Linguistic pragmatics studies the context-dependent use andinterpretation of expressions. Perhaps the most important notion inpragmatics is Grice’s (1967)conversationalimplicature. It is based on the insight that by means of generalprinciples of rational cooperative behavior we can communicate morewith theuse of a sentence than theconventional semanticmeaning associated with it. Grice has argued, for instance, thatthe exclusive interpretation of ‘or’—according towhich we infer from ‘John or Mary came’ that John and Marydidn’t come both—is not due to the semantic meaning of‘or’ but should be accounted for by a theory ofconversational implicature. In this particular example,—atypical example of a so-called Quantity implicature—thehearer’s implication is taken to follow from the fact that thespeaker could have used a contrasting, and informatively strongerexpression, but chose not to. Other implicatures may follow from whatthe hearer thinks that the speaker takes to be normal states ofaffairs, i.e., stereotypical interpretations. For both types ofimplicatures, the hearer’s (pragmatic) interpretation of anexpression involves what he takes to be the speaker’s reason forusing this expression. But obviously, this speaker’s reason mustinvolve assumptions about the hearer’s reasoning as well.

In this entry we will discuss formal accounts of conversationalimplicatures that explicitly take into account the interactivereasoning of speaker and hearer (e.g., what speaker and hearer believeabout each other, the relevant aspects of the context of utteranceetc.) and that aim to reductively explain conversational implicatureas the result of goal-oriented, economically optimised language use.For this entry, just as in traditional analyses of implicatures, wewill assume that sentences have a pre-existing semantic meaning andwill mostly focus on generalised conversational implicatures.

• 1. Bidirectional Optimality Theory
  • 1.1 Bidirectional OT and Quantity implicatures
  • 1.2 A Bi-OT analysis of Horn’s division
• 2. Implicatures and Game Theory
  • 2.1 Signaling games
  • 2.2 A game-theoretic explanation of Horn’s division
  • 2.3 Quantity implicatures and best responses
• 3. Conclusion
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Bidirectional Optimality Theory

1.1 Bidirectional OT and Quantity implicatures

Optimality Theory (OT) is a linguistic theory which assumes thatlinguistic choices are governed by competition between a set ofcandidates, or alternatives. In standard OT (Prince & Smolensky,1993) the optimal candidate is the one that satisfies best a set ofviolable constraints. After its success in phonology, OT has also beenused in syntax, semantics and pragmatics. The original idea ofoptimality-theoretic semantics was to model interpretation by takingthe candidates to be the alternative interpretations that the hearercould assign to a given expression, with constraints describinggeneral preferences over expression-interpretation pairs. Blutner(1998, 2000) extended this original version by taking also alternativeexpressions, or forms, into account that the speaker could have used,but did not. The reference to alternative expressions/forms isstandard in pragmatics to account for Quantity implicatures.Optimization should thus be thought of from two directions: that ofthe hearer, and that of the speaker. What is optimal, according toBlutner’s Bidirectional-OT (Bi-OT), is not just interpretationswith respect to forms, but rather form-interpretation pairs. In termsof a ‘better than’ relation ‘\(\gt\)’ betweenform-interpretation pairs, the pair \(\langle f,i\rangle\) is said tobe(strongly) optimal iff it satisfies the followingtwo conditions:

The first condition requires that \(i\) is an optimal interpretationof form \(f\). In Bi-OT this condition is thought of as optimizationfrom the hearer’s point of view. Blutner proposed that \(\langlef,i'\rangle \gt \langle f,i\rangle\) iff \(i'\) is a more likely, orstereotypical, interpretation of \(f\) than \(i\) is: \(P(i'\mid\llbracket f\rrbracket) \gt P(i\mid \llbracket f \rrbracket)\) (where\(\llbracket f\rrbracket\) denotes the semantic meaning of \(f\), and\(P(B\mid A)\) the conditional probability of \(B\) given \(A\),defined as \(P(A\cap B)/P(A))\). The second condition is taken toinvolve speaker’s optimization: for \(\langle f,i\rangle\) to beoptimal for the speaker, it has to be the case that she cannot use amore optimal form \(f'\) to express \(i. \langle f',i\rangle \gt\langle f,i\rangle\) iff either (i) \(P(i\mid \llbracket f'\rrbracket)\gt P(i\mid \llbracket f\rrbracket)\), or (ii) \(P(i\mid\llbracketf'\rrbracket = P(i\mid\llbracket f\rrbracket)\) and \(f'\) is a lesscomplex form to express \(i\) than \(f\) is.

