Investigating fragment usage with a gamified utterance selection task

Robin Lemke*

Abstract. Nonsentential utterances, or fragments, like A coffee, please! can often be
used to communicate a propositional meaning otherwise encoded by a complete sen-
tence (I’d like to order a coffee, please!). Previous research focused mostly on the
syntax and licensing of fragments, but the questions of why speakers use fragments
and how listeners interpret them are still underexplored. I propose a simple game-
theoretic account of fragment usage, which predicts (i) that listeners assign fragments
the most likely interpretation in context and (ii) that speakers are aware of this and
trade-off production cost and the risk of being misunderstood when choosing their ut-
terance. Using a corpus of production data, empirically founded and precise model
predictions are generated. These predictions are evaluated with two experiments us-
ing a novel gamified utterance selection paradigm. The experiments suggest that, as
predicted, speakers take into account both potential gains in efficiency and the risk of
being misunderstood when choosing their utterance.
Keywords. ellipsis, fragments, game theory

1. Introduction. Instead of a complete sentence like (1a), speakers often use nonsentential utter-
ances, or fragments (Morgan 1973), for this purpose (1b). Despite their reduced form, fragments
can be meaning-equivalent to their fully sentential counterparts in an appropriate context.

(1) [passenger asks conductor on the platform next to a waiting train:]
a. Does this train go to Paris?
b. To Paris?

While the syntax of fragments has been and is being extensively studied (see e.g. Ginzburg & Sag
2000, Merchant 2004, Reich 2007, Weir 2014, Ott & Struckmeier 2016, Lemke 2021), the question
of why speakers actually choose to use a fragment is relatively underexplored (but cf. Bergen &
Goodman 2015, Lemke 2021).

Intuitively, fragments have the advantage of being shorter than sentences, which reduces the
production effort for the speaker and the (syntactic) processing effort for the listener. This makes
communication more efficient, as the same information is transmitted in less time, a tendency that
also can be attributed to the Gricean maxim of manner “be brief” (Grice 1975). The downside
of fragments is their vagueness: A single fragment can often be used to communicate meanings
expressed by different complete sentences. For instance, the fragment in (1b) can communicate
not only (1a), but also any of the questions in (2). Due to this ambiguity, the listener might assign
the fragment a different interpretation than intended by the speaker, or they might not be able to
retrieve any interpretation at all. Both of these outcomes would result in communication failure
and require further clarifications, making communication less efficient.

(2) a. Are you traveling to Paris?
b. Have you ever been to Paris?

*Authors: Robin Lemke, Saarland University (robin.lemke@uni-saarland.de).

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I pursue the hypothesis that speakers counterbalance the gain in efficiency provided by fragments
with the risk of communication failure caused by their ambiguity when choosing how to encode
the message. If the gain in efficiency outweighs the risk, the speaker will use the fragment; if
it does not, a full sentence. This might explain the observation in Lemke (2021) that fragments
are preferred over sentences as answers to questions, but the opposite holds in discourse-initial
contexts, where the uncertainty about their intended interpretation is probably greater.

To formalize this idea, I use a simple game-theoretic model, which allows for including an
explicit utterance cost structure and which predicts how likely it actually is to communicate a
particular meaning in a context. Such models, which have been widely applied to pragmatic phe-
nomena, such as scalar implicature (Franke 2009), reference (Frank & Goodman 2012, Rohde et al.
2012) or the construction of social meaning (Burnett 2017), predict speaker and listener behavior
based on an explicit model of the utterance context. From the perspective of ellipsis research,
this has the advantage of explaining the interpretation of elliptical utterances based on pragmatic
inferences (i.e. relying on the same mechanisms as implicature generation), instead of having to
assume a specific processing mechanism, as has been suggested by e.g. Arregui et al. (2006) for
VP ellipsis. In order to generate model predictions, I rely on a corpus of production data collected
by Lemke (2021). This approach differs from most previous empirical investigations of game-
theoretic reasoning in pragmatics, which relied on controlled and balanced experimental setups in
which objects are referred to by their shape or color with one- or two-word utterances (e.g. Frank
& Goodman 2012, Rohde et al. 2012, Sikos et al. 2021). To my knowledge, my study is therefore
the first attempt in game-theoretic pragmatics to derive model predictions from a relatively large,
unbalanced, and diverse data set based on linguistic data produced by human subjects.

