communicating with cost-based implicature: a game-theoretic approach to ambiguity
TRANSCRIPT
Communicating with Cost-based Implicature:a Game-Theoretic Approach to Ambiguity
Hannah RohdeUniversity of Edinburgh
3 Charles StreetEdinburgh, UK
Scott SeyfarthUC San Diego
9500 Gilman DriveLa Jolla, CA, USA
Brady ClarkNorthwestern University
2016 Sheridan RoadEvanston, IL, USA
Gerhard JaegerUniversity of Tubingen
Wilhelmstraße 19Tubingen, Germany
Stefan KaufmannNorthwestern University
2016 Sheridan RoadEvanston, IL, USA
Abstract
A game-theoretic approach to linguistic com-munication predicts that speakers can mean-ingfully use ambiguous forms in a discoursecontext in which only one of several availablereferents has a costly unambiguous form andin which rational interlocutors share knowl-edge of production costs. If a speaker pro-duces a low-cost ambiguous form to avoid us-ing the high-cost unambiguous form, a ratio-nal listener will infer that the high-cost entitywas the intended entity, or else the speakerwould not have risked ambiguity. We reportdata from two studies in which pairs of speak-ers show alignment of their use of ambiguousforms based on this kind of shared knowledge.These results extend the analysis of cost-basedpragmatic inferencing beyond that previouslyassociated only with fixed lexical hosts.
1 Introduction
A growing body of work demonstrates that jointcommunication tasks yield alignment of referringexpressions, highlighting the role of interlocutors’experience of shared common ground in establish-ing conventions (Brennan & Clark, 1996; Garrod &Pickering, 2004). Less well-established, however,are predictions regarding which form∼meaningmappings interlocutors will converge on. To addressthis, we evaluate alignment in contexts where inter-locutors’ common ground includes the costs of pro-ducing particular forms.
Consider the shapes in Figure 1. In a context thatcontains only the first item, a speaker can efficiently
draw attention to it by saying “Look at the circle.” Ina context with all three shapes, a more specific refer-ring expression—such as “blue circle”—is requiredto unambiguously indicate that same item. However,if it is necessary to draw attention to the third item,the speaker may need to accept either inefficiencyor ambiguity. Since there is no efficient label (e.g.,“circle”) for the third item’s unique shape, it is costlyto unambiguously refer to it in the context of Fig-ure 1—a longer or more obscure expression is nec-essary (e.g., “the triangle-and-square thing” or “theblue shape that’s not a circle”). On the other hand,the speaker can avoid producing a costly expressionby instead using an ambiguous expression such as“the blue thing” or “the blue shape.” The question iswhether a listener can be expected to infer that theintended referent is the difficult-to-describe shape,even though “the blue shape” could in principle alsorefer to the blue circle.
An accurate inference about the intended referentof “the blue shape” requires the following chain ofreasoning: The listener would have to realize thathad the speaker intended to refer to the blue circle, arelatively short unambiguous expression would havesufficed (“the blue circle”); since the speaker used alow-cost ambiguous expression “the blue shape” in-stead of the available low-cost unambiguous name,
Figure 1: Variation in cost of referring expressions
the implicature is that the circle is not the intendedreferent, leaving the difficult-to-describe item as thepreferred target.
Using ambiguous forms to convey meaning de-pends in part on the listener’s ability to diagnose thesource of the ambiguity: Does the ambiguity likelyarise from the speaker’s own production decisions orfrom other factors that make the expression noisy orunclear (see Schlangen & Fernandez, 2007)? We fo-cus here on productions whose ambiguity listenerscan be confident originates with the speaker herself.
This paper presents two studies testing speakeralignment in dialog games with superimposed costsand rewards that are predicted to guide productionand comprehension of ambiguous forms. Our re-sults show that speakers’ use of ambiguous expres-sions reflects the relative costs of available forms.Rather than avoiding ambiguity, speakers show be-havior that is in keeping with theories of commu-nicative efficiency that posit that speakers make ra-tional decisions about redundancy and reduction.
2 Game Theory and Implicature
Game theory is an area of applied mathematics,which aims to provide a framework for analyzingthe behavior of individuals (players) in strategic sit-uations (games) (von Neumann and Morgenstern,1944; see Clark, 2011, Chapter 4 for an introduc-tion). It is used to describe games in which play-ers have choices regarding their behavior and pref-erences over possible outcomes. The outcomes de-pend on both players’ choices: While some gamesare zero-sum, meaning that success requires thatonly one player can win, in other types of games,both players can succeed if they coordinate their ac-tions. Linguistic communication is typically arguedto be an example of this second type (Lewis, 1969).
