Games and Quantity implicatures

Abstract

In this paper we seek to account for scalar implicatures and Horn's division of pragmatic labor in game‐theoretical terms by making use mainly of refinements of the standard solution concept of signaling games. Scalar implicatures are accounted for in terms of Farrell's (1993) notion of a ‘neologism‐proof’ equilibrium together with Grice's maxim of Quality. Horn's division of pragmatic labor is accounted for in terms of Cho and Kreps’ (1987) notion of ‘equilibrium domination’ and their ‘Intuitive Criterion’.

Keywords:

• conversational implicatures
• game theory
• Horn's division
• Intuitive Criterion
• neologism‐proofness
• pragmatics

Acknowledgements

I would like to thank Michael Franke, Kris de Jaegher, Tikitu de Jager, Brian Skyrms, and especially the anonymous reviewer for this journal for remarks and discussion.

Notes

1. The question whether f_some is credible never arises according to our simple definition, because the semantic meaning of this message contains more than one type, [f_some ]  =  {t_sbna , t_all }.

2. To implement this constraint, we require that ∀t ∈ T and ∀σ, t ∈ [σ(t)]. Though economists might find this an unnatural, or unmotivated, assumption, making such an assumption is uncontroversial among linguists and philosophers.

3. Based on the earlier mentioned result of Wärneryd (1993) that only separating equilibria are evolutionarily stable.

4. Cho and Kreps' (1987) Intuitive Criterion as used in section 4 of this paper is very close to Farrell's requirement. The only difference seems to be that the Intuitive Criterion relies on the cost of sending a certain message, not on the meaning. Thanks to the reviewer for making my earlier claim more precise.

5. To some readers this last sentence might conversationally implicate that the analysis of scalar implicatures proposed in the previous section cannot account for those more general Quantity₁ implicatures. This is indeed the case, but one can define a generalization of Farrell's condition of being neologism‐proof – call it communication‐proof – that an equilibrium has to satisfy. In terms of such a more general notion we could predict more general Quantity₁ implicatures, but the resulting analysis would be very similar to what will be proposed in this section.

6. This assumption is controversial. Those who don't accept this assumption should think of other examples where the semantic meanings of the alternative expressions form a linear chain with respect to inference. The scales ⟨and, or⟩ and ⟨all, most, some⟩ would do if ‘or’ is read inclusively and the quantifiers ‘all’ and ‘most’ give rise to an existential presupposition.

7. According to Blume et al. (1993), this means that we are still in the realm of cheap talk signaling games.

8. For a game‐theoretical treatment of something that is closely related to Horn's division of pragmatic labor, see Benz and van Rooij (in press).

9. By starting with this pooling equilibrium, my proposal is closely related to Kris de Jaegher's (n.d.) evolutionary approach to Horn's division of pragmatic labor. I profited from our discussion of my alternative way to proceed.

10. The pooling Nash equilibrium is also a pooling Bayesian equilibrium, because S's strategy of this pooling equilibrium doesn't put any constraint on the conditional beliefs R could have if the unused message would have been sent.

11. It is, of course, the two‐level interpretation that is crucial for our analysis, not the particular rewording that illustrates it.