Abstract
Language use and interpretation is heavily contingent on context. But human interlocutors need not always agree what the actual context is. In game theoretic approaches to language use and interpretation, interlocutors’ beliefs about the context are the players’ beliefs about the game that they are playing. Together this entails that we need to consider cases in which interlocutors have different subjective conceptualizations of the game they are in. This paper therefore extends iterated best response reasoning, as an established model for pragmatic reasoning, to games with unawareness. This extension not only leads to more plausible context models for many communicative situations, but also to improved predictions for otherwise problematic cases and an extension of the scope of pragmatic phenomena that can be captured by game theoretic analysis.
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Notes
\(\Updelta(X)\) is the set of all probability distributions over set X.
For simplicity, I will assume throughout that we only deal with signaling games in which T, M and A are finite sets, and where further for all \(t \in T\) we have Pr(t) > 0 and \(t \in [\![m]\!]\) for some m, and that for every message m we have \([\![m]\!]\! \neq \emptyset\).
Wherever feasible, functions are represented as “lists of mappings”.
This explicit epistemic formulation of a solution concept is also crucial when we turn to games with unawareness, which essentially place restrictions on the beliefs that agents may hold.
Bayesian conditionalization is not possible for a given message m if σ k+1(t, m) = 0 for all t. In that case, m is a surprise message that the receiver did not expect given his behavioral belief. To specify posterior beliefs for surprise messages, sophisticated belief revision strategies could be implemented. But for the purposes of this paper, we can keep things simple and assume that the receiver adopts an arbitrary posterior belief compatible with the semantic meaning of the surprise message.
Although certain features of it are variable, (something very much like) this signaling game model is the standard model assumed in game theoretic accounts of Horn’s division of pragmatic labor and similar phenomena (Parikh 1992, 2001; van Rooij 2004, 2006; Benz and van Rooij 2007; de Jaegher 2008; van Rooij 2008; Mühlenbernd 2011; Jäger 2013) and it is also in line with the standard formalization of the problem in Bidirectional Optimality Theory (c.f. Blutner 1998, 2000).
Usually it is assumed that the receiver’s subjective beliefs are an accurate reflection of the true occurrence frequencies, so that these two interpretative possibilities coincide.
To be clear, I do not mean to suggest that all cases of linguistic markedness should be treated in terms of unawareness, but merely that some cases of M-implicatures invite a treatment in terms of diverging subjective conceptualizations. The concluding Sect. 7 comes back to these more general issues.
The literature on games with unawareness has considered what I call pruning (e.g. Halpern and Rêgo 2006; Feinberg 2011a; Heifetz et al. 2011, 2012), but, as far as I know, not lumping. This extension is, however, particularly useful for linguistic applications in modeling unawareness from inattentiveness. More on the relation between the formalism proposed here and the existing literature is given in Sect. 7
Alternatively, we could restrict attention to cases where lumping goes together with an implicit belief about the likelihood of the lumped histories. Towards that end, we’d choose \(p \in [0;1]\) and required for all terminal nodes \(z^\sqcup = \bigcup\{z,f(z)\}\) in the merged branch that \(u_i'(z^\sqcup) = p \times u_i(z) + (1-p) \times u_i(f(z))\). However, since not all failures to discriminate possibilities need to involve such implicit beliefs, we are more liberal here.