Bi-OT accounts for classical Quantity implicatures. A convenient(though controversial) example is the‘exactly’-interpretation of number terms. Let us assume,for the sake of example, that number terms semantically have an‘at least’-meaning.^[1] Still, we want to account for the intuition that the sentence“Three children came to the party” is normally interpretedas saying thatexactly three children came to the party. Oneway to do this is to assume that the alternative expressions that thespeaker could use are of the form “(At least) \(n\)children came to the party”, while the alternativeinterpretations for the hearer are of type \(i_n\) meaning that“Exactly \(n\) children came to the party”.^[2] If we assume, again for the sake of example, that all relevantinterpretations are considered equally likely and that it is alreadycommonly assumed that some children came, but not more than four, thestrongly optimal form-interpretation pairs can be read off thefollowing table:

In this table the entry \(P(i_3 \mid\llbracket\)‘two’\(\rrbracket) = ⅓\) because\(P(i_3 \mid \{i_2,i_3,i_4\}) = ⅓\). Notice that according tothis reasoning ‘two’ is interpreted as ‘exactly2’ (as indicated by an arrow) because (i) \(P(i_2 \mid\llbracket\)‘two’\(\rrbracket) = ⅓\) is higher than\(P(i_2 \mid \llbracket\)‘\(n \textrm{'} \rrbracket)\) for anyalternative expression ‘\(n\)’, and (ii) all otherinterpretations compatible with the semantic meaning of the numeralexpression areblocked: there is, for instance, anotherexpression for which \(i_4\) is a better interpretation, i.e., aninterpretation with a higher conditional probability.

With numeral terms, the semantic meanings of the alternativeexpressions give rise to a linear order. This turns out to be crucialfor the Bi-OT analysis, if we continue to take the interpretations asspecific as we have done so far. Consider the following alternativeanswers to the question “Who came to the party?”:

1. John came to the party.
2. John or Bill came to the party.

Suppose that John and Bill are the only relevant persons and that itis presupposed that somebody came to the party. In that case the tablethat illustrates bidirectional optimality reasoning looks as follows(where \(i_x\) is the interpretation that only \(x\) came):

This table correctly predicts that (1) is interpreted as saying thatonly John came. But now consider the disjunction (2).Intuitively, this answer should be interpreted as saying that eitheronly John, or only Bill came. It is easy to see, however, that this ispredicted only if ‘John came’ and ‘Bill came’are not taken to be alternative forms. Bi-OT predicts that in casealso ‘John came’ and ‘Bill came’ are taken tobe alternatives, the disjunction is uninterpretable, because thespecific interpretations \(i_j, i_b\), and \(i_{jb}\) all can beexpressed better by other forms. In general, one can see that in casethe semantic meanings of the alternative expressions are not linearly,but only partially ordered, the derivation of Quantity implicaturessketched above gives rise to partially wrong predictions.

As it turns out, this problem for Bi-OT seems larger than it reallyis. Intuitively, an answer like (2) suggests that the speaker hasincomplete information (she doesn’t know who of John or Billcame). But the interpretations that we considered so far are worldstates that do not encode different amounts of speaker knowledge. So,to take this into account in Bi-OT (or in any other analysis ofQuantity implicatures) we should allow for alternative interpretationsthat represent different knowledge states of the speaker. Aloni (2007)gives a Bi-OT account of ignorance implicatures (inferences, like theabove, that the speaker lacks certain bits of possibly relevantinformation), alongsideindifference implicatures (that thespeaker does not consider bits of information relevant enough toconvey). Moreover, it can be shown that, as far as ignoranceimplicatures are concerned, the predictions of Bi-OT line up with thepragmatic interpretation function called ‘Grice’in various (joint) papers of Schulz and Van Rooij (e.g., Schulz &Van Rooij, 2006). In these papers it is claimed thatGriceimplements the Gricean maxim of Quality and the first maxim ofQuantity, and it is shown that in terms of it (together with anadditional assumption of competence) we can account for manyconversational implicatures, including the ones of (1) and (2).

