Why do you ask? Good questions provoke informative answers.
Robert X. D. Hawkins, Andreas Stuhlmüller, Judith Degen, Noah D. Goodman{rxdh,astu,jdegen,ngoodman}@stanford.edu
Department of Psychology, 450 Serra Mall
Stanford, CA 94305 USA
Abstract
What makes a question useful? What makes an answer appropriate? In this paper, we formulate a family of increasingly
sophisticated models of question-answer behavior within the
Rational Speech Act framework. We compare these models
based on three different pieces of evidence: first, we demonstrate how our answerer models capture a classic effect in psycholinguistics showing that an answerer’s level of informativeness varies with the inferred questioner goal, while keeping
the question constant. Second, we jointly test the questioner
and answerer components of our model based on empirical evidence from a question-answer reasoning game. Third, we examine a special case of this game to further distinguish among
the questioner models. We find that sophisticated pragmatic
reasoning is needed to account for some of the data. People
can use questions to provide cues to the answerer about theirinterest, and can select answers that are informative about inferred interests.
Keywords: language understanding; pragmatics; Bayesianmodels; questions; answers

Introduction
Q:“Are you gonna eat that?” A:“Go ahead.”In this (real life) example, Q strategically chooses a questionthat differs from her true interest, avoiding an impolite question, yet manages to signal to A what her interests are; A in
turn reasons beyond the overt question and provides an answer that addresses Q’s interests. This subtle interplay raises
two questions for formal models of language: What makes aquestion useful? What makes an answer appropriate?
A number of studies in psycholinguistics have provided
evidence that answerers are both sensitive to a questioner’s
goals and attempt to be informative with respect to thosegoals. For instance, in the classic study of Clark (1979), researchers called liquor merchants and opened the conversation with one of two sentences to set context: “I want to buy
some bourbon” (the uninformative condition) or “I’ve got $5
to spend” (the five dollar condition). They then asked, “Doesa fifth of Jim Beam cost more than $5?” Merchants gave a literal yes/no answer significantly more often in the latter condition than the former, where an exact price was more common.
When provided with the five dollar context, the merchantinferred that the questioner’s goal was literally to find out
whether or not they could afford the whiskey, hence a simple ‘yes’ sufficed. In the uninformative context, however, themerchant inferred that the questioner’s goal was just to buy
whiskey, so the exact price was the most relevant response(Clark, 1979). Context and questioner goals have also been
implicated in accounts of answers to identification questions
like “who is X?” (Boër & Lycan, 1975), and to questions like“where are you?” that permit answers at many levels of abstraction (Potts, 2012). While most of this work has focusedon answerer behavior, it suggests that the question itself isimportant in prompting a relevant answer.
Recent work on Rational Speech Act (RSA) models (Frank& Goodman, 2012; Goodman & Stuhlmüller, 2013) has
mathematically formalized pragmatic language understanding as a form of recursive Bayesian inference, where listeners
reason about speakers who choose utterances that maximize
information gained by an imagined listener. In this paper
we extend the RSA framework to address simple questionanswer dialogs. The immediate challenge in doing so is thatthe speaker utility in RSA is based on direct information provided by an utterance—since questions don’t provide direct
information, we must say what utility they do have.
We suggest, following Van Rooy (2003), that the value ofa question is the extent to which it can be expected to elicit
information relevant to the questioner later in the dialogue.
More specifically, for the questioner, the value of a questionis the expected information gained about her interests, given
the set of likely answers it may provoke. This diverges from
regular RSA in that the value of a question depends on information gained by the speaker (rather than listener), and that
this information comes later in the (very short) conversation.
To fully specify this questioner we need a model of theanswerer, which can serve as both the model assumed by aquestioner, and as a model of answer behavior itself. We explore three, increasingly sophisticated, answerer models. The
simplest answerer provides a literal answer to the question(without attempting to be informative); the explicit answerer
attempts to be informative with respect to the explicit question asked (without inferring the questioner’s underlying interests); the pragmatic answerer infers the most likely true
interests of the questioner, and then informatively addresses
those interests. The latter model extends RSA to reason about
the topic of conversation, as proposed by Kao, Wu, Bergen,
and Goodman (2014) to explain hyperbole; it goes beyond
previous work by using the explicit question as a (potentiallyindirect) cue to this topic.
The rest of this paper is structured as follows. First, we layout the details of our question-answer models. We show thatthe pragmatic answerer model can select different answers
to a question depending on context, as in Clark (1979), described above. We then use a communication game paradigm
that allows us to manipulate goals, potential questions, andpotential answers, testing the predictions of the different
models. We close with a brief discussion of related models
and future directions.