Games are characterized by shared knowledge,meaning that players know the overall structure ofthe game. They know what moves are possible inwhich situation by each player, what consequencesare associated with each move, and what preferenceseach player has. Crucially, all players know that theother players know these facts. A game-theoreticapproach makes an assumption of player rationality.
In a recent computational model that treats lan-guage use as a cooperative game, Golland, Liang,and Klein (2010) show that a rational speaker’sdecisions about whether or not to use ambigu-ous referring expressions can be significantly im-
proved by embedding a model of the listener thatrepresents the shared game knowledge describedabove. In their communication game with a humanlistener, unambiguous expressions were preferredover ambiguous-yet-accurate expressions when thespeaker selected an expression that optimized utilityfor the listener.
However, in that type of listener-oriented model,a speaker’s choice of expression is based solely onmaximizing the probability that the listener under-stands the speaker’s intent. In this paper, we con-sider how a game-theoretic approach offers furtherpredictions about the players’ behavior when thevarious (ambiguous and unambiguous) referring ex-pressions also have different costs, when the play-ers share knowledge of costs, and when the play-ers know that they share knowledge. In particular,this approach suggests cases in which literal ambi-guity might actually be preferred in pursuit of effi-cient communication.
The prediction from such an approach is that anambiguous form can be used to refer unambiguouslyif an unambiguous form is costly and other mean-ings can be conveyed at low cost (Clark, 2011; Jager,2008). In other words, a listener who knows the rel-ative costs of unambiguously referring to X (high-cost) or Y (low-cost) may reason that a speaker us-ing an ambiguous word “X-or-Y” (low-cost) intendsto convey X, or else she would have said “Y”.
This type of reasoning has been used to ex-plain the conventional use of “some” (Jager, 2008).Having heard a speaker use the word “some”,the listener is faced with two possible interpreta-tions: AT-LEAST-ONE-AND-POSSIBLY-ALL or elseAT-LEAST-ONE-AND-NOT-ALL. A rational listeneris said to reason that, had the speaker intended toconvey the meaning ALL, she would have used thelow-cost (short and easy to produce) form “all”. Thespeaker, having instead used a low-cost but ambigu-ous form “some”, can be taken to implicate that theintended interpretation is not ALL, but is instead ameaning that would have been costly to produce un-ambiguously: AT-LEAST-ONE-BUT-NOT-ALL.
This account of “some” formulates in game-theoretic terms the well-known Gricean account,which focuses on the amount of information con-veyed. In the Gricean version, the literal meaning of“some”, AT-LEAST-ONE-AND-POSSIBLY-ALL, con-veys less information than “all”. Its meaning isstrengthened to a more informative meaning of AT-
LEAST-ONE-BUT-NOT-ALL via implicature: A co-operative speaker who obeys the maxim of Quantityand intends to convey the more informative meaningALL would have said “all”, but since she did not, themeaning AT-LEAST-ONE-BUT-NOT-ALL is favored.1
Grice’s recognition of the importance of speaker in-tention echoes a game-theoretic approach to signal-ing and the calculation of what must be true in orderfor a rational speaker to have produced a particularutterance (Stalnaker, 2005).
The AT-LEAST-ONE-BUT-NOT-ALL implicatureassociated with the use of “some” is what Gricecalled a generalized conversational implicature: theimplicature AT-LEAST-ONE-BUT-NOT-ALL is typ-ically associated with the proposition expressed.What remains an open question is whether this typeof cost-based inferencing applies beyond a fixed lex-ical host like “some”. The next sections describe twostudies aimed at measuring alignment in a commu-nication game with explicit superimposed costs andrewards for production and comprehension.