This paper is organized as follows: Section 2 presents the game-theoretic account of fragment
usage and Section 3 the data set on which model predictions are based. Section 4 introduces the
experimental design, before Sections 5 and 6 present the results of the two experiments and Section
7 summarizes the main results and conclusions.

2. A game-theoretic account of fragment usage. I formalize the idea that fragment usage re-
sults from a trade-off between the gain in efficiency and the risk of being misunderstood within a
signaling game framework (Franke 2009, Frank & Goodman 2012, Sikos et al. 2021). The model
predicts speaker and listener behavior based on a (possibly partial) mapping between a set of utter-
ances and a set of messages, i.e. meanings the speaker could communicate. The components of the
model are mostly equivalent to those in Franke (2009), although I slightly adapt the terminology
to better fit the inference from (possibly nonsentential) utterances to complete sentences.

There is a set of messages M ∈ m comprising all of the meanings the speaker can commu-
nicate in a situation (e.g. (1a) and (2) in the train example), and a set of utterances u ∈ U that
they can use for this purpose. The speaker’s task is to send the utterance which is optimal to com-
municate the intended message. The listener receives this utterance and performs an interpretation
action a ∈ A, i.e. they assign the utterance a meaning. Since M contains all possible meanings,
A = M . The set of utterances U contains (i) sentences, which are equivalent to the m ∈ M 1 and

1This is a simplification, because there are also meaning-equivalent sentences. I address this concern by pooling
the production data to sentences differing in meaning as discussed in Section 3, but it would also be possible to include
all of the synonym sentences in U .

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(ii) all fragments which can be derived from the m ∈ M by grammatically licensed omission.2 I
consider only fragments that are constituents or sequences thereof (e.g. To Paris?, This (one) to
Paris?, but not, for example, the heads of a DP and a PP only (e.g. This to?). The reason for this
is that fragments are licensed by focus or givenness (e.g. Merchant 2004, Reich 2007) and these
concepts apply generally to constituents. Having established M and U , it is necessary to determine
whether an utterance u can be used to communicate a message m. In his account of implicature,
Franke (2009) relies on truth conditions, but fragments cannot be considered true or false before
having been enriched to a complete proposition. Given the computation of U described above,
I rely on whether u can be derived from m by grammatically licensed omission. Following the
notation in Franke (2009), assume a denotation function [[·]], which returns 1 if u can be derived
from m and 0 otherwise. Finally, not all messages are necessarily equally likely in a situation. This
might be due to world knowledge (people are more likely to ask the conductor whether the train
goes to Paris than whether they have been to Paris) or statistical knowledge about language use.
This potential difference in likelihood is captured by a prior probability distribution Pr(M).

Based on these components, following Franke (2009), chains of mutual reasoning about the
behavior, preferences and beliefs of the interlocutors are initialized: one starting from a “literal”
speaker S0 and one starting from a “literal” listener L0. Since I model the speaker’s perspective
in my experiments, I focus on the basic listener model, which the speaker should consider when
selecting their utterance. Given a message u sent by the speaker, L0 calculates the likelihood of
each m ∈ M given u. As equation 1 shows, this posterior p(m|u) is determined as the ratio of
Pr(m) and the probability mass of other possible interpretations m′ of u. This reweighs the prior
probability among those messages from which u could have been derived. Based on the resulting
probability distribution, the listener selects the most likely message as the interpretation of u. As
I discuss in Section 4.2, the reason for this is that successful communication is rewarded with a
payoff and maximizing L0p(m,u) also maximizes the payoff.