A Rational Speech Act model of question andanswer behavior
How should a questioner choose between questions? We startby assuming that the questioner aims to learn information relevant to a private goal. In order to choose a question that results in useful information, the questioner reasons about how
the answerer would respond, given different possible states of
the world; she selects a question that results in an answer thattends to provide goal-relevant information.
More formally, suppose there is a set of world states W ,a set of possible goals G , a set of possible questions Q ,and a set of possible answers A . These sets are taken to
be in common ground between the questioner and the answerer. An informational goal g ∈ G is a projection function that maps a world state to a particular feature or set offeatures that the questioner cares about; this is similar to thenotion of a question-under-discussion (Roberts, 1996). We
will use the notation Pg (w) to indicate the probability P̂(g(w))
of the g-relevant
aspect of w under the projected distributionRP̂(v) = W δv=g(w) P(w)dw.
The questioner takes a goal g ∈ G as input and returns a
distribution over questions q ∈ Q :∗P(q|g) ∝ eEP(w∗ ) [DKL (Pg (w|q,w ) k Pg (w))]−C(q)It trades off the cost of asking a question, C(q), and expected
information gain. The cost likely depends on question length,among other factors. Information gain is measured as the
Kullback-Leibler divergence between the prior distribution
over g-relevant worlds, Pg (w), and the posterior distribution
one would expect after asking a question q whose answer reflected true world state w∗ :Pg (w|q, w∗ ) = ∑ Pg (w|q, a)P(a|q, w∗ )a∈AThis distribution has two components: First, it depends on
P(a|q, w∗ ), a model of the answerer which we will explore
shortly. Second, it depends on (the goal projection of)P(w|q, a), an ‘interpreter’ that specifies the likelihood assigned to different worlds given question and answer pairs.
To define the interpreter function, which all agents use tocompute the literal interpretation of a question-answer pair,
we must assign questions a semantic meaning. We assume
that a question is an informational goal that projects fromworlds to the answer set A . This is equivalent to the morecommon partition semantics of Groenendijk and Stokhof(1984), as can be seen by considering the pre-image of sucha projection; an answer picks out an element of the partition
via q−1 (a). The interpreter constrains the prior on worlds tothe subset of its support that is consistent with the semantics
of a question-answer pair1 :P(w|q, a) ∝ P(w)δq(w)=a1 We should also have a semantic evaluation function that maps
an answer utterance to its value in A . For clarity we assume this is atrivial mapping and suppress it.

We next describe three different answerer models; thequestioner could assume any one of them, leading to threecorresponding versions of the questioner model. All answerers take a question q ∈ Q and a true world state w∗ ∈ W as
input and return a distribution over answers a ∈ A . The literal answerer simply chooses answers by trading off prioranswer probability and how well a question-answer pair conveys the true state of the world to an interpreter:P(a|q, w∗ ) ∝ P(a)P(w∗ |q, a)
For a fixed question, this is equivalent to the speaker in previous RSA models. The question enters only in specifying
the literal meaning of an answer. The explicit answerer additionally evaluates answers with respect to how well they
address the explicit question q:
P(a|q, w∗ ) ∝ P(a)Pq (w∗ |q, a)
The pragmatic answerer also evaluates answers with respect to how well they address the informational goal, butdoesn’t take the question’s explicit meaning at face value. Instead, the pragmatic answerer reasons about which goals g are
likely given that a question q was asked, and chooses answers
that are good on average:
P(a|q, w∗ ) ∝ p(a) ∑ P(g|q)Pg (w∗ |q, a)g∈GReasoning backwards from questions to goals is a simple
Bayesian inversion of the (explicit) questioner using a prioron goals:
P(g|q) ∝ P(q|g)P(g)
For all of the questioner and answerer models, we can vary
how strongly optimizing they are—that is, to what extent they
are sampling from the distributions defined above, and towhat extent they deterministically choose the most likely element. For any such distribution over utterances, we introduce
an optimality parameter α and transform it by P0 (x) ∝ P(x)α .
This concludes our specification of the model space, giving a set of three answerers and three corresponding questioners that reason about them. We have implemented thesemodels in WebPPL, a probabilistic programming language(Goodman & Stuhlmüller, electronic). The model predictions
shown throughout the rest of the paper are computed using
this implementation.

