Wu (2016) is that a mechanism which allows the participants to havemore flexibility in using context-dependent language dominates another one which allowsless.

On the experimental side, a few recent papers investigate how the availability of vaguemessages improves or preserves efficiency. Serra-Garcia et al. (2011) show that vaguelanguages help players preserve efficiency in a two-player sequential-move public goodsgame with asymmetric information. Wood (2016) explores the efficiency-enhancing role ofvague languages in a discretized version of canonical sender-receiver games a la Crawfordand Sobel (1982). Agranov and Schotter (2012) show that vague messages are useful inconcealing conflict between a sender and a receiver that, if it is publicly known, wouldprevent them from coordinating and achieving an efficient outcome. All these papers takethe availability of vague messages as given and study how it affects players’ coordinationbehavior. On the contrary, we explore how a message endogenously obtains its vaguemeaning. For more comprehensive discussion of the experimental literature on vaguelanguages, see the recent survey by Blume et al. (2017).

2 Experimental Games

We would like to know whether people actually do use languages in a literally vague fash-ion. This is a very important step towards understanding whether literal vagueness, dueto its efficiency advantage, stands firm as an explanation for some linguistic vaguenessthat we experience everyday, because the whole efficiency foundation for vagueness ispointless if people cannot make use of literally vague languages effectively. Of course,the finding that people can effectively make use of literally vague language alone does notsufficiently prove that the prevalence of linguistic vagueness is founded on the efficiencyadvantage of literal vagueness in our sense, yet it is a worthwhile first step. It is with thisquestion in mind we design our experiments.

We use the situation described in Example 4 to examine whether, and if so, how, peopleuse literally vague language to converse. This example has a number of important attrac-tive features which make it particularly suitable for our purpose. Firstly, as we shall showbelow, there is a moderate degree of efficiency advantage to the optimal literally vaguelanguage, which renders literal vagueness potentially useful but not entirely crucial for

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communication. Secondly, the situation is simple and straightforward, and thus should inprinciple shorten the period of learning and make the experimental results closer to theeventual stable language, or in terms of Lewis (1969) the “convention”. Thirdly, the situ-ation covers the most general environment in which the context, being the message fromAlice, is endogenously generated. Fourthly, given that for optimal communication Alicedoes not need to use a literally vague language whereas Bob does, having both senders inthe same game gives us an additional comparison regarding how people’s use of languagedepends on the conversational environment they face.

The Benchmark Game

There are three players: two senders, Alice and Bob, and one receiver, Charlie. Aliceprivately observes a number A and Bob a number B. A and B are independently anduniformly drawn from [0,100]. Alice sends a message to Bob, where the message is either“A is Low” or “A is High”. Alice’s message is unobservable to Charlie. Then Bob sendsa message to Charlie, where the message is either “B is Low” or “B is High”. Charliereceives Bob’s message and chooses an action: UP or DOWN. Players’ preferences areperfectly aligned. If A +B ≥ 100 and UP is chosen, or if A +B ≤ 100 and DOWN is chosen,then all receive a payoff of 1. Otherwise all receive a payoff of 0.

Equilibria of the game can be classified into three categories:5

1. Bob babbles: Bob uses a strategy given which Charlie’s posterior about A + B re-mains the same as the prior upon seeing any message chosen by Bob with positiveprobability. Whether Alice babbles or not does not have bearing on the outcome. Noinformation is transmitted and Charlie is indifferent between the two actions regard-less of the message he receives. Accordingly, the success rate, which is the probabilitythat Charlie chooses the optimal action, is 50%.

2. Only Alice babbles: Bob sends message “B is High” if B > 50 or “B is Low” if B < 50.6Charlie chooses UP seeing “B is High” and DOWN otherwise. Only Bob’s informationis transmitted. Accordingly, the success rate is 75%.

3. Neither babbles: Alice sends “A is High” if A > 50 and “A is Low” if A < 50.5A similar categorization of equilibria persists for variations of the benchmark game to be introduced.

For those variations we will skip the analysis of equilibria in which someone babbles, because they are of notheoretical consequence and do not correspond to the experiment results.

6Of course there are outcome-equivalent equilibria in which Bob uses the messages in the opposite way.We do not explicitly itemize such equilibria here and thereafter.

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• If Alice’s message is “A is High” then Bob sends “B is High” if B > 25 and “B isLow” if B < 25.

• If Alice’s message is “A is Low” then Bob sends “B is High” if B > 75 and “B isLow” if B < 75.

Charlie chooses UP seeing “B is High” and DOWN otherwise. Accordingly, the suc-cess rate is 87.5%.

Clearly any equilibrium in which no one babbles is Pareto-optimal. Because the mes-sages available to Bob explicitly refer to the value ofB alone, Bob’s language in any Pareto-optimal equilibrium is clearly literally vague, because the set of values of B a particularmessage, say “B is high”, describes depends on Alice’s message, which serves as the con-text.

To best test whether indeed the players effectively use the optimal literally vague lan-guage, we propose to consider the counterfactual in which they don’t. This can be formallymodeled as Bob being constrained to use the same messaging strategy regardless of Alice’smessage. In this case, the best Bob can do is to always use the cutoff of 50, and the corre-sponding success rate is 75%. To examine the results and test whether the counterfactualholds, we thus should pay particular attention to how Bob’s use of language depends onAlice’s message.

In addition, for a literally vague language to be effectively used, the listener should alsobe aware of the underlying context-dependence and correctly take that into considerationwhen making decisions. However, Charlie’s strategy in the counterfactual would be thesame as that in an optimal equilibrium so that considering the benchmark only will notallow us to identify whether the listener fully understand the optimal, context-dependentlanguage or not. We thus need further variations for sharper identification.

Variation 1 (Charlie hears Alice).

Consider a variation of the Benchmark: the only difference is that now Alice’s messagesis also observable to Charlie. The equilibria in which no one babbles remain the same asin the Benchmark. In particular, it is notable that Charlie’s strategy does not depend onAlice’s messages despite it being available.

In this variation, if Charlie believes that Bob uses his language in the optimal, literallyvague way, the former’s choice should not depend on Alice’s messages, because the infor-mation contained in Alice’s messages is fully incorporated into Bob’s messages throughcontext-dependence. Thus we can tease out whether Charlie correctly interprets Bob’smessages according to the optimal language by checking whether Charlie’s choice dependson Alice’s messages.