L0(m,u) =
Pr(m)× [[u]]m∑
m′ Pr(m′)× [[u]]m′

(1)

A speaker who takes listener behavior into account (a S1 speaker in Franke’s terminology)
will constrain their utterance choice based on the likelihood that the listener will interpret it as
intended. Since U has been derived by omission from M , sentential utterances fully disambiguate
between messages. Therefore, from the perspective of successful communication they should be
always at least as ideal as a fragment. However, sentences come with an additional production
cost, which might result in them being less ideal for the speaker.3 Since it is a priori unclear how to
quantify utterance cost and its relationship to the gain in efficiency, I use an explicit cost structure
in my experiments to ensure that sending fragments is cheaper than sending sentences.

2Note that I follow most of the theoretical literature cited above and treat fragments as grammatically well-formed,
but there are diverging views: Bergen & Goodman (2015) assume that fragments are ungrammatical but can still be
suitable means of communication if the listener can retrieve their meaning. The account I propose is in principle
independent from the question of whether fragments are grammatical, but assuming ungrammaticality would require
to include ungrammatical fragments in U , which I do not.

3Of course, unnecessary redundancy might be dispreferred by the listener (Schäfer et al. 2021), but since they have
no control over which utterance the speaker sends, this does not need to be modeled.

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3. Data set. Testing the predictions of the account described in Section 2 empirically requires
realistic estimates of the model’s components: (i) the set of messages M , (ii) the set of utterances
U , (iii) the mapping between M and U , and (iv) the prior probability distribution over messages
Pr(M). I calculate these parameters from a data set originally collected by Lemke (2021).4 Lemke
(2021) used a crowd-sourced written production task to elicit utterances with script-based context
stories like (3). The participants were instructed to produce the most natural utterance in that
situation by the person specified in the prompt Annika to Jenny:

(3) Annika and Jenny want to cook pasta. Annika has put a pot with water on the stove. Then
she has turned the stove on. After a few minutes, the water has started to boil. Annika to
Jenny:

The stories were based on event chains extracted from the DeScript corpus of script knowledge
(Wanzare et al. 2016). For each story, which represents a different script-based scenario, about
100 responses were collected. The answers (e.g. (4a)) were preprocessed into abstract represen-
tations like (4b), where each word corresponded to a constituent that could be freely omitted in
German. This ensures that all fragments derived from these representations are grammatical, i.e.
constituents or sequences thereof, as discussed in Section 2.5

(4) a. Pour the pasta into the pot!
b. pour pasta pot.GOAL

The set of unique representations for each scenario (e.g. cooking pasta) was taken to represent M
in this scenario. Since German has free word order, all of the words in the representations like
(4b) were ordered alphabetically. This ensures that the synonym pour pasta pot.GOAL and
pour pot.GOAL pasta are not treated as different meaning representations. The likelihood
of each representation within the 100 preprocessed responses determines Pr(M) in this scenario.
U is the union of M and all fragments that can be derived by grammatical omission from all
m ∈ M : For a message m, this includes the corresponding sentence, all of its constituents (i.e.
the individual “words” in the abstract representation) and all possible combinations of thereof. (5)
exemplifies this at the example of (4b).

(5) pour pasta pot.GOAL, pasta pot.GOAL, pour pot.GOAL, pour pasta,
pour, pasta, pot.GOAL

Finally, [[u]]m determined for each u ∈ U and each m ∈ M whether u could be derived from m
by omission. Applying equation 1, the likelihood of communicative success L0(m|u) was then
calculated for each m ∈ M and u ∈ U .

4. Experimental approach. The experiments presented below were designed to test the hypoth-
esis that the usage of fragment is conditioned by a trade-off between the gain in efficiency and the
risk of being misunderstood. Therefore, fragments are expected to be more strongly preferred if
(i) their cost relative to a sentence is lower, and if (ii) the likelihood of communicative success is
relatively high.