Whiskey pricing: a case study
Our model can provide different—sometimes over- or underinformative—answers to the same explicit question, depending on context. To illustrate, we model Clark’s (1979)
whiskey study. Recall that liquor merchants were more likely
to give over-informative answers (specifying exact price) tothe question “Does a fifth of Jim Beam cost more than $5?”
in the uninformative context (“I want to buy some bourbon”)
than in the five dollar context (“I’ve got $5 to spend”).

Our world state is a pair of the whiskey’s price ($1, $2, . . . ,$10) and a Boolean indicating whether the merchant takes
credit cards. There are three possible goals: learning the priceof whiskey, learning whether the price is greater than $5, andlearning whether the merchant takes credit cards. Note thatthe credit card question was not in the original study, but reflects the important fact that there exist alternative reasons forcalling a liquor store aside from price-related questions. The
set of answers includes exact prices as well as “yes” and “no”,
with lower cost for “yes” and “no” than the price statements.
We model the context sentence as affecting the answerer’s
goal prior. We assume that there is a fixed 40% probabilityof the credit card goal, with the remaining 60% split between
the two price-related goals. When the context is “I’d like tobuy some whiskey,” we assume that the split is even. When it
is “I only have $5 to spend,” we assume that it is 9:1 in favorof learning whether the price is greater than $5.

animal

pet
dog

d1

d2

b1

Figure 1: Stimulus hierarchy used in Exp. 1. The goal space
and answer space contained the four leaves. The question
space, however, was restricted to the highlighted nodes, proceeding up the hierarchy, allowing for indirect questions.

Results When the question is “Do you take credit cards?”,
the pragmatic answerer prefers to give the accurate Boolean
answer (with probability .76 and .78, weakly depending oncontext), with no preferential treatment for any of the numericanswers. When the question is “Does Jim Beam cost morethan $5?”, the correct Boolean answer is still the most probable choice, but more weakly (at probability .44 and .49). Critically, there is a context-dependence for answers to this question: when prefaced with “I’d like to buy some whiskey.”, thecorrect exact price answer is favored more strongly (at probability .18) than when the context is “I only have $5 to spend.”(probability .11). By contrast, the explicit answerer (which
has no natural way to account for context) does not make differential predictions in the two situations.
This suggests that our pragmatic answerer is consistent
with human behavior in psychologically interesting situations, passing a first, qualitative, test. However, we have notyet shown that the questioner behavior matches that of humans. Indeed, the questioner has been largely neglected instudies of answering (but see, e.g., Potts, 2012), even though,as our opening example illustrates, the choice of question isimportant for understanding answers. In the next section we
introduce an experimental paradigm that allows us to jointly
explore quantitative behavior of both questioners and answerers.

question, chosen from a restricted set of options, and the answerer responded by revealing the object behind a single gate.
This restriction was motivated by one of the key features ofour opening example: when the most direct question (“can I
eat your food?”) is suppressed due to politeness, utterance
length, complexity, or some other intervening factor, questioners must rely instead on an indirect question.
This set of restricted options was critical to distinguishing
between the pragmatic and explicit variants of our model. If
all questions were equally available, both our ‘explicit’ and‘pragmatic’ questioner models would prefer the most directone. To see how they make different predictions in the presence of restrictions, suppose ‘poodle?’ was not available thequestioner. If the questioner asked about a ‘dog?’, the poodle and dalmatian would be considered equally good options
by an explicit answerer because they are both dogs. However, the pragmatic answerer could reason that if the questioner was truly interested in the location of the dalmatian, hewould have asked about the dalmatian. Because he didn’t, hemust be interested in the other valid response that he lacks adirect question for: the poodle.