As with utilities, we are liberal here and leave the prior probabilities \(\hbox{Pr}_{h_{p}}^{\prime}\) unspecified on purpose. If we wanted to be more restrictive, a natural idea would be to conserve the values of \(\hbox{Pr}_{h_p}\) wherever possible, and to assign probability \(\hbox{Pr}'_{h_p}(h^\sqcup) = p \times \hbox{Pr}_{h_p}(h_1) + (1-p) \times \hbox{Pr}_{h_p}(h_2)\), where p is again an implicit belief (see Footnote 3). This, call it, additive assignment of probabilities to sets of states is systematic, in the sense that there is a unique prior that results from lumping (c.f. de Jager 2009; Franke and de Jager 2011, for more on this additive assignment of probabilities of coarse-grained states). But this may occasionally be too inflexible, especially in the light of certain empirical results on subjective probability assignments. Fox and Levav (2004) argue that human performance in probability assignment tasks, such as, for instance, the Monty-Hall puzzle, can be explained based on the assumption that subjects assign flat probabilities to either naïvely coarse-grained, or to more sophisticated fine-grained partitions of logical space. If we want to allow for this—flat probabilities irrespective of granularity—we should not further restrict the definition of a lumping step across the board.
It is clear that the classical interpretation of the game model G where everybody’s view coincides with the modeller’s view is easily modelled with an awareness structure with just one world w 0 with g(w 0) = G. Our constraints then require \(Q_v = \{\langle w_0,w_0 \rangle\}\) for all information states v in G. Awareness structures are thus a conservative extension of classical games.
Here and elsewhere, I omit arrows that obviously follow from reflexivity and transitivity constraints.
There is a form of unawareness that most likely does play a role in our linguistic choices, but that we unfortunately cannot model. We cannot model agents who are aware of their own unawareness (Halpern and Rêgo 2009). Halpern and Rêgo (2006) have introduced so-called virtual moves to model agents who are aware of their own unawareness, and we could simply adopt this idea (as did Heifetz et al. 2012). Adding nothing but virtual moves to a partial representation should count as the same partial representation in an awareness structure. Another possibility would be to explicitly mark some conglomerates of actions that are lumped together in the subjective conceptualization of some agent, to represent the fact that the agent is aware that this is a conglomerate without knowing what is lumped together.
The problem with integrating awareness of unawareness is that it has further strategic implications that I would like to side-step here. Rationality of agents is limited to the set of choices and beliefs that they can grasp by their possibly limited awareness. This is what sets unawareness apart from probability-zero beliefs: whereas the latter can be deemed irrational (and therefore ruled out by a solution concept, such as rationalizability), the former cannot unless, perhaps, the agent is aware of her unawareness. Then by rationality the agent might be required to respond to her unawareness differently from both a probability-zero belief and complete unawareness. This is why the fascinating issue of awareness of unawareness will be left for another occasion.
Strictly speaking, this definition of belief types is too encompassing, because it may contain redundant types, namely whenever for some world w and game g(w), the views v and v′ reoccur as the same information state (due to lumping). In that case, by the Lumping constraint of Definition 4, belief types \(\langle w,v \rangle\) and \(\langle w,v' \rangle\) are the same.
Here, π wt k+1 is the unique projection from G to G′ with \(G' \sqsubseteq G\).
Of course, there have been notable exceptions to this tendency, such as Benz (2009, 2012). Also, Parikh (1992, 2001)—one of the pioneers of game theoretic pragmatics—has argued for the use of a special game form in pragmatics, which he called games of partial information and which he defended against identification with signaling games (Parikh 2006). Readers unfamiliar with Parikh’s work will likely not be able to appreciate this, so I would like to mention only in passing that games of partial information may be conceived of as a special case of dynamic games with unawareness.
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Acknowledgments
I am heavily indebted to Tikitu de Jager for many heated and inspiring discussions about unawareness and its role in language. Many other colleagues have enriched my world with their sharp comments after hearing presentations of this material at Stanford, Tübingen, Paris, Amsterdam, esslli in Ljubljana and Tilburg. A grateful tip to the hat to all of you. The manuscript also benefitted from the help of two anonymous reviewers. Thanks finally to Alistair Isaac and Sven Lauer for their work, comments and encouragement.
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Franke, M. Pragmatic Reasoning About Unawareness. Erkenn 79 (Suppl 4), 729–767 (2014). https://doi.org/10.1007/s10670-013-9464-1
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DOI: https://doi.org/10.1007/s10670-013-9464-1