4The data set was collected in German, but I present the context stories in English for convenience.
5For details on the preprocessing procedure, see Lemke (2021; 202–206).

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I tested this with a pseudo-interactive gamified utterance selection task inspired by Rohde
et al. (2012). In my experiments, the participant took the speaker role and chose between differ-
ent sentential and fragment utterances to communicate a message determined by the experiment.
The listener was simulated to behave according to model predictions: For sentences, they always
selected the correct interpretation and for fragments, they maximized L0p(m|u). An explicit cost
structure ensured that fragments were cheaper than sentences.

4.1. MATERIALS: CONTEXT STORY, MESSAGES, AND UTTERANCES. The stimuli were derived
from the data set by Lemke (2021) introduced above. In each trial, the participant saw a context
story, three messages and six utterances, as shown in Figure 1. The context stories were based
on those used by Lemke (2021), but the roles of the characters were assigned to the participant
and their (simulated) partner. The messages were presented as states of affairs that the participant
might want to communicate to their partner. The message the participant had to communicate
was determined by the experiment depending on the experimental condition (See Section 4.2) and
highlighted in blue. Among the six utterances, three were sentences, each unambiguously encoding
one of the messages. One of the three fragments (into the water in Figure 1) is ambiguous and
can communicate two of the messages. These two messages were selected from the production
data so that one of them has a higher L0p(m|u) given the ambiguous fragment than the other
one. If participants base their utterance choice on the likelihood of communicative success, they
should be more likely to use a fragment when asked to refer to the more likely message than
when referring to the less likely one, because a listener maximizing L0p(m|u) will choose the
more likely interpretation more often. The second fragment (in the example: on the table) always
referred unambiguously to the third message. This establishes a baseline for the rate of fragment
usage when there is no risk of misunderstanding in the experimental setup. The third fragment (the
recipe!) had no corresponding message and served as a control to exclude inattentive participants
who randomly click on any of the utterances. All messages and utterances were based on the
abstract representations contained in the production data for the respective scenario.

4.2. CONDITIONS, UTTERANCE COST AND EXPECTATIONS. There were three conditions differ-
ing in the message to be communicated. In the critical condition, the most likely message given the
ambiguous fragment was highlighted. In the competitor condition, the less likely message com-
patible with the ambiguous fragment was highlighted. In the unambiguous condition, the message
to which the unambiguous fragment refers was to be communicated. Note that this fragment was
unambiguous with respect to the three messages displayed in the experimental setup, though not
necessarily in the production data set.6 As Table 1 shows, on average, the ambiguous fragment had
a higher L0p(m|u) in the critical condition than in the competitor condition within each scenario
(see Table 1). As the ranges show, there was some overlap in the probabilities, but within each
scenario, the message in the critical condition was always more likely.

Production cost was implemented with a system of virtual coins. In Experiment 1, partic-
ipants received a starting balance of 500 coins and the cost for sending a fragment (30 coins)
was lower than that for sentences (100 coins). Successful communication was rewarded with 120

6In some stimuli, it was also unambiguous in the production data, as the maximal L0p(m|u) of 1 in Table 1 shows,
but this was not always the case.

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Figure 1: Screenshot of Experiment 1 after revealing the messages and utterances. The experiment
was conducted in German, but has been translated here for convenience.

Condition Lowest p(m|u) Highest p(m|u) Mean p(m|u)
critical 0.12 0.69 0.38
competitor 0.03 0.22 0.09
unambiguous 0.15 1.0 0.75

Table 1: Range of L0(m|u) probabilities and means by conditions

coins.7 These quantities were set based on the expected utility (EU) (Franke 2009) of the utterances
in each condition. In the experiment, EU can be calculated as shown in equation 2 by subtracting
the cost c from the product of the likelihood of successful communication L0p(m|u) and the utility
U(m,u), i.e. the payoff in case of success.