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Variation 2 (Charlie chooses from three actions).

In this variation, Charlie has three actions to choose from: UP, MIDDLE and DOWN.UP is optimal if A +B ≥ 120, MIDDLE if 80 ≤ A +B ≤ 120, and DOWN if A +B ≤ 80. If theoptimal action is chosen the players all receive a payoff of 1, otherwise 0.

In equilibria in which no one babbles, Alice sends “A is High” if A > 50 and “A is Low”if A < 50.

• If Alice’s message is “A is High” then Bob sends “B is High” if B > 25 and “B is Low”if B < 25.

• If Alice’s message is “A is Low” then Bob sends “B is High” if B > 75 and “B is Low”if B < 75.

Charlie chooses UP seeing “B is High” and DOWN otherwise. Accordingly, the successrate is 63.5%. It should be noted that MIDDLE is never chosen in this equilibrium.

The purpose of introducing the third action is to make the conversational environmentmore complex, in particular for Bob and Charlie. The Benchmark and Variation 1 are rel-atively simple environments in which it is not supremely difficult to “compute” the optimalcutoffs.7 On the other hand, when people talk in real life they do not typically derive theoptimal language consciously. Thus we want to see, when it is more difficult to explic-itly derive the optimal language, whether people can still arrive at the optimal, literallyvague language and use it to effectively communicate, or whether they instead revert tocontext-independent language, which is simpler to use for the speaker and to understandfor the listener. Thus it is crucial to examine whether in this variation Bob’s message iscontext-dependent or not, and whether Charlie best responds or not.

Variation 3 (Charlie chooses from three actions and hears Alice).

This variation differs from Variation 2 in that Alice’s message is now observable toCharlie. In an optimal equilibrium, Alice sends “A is High” if A > 50 and “A is Low” ifA < 50. Bob’s strategy does not quite differ from that in Variation 2 qualitatively, but iswith different optimal cutoffs.

• If Alice’s message is “A is High” then Bob sends “B is High” if B > 45 and “B is Low”if B < 45.

7The key logic one can easily come with is that Alice would use a cutoff of 50 because of the symmetry ofthe problem. Thereafter the optimal cutoffs of 25 and 75 can be deduced simply by mind or at most by someback-of-envelope calculation.

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• If Alice’s message is “A is Low” then Bob sends “B is High” if B > 55 and “B is Low”if B < 55.

Charlie chooses UP seeing (“A is High”, “B is High”), DOWN seeing (“A is Low”, “B isLow”), and MIDDLE otherwise. The success rate is 78.5%. It should be noted that MID-DLE is chosen in this equilibrium when the messages from Alice and Bob disagree.

This variation serves two purposes. First, it allows us to study how the quantitativechange in the optimal cutoff values affects the use of languages. The optimal cutoff val-ues that are substantially closer each other generates a very minimal benefit of context-dependence relative to the context-independent counterfactual. In fact, the success ratefrom the context-independent counterfactual is 78%. Thus, this variation enables us tounderstand how individuals’ choice of context-dependent languages are guided by thesalience of incentives.

Variation 4 (Bob’s Messages are Imperative).

Consider a variation of the Benchmark in which we replace Bob’s messages “B is High”and “B is Low” by “Take UP” and “Take DOWN”. Clearly this change eradicates any pos-sibility of literal vagueness because Bob’s messages, now imperative, have unambiguousliteral interpretations with respect to the decision problem at hand. Apart from the differ-ence in literal interpretation of the messages the variation is the same as the Benchmark.Hence the variation serves as a nice control version of the Benchmark. In particular, itallows us to test whether literal vagueness may intimidate players from using the optimallanguage.

In the experiments, we create not only an “imperative messages” version of the Bench-mark, but also that of Variation 2.

3 Experimental Implementation

3.1 Experimental Design and Hypotheses

The benchmark game and its variants introduced in the previous section constitute ourexperimental treatments. Our experiment features a (2 × 2) + (2 × 1) treatment design(Table 1). The first treatment variable concerns the number of actions available to thereceiver (Charlie) and the second treatment variable concerns whether Alice’s messagesare observed by Charlie or not. The third treatment variable concerns whether Bob’s

15

messages are framed to be indicative or imperative.8 We consider the treatments withimperative messages as a robustness check so that we omit the corresponding treatmentsin which Alice’s messages are observed by Charlie.

Table 1: Experimental Treatments

Indicative Messages from Bob

Alice’s messages# of Actions

Two ThreeUnobservable 2A-U-IND 3A-U-INDObservable 2A-O-IND 3A-O-IND

+Imperative Messages from Bob

Alice’s messages# of Actions

Two ThreeUnobservable 2A-U-IMP 3A-U-IMP

Our first experimental hypothesis concerns the overall outcome of the communicationgames represented by the success rate. Let S(T ) denote the average success rate of Treat-ment T . Postulating that the optimal equilibria are played in each game, we have thefollowing hypothesis.

Hypothesis 1 (Success Rate). S(2A-O-IND) = S(2A-U-IND) = S(2A-U-IMP) > S(3A-O-IND)> S(3A-U-IND) = S(3A-U-IMP)

This hypothesis can be decomposed into two sub-hypotheses. First, the observability ofAlice’s message to Charlie does not affect the success rate in the treatments with twoactions (S(2A-O-IND) = S(2A-U-IND)) while the very same observability increases thesuccess rate in the treatments with three actions (S(3A-O-IND) > S(3A-U-IND)). Second,the imperativeness of Bob’s messages does not affect the success rate (S(2A-U-IND) = S(2A-U-IMP) and S(3A-U-IND) = S(3A-U-IMP)).

Our second hypothesis considers the counterfactual in which Bob is constrained to becontext-independent and thus always uses the cutoff of 50.9 In the counterfactual sce-nario, the success rates are 75% in Treatments 2A-O-IND, 2A-U-IND, and 2A-U-IMP,78% in Treatment 3A-O-IND, and 55% in Treatments 3A-U-IND and 3A-U-IMP. If theplayers effectively use the optimal, literally vague language, the success rates should besignificantly above the levels predicted by the counterfactual. Thus, we have the followinghypothesis.