1.2 A Bi-OT analysis of Horn’s division

Bi-OT can also account forHorn’s division of pragmaticlabor orM-implicatures, as they are alternativelysometimes called after Levinson (2000). From this division ofpragmatic labor it follows, according to Horn, that while(morphologically)unmarked expressions typically get anunmarked, or stereotypical interpretation via Grice’s maxim ofRelation, marked expressions—being morphologicallycomplex and less lexicalised—tend to receive a marked,non-stereotypical, interpretation. Horn (1984) claimed that thisfollows from the interaction between both Gricean submaxims ofQuantity, and the maxims of Relation and Manner, and that for theinterpretation of marked expressions it is crucial that an alternativeunmarked expression is available as well. To illustrate, consider thefollowing well-known example:

3. John killed the sheriff.
4. John caused the sheriff to die.

We typically interpret the unmarked (3) as meaning stereotypicalkilling (on purpose), while the marked (4) suggests that John killedthe sheriff in a more indirect way, maybe unintentionally. Blutner(1998, 2000) shows that this can be accounted for in Bi-OT. Take\(i_{st}\) to be the more plausible interpretation where John killedthe sheriff in the stereotypical way, while \(i_{\neg st}\) is theinterpretation where John caused the sheriff’s death in anunusual way. Because (3) is less complex than (4), and \(i_{st}\) isthe more stereotypical interpretation compatible with the semanticmeaning of (3), it is predicted that (3) is interpreted as \(i_{st}\).Thus, in terms of his notion ofstrong optimality, i.e.,optimality for both speaker and hearer, Blutner can account for theintuition that sentences typically get the most plausible, orstereotypical, interpretation. In terms of this notion of optimality,however, Blutner is not able yet to explain how the more complex form(4) can have an interpretation at all, in particular, why it will beinterpreted as non-stereotypical killing. The reason is that on theassumption that (4) has the same semantic meaning as (3), thestereotypical interpretation would be hearer-optimal not only for (3),but also for (4).

To account for the intuition that (4) is interpreted in anon-stereotypical way, Blutner (2000) introduces a weaker notion ofoptimality that also takes into account a notion ofblocking:one form’s pragmatically assigned meaning can take away, so tospeak, that meaning from another, less favorable form. In the presentcase, the stereotypical interpretation is intuitively blocked for thecumbersome form (4) by the cheaper alternative expression (3).Formally, a form-interpretation pair \(\langle f,i\rangle\) isweakly optimal^[3] iff there is neither a strongly optimal \(\langle f,i'\rangle\) suchthat \(\langle f,i'\rangle \gt \langle f,i\rangle\) nor a stronglyoptimal \(\langle f',i\rangle\) such that \(\langle f',i\rangle \gt\langle f,i\rangle\). All form-interpretation pairs that are stronglyoptimal are also weakly optimal. However, a pair that is not stronglyoptimal like \(\langle\)(4),\(i_{\neg st}\rangle\) can still be weaklyoptimal: since neither \(\langle\)(4),\(i_{st}\rangle\) nor\(\langle\)(3),\(i_{\neg st}\rangle\) is strongly optimal, there is noobjection for \(\langle\)(4),\(i_{\neg st}\rangle\) to be a (weakly)optimal pair. As a result, the marked (4) will get thenon-stereotypical interpretation. In general, application of the abovedefinition of weak optimality can be difficult, but Jäger (2002)gives a concise algorithm for computing weakly optimalform-interpretation pairs.

2. Implicatures and Game Theory

The first to systematically explore the use of game theory for anaccount of communication asstrategic inference was PrashantParikh in his 1988 dissertation (Parikh 1988). Parikh’s theoryis based on what he callsgames of partial information,revolves around the game-theoretic solution concept ofPareto-optimal Nash equilibrium and has been refined andapplied to various phenomena, including but not restricted toimplicature-strengthened utterance meaning, in a series of articlesand monographs (e.g., Parikh 1991, 1992, 2000, 2001, 2010, 2019).Another noteworthy early contribution is the work of Asher, Sher, andWilliams (2001), which tries to rationalize Grice’s Maxims ofConversation using ideas from game theory. In the following we focuson approaches to implicatures based on so-called iterative reasoningin signaling games. The relation between optimality theory and gametheory is explored by Dekker and van Rooij (2000) and Franke andJäger (2012). Stevens and Benz (2018) give another insightfuloverview of game-theoretic pragmatics.

