Abstract
Speakers’ choice between linguistic alternatives often depends on the situation, a prime example involving level of precision at which numerical information is communicated. We report on a production study in which participants report the time of an event in two different situations, and demonstrate that the results can be reproduced by a probabilistic gametheoretical model in which the speaker’s choice reflects a tradeoff between informativity, accuracy and heareroriented simplification. These findings shed light on the pragmatics of (im)precision, and the dynamics of situationally driven pragmatic variation more generally.
1 Introduction
It is well known that speakers’ choice of linguistic forms varies according to the utterance situation. This is particularly well documented in the case of classic sociolinguistic variables such as the velar versus alveolar realization of ING, e.g. working versus workin’ (Labov 1966, and ff.). But situationally driven variation is also observed for alternatives that differ in their semantic content, a domain more usually investigated in semantics and pragmatics rather than sociolinguistics.
A nice example of such pragmatic variation involves the level of precision at which numerical information is conveyed. A speaker who wishes to communicate the time at which a certain event occurred might do so in at least three ways: with a precise nonround number (1a); by rounding off to a multiple of 5 (1b); or by using an explicit approximator such as about or roughly (1c).
The accident happened at 8:31. 
The accident happened at 8:30. 
The accident happened at about 8:30. 
Intuitively, the choice between such alternatives depends on the situation: in a casual conversation, (1b) or (1c) would seem most appropriate, whereas when giving a statement to a police officer, (1a) might be a better choice. Indeed, Van der Henst et al. (2002) demonstrate empirically that speakers often round off when telling the time, but they do so less frequently when they perceive their interlocutor to need a more precise value.
In the theoretical literature on the pragmatics of imprecision (e.g. Krifka 2002, 2009; Lasersohn 1999), it is widely accepted that contexts differ with respect to the degree of precision that is required. But a puzzling question remains. Clearly, a speaker who has only imprecise knowledge of the relevant value must choose an approximate expression to convey that. But why would a speaker with precise knowledge choose to round off in communicating their knowledge, given that this is both less informative than reporting the precise value and seemingly requires greater effort? Two main sorts of explanations have been put forward. First, by giving up some precision, the speaker potentially gains in accuracy: by choosing a more approximate form, the speaker may avoid negative consequences of saying something false if their information turns out to be incorrect, e.g. if their wristwatch is off by a few minutes (Krifka 2002; Pinkal 1995). Secondly, it is proposed that round numerical expressions are in some way easier or less costly than precise ones. This might involve lower effort required on the part of the speaker, round numbers being on average shorter than nonround ones (Kao et al. 2014; Krifka 2002); alternately, it might reflect greater ease of comprehension or recall on the part of the hearer (Van der Henst et al. 2002); such an advantage for round numbers is demonstrated experimentally by Solt et al. (2017) for temporal expressions and by Nguyen et al. (2022) for other types of numerical expressions. Either way, the consequence is that in situations in which precise detail is not relevant, a round expression will be preferred.
Yet while the factors that underlie the choice of precision level have been investigated, there is to date no explicit formal model of how speakers weigh factors such as informativity, truthfulness, simplicity and processing ease to arrive at a choice between precise and approximate numerical forms, and how this choice depends on the context. Recently, the methods of probabilistic gametheoretic pragmatics have been successfully applied to model other instances of situationally driven linguistically variation. Two notable examples of such work are Burnett’s (2019) analysis of the variable realization of ING via the novel Social Meaning Games framework, and Yoon et al.’s (2020) modeling of the choice between evaluative adjectives – for example, whether a performance is characterized as terrible, not bad or amazing – as depending on whether the speaker’s goal is to be polite, informative or both. What these two analyses have in common is that they feature interlocutors who reason about each other’s beliefs and goals, with the speaker’s choice between forms optimized to meet certain potentially conflicting criteria. These characteristics are similar to those proposed to underlie the phenomenon of rounding, suggesting a similar approach to the present topic. A simple nonprobabilistic model of this sort is sketched out by Jäger (2013), but he does not test the model against actual speaker data. Somewhat relatedly, Egré et al. (2021) develop a Rational Speech Act model of the choice between around n and between n and m in cases of imprecise knowledge, but do not address situational variation, and likewise do not attempt an experimental validation.
In this paper, we extend this body of work. We first present a production study illustrating how speakers’ choice of precision level varies by context, and then fit a gametheoretic probabilistic choice model to the results. Through this modeling process we gain insights into the pragmatics of imprecision, and the dynamics of situationally driven pragmatic variation more generally.