Hypothesis 2 (Counterfactual Comparison).

1. S(2A-O-IND), S(2A-U-IND), S(2A-U-IMP) > 75%8For example, Bob’s message spaces in Treatments 2A-U-IND and 2A-U-IMP are “B is HIGH”, “B is

LOW” and “Take UP”, “Take DOWN”, respectively.9Charlie’s optimal strategy remains the same regardless of whether Bob is constrained to be context-

independent or not.

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2. S(3A-O-IND) > 78%

3. S(3A-U-IND), S(3A-U-IMP) > 55%

Note that the success rate predicted by the optimal, literally vague language in Treat-ment 3A-O-IND is 78.5%, which is not substantially different from the 78% predicted bythe counterfactual. The net benefit of the context-dependent language measured with re-spect to the success rates is only 0.5% (= 78.5% - 78%) in Treatment 3A-O-IND. The netbenefit of context-dependence becomes substantially larger in other treatments as it is12.5% (= 87.5% - 75%) in Treatments 2A-O-IND, 2A-U-IND, and 2A-U-IMP, and 6.5% (=63.5% - 55%) in Treatments 3A-U-IND and 3A-U-IMP.

Our third hypothesis concerns Alice’s message choices. For all treatments, the optimalequilibrium play predicts that Alice employs the simple cutoff strategy in which she sends“A is High” if A > 50 and “A is Low” if A < 50. Let PAlice(m∣A) denote the proportion ofAlice’s message m given the realized number A. Then we have the following hypothesis.

Hypothesis 3 (Alice’s Messages). Alice’s message choices observed in all treatments arethe same. Moreover, PAlice(“A is Low”∣A) = 1 for any A < 50 and PAlice(“A is High”∣A) = 1 forany A > 50.

Our next hypothesis concerns Bob’s message choices. The optimal equilibrium play pre-dicts that Bob’s message choices depend on the context, i.e., which message he receivedfrom Alice. To state our hypothesis clearly, let Pm′

Bob(m∣B) denote the proportion of Bob’smessagem given the realized numberB and Alice’s messagem′ ∈ “A is High”, “A is Low”.Define

CD(B) = PLBob(m∣B) − PH

Bob(m∣B)where m is “B is Low” for the treatments with indicative messages and “Take DOWN”for the treatments with imperative messages. CD(B) measures the degree of context-dependence of Bob’s message choices given the realized numberB. Bob’s optimal, context-dependent strategy implies that there is a range of number B under which Bob’s messagechoices differ depending on Alice’s messages, i.e., CD(B) = 1. Such intervals are [45,55]for Treatment 3A-O-IND and [25,75] for all other treatments. It is worthwhile to note thatCD(B) = 0 for any B ∈ [0,100] if Bob uses a context-independent strategy. Thus, we havethe following hypothesis:

Hypothesis 4 (Bob’s Messages). Bob’s messages are context-dependent in such a way thatis predicted by the optimal equilibrium of each game. More precisely,

1. For each treatment T , there exists an interval [XT , Y T ] with XT > 0 and Y T < 100 suchthat CD(B) > 0 for any B ∈ [XT , Y T ] and CD(B) = 0 otherwise.

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2. The length of the interval [XT , Y T ] is significantly smaller in Treatment 3A-O-INDthan in any other treatments.

Our last hypothesis concerns if the listener, Charlie, can correctly interpret the mes-sages. In particular, Charlie’s strategy in the optimal equilibrium does not depend onwhether or not Alice’s messages are observable to Charlie in the treatments with two ac-tions. In the games with three actions, however, the observability of Alice’s message toCharlie matters. Precisely, the optimal equilibrium predicts that MIDDLE should not betaken by Charlie in Treatments 3A-U-IND and 3A-U-IMP whereas MIDDLE is taken inTreatment 3A-O-IND when the messages from Alice and Bob do not coincide. Thus, wehave the following hypothesis.

Hypothesis 5 (Charlie’s Action Choices). In the treatments with two actions, Charlie’saction choices do not depend on whether Alice’s messages are observable or not. In thetreatments with three actions, MIDDLE is taken only in Treatment 3A-O-IND when themessages from Alice and Bob do not coincide.

3.2 Procedures

Our experiment was conducted in English using z-Tree (Fischbacher (2007)) at the HongKong University of Science and Technology. A total of 138 subjects who had no prior ex-perience with our experiment were recruited from the undergraduate population of theuniversity. Upon arrival at the laboratory, subjects were instructed to sit at separatecomputer terminals. Each received a copy of the experimental instructions. To ensurethat the information contained in the instructions was induced as public knowledge, theinstructions were read aloud, aided by slide illustrations and a comprehension quiz.

We conducted one session for each treatment. In all sessions, subjects participated in21 rounds of play under one treatment condition. Each session had 21 or 24 participantsand thus involved 7 or 8 fixed matching groups of three subjects, one Member A (Alice), oneMember B (Bob), and one Member C (Charlie). Thus, we used the fixed-matching protocoland between-subject design. As we regard each group in each session as an independentobservation, we have seven to eight observations for each of these treatments, which pro-vide us with sufficient power for non-parametric tests. At the beginning of a session, onethird of the subjects were randomly labeled as Member A, another one third labeled asMember B and the remaining one third labeled as Member C. The role designation re-mained fixed throughout the session.

We illustrate the instructions for Treatment 2A-U-IND. The full instructions can befound in Appendix A. For each group, the computer selected two integer numbers A and

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B between 0 and 100 (uniformly) randomly and independently. Subjects were presentedwith a two-dimensional coordinate system (with A in the horizontal coordinate and B inthe vertical coordinate) as in Figures 5(a) and 5(b) in Appendix A. The selected number Awas revealed only to Member A and the selected number B was revealed only to MemberB. Member A sent one of two messages, “A is Low” and “A is High”, to Member B but notto Member C. After observing both the selected number B and the message from MemberA, Member B sent one of two messages, “B is Low” and “B is High”, to Member C whothen took one of two actions, UP and DOWN. The ideal actions for all three players wereUP when A +B > 100 and DOWN when A +B < 100.10 Every member in a group received50 ECU if the ideal action was taken and 0 ECU otherwise.

