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

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 linguistic ambiguity due to language used in a context-dependent way. This broadened notion, termed as literal vagueness, necessarily arises in any Pareto-optimal equilibrium in many standard conversational situations. With controlled laboratory experiments we find that people can make use of literally vague language to effectively transmit information. Keywords: Communication Games, Context-Dependence, Vagueness, Laboratory Experiments JEL classification numbers: C91, D03, D83 This study is supported by a grant from the Research Grants Council of Hong Kong (Grant No. GRF16502015). † 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 still lack 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 explain 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 deviating from the full-rationality paradigm, we take an alternative perspective. In particular, we broaden the notion of vagueness and show that vagueness in this broader sense has an advantage in many standard conversational situations. Vagueness language admits a borderline case. For example, no one can claim that there is a clear cutoff between red and orange that allows him/her to separate the color spectrum into two disjoint sets. Thus, if a speaker uses a vague language, listener’s posterior beliefs cannot be concentrated on a precise subset of a state space. Such vague languages cannot create an efficiency improvement from a non-vague language. We argue that in many cases, there is an exogenous relation between messages and a particular sub-dimension of states they conventionally signify. The other dimension with which the literal meaning of messages does not have any exogenous relation can be regarded as context. One important observation from our daily life conversations is that the presence of context and the use of a language dependent upon the context may impress a listener that the language is vague. We call such context-dependent use of language and its consequential vagueness literal vagueness. We show that literal vagueness can indeed be a fundamental property of the optimal language in many standard conversations in which the context — either directly payoff-relevant or not — matters. Thus literal vagueness has a solid efficiency foundation, which makes it a plausible explanation for why language is often vague. We then experimentally investigate whether people could make use of literally vague languages to efficiently communicate. We consider a conversational environment in which two speakers speak sequentially to a listener, and the way the later speaker talks may rely on what the earlier speaker has said. In this simple environment in which the optimal language is necessarily literally vague, we observed that subjects indeed tended to do so. Several variations of the environment with varying degrees of complexity in coordinating on the optimal, literally vague language provide further supporting evidence. Although our experimental result alone does not suffice to identify the efficiency advantage of literal 2 vagueness as the explanation for the omnipresence of vague languages, we view them as an important first step to understanding vagueness of language. In the rest of the introduction, we shall discuss the theoretical aspects of vagueness and introduce our notion of literal vagueness. Vagueness Take the simplest case of communication: A speaker wishes to tell the state of the world ω ∈ Ω to a listener by sending the latter a message m ∈ M , where Ω is the set of all states and 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 other words, 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 disjoint supports. Otherwise λ is vague. What is vagueness meant to capture? Note that if a language λ is not vague then for each message m ever used there is a unique block π(m) ∈ Πλ such that π(m) contains the states in which m is drawn with positive probability. Therefore the message m helps the listener precisely narrow down the possible states to π(m) in the sense that the listener’s posterior is simply his prior concentrated on π(m). If λ is vague then it does not induce such sharp demarcation of Πλ : Upon receiving some message m that is used with positive probability in distinct blocks π ∈ Πλ and π ′ ∈ Πλ the listener remains uncertain which block obtains. Theorem 1 of Lipman (2009) establishes that under common interest there is always a Pareto-optimal language that is not vague, thus negating the necessity of vague language.1 1 The definition we give here is weaker than that given in Lipman (2009), as the latter rules out any randomization. The difference is minimal, though, because Condition V implies the weaker version only admits randomization among messages that are entirely synonymous. It is straightforward to verify that all results in Lipman (2009) hold under the weaker definition as well. 3 Literal Vagueness We would like to use an example to demonstrate that vagueness as defined above may exclude 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) or not (N L), and 2) whether she has time for further conversation (T ) or is in a hurry to the next session (N T ). If Alice does not have time she says “It’s interesting.” (l). If she has time she says “It’s interesting” if she likes the talk, or “The research can benefit from some further development” (nl) if she does not like the talk. Is Alice’s language vague? We can formalize the example with state space Ω = {L, N L}× {T, N T } and M = {l, nl}. Alice’s language is technically not vague as defined above, because she uses only nl in state (N L, T ) and only l in other states. However, Alice’s language may strike Bob as vague. For instance, if Alice says “It’s interesting”, Bob is uncertain whether Alice 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 messages “It’s interesting” and “The research can benefit from some further development” to the states. These messages have a literal interpretation that only concerns Alice’s opinion about Bob’s talk and is irrelevant whether Alice has time. Literal interpretation represents an exogenous relation between messages and the states they conventionally signify. Thus, an impression of vagueness would result if the language does not precisely demarcate the “literal state space” {L, N L} 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 because in the standard model of conversation, Ω and M are taken as abstract sets lacking structure and relation. No exogenous literal interpretation exists outside the speaker’s idiosyncratic use of the messages, and thus interpretation is purely endogenous. In the background of everyday conversation, on the other hand, there is a focal language which endows messages with exogenous literal interpretation, and the sense of vagueness often occurs because of the inconsistency between the common literal interpretation and an individual’s idiosyncratic language use, as is the case in Example 1. Thus it would be beneficial 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 has a two-dimensional product structure such that Ω = F × C, where a typical f ∈ F is called a feature and c ∈ C a context. The message space M has an exogenous literal relation with F , but not C, that is, messages in M are conventionally used to describe features but 4 has no literal relation with the context.2 For instance “I’m well” is conventionally used to describe wellness but not hurriedness. Given this setup we propose a new notion of vagueness. Specifically, a language λ is said to be not literally vague if Πλ satisfies the following conditions: L1. For any two states ω and ω ′ , if Πλ (ω) ≠ Πλ (ω ′ ) then λ(ω) and λ(ω ′ ) have disjoint supports. 