2 The imprecision experiment
2.1 Design and materials
A production experiment was conducted to obtain robust empirical data on the choice of precision level by context. Participants read a scenario in which they had witnessed an automobile accident; the time it occurred (as recalled by the participant) was displayed visually. Participants then read one of two continuations in which an interlocutor asks what time the accident took place, and typed in their answer in a text box. Subsequently, participants were asked the reasons for their answer.
Seven information states were tested: the precise states 8:30, 8:30 ± 1, 8:30 ± 2, 8:30 ± 3, 8:30 ± 4, 8:30 ± 5 (where 8:30 ± 1 etc. means that half of respondents saw 8:29 while half saw 8:31); and the approximate state 8:26–8:34. These were tested in two contexts: a police station, where a police officer takes a witness statement; and a party, where a neighbor asks about the accident. The first of these was intended to exemplify a situation where both precision and accuracy are important, and was predicted to elicit a high level of precise/nonround responses. The second was designed to represent a situation where precise detail and absolute accuracy are less important than facilitating hearer comprehension, and was predicted to elicit a higher level of approximate/rounded responses. This resulted in 14 (2 × 7) conditions in total, which were tested in a singleitem fully betweensubjects design. See Figure 1 for the full text of both contexts and Figure A.1 in the appendix for depictions of all information states.
The experiment was programmed in LabVanced; links are listed in Appendix Table A.1.
2.2 Participants
499 selfdeclared native speakers of English were recruited via Prolific, comprising approximately 30 participants/condition in each of the 12 precise information state conditions and 60 participants/condition in the 2 approximate information state conditions (see Appendix Table A.2 for details). Participants completed an informed consent approved by the Ethics Committee of the German Linguistic Society (DGfS) and were paid £0.50 for their participation.
2.3 Results
Responses to the primary ‘what time?’ question were coded according to the time value (e.g. 8:30, 8:34) and the presence of an approximator (e.g. about, approximately). Truthconditionally equivalent responses were collapsed together (e.g. eight thirty and half past eight were both coded as 8:30).^{[1]} Intervaldenoting expressions (e.g. between 8:26 am and 8:34 am) were coded separately. Data from 24 respondents could not be successfully coded; this consisted of cases where the participant had misread the clock representation (which occurred primarily in the approximate information state condition) or otherwise misunderstood the experimental task. These data points were excluded from further analysis and model fitting. Normalized results matrices are shown in Figure 2 (Matrices with absolute numbers are given in Appendix Figure A.2).
The primary findings can be summarized as follows:
Most importantly, participants in precise nonround information states (8:30 ± 1, 8:30 ± 2, 8:30 ± 3, 8:30 ± 4) frequently ‘rounded off’ their answers to values divisible by 15 (e.g. [about] eight thirty) or divisible by 5 (e.g. [approximately] 8:25). But as predicted, a difference in the frequency of rounding was observed between the two contexts (χ ^{2} = 13.198, p < 0.001), with more rounded answers in the neighbor context than the police context. Both bare round numbers and approximatormodified numbers were used in rounding (see Appendix Figure A.3 for details on choice of approximator by context).
Secondly, participants in the precise round information states 8:30 and 8:30 ± 5 almost universally used bare round numbers.
Finally, participants in the approximate information state 8:26–8:34 gave three types of answers: the bare round value 8:30; approximator +8:30; and interval expressions such as between 8:26 and 8:34; the latter expression type is not used in the other conditions. There was a nearsignificant difference in answer type between context conditions (χ ^{2} = 5.121, p = 0.0773), with interval expressions tending to be used more frequently in the police context, and bare round numbers more frequently in the neighbor context.
Participants’ justifications for their answers were likewise coded into broad categories; results are summarized in Appendix Table A.3. Here we observe that participants’ own reasoning reflects factors posited in the theoretical literature to underlie the choice between precise and approximate numerical expressions, including the level of detail required by the context and the interlocutor, the lack of precise information or possibility of incorrect information, the importance of accuracy or truthfulness, and considerations of what is easier to say and/or easier for the hearer to understand and remember. Furthermore, typical justifications tended to differ by context, with mentions of accuracy and possible information lack/misinformation more frequent in the police context, and mentions of habit/convention, contextual appropriateness and speaker/hearer ease more frequent in the neighbor context.
Complete participantlevel data are part or the supplementary material and also available at: https://www.doi.org/10.6084/m9.figshare.21629531.
3 The imprecision model
In this section we introduce the main aspects of the Imprecision Model in a semiformal manner. A complete formal definition including the full implementation is given in the Technical Report. The Python implementation of the model is available at: https://osf.io/ur5fp/?view_only=7d646308e246437caee7015a96650da3.
3.1 Model structure