For Rounds 1-20, we used the standard choice-method so that each participant firstencountered one possible contingency and specified a choice for the given contingency.For example, Member A decided what message to send after seeing the randomly selectednumberA. Member B decided what message to send after observing the randomly selectednumber B and the message from Member A. Similarly, Member C decided what action totake after receiving the message from Member B. For Round 21, however, we used thestrategy-method and elicited beliefs of players. For the belief elicitation, a small amountof compensation (in the range between 2 ECU and 8 ECU) was provided for each correctguess.11 For more details, see the selected sample scripts for the strategy-method and thebelief-elicitation provided in Appendix B.

We randomly selected two rounds out of the 21 total rounds for each subject’s payment.The sum of the payoffs a subject earned in the two selected rounds was converted intoHong Kong dollars at a fixed and known exchange rate of HK$1 per 1 ECU. In additionto these earnings, subjects also received a show-up payment of HK$30. Subjects’ totalearnings averaged HK$103.5 (≈ US$13.3).12 The average duration of a session was about1 hour.

4 Experimental Findings

We report our experimental results as a number of findings that address our hypothesesas set forth in Section 3.1.

10To make the likelihood of each action being ideal exactly equal across two actions, we set both actions tobe ideal when A +B = 100.

11Although we were aware of the fact that an appropriate incentive-compatible mechanism is needed toelicit beliefs correctly, we took this simple elicitation procedure because of its simplicity as well as the factthat the belief and strategy data were only secondary data mainly for the purpose of robustness checks.

12Under the Hong Kong’s currency board system, the Hong Kong dollar is pegged to the US dollar at therate of HK $7.8 = US$1.

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4.1 Overall Outcome

Note: The red bars depict the theoretical predictions from the optimal,literally vague equilibria. The red dotted lines depict the predictions fromthe counterfactual in which Bob is constrained to be context-independent.

Figure 1: Average Success Rate

Figure 1 reports the average success rates aggregated across all rounds and all match-ing groups for each treatment. It also presents the theoretical predictions from the optimalequilibrium depicted by the red bars and the predictions from the counterfactual in whichBob is constrained to be context-independent depicted by the dotted lines. A few observa-tions were apparent. First, non-parametric Mann-Whitney test reveals that the successrates in Treatment 2A-O-IND and in Treatment 2A-U-IND were not statistically differ-ent (81.6% vs. 83.6%, two-sided, p-value = 0.6973). On the contrary, the success rate inTreatment 3A-O-IND was 73.8%, which is significantly higher than 54.2% in Treatment3A-U-IND (Mann-Whitney test, p-value = 0.0267). This observation is consistent with Hy-pothesis 1 that the observability of Alice’s message to Charlie affects the success rate onlyin the treatments with three actions.

Second, there was no significant difference in the success rates between Treatment2A-U-IND and Treatment 2A-U-IMP (83.6% vs. 85.7%, two-sided Mann-Whitney test, p-value = 0.5989) and between Treatment 3A-U-IND and Treatment 3A-U-IMP (54.2% vs.51.2%, two-sided Mann-Whitney test, p-value = 0.2237). This observation is also consistentwith Hypothesis 1 that imperativeness of Bob’s message does not affect the success rateregardless of the number of available actions. Confirming Hypothesis 1, we thus have ourfirst finding as follows:

Finding 1. Observability of Alice’s message to Charlie affected the success rate only in thetreatments with three actions. Imperativeness of Bob’s message did not affect the successrate regardless of the number of available actions.

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Figure 1 seems to suggest that the success rates observed in the three treatments withtwo actions (hereafter Treatments 2A) are better approximated by the predictions fromthe optimal, context-dependent equilibrium languages than by the predictions from thecontext-independent counterfactual. Indeed, we cannot reject the null hypothesis that thesuccess rates are not different from 87.5%, the predicted value from the optimal equilib-rium (two-sided Mann-Whitney tests, p-values are 0.8262, 0.5076, 1.000 for Treatments2A-O-IND, 2A-U-IND and 2A-U-IMP, respectively). Even if we can reject the alterna-tive hypothesis that the success rates are significantly higher than the predicted levelof 75% from the context-independent counterfactual only for Treatment 2A-U-IND (one-sided Mann-Whitney tests, p-values are respectively 0.2551, 0.0610, 0.2174 for Treatments2A-O-IND, 2A-U-IND and 2A-U-IMP), the p-values resulted from the non-parametric anal-ysis suggest that the optimal, context-dependent equilibrium is a better predictor of theresults observed from these treatments.

On the other hand, we do not have the same observation from the three treatmentswith three actions (hereafter Treatments 3A), especially those with unobservable mes-sages from Alice. We cannot reject the null hypothesis that the success rates observed inthese treatments are the same as the success rates predicted by the context-independentcounterfactual (two-sided Mann-Whitney tests, p-values are 0.4347, 0.6936, 0.4347 respec-tively for Treatments 3A-O-IND, 3A-U-IND and 3A-U-IMP). Moreover, the success ratesobserved in Treatments 3A-U-IND and 3A-U-IMP were respectively 54.2% and 51.2%,which are substantially lower than the predicted level of 63.5% from the optimal, literallyvague equilibrium language. Although the difference is statistically insignificant (two-sided Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively), the p-valuesgenerated from the non-parametric analysis suggest that the context-independent coun-terfactual is a better predictor of the results from Treatments 3A-U-IND and 3A-U-IMP.13

Thus, we have the following result:

Finding 2. The average success rates observed in Treatments 2A-O-IND, 2A-U-IND, and2A-U-IMP were higher than the predicted level from the counterfactual in which Bob iscontext-independent. The average success rates observed in Treatments 3A-O-IND, 3A-U-IND, and 3A-U-IMP were lower than the predicted level from the counterfactual. However,the difference between the observed success rate and the prediction from the counterfactualis statistically significant only in Treatment 2A-U-IND.

Among Treatments 3A, more substantial deviations from the optimal, context-dependentequilibrium were observed in Treatments 3A-U-IND and 3A-U-IMP. This observed devi-

13The success rates observed in Treatments 3A-O-IND was 73.8%, which is not significantly different fromthe predicted level of 78.5% from the optimal, literally vague equilibrium language (two-sided Mann-Whitneytest, p-value = 0.4347).