2.1 Signaling games

David Lewis (1969) introduced signaling games to explain how messagescan be used to communicate something, although these messages do nothave a pre-existing meaning. In pragmatics we want to do somethingsimilar: explain what is actually communicated by an expression whoseactual interpretation is underspecified by its conventional semanticmeaning. To account for pragmatic inferences in game theory, Parikh(1991, 1992) introduced games of partial information. These games aremuch like Lewis’s original signalling games, except thatmessages are taken to have a conventional semantic meaning, andspeakers are assumed to say only messages that are true. In theinterest of a broad overview, we gloss over interesting but subtledifferences between Lewis' signaling games and Parikh's games ofpartial information.

A signaling game, then, is a game of asymmetric information between asender \(s\) and a receiver \(r\). The sender observes the state \(t\)that \(s\) and \(r\) are in, while the receiver has to perform anaction. Sender \(s\) can try to influence the action taken by \(r\) bysending a message. \(T\) is the set of states, \(F\) the set of forms,or messages. The messages already have a semantic meaning, given bythe semantic interpretation function \(\llbracket \cdot \rrbracket\)which assigns to each form a subset of \(T\). The sender will send amessage/form in each state, asender strategy \(S\) is thus afunction from \(T\) to \(F\). The receiver will perform an actionafter hearing a message with a particular semantic meaning, but forpresent purposes we can think of actions simply as interpretations. Areceiver strategy \(R\) is then a function that maps amessage to an interpretation, i.e., a subset of \(T\). Autilityfunction for speaker and hearer represents what interlocutorscare about, and so the utility function models what speaker and hearerconsider to be relevant information (implementing Grice’s Maximof Relevance). For simplicity, it is normally assumed that the utilityfunctions of \(s\) and \(r (U_s\) and \(U_r)\) are the same(implementing Grice’s cooperative principle), and that theydepend on (i) the actual state \(t\), (ii) the receiver’sinterpretation, \(i\), of the message \(f\) sent by \(s\) in \(t\)according to their respective strategies \(R\) and \(S\), i.e., \(i =R(S(t))\), and (iii) (in section 2.3) the form \(f = S(t)\) used bythe sender. Nature is taken to pick the state according to some(commonly known) probability distribution \(P\) over \(T\). Withrespect to this probability function, the expected, or average,utility of each sender-receiver strategy combination \(\langleS,R\rangle\) for player \(e \in \{s,r\}\) can be determined asfollows:

A signaling game is taken to be a (simplified, abstract) model of asingle utterance and its interpretation, which includes some of thearguably most relevant features of a context for pragmatic reasoning:an asymmetry of information (speaker knows the world state, hearerdoes not), a notion of utterance alternatives (in the set ofmessages/forms) with associated semantic meaning, and a flexiblerepresentation of what counts as relevant information (via utilityfunctions). If this is not enough, e.g., if we want the listener tohave partial information not shared by the speaker as well (such aswhen the speaker is uncertain about what is really relevant for thelistener), that can easily be accommodated into a more complex gamemodel, but we refrain from going more complex here. The strategies ofsender and receiver encode particular ways of using and interpretinglanguage. The notion of expected utility evaluates how good ways ofusing and interpreting language are (in the given context).Game-theoretic explanations of pragmatic phenomena aim to single outthose sender-receiver strategy pairs that correspond to empiricallyattested behavior as optimal and/or rational solution of the gameproblem.

The standard solution concept of game theory isNashequilibrium. A Nash equilibrium of a signaling game is a pair ofstrategies \(\langle\)S\(^*\),R\(^*\)\(\rangle\)which has the property that neither the sender nor the receiver couldincrease his or her expected utility by unilateral deviation. Thus,S\(^*\) is a best response toR\(^*\) andR\(^*\) is a best response toS\(^*\). There areplenty of refinements of Nash equilibrium in the game-theoreticliterature. Moreover, there are alternatives to equilibrium analyses,the two most prominent of which are: (i) explicit formalizations ofagents’ reasoning processes, such as is done in epistemic gametheory (e.g., Perea 2012), and (ii) variants of evolutionary gametheory (e.g., Sandholm 2010) that study the dynamic changes inagents’ behavioral disposition under gradual optimizationprocedures, such as by imitation or learning from parents. Theseissues are relevant for applications to linguistic pragmatics as well,as we will demonstrate presently with the example ofM-implicatures/Horn’s division of pragmatic labor.