The Imprecision Model predicts pragmatic speaker behavior, more concretely, the probability that a pragmatic speaker will use an utterance
We can then model a speaker’s probability
whereby U
_{tot} is the speaker’s total utility for using utterance
3.2 Speaker utilities
We posit that a speaker’s total utility contains distinct components that represent different goals that speakers may entertain. Formally, the speaker’s total utility U
_{tot} for using utterance
U
_{tot} is a combination of the informational utility U
_{inf}, and the weighted addends roundness utility U
_{rnd} and accuracy utility U
_{acc}, minus the utterance cost
We assume that a cooperative speaker follows the goal of informativity to a full degree. The rationale is that we see informativity as the most basic and general speaker goal, which should be present across situations; therefore, U
_{inf} is assumed to be fully weighted with factor 1 in both contexts.^{[6]} The goals of heareroriented simplification and accuracy may be only followed to some degree. Therefore, U
_{rnd} and U
_{acc} are weighted with the weight factors w
_{
R
} and w
_{
A
}, respectively, whereby
3.2.1 Informational utility
The informational utility U
_{inf} represents the speaker’s goal of informativity: it expresses how informative an utterance
whereby
More concretely,
whereby
In many models, the standard assumption for the payoff function is
whereby
3.2.2 Roundness utility
The roundness utility U _{rnd} represents the speaker’s goal of heareroriented simplification: the roundness utility of the utterance accommodates the hearer’s task to better perceive and memorize the utterance, reflecting the processing advantage for round numbers (see above).
We distinguish different levels of roundness with respect to a roundness hierarchy. We define a function rnd(n) that returns a roundness level from a numeric utterance (for details see the Technical Report). All utterances’ roundness levels are listed in Table A.4. The roundness utility
where the roundness level of utterance
3.2.3 Accuracy utility
The accuracy utility U
_{acc} represents the speaker’s goal of accuracy: it rewards the chance of the literal interpretation
^{[8]} of utterance
In the situation where the speaker’s knowledge is imprecise, there can be utterances which have a better chance to communicate the true state of the world (or short: to be accurate) than others. Let’s give an example: imagine that the speaker’s information state is
Following the line of thought above, we define the accuracy utility
4 Reproduction of empirical data
From the experiment we obtained empirical data on speaker behavior, where proportions (e.g. matrix entries in the matrices of Figure 2) can be interpreted as probabilities. The goal of the Imprecision Model is to reconstruct such probabilistic speaker behavior. Here, the probability of using utterance
The results of this analysis are as follows: the ‘police context’ matrix is best reconstructed with the parameter values
Figure 3 displays the juxtaposition of optimal parameter values across both matrices. These finding show that speaker goals are differently weighted across contexts: the pressure towards heareroriented simplification (weighted by
As a next step we want to evaluate the quality of the matrix reconstructions. In Figure 4 we juxtapose the empirical matrices (top) with the optimal reconstructions (bottom). This comparison gives a first visual impression. In a next step we conducted a statistical analysis of the reconstruction data. We computed the Pearson correlation coefficient r and the coefficient of determination r
^{2} between the empirical data (independent variable) and the reconstructed data (dependent variable), taking the identity function y = x as linear regression. The r
^{2} value with identity function as linear regression is also known as the ‘goodnessoffit’ value. The results over all 63 data points were:
Finally, we tested our model against a second version of the Imprecision model where both weights
All in all, the statistical analysis of our model yields correlation values and goodnessoffit values that are relatively close to 1, thus close to a perfect reconstruction. Moreover, with respect to mean square error, Pearson correlation coefficient and goodnessoffitvalue, our model outperforms a second version of the Imprecision model where weights
5 Conclusions
In this paper, we have demonstrated that speakers’ choice of how precisely to communicate numerical information depends on the utterance situation, and shown that these patterns of behavior can be reproduced by a probabilistic gametheoretic model featuring a multicomponent utility function. Through this modeling approach, we have found support for the view that situational variation in precision level derives from variation in the goals that the speaker pursues in different situations, with the relative importance of accuracy versus heareroriented simplification playing a crucial role in the two scenarios tested.
We see these findings as important in their own right, numerical expressions being frequent in language. More generally, we take these findings to demonstrate the value of probabilistic gametheoretic modeling as a route to understanding the dynamics of situationally driven pragmatic variation. In future work we plan to extend this approach to additional phenomena and parameters of situational variation.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: SFB 1412, 416591334
Acknowledgments
We thank Heather Burnett, Manfred Krifka, Uli Sauerland and the audiences at the ZAS and Humboldt University for helpful discussion, and Alexandra Fossa and Hadewych Versteegh for assistance with the data analysis of the experimental data.