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ation in the average success rates may imply that the context-dependent, literally vaguelanguages were not emerged in those treatments, probably due to the complexity of the en-vironment considered. However, another completely plausible scenario would be that theobserved deviation originates from a different source, such as Charlie’s choices not beingconsistent with the optimal equilibrium. Without taking a careful look at the individualbehavior, it is impossible to draw any meaningful conclusion. Hence, in the subsequentsections, we shall look at individual players’ choices.

4.2 Alice’s Behavior

Figure 2 reports Alice’s message strategies by presenting the proportion of each messageas a function of the number A where data were grouped into bins by the realization ofnumberA (e.g., [0,5), [5,10), ..., etc.). Figure 2(a) provides the aggregated data from Treat-ments 2A while Figure 2(b) provides the aggregated data from Treatments 3A. The samefigures separately drawn for each treatment can be found in Appendix C.

0

20

40

60

80

100

[0,5)

[5,10)

[10,15)

[15,20)

[20,25)

[25,30)

[30,35)

[35,40)

[40,45)

[45,50)

[50,55)

[55,60)

[60,65)

[65,70)

[70,75)

[75,80)

[80,85)

[85,90)

[90.95)

[95,100]

Prop

or%on

Number A

Alice's Message -­‐ Treatments 2A

LOW

HIGH

(a) Treatments 2A

0

20

40

60

80

100

[0,5)

[5,10)

[10,15)

[15,20)

[20,25)

[25,30)

[30,35)

[35,40)

[40,45)

[45,50)

[50,55)

[55,60)

[60,65)

[65,70)

[70,75)

[75,80)

[80,85)

[85,90)

[90.95)

[95,100]

Prop

or%on

Number A

Alice's Message -­‐ Treatments 3A

LOW

HIGH

(b) Treatments 3A

Note: The red dotted lines illustrate the optimal equilibrium strategy with cutoff of 50.

Figure 2: Alice’s Messages

From these two figures, it was immediately clear that the subjects whose designatedroles were Alice in our experiments tended to use cut-off strategies well approximated bythe optimal cutoff of 50. Using the matching-group level data from all rounds for eachbin of the realized number A (e.g. [0,5), [5,10), ..., etc.) as independent data points foreach treatment, a set of (two-sided) Mann-Whitney tests reveals that 1) for any bins ofA below 50, we cannot reject the null hypothesis that the proportion of message “A isLow” being sent was 100%, and 2) for any bins of A above 50, we cannot reject the nullhypothesis that the proportion of message “A is Low” being sent was 0%. Among 20 binsin each treatment, the p-values for 14-17 bins were 1.0 while the lowest p-value for each

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treatment was ranged between 0.2636 and 0.5637. Confirming our Hypothesis 3, we thushave the following result:

Finding 3. For any treatment, PAlice(“Low”∣A) = 1 for any A < 50 and PAlice(“High”∣A) = 1

for any A > 50.

The elicited strategies and beliefs reported in Figure 10 in Appendix C provided ad-ditional supports for Finding 3. We cannot reject the null hypothesis that the reportedcutoff values for Alice’s strategy and other player’s reported beliefs for Alice’s strategy inall treatments were the same as the optimal equilibrium cutoff of 50 (two-sided Mann-Whitney tests, p-values are in the range between 0.4533 and 1.000).

4.3 Bob’s Behavior

We now look at Bob’s behavior. Recall that PLBob(m∣B) denoted the proportion of Bob’s

message m given the realized number B and Alice’s message “A is Low”, and PHBob(m∣B)

denoted that given Alice’s message “A is High”. We introduced the measure for the context-dependence as CD(B) = PL

Bob(m∣B) − PHBob(m∣B) where m is “B is Low” for the treatments

with indicative messages and “Take DOWN” for the treatments with imperative messages.Figures 3(a)-(f) illustrate the distributions of CD(B) over the realization of number B,aggregated across all matching groups for each treatment. Again, the data from all roundswere grouped into bins by the realization of number B (e.g., [0,5), [5,10), ..., etc.).

The optimal, context-dependent equilibrium language implies that there exists an in-terval (strictly interior of the support of B) such that the value of CD(B) is 1 if B is inthe interval and 0 otherwise. Moreover, the boundaries of such intervals are determinedby the equilibrium cutoff strategy so that the interval is narrower in Treatment 3A-O-IND. The exact prediction of the distribution of CD(B) made by the optimal equilibriumis illustrated by the red-dotted lines in Figures 3(a)-(f).14

These figures convincingly visualize the fact that, in each treatment, there existed aninterval in which the value of CD(B) is strictly positive. For Treatment 2A-O-IND, forinstance, we cannot reject the null hypothesis that CD(B) = 0 for B ∈ [0,30) and for B ∈[60,100] (two-sided Mann-Whitney test, both p-values = 1.00). However, for B ∈ [30,60), wecan reject the null hypothesis that CD(B) = 0 in favor of the alternative that CD(B) > 0

(one-sided Mann-Whitney test, p-value = 0.04).15 Similarly, for Treatments 2A-U-IMP and14Figures 8(a)-(f) and 9(a)-(f) presented in Appendix C separately report the distributions of

PLBob(“B is Low”∣B) and of PH

Bob(“B is Low”∣B) for each treatment.15To conduct Mann-Whitney tests for Hypothesis 4, we first eyeball Figures 3(a)-(f) to identify the plausible

choices of the interval with CD(B) > 0 for each treatment. For example, for Treatment 2A-O-IND, relying

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(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines present the predicted distribution from the optimal equilibrium strategy.

Figure 3: Bob’s Message Strategy

3A-U-IMP, we cannot reject the null hypothesis that CD(B) = 0 for the intervals of [25,75)(p-values are respectively 0.052 and 0.059) in favor of the alternative that CD(B) > 0.Qualitatively the same but less significant results were obtained from Treatment 2A-U-IND with the non-zero interval of [25,70) (p-value = 0.136) and from Treatment 3A-U-INDwith the non-zero interval of [25,75) (p-value = 0.121). Thus, we have the following result:

on Figure 3(a), we divide the support of number B into three intervals – [0,30), [30,60), and [60,100]. Wenext calculate the value of CD(B) for each of the three intervals for each matching group. Taking thosevalues as group-level independent data points for each treatment, we conducted the non-parametric test.