3 Study 1: Communicating about Objectswith Divergent Costs
We created a networked interactive two-player chatenvironment (see Figure 2) in which pairs of playerscould communicate about a set of objects. Costs andrewards were made explicit via points, and playersshared both knowledge of the cost/reward structureas well as a shared goal of working together to com-municate successfully. In contrast to Study 2 in thenext section, the costs imposed in Study 1 served tohighlight a single high-cost entity in each category,creating a bias to conventionalize the meaning of anambiguous form to refer to that entity. In produc-tion, the prediction is that players will use a low-costambiguous word to refer to an object with a high-cost unambiguous name, as long as other objects canbe unambiguously referred to with relatively low-
1This logic is spelled out in the Stanford Encyclopedia ofPhilosophy (Davis, 2010) in terms of the interaction of cost(maxim of Manner) and information (maxim of Quantity):
Assuming that the accepted purpose of the conversa-tion requires the speaker to say whether or not all ath-letes smoke, a speaker who said “Some athletes smoke”would be infringing the Quantity maxim if she meantonly what she said. So she must have meant more. If shebelieved that all athletes smoke, she would have said so.Since she did not, she must have meant that some but notall athletes smoke. As a bonus, she achieved brevity, inconformity to the maxim of Manner.
cost names. In comprehension, the prediction is thatplayers will more often interpret ambiguous wordsto refer to objects with a costly unambiguous namethan to objects whose unambiguous name is associ-ated with a mid or low cost (henceforth ‘high-costobjects’, ‘mid-cost objects’, and ‘low-cost objects’).
3.1 Participants
10 pairs of English speakers from NorthwesternUniversity received $10 to participate in the study.
3.2 Methods
Materials The game involved a set of objects intwo categories—three flowers and three trees. Play-ers could communicate using a set of eight referringexpressions: six unambiguous names and two am-biguous words. The costs varied among the unam-biguous names, but the ambiguous words were bothlow-cost. Table 1 shows the point costs associatedwith the eight forms. The point values themselvesare less important than the relative values: In thisstudy, the cost of the most expensive name in eachcategory (“Tulip”/“Pine Tree”) was more than fourtimes the cost of the least expensive name and morethan twice the cost of the mid-cost name.2
Name Cost Name Cost“Rose” -60 “Apple Tree” -60“Daisy” -120 “Palm Tree” -120“Tulip” -280 “Pine Tree” -250“Flower -80 “Tree” -80
Table 1: Referring expressions and their costs (Study 1)
Task Players were seated in separate rooms in frontof computers showing the interactive game interfacedepicted in Figure 2. They were told that they wouldtake turns as Sender and Receiver in a game that in-volved communicating about a set of objects. Asthe Sender, a player would see a gnome highlightan object with a spotlight, and the Sender’s taskwas to send a message to the other player so thatthe other player (the Receiver) could guess what thehighlighted object must have been. Sending a wordconsisted of pressing a button on the screen which
2Alternate materials were constructed with abstract shapesand nonsense names, but a pilot study found that participantshad difficulty learning the names. Variants of Study 1 were con-ducted with first names (e.g., “Ann”) for unambiguous namesfor objects in plant and vessel categories and family names ornonsense names (e.g., “Puliniki”) for ambiguous words; the re-sults matched those presented here with flower/tree materials.
Rose [-60]
/1000
Apple Tree
Apple Tree
Rose
Match! 18 min 20 sec
Apple Tree
Flower
Match! Time remaining:
/1000Your points:115Your partner’s points:
Tulip
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Daisy [-120]
Tulip [-280]
Apple Tree [-60] Flower [-80]
Palm Tree [-120]
Pine Tree [-250]
Tree [-80]Daisy
Figure 2: Player’s view of the interactive chat environment for both Study 1 and 2 (point values from Study 1)
displayed the name and cost; pressing the buttonresulted in an immediate point decrement for theSender. If the Receiver successfully identified theintended object, then both players got an immediatereward (Sender and Receiver scores increased). Oth-erwise, there was no reward and the Sender had toretry until communication succeeded—the penaltyin that case being the continued point decrementsfor sending multiple words. The reward for success-ful communication was +85 points for each player.If either player reached the target score of 1000points, the game ended and both players were freeto leave. Otherwise, games continued for 20 min-utes or until the gnome had highlighted a total of 60objects. Scores could become negative if the pointdecrements of production outstripped the point in-crements for successful communication.
For each pair, we calculated a unique sequence ofobjects to be highlighted so that it would be impos-sible to reach 1000 points in less than 60 turns with-out successfully coordinating the use of ambiguouswords. The first 10 turns were intended as practiceturns, involving only low-cost and mid-cost objects.
Shared Knowledge Both players were able to seethe available referring expressions and their costsand were told that the other player could as well.Both players were informed of the reward struc-ture (+85 points each for successful communication,1000 points to end early) and told that the otherplayer had likewise been informed. The game in-terface showed the current scores of the two players,the time remaining, and a scrolling chat window list-ing previous words sent and received.