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 identical to Condition V. Thus literal vagueness is broader than vagueness. The additional Condition L2 further demands that, for a language to be not literally vague, every message m ever sent precisely narrows down the possible features for the listener in the sense that 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 interesting”, Bob’s prior does not become concentrated on any subset of F = {L, N L}. The Efficiency Advantage of Literal Vagueness We wish to demonstrate that, unlike strictly vague languages, literally vague languages may have an efficiency advantage. To show that we adopt the standard cheap talk model with 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 available actions A, and consequently both players receive a payoff of u(ω, a). To help the listener, the speaker sends a message m ∈ M . To incorporate literal vagueness we impose the product structure Ω = F × C on the state space as discussed above. We do not require a common prior. However, we do assume that each state is believed by at least one of the players as possible i.e. its probability is positive. The following proposition establishes the potential efficiency advantage of literally vague languages by showing that if the choice problem is not trivial and if messages are limited then there are payoff functions u given which the optimal language must be literally vague. 2 3 The notion of context in our framework resembles the “prior collateral information” in Quine (1960). An earlier, non-formal study on language use in this situation is due to Lewis (1969). 5 Proposition 1. If ∣F ∣ ≥ 2, ∣C∣ ≥ 2, ∣A∣ ≥ 2 and min(∣F ∣∣C∣ − 1, ∣A∣) ≥ ∣M ∣ ≥ 2 then there is a payoff function u ∶ Ω × A ↦ R given which the sender’s language in any Pareto-optimal perfect 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 that a∗ 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 such that: 1. For any p ∈ P there is some α(p) ∈ A∗ ∪ {a, a′ } where α(p) is the unique optimal action in 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)). Message µ(P (ω)) can be interpreted as the recommendation “the optimal action is α(P (ω)) in the current state”. Therefore Πλ = P . Thus {(f, c), (f ′ , c′ )} ∈ Πλ since only for these two states the optimal action is a. λ is literally vague because {(f, c), (f ′ , c′ )} is not a product of any subsets of F and C. Remark 1. The conditions ∣F ∣ ≥ 2, ∣C∣ ≥ 2 and ∣A∣ ≥ 2 reflect that the state space and the 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 the state space, that is, it is not possible to use a distinct message to denote each state. It is noteworthy that the whole issue of vagueness would be less of a concern if messages were not limited, for obviously full description is best if feasible. Remark 2. That ∣A∣ ≥ ∣M ∣ is not a necessary condition. The assumption is made to simplify the proof, which is based on a particular construction. Although the construction 6 may strike us as contrived and peculiar, the spirit of the idea underlying optimality of a 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 may be used in a context-dependent way, because the messages, albeit literally descriptive only of features, need to contain information about the context as well. Therefore, to correctly interpret which subset of features a message points to, it is necessary to know the corresponding context; in contrast, without knowledge of the context, the message is vague in terms of which subset of features it points to. Remark 3. That the result holds without the common prior assumption is because the constructed payoff function u implies that ex post optimality is achieved in a Paretooptimal equilibrium. Thus the potential conflict of interest due to heterogeneous priors becomes irrelevant concerning the Pareto-optimal equilibrium because it is the most preferred 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 nature of the decision problem being context-dependent, that is, the payoff depends on the 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 message refers to different subsets of features depending on the context that obtains, because information about the context is also worth communicating. It is not unusual that the meaning of a word depends on the context. Indeed, the whole linguistic field of pragmatics is dedicated to the study of context-dependence. It is important to distinguish between two types of context-dependence embedded in a language. The first type refers to the case that the sender’s choice of message depends on the 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 contrast, “do-ze” is mostly used to thank for something materialistic, e.g., a gift.4 This type of context-dependent use of language needs not be literally vague, because distinct messages may simply convey distinct information about the context without differentiating between information about the feature. The second type of context-dependence in a language refers to the case that the correct interpretation about which subset of features is signified by 4 However, no such distinction in expressing gratitude exists in Mandarin Chinese. 7 a message depends on the context. In this case, not only the speaker’s choice of message depends on the context, but he also chooses the same message to signify different subsets of features depending on the context. In Example 1, Alice’s use of “It’s interesting” corresponds to this case. Thus the second type is stronger than the first type, and it necessarily implies literal vagueness. When thereafter we discuss context-dependence in a language we mean the second type. A crucial source of context-dependence and thus literal vagueness is that the focal language is essentially one-dimensional: It only has terms literally relating to the featuredimension of the state. Why would people use such an overly parsimonious language? One important reason is its simplicity. After all, using a richer language is more costly, and often the richer language may not be feasible at all — for instance, if the common language shared by people from different linguistic backgrounds is only descriptive of one aspect of the state. People with different native tongues can use facial expression in person, or emoji , and emoticon :-) on the internet, as a common language to describe emotion, but this 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 is not directly payoff-relevant, people still prefer to use a literally vague language. This is particularly the case if the context is meta-linguistic. Here we give two examples, both variants 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 next dress. 