Exp. 1: Hierarchical questions and answers

Stimuli & Procedure In terms of our model specification,
the world space W was the set of 4! = 24 possible assignments of four objects to four gates. The goal space G was theset of four objects that the questioner could be trying to find(the leaves of the tree in Fig. 1). The answer space A was theset of four gates that the answerer could reveal. The restrictedquestion space Q contained the set of highlighted nodes in thehierarchy: ‘dalmatian?’, ‘dog?’, ‘pet?’, and ‘animal?’.
Each participant provided responses for four trials in therole of the questioner (corresponding to the four goals), andfour trials in the role of the answerer (corresponding to thefour possible questions). In the questioner block, players

In order to simultaneously test how questioners choose questions when faced with a particular goal and how answerers respond under uncertainty about this goal, we used a guessinggame task played by two players: a questioner and an answerer. In this game, 4 animals (a dalmatian, a poodle, acat, and a whale) were hidden behind 4 gates. These animals corresponded to different levels in a class hierarchy (seeFig. 1). The questioner received a private goal of finding oneof the objects (e.g. ‘find the poodle’), and the answerer (butnot the questioner) knew the location of each object. Before
choosing a gate, the questioner asked the answerer a single

Participants We recruited 125 participants from Amazon’s
Mechanical Turk to participate in this task. Eleven participants were excluded due to self-reported confusion about thetask instructions or due to being non-native English speakers.

Pragmatic_Questioner
goal: dalmatian

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poodleanimalpetdogdalmatiananimalpetdogdalmatian

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Figure 2: Exp. 1 results, compared with the predictions of the best-performing model for questioner (left) and answerer (right).
The explicit and pragmatic questioner models do not make different predictions in this task, but the pragmatic answerer betteraccounts for the qualitative patterns in the response data than the explicit answerer.
were presented with a private goal from G , like “find thepoodle!” and were prompted to select a question from adrop-down menu containing elements of Q that would best
help them find it. In the answerer block, players were shown
which items were behind which gates and were told that theother player had asked a particular question from Q . They
were prompted to select a gate from a drop-down menu thatwould be most helpful for the questioner, keeping in mind hisor her constraints. (To minimize learning effects, questioners did not receive answers and neither role saw the outcomeof the game.) In order to collect responses for all elementsof G and Q , the order of the questioner and answerer blocks
was randomly assigned for each participant, and the order ofstimuli within these blocks was also randomized2 .
Results Results for the questioner role are shown alongside model predictions in Fig. 2 (left). We find that questioners systematically prefer to ask different questions given
different goals, even as those questions become more indirect. χ2 tests over each of the four response distributions
show a significant divergence from uniform. Questioners
preferentially ask about the ‘dalmatian’ given the dalmatiangoal, χ2 (3) = 137, p < .001, about the ‘dog’ given the poodle goal, χ2 (3) = 152, p < .001, about the ‘pet’ given thecat goal, χ2 (3) = 120, p < .001, and about the ‘animal’ when
given the whale goal, χ2 (3) = 150, p < .001.
Results for the answerer role are shown in Fig. 2 (right).
Answerers are highly sensitive to the constraints of the questioner, giving information about the dalmatian when asked
about a ‘dalmatian’, χ2 (3) = 281, p < .001, about the poodle when asked about a ‘dog’, χ2 (3) = 137, p < .001, about
the cat when asked about a ‘pet’, χ2 (3) = 57, p < .001,
2 The experiment is online at http://cocolab.stanford.edu/
cogsci2015/Q and A/experiment1/experiment1.html