3.3 Results and Discussion
Of the 10 pairs, 5 consistently coordinated their re-ferring expressions, allowing an early exit from thegame. Two pairs played for the full 20 minutes,struggling to coordinate their efforts, and their lim-ited attempts at coordination failed to yield an earlyexit to the game. Of the remaining 3 pairs, 1 pairdid not make a serious attempt at using the ambigu-ous words, and the other 2 pairs used them repeat-edly but had difficulty finding a strategy while doingso, although all 3 did eventually manage to exit thegame before reaching the time or turn limit.
Table 2 shows a transcript from one pair of play-ers, listing the first 26 moves of their game. Thetranscript demonstrates how the players developed acoordinated strategy for using the ambiguous words:the use of an ambiguous word by Player 1 when theintended referent was not a high-cost object (whichled to Player 2’s initial guess that the high-cost ob-ject was the target), the use of an ambiguous wordby Player 2 (which Player 1 failed to interpret asa reference to the high-cost object), and eventuallytheir convergence. After the success shown in thelast line of the table, the players continued to reli-ably use “Flower” and “Tree” to refer to the tulipand pine tree, and the game ended after 44 moveswhen Player 2 reached 1000 points.
For the analysis, we measured the effect of onewithin-players factor (the target object’s unambigu-ous cost) on two binary outcomes: whether theSender sent an ambiguous word (production out-come) and whether an ambiguous word resultedin successful communication (comprehension out-
SenderHighlighted Form Receiver’s
Target Used Guess1 daisy (mid) ‘Flower’ tulip (high)1 daisy (mid) ‘Daisy’ daisy (mid)2 palm (mid) ‘Palm’ palm (mid)1 palm (mid) ‘Tree’ pine (high)1 palm (mid) ‘Palm’ palm (mid)2 apple (low) ‘Apple’ apple (low)1 palm (mid) ‘Tree’ pine (high)1 palm (mid) ‘Palm’ palm (mid)2 daisy (mid) ‘Daisy’ daisy (mid)1 tulip (high) ‘Tulip’ tulip (high)2 apple (low) ‘Apple’ apple (low)1 pine (high) ‘Pine’ pine (high)2 tulip (high) ‘Flower’ daisy (mid)2 tulip (high) ‘Flower’ daisy (mid)2 tulip (high) ‘Flower’ tulip (high)1 pine (high) ‘Pine’ pine (high)2 pine (high) ‘Tree’ pine (high)1 rose (low) ‘Rose’ rose (low)2 rose (low) ‘Rose’ rose (low)1 palm (mid) ‘Palm’ palm (mid)2 tulip (high) ‘Flower’ rose (low)2 tulip (high) ‘Flower’ daisy (mid)2 tulip (high) ‘Flower’ tulip (high)1 pine (high) ‘Tree’ apple (low)1 pine (high) ‘Tree’ pine (high)2 pine (high) ‘Tree’ pine (high)...
......
...
Table 2: Excerpt of a game transcript from two successfulplayers in Study 1 (‘Apple Tree’, ‘Palm Tree’, and ‘PineTree’ are abbreviated without the word ‘Tree’). Consec-utive rows with the same Sender show retries.
come). We also measured the effect of trial num-ber on the interpretation of ambiguous words in or-der to evaluate the time course of the Receivers’cost-based inferencing. For that, the three-way out-come of Receiver guess was treated as three binaryvariables (high-cost-or-not, mid-cost-or-not, low-cost-or-not). For all analyses, means, and figures,we consider only non-retry moves. We report thelogistic-regression coefficient estimate and p-value(based on the Wald Z statistic; Agresti, 2002) forthe factors cost and trial number (both treated as nu-meric factors) with random participant-specific in-tercepts and slopes.
Additionally, Figure 3 shows the overall rates of
low mid high
Rate of ambiguity (and rate of success)
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Figure 3: The percentage of trials with a low-, mid-, orhigh-cost target in which a Sender produced an ambigu-ous word. The lower part of each bar represents the per-centage of trials in which the use of an ambiguous wordwas successful.
use and success for the ambiguous words, measuredover proportions of trials. The height of each barin the graph shows the proportion of trials for eachcost—low, mid, high—where an ambiguous wordwas used. The shaded (lower) portion of the barshows the proportion of those ambiguous words thatresulted in successful communication.