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Finding 4. In Treatments 2A-O-IND, 2A-U-IND, 2A-U-IMP, 3A-U-IND, and 3A-U-IMP,there existed an interval [X,Y ] with X > 0 and Y < 100 such that CD(B) > 0 if B ∈ [X,Y ]and CD(B) = 0 otherwise.

It is necessary to discuss the data from Treatment 3A-O-IND in Figure 3(b) more care-fully. First, as predicted by the optimal equilibrium strategy, the interval that has non-zero value of CD(B) seemed to shrink significantly compared to any other treatments. In-deed, we cannot reject the null hypothesis thatCD(B) = 0 forB ∈ [0,45) and forB ∈ [50,100](two-sided Mann-Whitney test, p-values are 1.00 and 0.7237, respectively). However, asubstantial deviation from the theoretical prediction was observed such that the reportedvalue for the bin [50,55) was negative.16 This observation was driven by the fact that theobserved cutoff value from Bob’s strategy conditional on Alice’s message “A is Low” was50, which may look more focal than the theoretically optimal cutoff of 45.17

The elicited strategies and beliefs reported in Figure 11 in Appendix C provide furthersupporting evidence. Wilcoxon signed-rank tests reveal that the reported cutoff valuesgiven Alice’s message “A is Low” were significantly higher than the cutoff values givenAlice’s message “A is High” for most of the treatments (p-values are ranged between 0.0004and 0.0204) except for Treatment 3A-O-IND and Treatment 3A-U-IND.18 The fact that thereported cutoff values in Treatment 3A-O-IND were not significantly different (two-sided,p-value = 0.1924) is not surprising at all because the optimal equilibrium cutoff values are45 and 55, distinctively closer each other than the predicted values for all other treatments.The insignificant result for Treatment 3A-U-IND (two-sided, p-value = 0.3561) mainlyoriginated from two observations that the reported cutoffs from two Charlie-subjects were50 and 45 given “A is Low” and 85 and 80 given “A is High”.

4.4 Charlie’s Behavior

Figure 4 reports Charlie’s action choices by presenting the proportion of each action asa function of information available to Charlie. Figure 4(a) presents the data aggregatedacross all matching groups of Treatments 2A while Figure 4(b) presents the data aggre-gated across all matching groups of Treatments 3A.

16For Treatment 3A-O-IND, we cannot conduct any meaningful statistical analysis for B ∈ [45,50) becausethere are only two group-level independent data points.

17Similarly, two substantial deviations were observed in Treatment 3A-U-IND - in the first bin of [0,5)and the ninth bin of [40,45) in Figure 3(d). The first deviation was solely driven by the single data pointwith B ∈ [0,5) in which Bob sent “B is High” after receiving “A is High” from Alice. The second deviationwas solely driven by the single data point with B ∈ [40,45) in which Bob sent “B is Low” after receiving “Ais High” from Alice.

18To conduct Wilcoxon signed-rank tests, we pooled the data from the reported cutoff values for Bob’sstrategy and other players’ reported beliefs.

25

(a) Treatments 2A (b) Treatments 3A

Note: The red bars present the theoretical predictions from the Pareto-optimal equilibria.

Figure 4: Charlie’s Actions

A few observations emerged immediately from these figures. First, Figure 4(a) revealsthat observed strategy by Charlie depended, to some but limited extent, on whether or notAlice’s messages were observable to him in Treatments 2A. In Treatment 2A-O-IND, theproportion of UP being chosen given Bob’s message “B is High” was about 52% and theproportion of DOWN being chosen given Bob’s message “B is Low” was about 75%, both aresubstantially and significantly different from 100% predicted by the optimal equilibrium.However, the observed proportions became 63% and 100% if we took the last three roundsdata only, showing that learning took place toward the right direction.19

Second, Figure 4(b) reveals that MIDDLE was taken by Charlie in Treatment 3A-O-IND when the messages from Alice and Bob did not coincide. The proportions of MIDDLEbeing chosen by Charlie given the message combinations (“A is High”, “B is Low”) and(“A is Low”, “B is High”) were higher than 90%. However, inconsistent with the predictionfrom the optimal equilibrium strategy, MIDDLE was taken even in Treatments 3A-U-INDand 3A-U-IMP. The proportions of MIDDLE being taken observed in these two treatmentsvaried between 24% and 43% which are substantially larger than 0%. This observed devi-ation seemed persistent as it did not disappear even when we took the data from the lastthree rounds only.20 Thus, we have the following result:

Finding 5. Charlie’s observed action choices in Treatment 2A-O-IND were not the sameas those observed in Treatment 2A-U-IND, showing that observability of Alice’s message to

19In an early stage of the project, we have conducted two sessions in which subjects participated in thefirst 20 rounds with the treatment condition of 2A-U-IND and in the second 20 rounds with the treatmentcondition of 2A-O-IND. The data from the second 20 rounds were almost perfectly consistent with the the-oretical prediction, showing another convincing evidence of learning. Data from this additional treatmentare available upon request.

20The elicited strategies and beliefs presented in Figure 12 in Appendix C were highly consistent with theresults in Finding 5.

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Charlie mattered. For Treatments 3A, MIDDLE was taken in Treatment 3A-O-IND whenCharlie received different messages from Alice and Bob. However, a substantial proportionof MIDDLE was observed even in Treatments 3A-U-IND and 3A-U-IMP.

This observed discrepancy in Charlie’s action choices between our data and the pre-diction from the optimal equilibrium was the main source of the lower success rates wehad in Treatments 3A-U-IND and 3A-U-IMP (see Figure 1). The success rates observedin Treatments 3A-U-IND and 3A-U-IMP were respectively 54.2% and 51.2%, which aresubstantially lower than the predicted level of 63.5%, although the difference is statisti-cally insignificant (Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively). Ifwe replace the observed empirical choices by Charlie with the hypothetical choices fromthe optimal strategy, the success rates in Treatments 3A-U-IND and 3A-U-IMP become58.3% and 56.0% respectively, both are substantially higher than the observed levels.

Note that MIDDLE was taken slightly more often in Treatment 3A-U-IMP than inTreatment 3A-U-IND (31% and 24% vs. 33% and 43%). The elicited strategies and be-liefs presented in Figure 12 in Appendix C demonstrate the difference more vividly. Thisdifference may come from the fact that we framed Bob’s messages as “Don’t take UP” and“Don’t take DOWN” to impose imperativeness to the messages for Treatment 3A-U-IMP.