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 terms denoting 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 may be literally vague. This is an example of the more general model studied in Blume and Board (2013). In their model, the available messages for the speaker may vary depending on the speaker’s language competence. Consequently, in optimal communication speakers of different language competence may use the same message to indicate different sets of payoff-relevant states, implying that the optimal language is vague in their model. Example 2 shows that language competence can be incorporated as the context-dimension of the state, and in the model with the enriched state space the optimal language is no longer vague, but is instead literally vague. Example 3. Alice wishes to describe the height of Charlie to Bob, so that Bob can recognize Charlie at the airport and pick him up. Moreover, 8 • 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 professional basketball player or not. Alice can also describe Charlie as “tall” or “short”. Her optimal choice of the word can depend both on Charlie’s height and on whether she believes Bob knows Charlie is a professional basketball player or not. For instance, if she believes Bob knows Charlie is a professional basketball player then she says Charlie is “tall” only if his height is above 6 foot 10, whereas if she believes Bob does not know then Charlie is “tall” only if his height is above 6 foot 2. If we take Alice’s belief about Bob’s knowledge as the context, then her language is literally vague. This example is adapted from a more casual discussion in Lipman (2009). A more formal model of the example is as follows: The listener’s knowledge about the payoff-relevant state space is not common knowledge. In particular, prior to the conversation the listener may receive with some probability an informative private signal which narrows down the possible 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 informative signal or not. It is very easy to construct a typical decision problem under which, in the optimal equilibrium, the meaning of a message from the speaker depends on the signal that he receives. If we do not incorporate the speaker’s signal as part of the state then the optimal language is vague. However, in the enriched model in which the speaker’s signal is considered as the context and the payoff-relevant state as the feature, the optimal language is not vague but literally vague. Finally, we show an example demonstrating that when there are multiple speakers who speak sequentially, the way later speakers talk may rely on what earlier speakers have said. Earlier messages become the context on which later messages depend – the context of the bilateral conversation between a later speaker and the listener is then endogenously generated in the larger-scale multilateral conversation. The following example shows one of such situations. Example 4. Alice and Bob interviewed a job candidate. Alice observes the candidate’s ability A and Bob observes his personality B. A and B are independently and uniformly distributed 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 committee chair Charlie who is responsible for the recruitment decision, with Alice speaking first. The 9 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 is high” 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 since generated on this subject, only until recently was linguistic vagueness given the academic attention 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 à la Crawford and 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 some of the conversing parties are unknown and show that this uncertainty could make the optimal language vague. The same authors further investigate the effect of higher-order uncertainty about linguistic ability on communication in Blume and Board (2014a), and find that, in the common interest case, vagueness persists but efficiency loss due to higherorder uncertainty is small. The relation between our paper and the above papers has been discussed in depth in the Introduction. It is well known that even in the presence of conflict of interest endogenous vague language still has no efficiency advantage in the cheap talk framework. However, Blume et al. (2007) show that exogenous noise in communication, which forces vagueness upon the language, can bring Pareto improvement. Blume and Board (2014b) further confirm that the speaker may intentionally take advantage of the noise to introduce more vagueness in the language. These papers differ from ours in that we focus on the common interest environment, and moreover we study the efficiency-foundation of endogenous vagueness. Context-dependence is the source of literal vagueness in our paper. Given common interest, context-dependence arises when available messages are not sufficient to fully communicate the complexity of the situation. Within the cheap talk framework, Tian (2016) discusses that, when the message space is small, how the optimal language changes with the common prior, that is, the context. In the framework of experimentation, formally 10 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 his language conditioning on contexts, which are messages received from earlier participants. Indeed, an implication of the characterization of the Pareto-ranking of communication mechanisms due to Wu (2016) is that a mechanism which allows the participants to have more flexibility in using context-dependent language dominates another one which allows less. On the experimental side, a few recent papers investigate how the availability of vague messages improves or preserves efficiency. Serra-Garcia et al. (2011) show that vague languages help players preserve efficiency in a two-player sequential-move public goods game with asymmetric information. Wood (2016) explores the efficiency-enhancing role of vague languages in a discretized version of canonical sender-receiver games à la Crawford and Sobel (1982). Agranov and Schotter (2012) show that vague messages are useful in concealing conflict between a sender and a receiver that, if it is publicly known, would prevent them from coordinating and achieving an efficient outcome. All these papers take the availability of vague messages as given and study how it affects players’ coordination behavior. On the contrary, we explore how a message endogenously obtains its vague meaning. For more comprehensive discussion of the experimental literature on vague languages, 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 fashion. This is a very important step towards understanding whether literal vagueness, due to its efficiency advantage, stands firm as an explanation for some linguistic vagueness that we experience everyday, because the whole efficiency foundation for vagueness is pointless 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 not sufficiently prove that the prevalence of linguistic vagueness is founded on the efficiency advantage of literal vagueness in our sense, yet it is a worthwhile first step. It is with this question in mind we design our experiments. We use the situation described in Example 4 to examine whether, and if so, how, people use literally vague language to converse. This example has a number of important attractive features which make it particularly suitable for our purpose. Firstly, as we shall show below, there is a moderate degree of efficiency advantage to the optimal literally vague language, which renders literal vagueness potentially useful but not entirely crucial for 11 communication. Secondly, the situation is simple and straightforward, and thus should in principle shorten the period of learning and make the experimental results closer to the eventual stable language, or in terms of Lewis (1969) the “convention”. Thirdly, the situation covers the most general environment in which the context, being the message from Alice, is endogenously generated. Fourthly, given that for optimal communication Alice does not need to use a literally vague language whereas Bob does, having both senders in the same game gives us an additional comparison regarding how people’s use of language depends on the conversational environment they face. The Benchmark Game There are three players: two senders, Alice and Bob, and one receiver, Charlie. Alice privately observes a number A and Bob a number B. A and B are independently and uniformly 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 sends a message to Charlie, where the message is either “B is Low” or “B is High”. Charlie receives Bob’s message and chooses an action: UP or DOWN. Players’ preferences are perfectly 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 remains the same as the prior upon seeing any message chosen by Bob with positive probability. Whether Alice babbles or not does not have bearing on the outcome. No information is transmitted and Charlie is indifferent between the two actions regardless of the message he receives. Accordingly, the success rate, which is the probability that 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 < 50.6 Charlie chooses UP seeing “B is High” and DOWN otherwise. Only Bob’s information is 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. 