As predicted, Senders produced an ambiguousword more often if the gnome had highlighted atarget object whose unambiguous name was highcost (βcost=1.94, p<0.001): 58.9% of high-cost-target trials yielded an ambiguous word, whereasonly 24.8% of mid-cost-target trials and 4.6% oflow-cost-target trials yielded ambiguity.
Receivers likewise paid attention to cost, correctlyguessing the target more often when an ambigu-ous word was used for a high-cost target than for amid- or low-cost target (βcost=1.68, p<0.005): Tri-als with an ambiguous word yielded successful com-munication 74.1% of the time if the intended targetwas high cost, compared to 37.8% and 37.5% suc-cess when an ambiguous word was used for mid-costand low-cost targets, respectively. In other words, ofthe 58.9% of trials in which an ambiguous word wasproduced for a high-cost target, 74.1% of those tri-als yielded successful communication (compared toless than half the time for trials in which an ambigu-ous word was used for a low- or mid-cost target), asdepicted in the ‘high’ bar of Figure 3.
We also ask whether the interpretation of an am-biguous word changes over successive trials. Re-stricting the analysis to trials in which an ambigu-ous word was used, we find that the interpreta-
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Figure 4: Rate at which Receivers in Study 1 interpreted ambiguous words as referring to objects with high-, mid-,and low-cost unambiguous names. Errors bars show standard error of the mean.
tion of ambiguous words favors high-cost objectsover time (βtrial=0.29, p<0.01) and disfavors, al-beit not significantly, mid-cost and low-cost ob-jects (mid: βtrial=-0.22, p=0.09; low: βtrial=-0.23,p=0.11). Figure 4 shows the probability of a Re-ceiver interpreting an ambiguous word as referringto a high, mid, or low-cost object. Trial number inFigure 4 (and in the regression) represents the num-ber of trials for which an object of that cost hasbeen highlighted—e.g., the datapoints at Trial=10are the 10th trials, within each game, in which thegnome highlighted a high-cost object (either a tulipor a pine tree) and the Sender sent an ambiguousword (‘Flower’ or ‘Tree’) and the Receiver guesseda high-cost object (solid line), a mid-cost object(dashed line), or a low-cost object (dotted line).
What is apparent in Figure 4 is that the data af-ter Trial=15 becomes noisier (more fluctuation) andmore sparse (limited/no error bars). This drop-offcorresponds to the point in the game when most suc-cessful players had reached 1000 points and left, sothe data for the later trials represents the behavior ofpairs of players who had failed to coordinate theirreferring expressions. These players overall usedfewer ambiguous words, and because of this, manywere watching their scores become more and morenegative. Data from all players up through Trial=15was analyzed in the time course logistic regression.
These results show that players can and do coor-dinate their use of referring expressions, convention-alizing the meaning of an ambiguous form to asso-ciate it with the object whose unambiguous name isthe most costly to produce.
4 Study 2: Communicating about Objectswith Similar Costs
As a further test of the predictions of a game-theoretic model, a second study was conducted thatshifted the cost structure within each category. Com-pared with the costs in Study 1, the high-cost namesin Study 2 were less costly than before, and the pointdifference between the low-cost, mid-cost, and high-cost names was smaller. The revised costs were pre-dicted to reduce the likelihood that players wouldcoordinate their use of referring expressions. Thetarget score and the reward for successful commu-nication stayed the same, but the stakes were lower(i.e., the cost structure imposed lower costs overall)so that it was possible for players to reach the targetscore in less time without making recourse to theambiguous words. Rational players could chooseto waste less effort attempting to align their use ofreferring expressions because the benefit of align-ment was potentially outweighed by the points lostin rounds in which successful communication re-quired the Sender to send more than one word.
4.1 Participants10 pairs of English speakers from NorthwesternUniversity received $10 to participate. None hadparticipated in Study 1.
4.2 MethodsThe game environment contained the same set of sixobjects. The game rules and shared knowledge ofthose rules matched Study 1. The only differencewas the costs (Table 3): The most expensive name
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Figure 5: The percentage of trials with a low-, mid-, orhigh-cost target in which a Sender produced an ambigu-ous word. The lower part of each bar represents the per-centage of trials in which the use of an ambiguous wordwas successful.
in each category cost slightly more than two timesthe least expensive name and not more than one anda half times the mid-cost name. Successful com-munication was still rewarded with +85 points toboth players and the game ended when either playerreached 1000 points. Since the absolute costs for thelow- and mid-cost objects were similar to Study 1while the absolute cost for the high-cost object wasreduced, the average point cost in Study 2 was re-duced and therefore it was possible to end the gameafter 48 turns rather than 60 without coordination.