4.5 Emergence of Context-dependent, Literally Vague Languages

In this section, we combine our findings presented in the previous sections to establishthe emergence of context-dependent, literally vague languages. Admittedly, we did notpresent a perfect match between our data and the prediction from the optimal, context-dependent equilibrium language. However, we provided convincing evidence that overallbehavior observed in our laboratory was qualitatively consistent with the prediction. Moreprecisely,

1. Finding 3 illustrated that Alice tended to use cut-off strategies well approximated bythe cut-off value of 50. Thus, contexts are properly defined.

2. Finding 4 showed that Bob tended to use context-dependent strategies.

3. Finding 5 suggested that Charlie understood the messages from the speaker(s) welland took actions in a manner that is qualitatively consistent with the optimal equi-librium strategy.

4. Finding 5 also revealed that the lower success rates observed in Treatments 3A-U-IND and 3A-U-IMP reported in Finding 2 were largely driven by Charlie’s behavior

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being partially inconsistent with the theoretical predictions from the optimal equi-librium.

Taking these findings together, we establish the following result:

Finding 6. Literally vague languages emerged in our experiment. The way it was used bythe speakers and understood by the listener was all consistent with the prediction from thePareto-optimal equilibria of the communication games.

5 Concluding Remarks

In this paper we introduce the notion of context-dependence and literal vagueness, andoffer our explanation of why language is vague. We show that literal vagueness arises in aPareto-optimal equilibrium in many standard conversational situations. Our experimen-tal data provide supporting evidence for the emergence of literally vague languages.

Although our discussion of linguistic vagueness focuses on the environment in whichplayers’ preferences are perfectly aligned, the theoretical discussions presented in Blumeet al. (2007) and Blume and Board (2014b) suggest that the communicative advantage ofliteral vagueness would be extended to the environment with conflicts of interests. Webelieve that experimentally investigating the role of vagueness in the presence of conflictof interests is an interesting avenue for future research.

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ReferencesAgranov, Marina and Andrew Schotter (2012), “Ignorance is bliss: An experimental study

of the use of ambiguity and vagueness in the coordination games with asymmetric pay-offs.” American Economic Journal: Microeconomics, 4, 77–103.

Blume, Andreas and Oliver Board (2013), “Language barriers.” Econometrica, 81, 781–812.

Blume, Andreas and Oliver Board (2014a), “Higher-order uncertainty about language.”Working paper.

Blume, Andreas and Oliver Board (2014b), “Intentional vagueness.” Erkenntnis, 79, 855–899.

Blume, Andreas, Oliver J. Board, and Kohei Kawamura (2007), “Noisy talk.” TheoreticalEconomics, 2, 395–440.

Blume, Andreas, Ernest K. Lai, and Wooyoung Lim (2017), “Strategic information trans-mission: A survey of experiments and theoretical foundations.” Working paper.

Crawford, Vincent and Joel Sobel (1982), “Strategic information transmission.” Economet-rica, 50, 1431–51.

Fischbacher, Urs (2007), “z-tree: Zurich toolbox for ready-made economic experiments.”Experimental Economics, 10, 171–178.

Lewis, David (1969), Convention: A Philosophical Study. Harvard University Press.

Lipman, Barton L. (2009), “Why is language vague.” Working paper.

Quine, Willard V. O. (1960), Word and object. Technology Press of the Massachusetts In-stitute of Technology, Cambridge.

Serra-Garcia, Marta, Eric van Damme, and Jan Potters (2011), “Hiding an inconvenienttruth: Lies and vagueness.” Games and Economic Behavior, 73, 244 – 261.

Smith, Lones, Peter N. Sørensen, and Jianrong Tian (2016), “Informational herding, op-timal experimentation, and contrarianism.”

Tian, Jianrong (2016), “Monotone comparative statics for cut-offs.” Working paper.

Wilson, Andrea (2014), “Bounded memory and biases in information processing.” Econo-metrica, 82, 2257–2294.

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Wood, Daniel H. (2016), “Communication-enhancing vagueness.” Working paper.

Wu, Qinggong (2016), “Coarse communication and institution design.” Working paper.

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Appendices

A Experimental Instructions - Treatment 2A-U-IND

INSTRUCTION

Welcome to the experiment. This experiment studies decision making between threeindividuals. In the following two hours or less, you will participate in 21 rounds of decisionmaking. Please read the instructions below carefully; the cash payment you will receiveat the end of the experiment depends on how well you make your decisions according tothese instructions.

Your Role and Decision Group

There are 24 participants in today’s session. One third of the participants will be ran-domly assigned the role of Member A, another one third the role of Member B, and theremaining the role of Member C. Your role will remain fixed throughout the experiment.At the beginning of the first round, three participants, one Member A, one Member B andone Member C, will be matched to form a group of three. The three members in a groupmake decisions that will affect their rewards in all 21 rounds. That is, you will stay in thesame group so that you will interact with the same two other participants throughout the21 rounds. You will not be told the identity of the participants in your group, nor will theybe told your identity—even after the end of the experiment.

Your Decision and Earning in Each of Round 1-20

In each round and for each group, the computer will select two integer numbers A and Bbetween 0 and 100 randomly and independently. Each possible number has equal chanceto be selected. The selected number A will be revealed only to Member A and the selectednumber B will be revealed only to Member B. Member C, without seeing any of thesenumbers, will have to choose one of two actions UP and DOWN.

The amount of Experimental Currency Unit (ECU) you earn in a round depends on thetwo numbers A and B as well as the action chosen by Member C. In particular,

1. When A +B > 100, if Member C chooses

(a) UP, every member in your group will receive 50 ECU.

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(b) DOWN, every member in your group will receive 0 ECU.

2. When A +B < 100, if Member C chooses

(a) DOWN, every member in your group will receive 50 ECU.

(b) UP, every member in your group will receive 0 ECU.

3. When A+B = 100, every member in your group will receive 50 ECU regardless of theaction chosen by Member C.

Member A’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 5(a). The horizontal axis represents the number A and the vertical axis representsthe number B. You will see a blue vertical line, which represents the actually selectednumber A in the horizontal axis. The red diagonal line represents the cases with A +B =100.