2.2 A game-theoretic explanation of Horn’s division

Parikh (1988, 1991, 1992) proposed the use of game theory to accountfor disambiguation and implicature. As it turns out, the same proposalcan be used to account for Horn’s division of pragmatic labouras well. Parikh assumes that a message, or form, \(f_u\), issemantically ambiguous, and can have a stereotypical interpretation,\(t_{st}\), and a non-stereotypical interpretation \(t_{\neg st}\),with \(P(t_{st}) \gt P(t_{\neg st})\). Both meanings, however, couldbe expressed unambiguously as well by a more complex form.Interpretation \(t_{\neg st}\), for instance, can be expressed notonly by \(f_u\), but by form \(f_m\) as well that literally expressesit. As is usual in so-called costly signalling games, and alsoconsistent with Parikh’s proposal, the sender’s utilityfunction can be decomposed into a benefit and a cost function, \(U_s(t,f,i) = B_s (t,i) - C(f)\), where \(i\) is an interpretation. Weadopt the following benefit function: \(B_s (t,i) = 1\) if \(i = t\),and \(B_s (t,i) = 0\) otherwise. The cost of the ambiguous message\(f_u\) is lower than the cost of the unambiguous message \(f_m\). Wecan assume without loss of generality that \(C(f_u) = 0 \lt C(f_m)\).In the use of signalling games for the analysis of human communicationit is standard to assume that it is always better to have successfulcommunication with a costly message than unsuccessful communicationwith a cheap message, which means that \(C(f_m)\), though greater than\(C(f_u)\), must remain reasonably small. If we limit ourselves to agame with 2 states, \(t_{st}\) and \(t_{\neg st}\), and threemessages, \(f_u\) and the two forms that literally express the twostates, and think of sender and receiver strategies as before, thecombination of sender and receiver strategies that gives rise to thebijective mapping \(\{\langle t_{st},f_u\rangle , \langle t_{\negst},f_m\rangle \}\) is a Nash equilibrium of this game. Intuitively,this is also the correct solution of the disambiguation game, becausethe ambiguous message \(f_u\) expresses the stereotypicalinterpretation \(t_{st}\), while the non-stereotypical state \(t_{\negst}\) is expressed by the marked and costlier message \(f_m\).Unfortunately, also the mapping \(\{\langle t_{st},f_m\rangle ,\langle t_{\neg st},f_u\rangle \}\)—where the lighter messagedenotes the non-stereotypical situation—is a Nash equilibrium ofthe game, which means that on the present implementation the standardsolution concept of game theory cannot yet single out the desiredoutcome.

This is were considerations of equilibrium refinements and/oralternative solution concepts come in. Parikh (1988, 1991, 1992)observes that of the two equilibria mentioned above, the first onePareto-dominates the second—being the unique Pareto-Nashequilibrium—and that for this reason the former should bepreferred. Van Rooij (2004) suggests that because Horn’sdivision of pragmatic labor involves not only language use but alsolanguage organization, one should look at signaling games from anevolutionary point of view, and make use of those variants ofevolutionary game theory that explain the emergence of Pareto-optimalsolutions. As a third alternative, following some ideas of De Jaegher(2008), van Rooij (2008) proposes that one could also make use offorward induction (a particular game-theoretic way of reasoning aboutsurprising moves of the opponent) to single out the desiredequilibrium. As an example of an approach that draws on detailedmodelling of the epistemic states of interlocutors, Franke (2014a)suggests that we should distinguish cases of M-implicature thatinvolve rather clearad hoc reasoning, such as (5) and (6),from cases with a possibly more grammaticalized contrast, such asbetween (3) and (4).

5. Mrs T sang ‘Home Sweet Home’.
6. Mrs T produces a series of sounds roughly corresponding to thescore of ‘Home Sweet Home’.

Franke suggests that the game model for reasoning about (5) and (6)should contain an element of asymmetry of alternatives: whereas it isreasonable (for a speaker to expect that) a listener would consider(5) to be an alternative utterance when hearing (6), it is quiteimplausible that (a speaker believes that) a listener will consider(6) a potential alternative utterance when hearing (5). This asymmetryof alternatives translates into different beliefs that the listenerwill have about the context after different messages. The speaker cananticipate this, and a listener who has actually observed (6) canreason about his own counterfactual context representation that hewould have had if the speaker had said (5) instead. Franke shows that,when paired with this asymmetry in context representation, a simplemodel of iterated best response reasoning, to which we turn next,gives the desired result as well.