5 A 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 no theoretical consequence and do not correspond to the experiment results. 6 Of 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. 12 • 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 success rate is 87.5%. Clearly any equilibrium in which no one babbles is Pareto-optimal. Because the messages available to Bob explicitly refer to the value of B alone, Bob’s language in any Paretooptimal equilibrium is clearly literally vague, because the set of values of B a particular message, say “B is high”, describes depends on Alice’s message, which serves as the context. To best test whether indeed the players effectively use the optimal literally vague language, we propose to consider the counterfactual in which they don’t. This can be formally modeled as Bob being constrained to use the same messaging strategy regardless of Alice’s message. In this case, the best Bob can do is to always use the cutoff of 50, and the corresponding success rate is 75%. To examine the results and test whether the counterfactual holds, we thus should pay particular attention to how Bob’s use of language depends on Alice’s message. In addition, for a literally vague language to be effectively used, the listener should also be aware of the underlying context-dependence and correctly take that into consideration when making decisions. However, Charlie’s strategy in the counterfactual would be the same as that in an optimal equilibrium so that considering the benchmark only will not allow us to identify whether the listener fully understand the optimal, context-dependent language 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 messages is also observable to Charlie. The equilibria in which no one babbles remain the same as in the Benchmark. In particular, it is notable that Charlie’s strategy does not depend on Alice’s messages despite it being available. In this variation, if Charlie believes that Bob uses his language in the optimal, literally vague way, the former’s choice should not depend on Alice’s messages, because the information contained in Alice’s messages is fully incorporated into Bob’s messages through context-dependence. Thus we can tease out whether Charlie correctly interprets Bob’s messages according to the optimal language by checking whether Charlie’s choice depends on Alice’s messages. 13 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 the optimal 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 success rate 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 environment more complex, in particular for Bob and Charlie. The Benchmark and Variation 1 are relatively simple environments in which it is not supremely difficult to “compute” the optimal cutoffs.7 On the other hand, when people talk in real life they do not typically derive the optimal language consciously. Thus we want to see, when it is more difficult to explicitly derive the optimal language, whether people can still arrive at the optimal, literally vague language and use it to effectively communicate, or whether they instead revert to context-independent language, which is simpler to use for the speaker and to understand for the listener. Thus it is crucial to examine whether in this variation Bob’s message is context-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 to Charlie. In an optimal equilibrium, Alice sends “A is High” if A > 50 and “A is Low” if A < 50. Bob’s strategy does not quite differ from that in Variation 2 qualitatively, but is with 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. 7 The key logic one can easily come with is that Alice would use a cutoff of 50 because of the symmetry of the problem. Thereafter the optimal cutoffs of 25 and 75 can be deduced simply by mind or at most by some back-of-envelope calculation. 14 • 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 is Low”), and MIDDLE otherwise. The success rate is 78.5%. It should be noted that MIDDLE 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 quantitative change in the optimal cutoff values affects the use of languages. The optimal cutoff values that are substantially closer each other generates a very minimal benefit of contextdependence relative to the context-independent counterfactual. In fact, the success rate from the context-independent counterfactual is 78%. Thus, this variation enables us to understand how individuals’ choice of context-dependent languages are guided by the salience 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 possibility of literal vagueness because Bob’s messages, now imperative, have unambiguous literal interpretations with respect to the decision problem at hand. Apart from the difference 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, it allows us to test whether literal vagueness may intimidate players from using the optimal language. In the experiments, we create not only an “imperative messages” version of the Benchmark, 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 our experimental treatments. Our experiment features a (2 × 2) + (2 × 1) treatment design (Table 1). The first treatment variable concerns the number of actions available to the receiver (Charlie) and the second treatment variable concerns whether Alice’s messages are 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 with imperative messages as a robustness check so that we omit the corresponding treatments in which Alice’s messages are observed by Charlie. Table 1: Experimental Treatments Indicative Messages from Bob Alice’s messages # of Actions Two Three Unobservable 2A-U-IND 3A-U-IND Observable 2A-O-IND 3A-O-IND Imperative Messages from Bob + Alice’s messages Unobservable # of Actions Two Three 2A-U-IMP 3A-U-IMP Our first experimental hypothesis concerns the overall outcome of the communication games represented by the success rate. Let S(T ) denote the average success rate of Treatment T . Postulating that the optimal equilibria are played in each game, we have the following 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 of Alice’s message to Charlie does not affect the success rate in the treatments with two actions (S(2A-O-IND) = S(2A-U-IND)) while the very same observability increases the success 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(2AU-IMP) and S(3A-U-IND) = S(3A-U-IMP)). Our second hypothesis considers the counterfactual in which Bob is constrained to be context-independent and thus always uses the cutoff of 50.9 In the counterfactual scenario, 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 the players effectively use the optimal, literally vague language, the success rates should be significantly above the levels predicted by the counterfactual. Thus, we have the following hypothesis. Hypothesis 2 (Counterfactual Comparison). 1. S(2A-O-IND), S(2A-U-IND), S(2A-U-IMP) > 75% 8 For 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. 