Research Funding: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1412, 416591334.
Appendix A: Supplementary tables and figures
ID  Date  Participants  Link 
1  May 3rd 2021  250  www.labvanced.com/player.html?id=22972 
2  May 17th 2021  252  www.labvanced.com/player.html?id=23753 
Situation  Information  Participants 

Police context  8:30  32 
Police context  8:30 ± 1  27 
Police context  8:30 ± 2  28 
Police context  8:30 ± 3  35 
Police context  8:30 ± 4  29 
Police context  8:30 ± 5  30 
Police context  8:26–8:34  65 
Neighbor context  8:30  31 
Neighbor context  8:30 ± 1  32 
Neighbor context  8:30 ± 2  35 
Neighbor context  8:30 ± 3  31 
Neighbor context  8:30 ± 4  32 
Neighbor context  8:30 ± 5  32 
Neighbor context  8:26 − 8:34  60 


499 
Answer category  Police  Neighbor 

Level of precision/detail  51  61 
Accuracy/truthfulness  28  20 
Possible lack of information  17  13 
Possible misinformation  9  3 
Safe choice  2  2 
Hearer needs  19  13 
Appropriateness for context  15  33 
Speaker ease  7  22 
Hearer ease  6  14 
Habit/convention  35  52 
How it sounds  2  7 
Other/irrelevant  108  105 


Total # of respondents  231  244 
v

Core semantic meaning 〚v〛  rnd(v)  Sample utterance 



1  ‘ … at 8:25.’ 


0  ‘ … at 8:26.’ 


0  ‘ … at 8:27.’ 


0  ‘ … at 8:28.’ 


0  ‘ … at 8:29.’ 


2  ‘ … at 8:30.’ 


0  ‘ … at 8:31.’ 


0  ‘ … at 8:32.’ 


0  ‘ … at 8:33.’ 


0  ‘ … at 8:34.’ 


1  ‘ … at 8:35.’ 


1  ‘ … around 8:25.’ 


2  ‘ … about 8:30.’ 


1  ‘ … approximately at 8:35.’ 


0  ‘ … between 8:25 and 8:35.’ 
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Supplementary Material
The online version of this article offers supplementary material (https://doi.org/10.1515/lingvan20220035).
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