2.3 Quantity implicatures and iterated reasoning

Unlike the case of M-implicatures, many Quantity implicatures hinge onthe fact that alternative expressions differ with respect to logicalstrength: the inference from ‘three’ to the pragmaticallystrengthed ‘exactly three’-reading, that we sketched inSection 1.1, draws on the fact that the alternative expression‘four’ is semantically stronger, i.e., ‘four’semantically entails ‘three’, but not the other wayaround, under the assumed ‘at least’-semantics. In orderto bring considerations of semantic strength to bear on game-theoreticpragmatics, we must assign conventional meaning some role in eitherthe game model or the solution concept. In the following, we look attwo similar, but distinct possibilities of treating semantic meaningin approaches that spell out pragmatic reasoning as chains of(higher-order) reasoning about interlocutors’ rationality.

A straightforward and efficient way of bringing semantic meaning intogame-theoretic pragmatics is to simply restrict the set of viablestrategies of sender and receiver in a signaling game to thosestrategies that conform to conventional meaning: a sender can onlyselect forms that are true of the actual state, and the receiver canonly select interpretations which are in the denotation of an observedmessage. This may seem crude and excludes cases of non-literallanguage use, lying, cheating and error from the start, but it mayserve to rationalize common patterns of pragmatic reasoning amongcooperative, information-seeking interlocutors. Based on such arestriction to truth-obedient strategies, it has been shownindependently by Pavan (2013) and Rothschild (2013) that there is anestablished non-equilibrium solution concept that nicely rationalizesQuantity implicatures, namelyiterated admissibility, alsoknown asiterated elimination of weakly dominated strategies.Without going into detail, the general idea of this solution conceptis to start with the whole set of viable strategies (all conforming tosemantic meaning) and then to iteratively eliminate all strategies\(X\) for which there is nocautious belief about which ofthe opponent’s remaining strategies the opponent will likelyplay that would make \(X\) a rational thing to do. (A cautious beliefis one that does not exclude any opponent strategy that has not beeneliminated so far.) The set of strategies that survive repeatediterations of elimination are then compatible with (a particular kindof) common belief in rationality. In sum, iterated admissibility is aneliminative approach: starting from the set of all (truth-abiding)strategies, some strategies are weeded out at every step until weremain with a stable set of strategies from which nothing can beeliminated anymore.

An alternative to restricting attention to only truthful strategies isto use semantic meaning to constrain the starting point of pragmaticreasoning. Approaches that do so are theoptimal assertionsapproach (Benz 2006, Benz & van Rooij 2007),iteratedbest response models (e.g., Franke 2009, 2011, Jäger 2014),and relatedprobabilistic models (e.g., Frank & Goodman2012, Goodman & Stuhlmüller 2013, Franke & Jäger2014). The general idea that unifies these approaches can be traceddirectly to Grice, in particular the notion that speakers shouldmaximize the amount of relevant information contained in theirutterances. Since information contained in an utterance is standardlytaken to be semantic information (as opposed to pragmaticallyrestricted or modulated meaning), a simple way of implementing Griceanspeakers is to assume that they choose utterances by considering how aliteral interpreter would react to each alternative. Pragmaticlisteners then react optimally based on the belief that the speaker isGricean in the above sense. In other words, these approaches define asequence of Theory-of-Mind reasoning: starting with a (non-rational,dummy) literal interpreter, a Gricean speaker acts (approximately)rationally based on literal interpretation, while a Gricean listenerinterprets (approximately) rationally based on the behavior of aGricean speaker. Some contributions allow for further iterations ofbest responses, others do not; some contributions also look atreasoning sequences that start with literal senders; somecontributions assume that agents are strictly rational, others allowfor probabilistic approximations to classical rational choice (seeFranke & Jäger 2014 for overview and comparison).

