Modeling (im)precision in context

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 non-round number (1a); by rounding off to a multiple of 5 (1b); or by using an explicit approximator such as about or roughly (1c).

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 non-round 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 game-theoretic 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 non-probabilistic 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 game-theoretic 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/non-round 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 single-item fully between-subjects design. See Figure 1 for the full text of both contexts and Figure A.1 in the appendix for depictions of all information states.

Figure 1:

Screenshot of experimental tasks. Above: police context and information state 8:30 ± 3 (here 8.27); below: neighbor context and information state 8.30 ± 5 (here 8:35) (images from Freepik.com).

The experiment was programmed in LabVanced; links are listed in Appendix Table A.1.

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). Truth-conditionally equivalent responses were collapsed together (e.g. eight thirty and half past eight were both coded as 8:30).^[1] Interval-denoting 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).

Figure 2:

Normalized color-coded matrices representing speaker behavior, left for the police context, right for the neighbor context. Each entry represents proportion of participants in a particular information state (row units) using a particular (categorized) response expression (column units). Note that response expressions with subscript ‘a’ (e.g. v a 30 ) involve an approximator term, and v I n represents interval-denoting expressions.

The primary findings can be summarized as follows:

1. Most importantly, participants in precise non-round 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 approximator-modified numbers were used in rounding (see Appendix Figure A.3 for details on choice of approximator by context).

2. Secondly, participants in the precise round information states 8:30 and 8:30 ± 5 almost universally used bare round numbers.

3. 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 near-significant 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 participant-level 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.

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 v for information state s is given by the probability function P S ( v | s , P H , w ) . Moreover, we assume that the difference in probabilities across contexts is mainly due to the fact that speakers follow particular goals with different strengths in different situations. More concretely, we assume that the difference in speaker behavior in the police versus neighbor contexts is mainly due to the fact that they weight the goals of speaker-oriented simplification (weighted by w R ) and accuracy (weighted by w A ) differently. By calibrating the weight parameters w R and w A , we are aiming to find the optimal reconstruction matrix for each context: we are searching for the optimal parameters w R * and w A * that minimize the so-called mean square error (MSE) between empirical data and model predictions (for more details see the Technical Report).

The results of this analysis are as follows: the ‘police context’ matrix is best reconstructed with the parameter values w R * = 0.46 , w A * = 0.22 , achieving an MSE of 0.00139. The ‘neighbor context’ matrix is best reconstructed with the parameter values w R * = 0.66 , w A * = 0.02 , achieving an MSE of 0.00114. These parameter values are global optima, as illustrated in Appendix Figure A.4, which shows the MSE heat map over the ( w R , w A ) -space for each context.

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 hearer-oriented simplification (weighted by w R ) is stronger in the neighbor context than in the police context, whereas for the pressure towards accuracy (weighted by w A ), it is exactly the other way around. Further model checks showed that this cross-contextual difference of optimal weights w R * and w A * is very robust across alternative model settings.^[10]

Figure 3:

Optimal parameter values for w R and w A that minimize the mean square error between empirical data and model prediction.

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 ‘goodness-of-fit’ value. The results over all 63 data points were: r = 0.985 , r 2 = 0.971 for the police context, and r = 0.985 , r 2 = 0.967 for the neighbor context.^[11] The presentation of data points and regression line, including an overview of the statistical results, is given in Figure 5.

Figure 4:

The empirical data (top) and the best reconstructions of the model (bottom) of the police context (left) and the neighbor context (right).

Figure 5:

Representation of all data points of the police context matrix (left) and the neighbor context matrix (right), positioned with respect to the empirical data from the experiment (x-axis) and the model prediction data from the best reconstruction (y-axis). The red line represents the identity function (with x=y), used as linear regression for r ² calculations. The top left corner lifts the Pearson correlation coefficient r, the coefficient of determination (goodness-of-fit) for all data points r 2 ( a l l ) , and the goodness-of-fit for ’non-near-zero’ values r 2 ( n n z ) , where x and y both are not below 0.01.

Finally, we tested our model against a second version of the Imprecision model where both weights w R and w A are assumed to be invariant across contexts. For this second version, the best reconstruction has a clearly worse mean square error (around 0.00351, in comparison to 0.00139/0.00114), and achieves a lower Pearson correlation (0.969 vs 0.985) and a lower r-square value (0.935 vs 0.967/0.971). The results are juxtaposed in the Appendix, Figure A.5; more details are given in the Technical Report.

All in all, the statistical analysis of our model yields correlation values and goodness-of-fit values that are relatively close to 1, thus close to a perfect reconstruction. Moreover, with respect to mean square error, Pearson correlation coefficient and goodness-of-fit-value, our model outperforms a second version of the Imprecision model where weights w R and w A are assumed to be invariant across contexts. This shows that the central underlying assumption, namely that cross-situational variation in a speaker’s precision level is to a great extent due to different weighting of the two speaker goals of hearer-oriented simplification and accuracy, is strongly supported by the model comparison analysis.

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 game-theoretic model featuring a multi-component 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 hearer-oriented 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 game-theoretic 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.