9 Charlie’s optimal strategy remains the same regardless of whether Bob is constrained to be contextindependent or not. 16 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 Treatment 3A-O-IND is 78.5%, which is not substantially different from the 78% predicted by the counterfactual. The net benefit of the context-dependent language measured with respect to the success rates is only 0.5% (= 78.5% - 78%) in Treatment 3A-O-IND. The net benefit of context-dependence becomes substantially larger in other treatments as it is 12.5% (= 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 optimal equilibrium 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 of Alice’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 are the same. Moreover, PAlice (“A is Low”∣A) = 1 for any A < 50 and PAlice (“A is High”∣A) = 1 for any A > 50. Our next hypothesis concerns Bob’s message choices. The optimal equilibrium play predicts that Bob’s message choices depend on the context, i.e., which message he received m′ (m∣B) denote the proportion of Bob’s from Alice. To state our hypothesis clearly, let PBob message m given the realized number B and Alice’s message m′ ∈ {“A is High”, “A is Low”}. Define H L CD(B) = PBob (m∣B) − PBob (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 contextdependence of Bob’s message choices given the realized number B. Bob’s optimal, contextdependent strategy implies that there is a range of number B under which Bob’s message choices 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 that CD(B) = 0 for any B ∈ [0, 100] if Bob uses a context-independent strategy. Thus, we have the following hypothesis: Hypothesis 4 (Bob’s Messages). Bob’s messages are context-dependent in such a way that is predicted by the optimal equilibrium of each game. More precisely, 1. For each treatment T , there exists an interval [X T , Y T ] with X T > 0 and Y T < 100 such that CD(B) > 0 for any B ∈ [X T , Y T ] and CD(B) = 0 otherwise. 17 2. The length of the interval [X T , Y T ] is significantly smaller in Treatment 3A-O-IND than in any other treatments. Our last hypothesis concerns if the listener, Charlie, can correctly interpret the messages. In particular, Charlie’s strategy in the optimal equilibrium does not depend on whether or not Alice’s messages are observable to Charlie in the treatments with two actions. In the games with three actions, however, the observability of Alice’s message to Charlie matters. Precisely, the optimal equilibrium predicts that MIDDLE should not be taken by Charlie in Treatments 3A-U-IND and 3A-U-IMP whereas MIDDLE is taken in Treatment 3A-O-IND when the messages from Alice and Bob do not coincide. Thus, we have the following hypothesis. Hypothesis 5 (Charlie’s Action Choices). In the treatments with two actions, Charlie’s action choices do not depend on whether Alice’s messages are observable or not. In the treatments with three actions, MIDDLE is taken only in Treatment 3A-O-IND when the messages from Alice and Bob do not coincide. 3.2 Procedures Our experiment was conducted in English using z-Tree (Fischbacher (2007)) at the Hong Kong University of Science and Technology. A total of 138 subjects who had no prior experience with our experiment were recruited from the undergraduate population of the university. Upon arrival at the laboratory, subjects were instructed to sit at separate computer terminals. Each received a copy of the experimental instructions. To ensure that the information contained in the instructions was induced as public knowledge, the instructions were read aloud, aided by slide illustrations and a comprehension quiz. We conducted one session for each treatment. In all sessions, subjects participated in 21 rounds of play under one treatment condition. Each session had 21 or 24 participants and thus involved 7 or 8 fixed matching groups of three subjects, one Member A (Alice), one Member B (Bob), and one Member C (Charlie). Thus, we used the fixed-matching protocol and between-subject design. As we regard each group in each session as an independent observation, we have seven to eight observations for each of these treatments, which provide us with sufficient power for non-parametric tests. At the beginning of a session, one third of the subjects were randomly labeled as Member A, another one third labeled as Member B and the remaining one third labeled as Member C. The role designation remained fixed throughout the session. We illustrate the instructions for Treatment 2A-U-IND. The full instructions can be found in Appendix A. For each group, the computer selected two integer numbers A and 18 B between 0 and 100 (uniformly) randomly and independently. Subjects were presented with a two-dimensional coordinate system (with A in the horizontal coordinate and B in the vertical coordinate) as in Figures 5(a) and 5(b) in Appendix A. The selected number A was revealed only to Member A and the selected number B was revealed only to Member B. Member A sent one of two messages, “A is Low” and “A is High”, to Member B but not to Member C. After observing both the selected number B and the message from Member A, Member B sent one of two messages, “B is Low” and “B is High”, to Member C who then took one of two actions, UP and DOWN. The ideal actions for all three players were UP when A + B > 100 and DOWN when A + B < 100.10 Every member in a group received 50 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 first encountered one possible contingency and specified a choice for the given contingency. For example, Member A decided what message to send after seeing the randomly selected number A. Member B decided what message to send after observing the randomly selected number B and the message from Member A. Similarly, Member C decided what action to take after receiving the message from Member B. For Round 21, however, we used the strategy-method and elicited beliefs of players. For the belief elicitation, a small amount of compensation (in the range between 2 ECU and 8 ECU) was provided for each correct guess.11 For more details, see the selected sample scripts for the strategy-method and the belief-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 into Hong Kong dollars at a fixed and known exchange rate of HK$1 per 1 ECU. In addition to these earnings, subjects also received a show-up payment of HK$30. Subjects’ total earnings averaged HK$103.5 (≈ US$13.3).12 The average duration of a session was about 1 hour. 4 Experimental Findings We report our experimental results as a number of findings that address our hypotheses as set forth in Section 3.1. 10 To make the likelihood of each action being ideal exactly equal across two actions, we set both actions to be ideal when A + B = 100. 11 Although we were aware of the fact that an appropriate incentive-compatible mechanism is needed to elicit beliefs correctly, we took this simple elicitation procedure because of its simplicity as well as the fact that the belief and strategy data were only secondary data mainly for the purpose of robustness checks. 12 Under the Hong Kong’s currency board system, the Hong Kong dollar is pegged to the US dollar at the rate of HK $7.8 = US$1. 