Name Cost Name Cost“Rose” -80 “Apple Tree” -80“Daisy” -140 “Palm Tree” -135“Tulip” -165 “Pine Tree” -170“Flower -80 “Tree” -80
Table 3: Referring expressions and their costs (Study 2)
4.3 Results and DiscussionOf the 10 pairs, 8 coordinated their referring expres-sions, allowing an early exit from the game. Con-trary to prediction, the imposition of lower costs didnot reduce players’ motivation to conventionalize.This can be seen in reliable effects of cost on pro-duction and comprehension, as in Study 1. Figure5 shows the overall rates of use and success for am-biguous words in Study 2.
Senders produced an ambiguous word most oftenif the highlighted object was high cost (βcost=2.56,p<0.001): 72.6% of high-cost-target trials yieldedan ambiguous word, whereas only 25.6% of mid-cost-target trials and 6.4% of low-cost-target trials
yielded ambiguity. Receivers likewise paid atten-tion to cost, correctly guessing the target more oftenwhen an ambiguous word was used for a high-costtarget than for a mid- or low-cost target (βcost=1.18,p<0.001): Trials with an ambiguous word yieldedsuccessful communication 82.5% of the time if theintended target was high cost, compared to 51.0%and 21.4% success when an ambiguous word wasused for mid-cost and low-cost targets, respectively.
The time course differs slightly from Study 1,however. Receivers in Study 2 did not show a re-liable rise in their preference to interpret ambiguouswords as referring to high-cost targets. Again re-stricting the analysis to trials in which an ambiguousword was used (see Figure 6), the only reliable effectis that ambiguous words become less likely to be in-terpreted as referring to low-cost objects over time(βtrial=-0.65, p=0.05). The effect of trial numberon the use of ambiguity for mid-cost targets is againnot significant, though the coefficient is positive here(it was negative in Study 1), meaning that ambigu-ity tended to favor the mid-cost target slightly overtime (βtrial=0.13, p=0.32). The slight increase herein the rate at which ambiguous words were inter-preted to refer to high-cost objects is not significant(βtrial=0.09, p=0.22), unlike in Study 1. Given thedifferent cost structure, the point in the game whenmost successful players had reached 1000 points andleft comes at Trial=11. Figure 6 shows the subse-quent sparseness after that point, and the time courseregression includes data only up to Trial=11.
Across the two studies, Sender/Receiver pairswho coordinated their use of ambiguous forms werebetter able to communicate. Two pairs in Study 2converged on a pattern of usage whereby an ambigu-ous word was used to refer to the object with themid-cost unambiguous name. This could explain thetime course result whereby the slope for mid-costguesses for ambiguous words was positive (thoughnot significantly so) in this study but not in Study 1.Another difference between the two studies is that inStudy 2, convergence in the use of ambiguity in onecategory did not guarantee convergence in the other:two pairs used ‘Tree’ reliably but not ‘Flower’.
In terms of our predictions, players did show sen-sitivity to the differences in the cost structure, butwhat characterized the behavior of players in Study2 was a more immediate and sustained use of am-biguous words as referring to high-cost objects formost pairs and an openness to assign the ambigu-
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Figure 6: Rate at which Receivers in Study 2 interpreted ambiguous words as referring to objects with high-, mid-,and low-cost unambiguous names. Interpretation of axes and error bars as in Figure 3.
ous word to a mid-cost object for a small subsetof pairs. The players’ behavior seems to point to agreater willingness to experiment with the ambigu-ous words in a context like Study 2 where, despitetheir experimentation, they could see their scores in-creasing more quickly to the target value. Alterna-tively, rather than casting Study 2 as the context withincreased experimentation, one can ask why play-ers did not experiment more in Study 1. Perhapsthe higher production costs in Study 1 made playersavoid risking ambiguity and possibly having to retry.