EU(u,m) = L0p(m|u)× U(m,u)− c (2)

7See Section 6 for the cost structure used in Experiment 2.

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Since the EU of an (unambiguous) sentence is EU(u,m) = 1 × 120 − 100 = 20 and frag-
ments yield a higher EU if their L0p(m|u) > 0.42, the model predicts an absolute preference for
fragments beyond this threshold. Given the data in Table 1, in the range of L0p(m|u) it should be
possible to observe differences in the ratio of fragments produced. On average, the fragment ratio
is expected to be highest in the unambiguous condition and lowest in the competitor condition,
with the critical condition in between.

4.3. PROCEDURE. Each of the two experiments was completed by 60 self-reported native speak-
ers of German recruited on the crowd-sourcing platform Prolific. The participants were rewarded
with £2.67 (the equivalent of the German minimum wage of 12.41C for the projected duration of
15 minutes). The materials were distributed across three lists, so that each participant saw each
of the 15 stimuli in one of three conditions, with each condition appearing equally often. Stimuli
were presented in individual fully randomized order. 8

After giving informed consent by marking a checkbox, reading the instructions and provid-
ing their Prolific ID, participants were asked to enter a nickname, which should not be their real
name and which would not be published, to make the experiment appear more interactive. They
were then asked to wait for the connection to their partner to be established. The waiting time was
randomly sampled from an array of durations between 10 and 25 seconds. After this, the partici-
pants were notified that a partner had been found, informed about their partner’s name (randomly
sampled from an array of names), and asked to begin a practice phase of two (Experiment 1) or
three (Experiment 2) trials.9 The setup in the practice phase was identical to the main experiment
in terms of the number of messages and utterances, as well as the costs. The only difference was
that each of the three fragments referred unambiguously to one of the messages. After the practice
phase, the score was reset to the initial amount and participants were informed in advance about
this.

Each trial began with a display of the context story only. Participants were told to press the
space bar to display each of the other elements: the messages, the highlighted message, and the ut-
terances. This intended to make subjects read all of the messages before seeing which one would be
highlighted, which is critical to realize that the ambiguous fragment is ambiguous. After selecting
an utterance, participants received feedback on the partner’s interpretation. As anticipated above,
the partner always maximized L0p(m|u) when selecting an utterance: For correct sentential utter-
ances and fragments in the critical condition (where the message had a high L0p(m|u)) this results
always in correct interpretations. Fragment utterances in the (low L0p(m|u)) competitor condition
were never successful, as maximizing L0p(m|u) led to selecting the more likely interpretation.
If communication was successful, a notice “⟨partner⟩ understood you correctly” appeared in the
center of the screen, otherwise it read “⟨partner⟩ understood something else”. Additionally, the
message selected by the fake partner was highlighted in green (success) or red (failure) to indicate
what the partner understood. If the participant had selected the attention check fragment, which

8The experiment was followed by a questionnaire to evaluate the experiment, which included questions about the
participants’ affinity to games and self-assessed likelihood of taken risks to investigate potential individual differences.
Since there were no theoretically interesting correlations between the answers to these questions and behavioral data,
the questionnaire is not discussed further here.

9Since there were almost no errors in the practice phase, it was shortened to two trials in Experiment 2.

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0.00

0.25

0.50

0.75

1.00

competitor critical unambiguous

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competitor critical unambiguous

0.0

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0.00 0.25 0.50 0.75 1.00

L0 p(m|u)

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Figure 2: The left facet shows the ratio of fragments across the three conditions, the right facet il-
lustrates the continuous effect of L0(m|u) on the ratio of fragments with a Loess smooth (span=1).

did not refer to any of the three messages, the text field “⟨partner⟩ is not sure” was highlighted.
To simulate the partner’s reasoning, a delay was introduced between the participant’s sending the
message and the feedback. The delay time was randomly sampled from an array of values be-
tween 0.8 and 4.4 seconds. Since the participant was told that their partner could see the complete
screen throughout the trial, and consequently have read all of the messages and utterances in the
meantime, quick responses to the participants’ choice seemed natural.