19 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 from the 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 matching groups for each treatment. It also presents the theoretical predictions from the optimal equilibrium depicted by the red bars and the predictions from the counterfactual in which Bob is constrained to be context-independent depicted by the dotted lines. A few observations were apparent. First, non-parametric Mann-Whitney test reveals that the success rates in Treatment 2A-O-IND and in Treatment 2A-U-IND were not statistically different (81.6% vs. 83.6%, two-sided, p-value = 0.6973). On the contrary, the success rate in Treatment 3A-O-IND was 73.8%, which is significantly higher than 54.2% in Treatment 3A-U-IND (Mann-Whitney test, p-value = 0.0267). This observation is consistent with Hypothesis 1 that the observability of Alice’s message to Charlie affects the success rate only in the treatments with three actions. Second, there was no significant difference in the success rates between Treatment 2A-U-IND and Treatment 2A-U-IMP (83.6% vs. 85.7%, two-sided Mann-Whitney test, pvalue = 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 consistent with Hypothesis 1 that imperativeness of Bob’s message does not affect the success rate regardless of the number of available actions. Confirming Hypothesis 1, we thus have our first finding as follows: Finding 1. Observability of Alice’s message to Charlie affected the success rate only in the treatments with three actions. Imperativeness of Bob’s message did not affect the success rate regardless of the number of available actions. 20 Figure 1 seems to suggest that the success rates observed in the three treatments with two actions (hereafter Treatments 2A) are better approximated by the predictions from the optimal, context-dependent equilibrium languages than by the predictions from the context-independent counterfactual. Indeed, we cannot reject the null hypothesis that the success rates are not different from 87.5%, the predicted value from the optimal equilibrium (two-sided Mann-Whitney tests, p-values are 0.8262, 0.5076, 1.000 for Treatments 2A-O-IND, 2A-U-IND and 2A-U-IMP, respectively). Even if we can reject the alternative hypothesis that the success rates are significantly higher than the predicted level of 75% from the context-independent counterfactual only for Treatment 2A-U-IND (onesided Mann-Whitney tests, p-values are respectively 0.2551, 0.0610, 0.2174 for Treatments 2A-O-IND, 2A-U-IND and 2A-U-IMP), the p-values resulted from the non-parametric analysis suggest that the optimal, context-dependent equilibrium is a better predictor of the results observed from these treatments. On the other hand, we do not have the same observation from the three treatments with three actions (hereafter Treatments 3A), especially those with unobservable messages from Alice. We cannot reject the null hypothesis that the success rates observed in these treatments are the same as the success rates predicted by the context-independent counterfactual (two-sided Mann-Whitney tests, p-values are 0.4347, 0.6936, 0.4347 respectively for Treatments 3A-O-IND, 3A-U-IND and 3A-U-IMP). Moreover, the success rates observed 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, literally vague equilibrium language. Although the difference is statistically insignificant (twosided Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively), the p-values generated from the non-parametric analysis suggest that the context-independent counterfactual 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, and 2A-U-IMP were higher than the predicted level from the counterfactual in which Bob is context-independent. The average success rates observed in Treatments 3A-O-IND, 3A-UIND, 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 counterfactual is statistically significant only in Treatment 2A-U-IND. Among Treatments 3A, more substantial deviations from the optimal, context-dependent equilibrium were observed in Treatments 3A-U-IND and 3A-U-IMP. This observed devi13 The success rates observed in Treatments 3A-O-IND was 73.8%, which is not significantly different from the predicted level of 78.5% from the optimal, literally vague equilibrium language (two-sided Mann-Whitney test, p-value = 0.4347). 21 ation in the average success rates may imply that the context-dependent, literally vague languages were not emerged in those treatments, probably due to the complexity of the environment considered. However, another completely plausible scenario would be that the observed deviation originates from a different source, such as Charlie’s choices not being consistent with the optimal equilibrium. Without taking a careful look at the individual behavior, it is impossible to draw any meaningful conclusion. Hence, in the subsequent sections, 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 message as a function of the number A where data were grouped into bins by the realization of number A (e.g., [0, 5), [5, 10), ..., etc.). Figure 2(a) provides the aggregated data from Treatments 2A while Figure 2(b) provides the aggregated data from Treatments 3A. The same figures separately drawn for each treatment can be found in Appendix C. Alice's  Message  -­‐  Treatments  3A   100   80   80   60   LOW   40   Propor%on   Propor%on   Alice's  Message  -­‐  Treatments  2A   100   60   LOW   40   HIGH   HIGH   0   0   [0 ,5 ) [5   ,1 0 [1 )   0, 15 [1 )   5, 20 [2 )   0, 25 [2 )   5, 30 [3 )   0, 35 [3 )   5, 40 [4 )   0, 45 [4 )   5, 50 [5 )   0, 55 [5 )   5, 60 [6 )   0, 65 [6 )   5, 70 [7 )   0, 75 [7 )   5, 80 [8 )   0, 85 [8 )   5, 90 [9 )   0. 9 [9 5)   5, 10 0]   20   [0 ,5 ) [5   ,1 0 [1 )   0, 15 [1 )   5, 20 [2 )   0, 25 [2 )   5, 30 [3 )   0, 35 [3 )   5, 40 [4 )   0, 45 [4 )   5, 50 [5 )   0, 55 [5 )   5, 60 [6 )   0, 65 [6 )   5, 70 [7 )   0, 75 [7 )   5, 80 [8 )   0, 85 [8 )   5, 90 [9 )   0. 9 [9 5)   5, 10 0]   20   Number  A   Number  A   (a) Treatments 2A (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 designated roles were Alice in our experiments tended to use cut-off strategies well approximated by the optimal cutoff of 50. Using the matching-group level data from all rounds for each bin of the realized number A (e.g. [0, 5), [5, 10), ..., etc.) as independent data points for each treatment, a set of (two-sided) Mann-Whitney tests reveals that 1) for any bins of A below 50, we cannot reject the null hypothesis that the proportion of message “A is Low” being sent was 100%, and 2) for any bins of A above 50, we cannot reject the null hypothesis that the proportion of message “A is Low” being sent was 0%. Among 20 bins in each treatment, the p-values for 14-17 bins were 1.0 while the lowest p-value for each 22 treatment was ranged between 0.2636 and 0.5637. Confirming our Hypothesis 3, we thus have 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 additional supports for Finding 3. We cannot reject the null hypothesis that the reported cutoff values for Alice’s strategy and other player’s reported beliefs for Alice’s strategy in all treatments were the same as the optimal equilibrium cutoff of 50 (two-sided MannWhitney tests, p-values are in the range between 0.4533 and 1.000). 4.3 Bob’s Behavior L We now look at Bob’s behavior. Recall that PBob (m∣B) denoted the proportion of Bob’s H message m given the realized number B and Alice’s message “A is Low”, and PBob (m∣B) denoted that given Alice’s message “A is High”. We introduced the measure for the contextH L (m∣B) where m is “B is Low” for the treatments dependence as CD(B) = PBob (m∣B) − PBob 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 rounds were 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 interval (strictly interior of the support of B) such that the value of CD(B) is 1 if B is in the interval and 0 otherwise. Moreover, the boundaries of such intervals are determined by the equilibrium cutoff strategy so that the interval is narrower in Treatment 3A-OIND. The exact prediction of the distribution of CD(B) made by the optimal equilibrium is illustrated by the red-dotted lines in Figures 3(a)-(f).14 These figures convincingly visualize the fact that, in each treatment, there existed an interval in which the value of CD(B) is strictly positive. For Treatment 2A-O-IND, for instance, 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), we can 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 