A crucial difference between iterated best response approaches and thepreviously mentioned approach based on iterated admissibility is thatthe former does not shrink a set of strategies but allows for adifferent set of best responses at each step. This also makes it sothat (some) iterated best response approaches can deal with pragmaticreasoning in cases where interlocutors’ preferences are notaligned, i.e., where the Gricean assumption of cooperativity does nothold, or where there are additional incentives to deviate fromsemantic meaning (for more about game models for reasoning innon-cooperative contexts, see, e.g., Franke, de Jager & van Rooij2012, de Jaegher & van Rooij 2014, Franke & van Rooij 2015).Another difference between iterated best response models and iteratedadmissibility is that the latter do not by itself account forHorn’s division of pragmatic labor (see Franke 2014b and Pavan2014 for discussion).

To illustrate how iterated best response reasoning works in a simple(cooperative) case, let us look briefly at numerical expressionsagain. Take a signaling game with 4 states, or worlds, \(W = \{w_1,w_2, w_3, w_4\}\) where the indices give the exact/maximal number ofchildren that came to our party, and four messages \(F =\{\)‘one’,‘two’,‘three’,‘four’\(\}\), as shorthand for ‘\(n\) children cameto our party’. On a neo-Gricean ‘atleast’-interpretation of numerals, the meanings of the numeralexpressions form an implication chain:

because, for instance, \(\llbracket\)‘three’\(\rrbracket =\{w_3, w_4\}\). A literal interpreter, who is otherwise oblivious tocontextual factors, would respond to every message by choosing anytrue interpretation with equal probability. So, for instance, if theliteral interpreter hears ‘three’, he would choose \(w_3\)or \(w_4\), each with probability \(½\). But that means that anoptimal choice of expression for a speaker who wants to communicatethat the actual world is \(w_3\) would be ‘three’, becausethis maximizes the chance that the literal interpreter selects\(w_3\). Concretely, if the speaker chooses ‘one’, thechance that the literal listener chooses \(w_3\) is ¼; for‘two’ it’s ⅓; for ‘three’it’s \(½\), and for ‘four’ it’s zero,because \(w_3\) is not an element of\(\llbracket\)‘three’\(\rrbracket\). So, a rationalGricean speaker selects\(\llbracket\)‘three’\(\rrbracket\) in \(w_3\) and nowhereelse, as is easy to see by a parallel argument for all other states.But that means that a Gricean interpreter who hears‘three’ will infer that the actual world must be\(w_3\).

In recent years, a few promising extensions of this pragmaticreasoning scheme have been proposed. One is to include probabilisticchoice functions to model agents’ approximately rationalchoices, so as to allow for a much more direct link with experimentaldata (for overview see Franke & Jäger 2016, Goodman &Frank 2016). Such probabilistic pragmatic models have been applied toa number of phenomena of interest, including reasoning aboutreferential expressions in context (Frank & Goodman 2012),ignorance implicatures (Goodman & Stuhlmüller 2013), contexteffects on scalar implicatures (Degen et al. 2015), non-literalinterpretation of number terms (Kao et al. 2014), pragmatic meaningmodulation associated with particular intonation (Bergen & Goodman2015, Stevens 2016), and Quantity implicatures in complex sentences(Bergen et al. 2016, Potts et al. 2016). These models of pragmaticreasoning have furthermore been used successfully also in naturallanguage processing applications (Andreas & Klein 2016, Monroe etal. 2017) and other areas of linguistic research, such associolinguistics (Burnett 2019). Another recent approach is to allowfor uncertainty of the semantic meanings of the involved expressions,and for uncertainty of the beliefs and preferences of the participantsof thee conversation and of other aspects of the ‘commonground’ (e.g., Brochhagen 2017). Finally, in recent work morepragmatic phenomena are accounted for than just generalisedconversational implicatures, such as particular conversationalimplicature, explicatures or free enrichment (e.g., Parikh 2010), theanalysis of presupposition (Qing et al. 2016), and politeness (e.g.Clark 2012, Yoon et al. 2020).

3. Conclusion

Bidirectional Optimality Theory and Game Theory are quite natural, andsimilar, frameworks to formalize Gricean ideas about interactive,goal-oriented pragmatic reasoning in context. Recent developments turntowards epistemic or evolutionary game theory or to probabilisticmodels for empirical data. Finally, we remark that formal techniquessimilar to what is explored here are applicable to a wider range ofproblems in diverse areas, such as philosophy (e.g., Parikh 2019),language evolution (e.g., Ahern and Clark 2014; Brochhagen, Franke, andRooij 2018; Carcassi 2020), as well as computational linguistics (e.g.Vogel et al. 2014, Cohn-Gordon, Goodman, and Potts 2019).

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