and about the whale when asked about an ‘animal’,χ2 (3) = 121, p < .001. Note that, under an explicit interpretation of the question, revealing the dalmatian and the poodle would both be perfectly acceptable answers to a question
about a ‘dog’, but answerers strongly prefer to give the location of the poodle. In the next section, we compare theseresults to the predictions of our family of models (Fig. 3).
Model comparison Each model was run with uniform priorprobability over worlds, goals, questions, and answers, andwith equal cost for all utterances. For each model, a single
optimality parameter, which applied to all agents as describedabove, was fit to maximize correlation with the data.
We can rule out both the literal answerer and literal questioner. The literal answerer yields a uniform distribution overthe four answers. This has consequences for the corresponding literal questioner model: when this questioner reasons
about which question would generate the most helpful answer
from the literal answerer, it finds no differences in response
probabilities, and therefore has no preference for which question to ask. The predictions of these model, plotted against
our empirical results, are shown in the left-hand column ofFig. 3.
The two remaining questioner models make roughly thesame predictions for this task, and we are not able to distinguish them on the basis of these data. We found a model-datacorrelation of r = 0.96 for the explicit questioner and correlation of r = 0.99 for the pragmatic questioner. Although thepragmatic model has a slightly better fit, the two models only
differ slightly in the magnitude of predictions, not in qualitatively important ways such as the rank ordering of response.
The pragmatic questioner model’s predictions for each response distribution are shown in Fig. 2 (left). Although themagnitude of its predictions are not in perfect alignment with

sented above. Ten participants were excluded on the basis of
having a non-English native language, or reporting confusion
about the instructions.

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Figure 3: Full space of models, and their correlations with thedata from Exp. 1. Questioner models in the first row reason
about the answerers directly below them, and the pragmatic
answerer reasons about the explicit questioner.
the magnitude of the empirical data (because it is strongly
optimizing), it captures most of the interesting qualitative patterns of the data, particularly the modal responses.
The pragmatic answerer provides a much better fit to thedata than the explicit answerer: a model-data correlation ofr = 0.8 for the explicit answerer and r = 0.95 for the pragmatic answerer. Only the pragmatic answerer can accountfor essential qualitative features of the response data. For
example, the explicit answerer predicts that participants will
be equally likely to show the ‘dalmatian,’ ‘poodle,’ and ‘cat’
when asked about a pet. Instead, the data show a significant
preference for revealing the cat, leaving ‘dalmatian’ and ‘poodle’ at the same level as the other alternative. The pragmatic
answerer correctly predicts this pattern (see Fig. 2 (right)).
Even more dramatically, the explicit answerer predicts a uniform distribution over responses to the ‘animal?’ question.
However, the empirical distribution was significantly different from uniform. Thus, the pragmatic answerer is necessary
to account for these data.
These data provide strong evidence for a pragmatic answerer, but are more equivocal with respect to the explicit
and pragmatic questioner. Because the two models did not
make significantly different predictions for this experiment(and both work quite well), we ran a follow-up study on a
special case of the guessing-game paradigm in which the explicit and pragmatic questioners make different predictions.

Exp. 2: A Critical Test of Questioner Models
Participants We recruited 50 participants to participate
only in the questioner scenario of the guessing game pre-Stimuli & Procedure The procedure was the same as before with some changes to the stimuli. The world space W
consisted of possible assignments of the three pets to threegates. The possible goals G were the dalmatian and poodle(not the cat). The possible questions G were ‘dalmatian?’ or‘cat?’. The possible answers A were the three gates. Eachparticipant was given the two goals in a random order3 .
Results When the goal was to find the dalmatian, participants were significantly more likely to ask about the dalmatian than the cat, χ2 (1) = 12, p < 0.001. When the goal was
to find the poodle, participants were marginally more likely to
ask about the cat than the dalmatian, χ2 (1) = 3.6, p = 0.058.
When looking only at the first of the two trials, the dalmatian
result held, χ2 (1) = 14.4, p < 0.001, but participants’ preference for asking about the cat disappeared, χ2 (1) = 0.07, p =0.79. These results are shown in Fig. 4.
Model comparison The explicit questioner predicts thatparticipants should have no preference for a question given
the ‘poodle’ goal, since an explicit answerer would be equallyunlikely to give the desired answer for both. The pragmaticquestioner model, however, predicts that participants should
prefer to ask about the cat. This is because the (internal) pragmatic answerer would reason that if the questioner was interested in the dalmatian, they would ask about the dalmatian; if
they didn’t, they must be interested in the other possible goal.
It is again unclear which questioner model is best. Overall,
the response distribution matches the predictions of the pragmatic model: questioners prefer to ask about the cat. However, participants don’t show this behavior if we look at only
the first trial. This could be due to a number of reasons. Interestingly, the pragmatic model predicts a more explicit-like
response distribution if the questioner does not take into account the constraint on possible goals: if participants thought
the poodle was the only goal (counter to the instructions),
then asking about the dog would be consistent with the pragmatic model as well. It is possible that participants only
fully-processed the alternative (dalmatian) goal if they had
first done the trial where that was the goal.