and 14 Figures 8(a)-(f) and 9(a)-(f) presented in Appendix C separately report the distributions of H is Low”∣B) and of PBob (“B is Low”∣B) for each treatment. To 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 L PBob (“B 15 23 (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-UIND with the non-zero interval of [25, 70) (p-value = 0.136) and from Treatment 3A-U-IND with 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]. We next calculate the value of CD(B) for each of the three intervals for each matching group. Taking those values as group-level independent data points for each treatment, we conducted the non-parametric test. 24 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 carefully. First, as predicted by the optimal equilibrium strategy, the interval that has nonzero value of CD(B) seemed to shrink significantly compared to any other treatments. Indeed, we cannot reject the null hypothesis that CD(B) = 0 for B ∈ [0, 45) and for B ∈ [50, 100] (two-sided Mann-Whitney test, p-values are 1.00 and 0.7237, respectively). However, a substantial deviation from the theoretical prediction was observed such that the reported value for the bin [50, 55) was negative.16 This observation was driven by the fact that the observed cutoff value from Bob’s strategy conditional on Alice’s message “A is Low” was 50, 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 further supporting evidence. Wilcoxon signed-rank tests reveal that the reported cutoff values given Alice’s message “A is Low” were significantly higher than the cutoff values given Alice’s message “A is High” for most of the treatments (p-values are ranged between 0.0004 and 0.0204) except for Treatment 3A-O-IND and Treatment 3A-U-IND.18 The fact that the reported 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 are 45 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) mainly originated from two observations that the reported cutoffs from two Charlie-subjects were 50 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 as a function of information available to Charlie. Figure 4(a) presents the data aggregated across all matching groups of Treatments 2A while Figure 4(b) presents the data aggregated across all matching groups of Treatments 3A. 16 For Treatment 3A-O-IND, we cannot conduct any meaningful statistical analysis for B ∈ [45, 50) because there are only two group-level independent data points. 17 Similarly, 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 point with B ∈ [0, 5) in which Bob sent “B is High” after receiving “A is High” from Alice. The second deviation was solely driven by the single data point with B ∈ [40, 45) in which Bob sent “B is Low” after receiving “A is High” from Alice. 18 To conduct Wilcoxon signed-rank tests, we pooled the data from the reported cutoff values for Bob’s strategy 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) reveals that observed strategy by Charlie depended, to some but limited extent, on whether or not Alice’s messages were observable to him in Treatments 2A. In Treatment 2A-O-IND, the proportion of UP being chosen given Bob’s message “B is High” was about 52% and the proportion of DOWN being chosen given Bob’s message “B is Low” was about 75%, both are substantially and significantly different from 100% predicted by the optimal equilibrium. However, the observed proportions became 63% and 100% if we took the last three rounds data 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-OIND when the messages from Alice and Bob did not coincide. The proportions of MIDDLE being 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 prediction from the optimal equilibrium strategy, MIDDLE was taken even in Treatments 3A-U-IND and 3A-U-IMP. The proportions of MIDDLE being taken observed in these two treatments varied between 24% and 43% which are substantially larger than 0%. This observed deviation seemed persistent as it did not disappear even when we took the data from the last three rounds only.20 Thus, we have the following result: Finding 5. Charlie’s observed action choices in Treatment 2A-O-IND were not the same as those observed in Treatment 2A-U-IND, showing that observability of Alice’s message to 19 In an early stage of the project, we have conducted two sessions in which subjects participated in the first 20 rounds with the treatment condition of 2A-U-IND and in the second 20 rounds with the treatment condition of 2A-O-IND. The data from the second 20 rounds were almost perfectly consistent with the theoretical prediction, showing another convincing evidence of learning. Data from this additional treatment are available upon request. 20 The elicited strategies and beliefs presented in Figure 12 in Appendix C were highly consistent with the results in Finding 5. 26 Charlie mattered. For Treatments 3A, MIDDLE was taken in Treatment 3A-O-IND when Charlie received different messages from Alice and Bob. However, a substantial proportion of 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 prediction from the optimal equilibrium was the main source of the lower success rates we had in Treatments 3A-U-IND and 3A-U-IMP (see Figure 1). The success rates observed 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%, although the difference is statistically insignificant (Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively). If we replace the observed empirical choices by Charlie with the hypothetical choices from the optimal strategy, the success rates in Treatments 3A-U-IND and 3A-U-IMP become 58.3% 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 in Treatment 3A-U-IND (31% and 24% vs. 33% and 43%). The elicited strategies and beliefs presented in Figure 12 in Appendix C demonstrate the difference more vividly. This difference 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 establish the emergence of context-dependent, literally vague languages. Admittedly, we did not present a perfect match between our data and the prediction from the optimal, contextdependent equilibrium language. However, we provided convincing evidence that overall behavior observed in our laboratory was qualitatively consistent with the prediction. More precisely, 1. Finding 3 illustrated that Alice tended to use cut-off strategies well approximated by the 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) well and took actions in a manner that is qualitatively consistent with the optimal equilibrium strategy. 4. Finding 5 also revealed that the lower success rates observed in Treatments 3A-UIND and 3A-U-IMP reported in Finding 2 were largely driven by Charlie’s behavior 27 being partially inconsistent with the theoretical predictions from the optimal equilibrium. Taking these findings together, we establish the following result: Finding 6. Literally vague languages emerged in our experiment. The way it was used by the speakers and understood by the listener was all consistent with the prediction from the Pareto-optimal equilibria of the communication games. 5 Concluding Remarks In this paper we introduce the notion of context-dependence and literal vagueness, and offer our explanation of why language is vague. We show that literal vagueness arises in a Pareto-optimal equilibrium in many standard conversational situations. Our experimental data provide supporting evidence for the emergence of literally vague languages. Although our discussion of linguistic vagueness focuses on the environment in which players’ preferences are perfectly aligned, the theoretical discussions presented in Blume et al. (2007) and Blume and Board (2014b) suggest that the communicative advantage of literal vagueness would be extended to the environment with conflicts of interests. We believe that experimentally investigating the role of vagueness in the presence of conflict of interests is an interesting avenue for future research. 