Contribution
Horn’s division of pragmatic labour, that marked or costlier forms carry less probable meanings, had resisted derivation: in a signalling game where two signals differ only in cost, the efficient and the inefficient assignment are both equilibria, and game-theoretic pragmatics needed a selection rule or an evolutionary story to pick one. This paper derives it in one shot from recursive cooperative reasoning, by adding uncertainty about what the signals literally mean. It also shows, with novel symbols and no conventions, that people draw both specificity and Horn implicatures.
Key insight
A costly signal is worth paying for only to a speaker whose meaning would otherwise be misread. If the listener allows that the signal might mean anything, the only speaker with a reason to pay is one who means the unlikely thing, and that asymmetry, once in the listener’s model, is amplified by each level of reasoning.
Assumptions
- Common knowledge of communicative goals (Lewis 1969; Clark 1996); the speaker wants to convey a specific meaning, the listener to recover it.
- Literal listener L₀(m|u, L) ∝ L_u(m)P(m), with L_u a truth function; a signal with no conventional or iconic meaning is all-true in the base model.
- Softmax speaker with gain λ > 0 and utility log L_{n−1}(m|u) − c(u).
- Lexical uncertainty model: the prior over lexicons is uniform over all lexicons in which every utterance is true of at least one meaning and every meaning has at least one true utterance (seven in the 2 × 2 case).
- The set of alternative utterances is fixed and known to both.
Key results
- Eqs. 2–5 (base model). S_n(u|m) ∝ exp(λU_n(u|m)), U_n(u|m) = log L_{n−1}(m|u) − c(u), L_n(m|u) ∝ P(m)S_{n−1}(u|m). Eq. 4 is printed as S_n ∝ (L_{n−1}(m|u) e^{c(u)})^λ; from Eq. 3 the exponent should be e^{−c(u)}, a sign slip in the printed paper.
- Specificity implicature. Meanings pyramid and cube, utterances “pyramid” and “shape”, equal costs: S₁ prefers “pyramid” for pyramid and must say “shape” for cube, so L₂ reads “shape” as cube; with depth both tend to probability 1, for λ > 1 and P(m) > 0 (Fig. 2).
- Failure on costs. With two all-true signals of different cost, L₀ reads both as the prior and no level breaks the symmetry.
- Eqs. 6–8 (lexical uncertainty). S_n(u|m, L) ∝ exp(λU_n(u|m, L)); L_n(m|u) ∝ Σ_L P(m)P(L)S_{n−1}(u|m, L); U₁ uses L₀(m|u, L), U_n for n > 1 uses the marginal L_{n−1}(m|u). The model reduces to the base model when one lexicon is fixed, so specificity implicature survives; and it derives the efficient Horn mapping, for two meanings and for three (Fig. 3, averaged over a parameter grid).
- Experiment 1. 40 participants, five rounds each, random partners, $0.06 per success. Listeners read the alien symbol as the unnamed object on every trial; speakers chose it for the unnamed object on every trial and chose the name for the named object on all but two.
- Experiment 2. 140 participants; objects at 60/30/10%; messages free, 0.02; frequencies and costs randomized between subjects. All six efficient-mapping comparisons hold in mixed logit regressions (t from 5.31 to 8.27, p < .001). By-subject ANOVAs find no first-round main effect or interaction for five of six response types; the intermediate-cost message shows a small first-round interaction (p < .05).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Basic recursive reasoning yields specificity implicatures, converging with depth | strong (derivation) | the two-meaning argument and Fig. 2 |
| C2 | Basic recursive reasoning cannot derive Horn’s principle from costs | strong (derivation) | the symmetry argument |
| C3 | Lexical uncertainty derives the efficient Horn mapping in one-shot games without equilibrium refinements | moderate | two-lexicon argument; three-meaning case by simulation over a parameter grid |
| C4 | It is the first such derivation | weak | the authors’ survey of van Rooij and Franke |
| C5 | People draw specificity implicatures among novel signals without conventions | moderate | Experiment 1, ceiling behaviour, 40 participants |
| C6 | People coordinate on Horn’s mapping in one shot, by reasoning rather than learning | moderate | Experiment 2 regressions; first-round analysis, one exception; five rounds per participant, with feedback after each |
Method
Model predictions are computed by recursion to a chosen depth; for three meanings they are averaged over a fine grid of λ and depth. The games are web interfaces in which a speaker sees an object and clicks one of the messages; the listener clicks an object; both learn whether they succeeded. Ten practice rounds without partner or feedback precede five scored rounds with re-randomized partners.
Concepts
- specificity implicature: strengthening a less specific utterance to the negation of a more specific available one, with or without a canonical scale (the authors’ term, for scalar implicature generalized).
- Horn implicature / Horn’s principle: costlier forms are associated with less probable meanings (Horn’s Q- and R-principles).
- lexicon: a map from utterances to truth functions on meanings.
- lexical uncertainty: the listener and speaker reason over a distribution of lexicons rather than one.
Connections
The base model is “very similar” to Jäger and Ebert’s iterated best response; the informativeness utility is credited to Frank, Goodman, Lai and Tenenbaum (2009, CogSci), not to the Psychological Science paper of the same year (LIT-815). The equilibrium-selection problem is Cho and Kreps’s and Chen, Kartik and Sobel’s; evolutionary or hybrid derivations of Horn’s principle are van Rooij’s and Franke’s. Goodman and Frank (LIT-816) cite this paper for cost sensitivity, and its lexical-uncertainty model is the ancestor of the uRSA lexical-uncertainty treatment of embedded implicature they describe.
Bearing on the record
- Produces, with LIT-821 and LIT-816, THEORY-172. Its specific contribution there is that interpretation depends on the alternatives and their costs, not only on the uttered form, and that this holds for signals with no prior meaning.
- On meaning by contrast. In Experiment 1 the alien symbol acquires a meaning only through its contrast with the iconic alternative. That is an experimental case of a sign’s interpretation being fixed by what else could have been said; it is not a case of meaning without reference, since the contrast works only because one alternative is iconically tied to an object.
- On LIT-473. The paper assumes Lewis’s common knowledge but argues that its two conventions need no precedent; LIT-473 is unread, so the record cannot yet say whether that departs from Lewis’s account.
- No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Proceedings paper, six pages; the three-meaning predictions are shown but not analysed.
- Experiment 1 is at ceiling with 40 participants and one design; it cannot distinguish models that all predict the same mapping, and the displayed predictions are said to be insensitive to parameters.
- Experiment 2’s players get feedback after every round, so learning within five rounds is possible; the first-round analysis tests for it with null results in five of six cases, not a direct one-shot test.
- Costs are money, made explicit; whether the same reasoning governs costs that are length, effort or frequency is not tested.
- The lexicon prior is uniform over a hand-specified set.
Open questions
- Does lexical uncertainty predict Horn mappings when costs are implicit and graded, as in natural language?
- How sensitive is the derived mapping to the lexicon prior, beyond the uniform one used?
- A strictly single-trial version of Experiment 2 would separate reasoning from fast learning.