Lastly, one can ask if players simply used a trial-and-error strategy for finding an efficient alignmentwithout recourse to the pragmatic inference requiredfor the cost-based implicature. To rule this out, weconsidered the trials in which a Sender first sent anambiguous word. We pooled the data from the twostudies since each participant could only contributeone datapoint. In keeping with the cost-based impli-cature account, Receivers inferred, more often thanchance, that the high-cost object was the intendedobject (χ2(2)=11.54, p<0.005).
5 General Discussion
In keeping with the game-theoretic prediction, wesaw that ambiguous words can be used meaning-fully to refer to entities with costly unambiguousnames, crucially if other referents can be identifiedwith low-cost unambiguous names. This extends theclaim that listeners draw cost-based implicature be-yond the case of a fixed lexical host like “some”.
We also saw sensitivity to relative costs: In com-paring the two studies, the trajectory for how am-
biguous words were interpreted over time in Study 1(where the unambiguous names had more divergentcosts) differs from the trajectory in Study 2 (wherethe unambiguous names had more similar costs).Only in Study 2 did players ever assign the ambigu-ous word to a mid-cost item, and only in Study 2 dida pair of successful players coordinate their use ofone ambiguous word but not both. Contrary to pre-dictions, however, the lower stakes in Study 2 didnot yield a reduction in the players’ overall ability tocoordinate referring expressions.
This line of research raises an important questionabout how production cost should be measured. Forthe studies here, we imposed our costs arbitrarily—i.e., we could just as well have assigned the cost of‘Rose’ to ‘Tulip’ or to ‘Daisy’. One could imag-ine instead a cost metric that reflects properties suchas length (as in Figure 1) or the presence of non-native phonemes or other articulation-based com-plexity. Alternatively, there is evidence that fre-quency contributes to production difficulty: Speak-ers are slower to name objects with low-frequencynames than high-frequency names (Oldenfield &Wingfield, 1965). Speakers also experience diffi-culty, as measured by their disfluency, when de-scribing objects which have not yet been mentioned(Arnold & Tanenhaus, 2007), are unfamiliar or lacka name (Arnold, Kam, & Tanenhaus, 2007). Toavoid the inference step of assessing what phonolog-ical/lexical/pragmatic properties speakers perceiveas costly, the proof-of-concept studies presentedhere used externally imposed costs to test the use ofcost-based implicatures. The next stage of this re-
search will extend the experimental framework de-scribed here to the kinds of costs that are imposednaturally in regular conversation.
If cost does influence choice of referring expres-sion, one must still ask whether its role is automaticor strategic (Horton, 2008). By presenting this workfrom the standpoint of calculable implicatures, wehave framed the questions in strategic terms. Thefactors that guide speakers’ strategic selection of re-ferring expressions may depend not only on the costsassociated with production (as we have shown here)but also on speakers’ estimates of the costs and ben-efits of successful communication and of the degreeof coordination between speaker and hearer (vanDeemter, 2009).
It is also possible that our participants had a moreautomatic reaction to the salience of high-cost ob-jects — maybe they just associated an ambiguousform with the most salient object of that category,where cost indicated salience. This scenario is com-patible with a game-theoretic account — the reason-ing being that it would be unnecessarily costly torefer to a prominent entity with a full name when areduced or ambiguous form could be used.3
Lastly, these results fit with existing work on therole of reduction in communication — namely, workshowing that speakers make rational decisions aboutredundancy and reduction and do not necessarilyavoid ambiguity (Levy & Jaeger, 2007; Jaeger,2010; Piantadosi, Tily, & Gibson, 2011). Like thisprevious work, we argue that ambiguity arises froma rational communication process. In our case, am-biguity arises in contexts in which the explicit costsof production are part of speakers’ common ground.
Acknowledgments
This research was supported by an Andrew W.Mellon postdoctoral fellowship to the first author.The results from this paper have been presentedat the 25th Annual CUNY Conference on HumanSentence Processing in New York and the 36thPenn Linguistics Colloquium in Philadelphia, bothin March 2012. We thank Judith Degen, Roger
3Clark (2011) characterizes the role of salience in reduction(here, pronominalization) in a game-theoretic framework:
It is cheaper to refer to a prominent element with apronoun. It is correspondingly more marked (hence,costlier) to refer to a more prominent element with a de-scription or name when a pronoun could be used. Herethe speaker and hearer are presumably using some notionof salience to guide their choice. (p. 252)
Levy, and Kenny Smith for helpful discussion. Wealso thank research assistants Elizabeth Mazzocco,Anant Shah, Alex Djalali, and John Lee.
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