With all this information on your screen, you will be asked to send one of two messages“A is LOW” and “A is HIGH” to Member B in your group. Once you click one of the messagebuttons, your decision in the round is completed and your message will be transmitted toMember B in your group.

(a) Member A’s Screen (b) Member B’s Screen

Figure 5: Screen Shots

Member B’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 5(b). The horizontal axis represents the number A and the vertical axis representsthe number B. You will see a blue horizontal line, which represents the actually selectednumber B in the vertical axis. The red diagonal line represents the cases with A+B = 100.

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You will also receive a message from Member A in your group. With all this informationon your screen, you will be asked to send one of two messages “B is LOW” and “B is HIGH”to Member C in your group. Once you click one of the message buttons, your decision inthe round is completed and your message will be transmitted to Member C in your group.

Member C’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 6. The horizontal axis represents the number A and the vertical axis representsthe number B. The red diagonal line represents the cases with A +B = 100.

You will receive a message from Member B in your group. With all this information onyour screen, you will be asked to take one of two actions DOWN and UP. Once you clickone of the action buttons, your decision in the round is completed.

Figure 6: Member C’s Screen

Information Feedback

At the end of each round, the computer will provide a summary for the round: actuallyselected numbers A and B, Member A’s message, Member B’s message, Member C’s actionchoice, and your earning in ECU.

Your Decision in Round 21

After the 20th round, your screen will provide further instructions for your decisions inRound 21. The game you are going to play in this round is essentially the same as before,but you need to follow some new procedures. Please read the instructions carefully beforeyou start the 21st round. You will have an opportunity to ask questions if anything isunclear about the new instructions.

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Your Cash Payment

To calculate your cash payment, the experimenter will randomly select two rounds tocalculate your cash payment. Each round between Rounds 1 and 21 has an equal chanceto be selected. So it is in your best interest to take each round seriously. Your total cashpayment at the end of the experiment will be the sum of ECU you earned in the twoselected rounds, translated into HKD with the exchange rate of 1 ECU = 1 HKD, plus a30 HKD show-up fee.

Quiz and Practice

To ensure your understanding of the instructions, we will provide you with a quiz andpractice round. We will go through the quiz after you answer it on your own.

You will then participate in 1 practice round. The practice round is part of the in-structions which is not relevant to your cash payment; its objective is to get you familiarwith the computer interface and the flow of the decisions in each round. Once the practiceround is over, the computer will tell you “The official rounds begin now!”

Administration

Your decisions as well as your monetary payment will be kept confidential. Rememberthat you have to make your decisions entirely on your own; please do not discuss yourdecisions with any other participants. Upon finishing the experiment, you will receiveyour cash payment. You will be asked to sign your name to acknowledge your receipt ofthe payment. You are then free to leave. If you have any question, please raise your handnow. We will answer your question individually.

1. Suppose you are assigned to be a Member A. The computer chooses the random numbers A = 25 andB = 50. Which of the following is true?

(a) Both you and Member B know the chosen numbers A and B but Member C does not know anyof the numbers.

(b) Neither you nor Member B knows the chosen numbers A and B.

(c) You are the only person in your group who knows the chosen number A and Member B is theonly person in your group who knows the chosen number B.

2. Suppose that the computer chooses the random numbers A = 25 and B = 50. Member C in your grouptakes action DOWN. Please calculate the earning for each player:

• Member A’s payoff:

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• Member B’s payoff:

• Member C’s payoff:

3. Suppose that the computer chooses the random numbers A = 60 and B = 73. Member C in your grouptakes action DOWN. Please calculate the earning for each player:

• Member A’s payoff:

• Member B’s payoff:

• Member C’s payoff:

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B Scripts for Strategy-method and Belief-elicitation -Treatment 2A-U-IND

1. Strategy - Member A

In this round, we ask you to report your plan. After you specify your plan below, A will be realized and your plan will beimplemented accordingly.

Your plan: Send “A is LOW” if A is less than or equal to , and otherwise, send “A is HIGH”.What is the number for you in the blank above?

2. Belief - Member A

In this round, Member A is going to report his/her plan according to the following form:Send “A is LOW” if A is less than or equal to , and otherwise, send “A is HIGH”.

What do you think is the number for him/her in the blank above?If your guess is in the range of the actual value (chosen by Member A) plus-minus 5, then you will receive extra 8 ECU.

3. Strategy - Member B

In this round, we ask you to report your plan. After you specify your plan below, A and B will be realized and your plan willbe implemented accordingly.

(a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to , andotherwise, send “B is HIGH”.

(b) When receiving “A is HIGH” from Member A, send “B is LOW” if B is less than or equal to , andotherwise, send “B is HIGH”.

What is the number for you in the first blank above?What is the number for you in the second blank above?

4. Belief - Member B

In this round, Member B is going to report his/her plan according to the following form:(a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to , and

otherwise, send “B is HIGH”.(b) When receiving “A is HIGH” from Member A, send “B is LOW” if B is less than or equal to , and

otherwise, send “B is HIGH”.What do you think is the number for him/her in the blank in (a)?What do you think is the number for him/her in the blank in (b)?

If each of your guesses for (a) and (b) is in the range of the actual value (chosen by Member B) plus-minus 5, then you willreceive extra 4 ECU.

5. Strategy - Member C

In this round, we ask you to report your plan. After you specify your plan below, A and B will be realized and your plan willbe implemented accordingly.

What action would you like to take if the message from Member B is(a) B is LOW (b) B is HIGH

6. Belief - Member C

In this round, we ask Member C to report his/her plan about what action to take for each possible message.What action do you think would Member C like to take if the message from Member B is

(a) B is LOW (b) B is HIGHIf each of your guesses for (a) and (b) is correct, then you will receive extra 4 ECU.

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C Figures and Tables

(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 7: Alice’s Messages

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(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 8: Bob’s Messages ∣ “Low” from Alice

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(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 9: Bob’s Messages ∣ “High” from Alice

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Figure 10: Alice’s Elicited Strategy and Other Players’ Beliefs

Figure 11: Bob’s Elicited Strategy and Other Players’ Beliefs

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(a) Treatment 2A-O-IND

(b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Figure 12: Charlie’s Elicited Strategy and Other Players’ Beliefs’

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