5. Experiment 1. Experiment 1 was conducted using the cost structure described in Section 4.2.
Due to an error in preparing the stimulus table, which resulted in the target fragment being un-
ambiguous, this stimulus had to be discarded before the analysis, leaving 14 stimuli for analysis.
One participant, who provided more utterances incompatble with the target message (n=8) than
compatible ones (n=6), was also excluded from further analysis.

5.1. RESULTS. The participants were very accurate: Only in three trials, the distractor was chosen
and in 16 trials an utterance that could not communicate the highlighted message. These 19 trials
(2.3% of the data) were excluded before the analyses reported in what follows.

The responses across the three conditions and as a function of L0p(m|u) are summarized in
Figure 2. Overall, the participants had a strong preference for complete sentences: In the am-
biguous conditions, the fragment was selected in only about 12-13% of the trials, and even in the
unambiguous condition, participants preferred the fragment only in half of the trials. Since the
game-theoretic account predicts a gradual effect of an utterance’s expected utility on speakers’
choices, and there are notable differences in L0p(m|u) between the scenarios tested (see Table 1),
in the statistical analyses, the PROBABILITY of successful communication was used as a continu-
ous predictor in the statistical analyses rather than investigating categorical contrasts between the

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three conditions. Descriptively, the right facet of Figure 2 supports the prediction that a higher
likelihood of successful communication with fragments increases the fragment ratio.

The data were analyzed with mixed effects logistic regressions (Bates et al. 2015) in R (R Core
Team 2024; version 4.4.1) using a backward model selection procedure. The models predicted a
binary dependent variable ELLIPSIS (sentence/fragment) from the PROBABILITY (L0p(m|u)) of
the fragment and the scaled and centered POSITION of the trial in the experiment. The full model
contained the maximal random effects structure supported by the data, which consisted of by-
subject and by-item random intercepts and by-item random slopes for PROBABILITY.10 Starting
from this full model, fixed effects that did not significantly improve model fit were subsequently
removed. This was determined using Likelihood ratio tests conducted with the anova function (R
Core Team 2024).

The final model contained only a significant main effect of PROBABILITY (χ2 = 6.5, p <
0.05), which suggests that, as predicted by the game-theoretic account, the ratio of fragments
increases with their L0p(m|u). This finding is consistent with the game-theoretic approach. How-
ever, the left facet of Figure 2 suggests that there may be no difference between the ambiguous
(critical and competitor) conditions, despite the higher L0p(m|u) in the critical condition pre-
dicting a higher fragment ratio. Consequently, the PROBABILITY effect might be driven by the
unambiguous condition alone. This aligns with the account’s predictions, because the fragments
in the unambiguous condition had a higher average L0p(m|u). However, this does not provide
genuine evidence for probabilistic rational reasoning, as participants may have simply avoided
ambiguity in the experiment. Therefore, I conducted a further regression analysis using only the
data for the ambiguous conditions. If participants relied on game-theoretic reasoning, the effect of
PROBABILITY should be replicated in this subset. Note that the opposite result would not prove
that participants did not rely on game-theoretic reasoning, as they might it and be but less likely
to assume risks than predicted by the current model. The analysis followed the same procedure as
the main analysis reported above in this section. However, there was no significant main effect of
PROBABILITY (χ2 = 0.01, p > 0.9), failing to confirm the game-theoretic prediction of a gradual
effect.

5.2. DISCUSSION. The results of Experiment 1 are in line with the prediction of the game-
theoretic account that speakers choose ambiguous fragments more often if the risk of misunder-
standing is reduced due to a high L0p(m|u). This gradual effect was not confirmed when excluding
the unambiguous condition, which could have at least two reasons. First, fragments had the highest
L0p(m|u) in this condition, so that the preference for using fragments might simply be stronger.
Second, if participants did not rely on game-theoretic reasoning and prior message probabilities,
they could simply avoid any expression which is ambiguous in the context of the experiment.
While the former explanation would support the game-theoretic account, the second would pro-
vide an explanation for the results which does not require probabilistic pragmatic reasoning.