General discussion
Perhaps the most important formal advance of the models
considered here is to move the Rational Speech Act framework beyond interpretation of single utterances (in context),
to consider the dynamics of simple dialogs (albeit consisting of a single question and its answer). Doing so requires
replacing the immediate motive to convey true information
with the more distant motive to provoke useful informationfrom one’s interlocutor. On the answerer side, sophisticated
3 The experiment is online at http://cocolab.stanford.edu/
cogsci2015/Q and A/experiment2/experiment2.html

Exp 2 Results (Pooled)goal: dalmatian goal: poodle

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response

Figure 4: The overall response distribution in Exp. 2 (left)
and the same distribution split into first- and second-trial data(right).
inference was required to account for the implicit interestsof the questioner. This provides a useful connection to current game-theoretic and decision-theoretic models (Vogel,Bodoia, Potts, & Jurafsky, 2013; Van Rooy, 2003), which
also emphasize the importance of goals and speaker beliefs
in communication but emphasize less the complex interplay
of inference between questioner and answerer.
We have presented evidence that answerer behavior is best
described by a pragmatic model that does reason about questioner intentions, using the question utterance as a signal.
The superiority of pragmatic answerer predictions over theother answerer models was robust. Questioner behavior in
Exp. 2, however, seemed to be much more dependent on
experience. In another version of Exp. 1, we did not emphasize certain aspects of the game in the instructions, suchas the fact that the answerer knows about the restricted answer set, which might prompt perspective-taking. Our datain this pilot experiment appeared to contain a mixture of explicit and pragmatic answerers and questioners (though otherconfounds were present in this version). Overall, it will be
important to explore the mixture of explicit- and pragmaticquestioning across a larger range of situations: these issues
may be a product of our artificial game paradigm, or they may
be reflective of real tendencies in language use, raising novel
questions about audience design in question-answer behavior.
While the artificiality of our question-answer game may
distance the behavior of participants from the natural use oflanguage, there are also some benefits to this design. In particular, it is easy in this setting to control the exact spaceof questions, goals, and answers. While the restrictions onquestion space may seem peculiar, it is directly motivated byconversational scenarios in everyday usage which feature restrictions on the set of things one can ask about, due to politeness, salience, time cost, and other factors. In future work, wewill explore the extent to which the proposed model can scale
up to real-time, multiplayer games, extended dialogues, andother more naturalistic language settings. To deal with dialogues lasting longer than a single exchange, for instance, wemust specify the way in which the contributions of questioner
and answerer affect the context in which later utterances operate.
Humans are experts at inferring the intentions of otheragents from their actions (Tomasello, Carpenter, Call, Behne,& Moll, 2005). Given simple motion cues, for example, we
are able to reliably discern high-level goals such as chasing, fighting, courting, or playing (Barrett, Todd, Miller, &Blythe, 2005; Heider & Simmel, 1944). Experiments in
psycholinguistics have shown that this expertise extends tospeech acts. Behind every question lies a goal or intention.
This could be an intention to obtain an explicit piece of information (“Where can I get a newspaper?”), signal some common ground (“Did you see the game last night?”), test the answerer’s knowledge (“If I add these numbers together, what
do I get?”), politely request the audience to take some action (“Could you pass the salt?”), or just to make open-ended
small talk (“How was your weekend?”). These wildly different intentions seem to warrant different kinds of answers.
By formalizing the computational process by which answerers infer these different intentions, our model framework provides a unifying way to accommodate this diversity.

Acknowledgements
This work was supported by ONR grants N00014-13-1-0788 and
N00014-13-1-0287, and a James S. McDonnell Foundation ScholarAward to NDG. RXDH was supported by the National Science
Foundation Graduate Research Fellowship under Grant No. DGE114747.

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