28 References Agranov, Marina and Andrew Schotter (2012), “Ignorance is bliss: An experimental study of the use of ambiguity and vagueness in the coordination games with asymmetric payoffs.” 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.” Theoretical Economics, 2, 395–440. Blume, Andreas, Ernest K. Lai, and Wooyoung Lim (2017), “Strategic information transmission: A survey of experiments and theoretical foundations.” Working paper. Crawford, Vincent and Joel Sobel (1982), “Strategic information transmission.” Econometrica, 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 Institute of Technology, Cambridge. Serra-Garcia, Marta, Eric van Damme, and Jan Potters (2011), “Hiding an inconvenient truth: Lies and vagueness.” Games and Economic Behavior, 73, 244 – 261. Smith, Lones, Peter N. Sørensen, and Jianrong Tian (2016), “Informational herding, optimal experimentation, and contrarianism.” Tian, Jianrong (2016), “Monotone comparative statics for cut-offs.” Working paper. Wilson, Andrea (2014), “Bounded memory and biases in information processing.” Econometrica, 82, 2257–2294. 29 Wood, Daniel H. (2016), “Communication-enhancing vagueness.” Working paper. Wu, Qinggong (2016), “Coarse communication and institution design.” Working paper. 30 Appendices A Experimental Instructions - Treatment 2A-U-IND INSTRUCTION Welcome to the experiment. This experiment studies decision making between three individuals. In the following two hours or less, you will participate in 21 rounds of decision making. Please read the instructions below carefully; the cash payment you will receive at the end of the experiment depends on how well you make your decisions according to these instructions. Your Role and Decision Group There are 24 participants in today’s session. One third of the participants will be randomly assigned the role of Member A, another one third the role of Member B, and the remaining 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 and one Member C, will be matched to form a group of three. The three members in a group make decisions that will affect their rewards in all 21 rounds. That is, you will stay in the same group so that you will interact with the same two other participants throughout the 21 rounds. You will not be told the identity of the participants in your group, nor will they be 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 B between 0 and 100 randomly and independently. Each possible number has equal chance to be selected. The selected number A will be revealed only to Member A and the selected number B will be revealed only to Member B. Member C, without seeing any of these numbers, 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 the two 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. 31 (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 the action chosen by Member C. Member A’s Decisions You will be presented with a two-dimensional coordinate system on your screen as in Figure 5(a). The horizontal axis represents the number A and the vertical axis represents the number B. You will see a blue vertical line, which represents the actually selected number 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 message buttons, your decision in the round is completed and your message will be transmitted to Member 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 in Figure 5(b). The horizontal axis represents the number A and the vertical axis represents the number B. You will see a blue horizontal line, which represents the actually selected number B in the vertical axis. The red diagonal line represents the cases with A + B = 100. 32 You will also receive a message from Member A in your group. With all this information on 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 in the 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 in Figure 6. The horizontal axis represents the number A and the vertical axis represents the 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 on your screen, you will be asked to take one of two actions DOWN and UP. Once you click one 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: actually selected numbers A and B, Member A’s message, Member B’s message, Member C’s action choice, and your earning in ECU. Your Decision in Round 21 After the 20th round, your screen will provide further instructions for your decisions in Round 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 before you start the 21st round. You will have an opportunity to ask questions if anything is unclear about the new instructions. 33 Your Cash Payment To calculate your cash payment, the experimenter will randomly select two rounds to calculate your cash payment. Each round between Rounds 1 and 21 has an equal chance to be selected. So it is in your best interest to take each round seriously. Your total cash payment at the end of the experiment will be the sum of ECU you earned in the two selected rounds, translated into HKD with the exchange rate of 1 ECU = 1 HKD, plus a 30 HKD show-up fee. Quiz and Practice To ensure your understanding of the instructions, we will provide you with a quiz and practice 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 instructions which is not relevant to your cash payment; its objective is to get you familiar with the computer interface and the flow of the decisions in each round. Once the practice round 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. Remember that you have to make your decisions entirely on your own; please do not discuss your decisions with any other participants. Upon finishing the experiment, you will receive your cash payment. You will be asked to sign your name to acknowledge your receipt of the payment. You are then free to leave. If you have any question, please raise your hand now. We will answer your question individually. 1. Suppose you are assigned to be a Member A. The computer chooses the random numbers A = 25 and B = 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 any of 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 the only 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 group takes action DOWN. Please calculate the earning for each player: • Member A’s payoff: 34 • 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 group takes action DOWN. Please calculate the earning for each player: • Member A’s payoff: • Member B’s payoff: • Member C’s payoff: 35 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 be implemented accordingly. , and otherwise, send “A is HIGH”. Your plan: Send “A is LOW” if A is less than or equal to 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 will be implemented accordingly. , and (a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to 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 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: , and (a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to 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 will receive 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 will be 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 HIGH If each of your guesses for (a) and (b) is correct, then you will receive extra 4 ECU. 36 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 37 (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 38 (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 39 Figure 10: Alice’s Elicited Strategy and Other Players’ Beliefs Figure 11: Bob’s Elicited Strategy and Other Players’ Beliefs 40 (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 ’ 41