Another result of the experiment is the overall relatively low fragment ratio. Even for some
fragments with a L0p(m|u) of 1 in the unambiguous condition, participants selected the fragments
only in 30–50% of the trials. This might be at least partially due to the cost structure, which
guaranteed participants a net benefit of 20 coins per trial when using only unambiguous sentences.

10Ellipsis ∼ Probability * Position + (1 | Subject) + (1 + Probability | Item)

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0.00

0.25

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competitor critical unambiguous

Condition

F
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L0 p(m|u)

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competitor critical unambiguous

Figure 3: The left facet shows the ratio of fragments across the three conditions, the right facet il-
lustrates the continuous effect of L0(m|u) on the ratio of fragments with a Loess smooth (span=1).

When looking at the individual participants’ data, it turns out that 15 of them correctly completed
all trials by selecting only sentences. This suggests that the cost structure used in Experiment 1
biased participants toward adopting a risk-avoiding strategy, which might have obscured gradual
differences between the ambiguous conditions.

6. Experiment 2. Experiment 2 addressed the concern that the low fragment ratio in Experiment
1 might have occurred due to a cost structure favoring a risk-avoiding strategy which made about
25% of the participants to select sentences only. To address this, the cost structure was modified to
reduce the benefits obtained in Experiment 1 through this strategy.

6.1. MATERIALS AND COST STRUCTURE. The materials used in Experiment 2 were identical
to Experiment 1, except that the error resulting in the exclusion of one experimental item was
resolved. To make a sentence-only strategy less rewarding, the utterance costs and the starting bal-
ance were adjusted: The reward for successful communication was reduced to 100 coins (instead
of 120), the cost of sentences increased to 130 (instead of 100) and the cost of fragments increased
to 40 (instead of 30). Unlike in Experiment 1, selecting sentences now resulted in a loss of 30 coins
per trial instead of a benefit of 20 and the lower starting balance should increase the motivation to
keep the score high (even though a negative score had no consequences for the subjects’ reward
and they were informed about this in advance).

6.2. PROCEDURE. The procedure was identical to Experiment 1. 60 further participants, who did
not participate in Experiment 1, were recruited on Prolific and rewarded with £2.67.

6.3. RESULTS. Figures 3 shows the ratio of fragments by condition and as a function of L0p(m|u).
The overall fragment ratio was higher than in Experiment 2 (35.01% instead of 25.32%), which

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Predictor Est. Std. error χ2 p-value
INTERCEPT 2.14 0.36 17.71 < 0.001
PROBABILITY −2.56 0.67 9.23 < 0.01
EXPERIMENT −0.50 0.12 18.02 < 0.001

Table 2: Fixed effects in the final model for the joint analysis of experiments 1 and 2.

was also reflected in participants’ individual behavior: Unlike in Experiment 1, only three partici-
pants followed a sentence-only strategy, which resulted in a score of -150. This suggests that the
smaller but secure reward that this strategy returned in Experiment 1 was (at least partially) respon-
sible for the higher relatively low fragment usage in Experiment 1). Besides the higher fragment
ratio, the pattern looks similar to Experiment 1: Fragment ratio seems to increase as a function of
L0p(m|u), but there is no obvious difference between the ambiguous conditions.

I first conducted an analysis in parallel with Experiment 1 using mixed effects logistic re-
gressions (Bates et al. 2015; version 1.1-35.3) in R (R Core Team 2024; version 4.4.1). The full
model contained by-subject and by-item random intercepts and by-item slopes for PROBABILITY,
as well as fixed effects for PROBABILITY, POSITION (scaled and centered) and their interaction.
The overall pattern was identical to Experiment 1: In the analysis of the complete data set, there
was a significant main effect of PROBABILITY (χ2 = 11.04, p < 0.001). As in Experiment 1, this
effect was not significant within the ambiguous conditions only (χ2 = 0.4, p > 0.5).

To quantify the difference between the experiments, I then conducted a joint analysis of the
data from both studies. The procedure was identical to the regression analyses reported above, ex-
cept that I included an additional predictor EXPERIMENT (sum-coded, exp.1 = −0.5, exp.2 = 0.5)
and its interactions with PROBABILITY (numeric) and POSITION (numeric, scaled and centered).
A significant main effect of EXPERIMENT would statistically confirm the higher fragment ratio
in Experiment 2 and an EXPERIMENT:PROBABILITY interaction would show whether the effect
of PROBABILITY is also stronger. The full model contained main effects for all three predictors
and all interactions between them, as well as by-items random intercepts and random slopes for
PROBABILITY. The final model (See Table 2) contains a main effect of PROBABILITY, which
had been also found in the analysis of the individual experiments (χ2 = 9.23, p < 0.01), a main
effect of EXPERIMENT (χ2 = 18.02, p < 0.001). The latter effect confirms the intuition that the
fragment ratio is higher in Experiment 2. The interaction between both predictors is not significant
(χ2 = 2.14, p > 0.1).

6.4. DISCUSSION. Experiment 2 investigated whether the low fragment ratio in Experiment 1
was (partially) due to an overall bias toward sentences in order to avoid risks. This could have ob-
scured gradual differences between the ambiguous conditions, which would have provided genuine
evidence for pragmatic reasoning in fragment usage. The significant difference in fragment ratio
between both experiments, confirmed by the joint analysis, suggests that the cost structure had an
effect: If fragments yield a higher potential benefit, their ratio is increased, even if the likelihood
of communicative success remains constant. However, it is also clear that the stronger preference
for sentences in Experiment 1 was not the only cause for the lack of a gradual effect within the
ambiguous conditions, since this effect was also not found in Experiment 2.

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7. General discussion. I conducted two pseudo-interactive utterance selection experiments to in-
vestigate whether the usage of fragments follows the predictions of a game-theoretic account.
Unlike previous studies in the field, I did not use tightly restricted and balanced sets of utterances
and messages; instead, I constructed the stimuli based on a diverse, crowd-sourced data set, with a
much larger variety of messages and utterances, as well as unbalanced priors over messages.

The results are overall in line with the trade-off between reducing production cost and maxi-
mizing the likelihood of successful communication predicted by the game-theoretic account: In the
main analyses of both experiments, participants preferred fragments more often when the model
predicted a high likelihood of getting the message across. They also selected more fragments in
Experiment 2, where the modified cost structure increased the gain in efficiency as compared to
Experiment 1. However, the effect observed in the complete data set was not found when examin-
ing only the data from the ambiguous conditions. Experiment 2 investigated whether it had been
obscured by the bias toward using sentences in Experiment 1, but this was not the case. Therefore,
at this point it cannot be ruled out that participants simply avoided ambiguity in the experimental
setup, which does not require pragmatic reasoning. A possible reason for the absence of the grad-
ual effect might be the method used to collect the production data set. Since participants provided
only a single most likely utterance, less likely utterances are probably underrepresented in the data
set. This could result in low L0p(m|u) messages being relatively predictable, which may have
biased subjects to prefer sentences more often.

Taken together, the experiments show how a game-theoretic account of language production
and interpretation can also be applied to more realistic and diverse communication situations than
the previous tightly controlled studies on reference. They also suggest that the choice between
an elliptical and a sentential utterance depends on a trade-off between production cost and the
likelihood of communicative success, which can be explicitly captured by a game-theoretic model
of pragmatic inference. Future research could explore the extent to which this reasoning is re-
quired only for interpreting discourse-initial fragments tested in the experiments presented here, or
whether it also extends to other types of ellipsis.

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