@cite{frank-goodman-2012} #
@cite{degen-2023} @cite{tenenbaum-griffiths-2001} @cite{heim-kratzer-1998} @cite{grice-1975}
"Predicting Pragmatic Reasoning in Language Games" Science 336(6084): 998
Paradigm #
Three objects varying on two dimensions (color × shape) in a reference game. Speaker produces a single feature word; listener identifies the target object.
Context: {blue_square, blue_circle, green_square} Utterances: {blue, green, square, circle}
Architecture (SM Eq. S1–S4) #
⟦w⟧(o) Boolean denotation: does word w apply to object o?
L0(rₛ|w) = ⟦w⟧(rₛ) / Σ_o ⟦w⟧(o) (Eq. S3)
U(w; rₛ, C) = log L0(rₛ|w) − D(w) (Eq. S2)
P_S1(w|rₛ, C) ∝ e^{α · U(w; rₛ, C)} (Eq. S1)
L1(rₛ|w) ∝ P_S1(w|rₛ, C) · P(rₛ) (Eq. 1)
With D(w) = 0 (no word cost) and α = 1 (Luce choice rule). No latent variables, uniform priors.
Qualitative Findings #
The paper introduced the RSA framework and demonstrated that pragmatic inferences in reference games can be predicted by modeling listeners as reasoning about rational, informative speakers.
| # | Finding | Word | Comparison | Mechanism |
|---|---|---|---|---|
| 1 | Pragmatic narrowing | "square" | blue_sq > green_sq | S1 prefers "green" for green_sq |
| 2 | Pragmatic narrowing | "blue" | blue_sq > blue_circ | S1 prefers "circle" for blue_circ |
| 3 | Unique reference | "green" | green_sq > blue_sq | "green" applies only to green_sq |
| 4 | Unique reference | "circle" | blue_circ > blue_sq | "circle" applies only to blue_circ |
Model predictions correlate r = 0.99 with human judgments (combined
salience + informativeness; r = 0.98 for speaker alone; r = 0.87 for
listener excluding trivial predictions). N = 745 (206 speaker, 276
salience, 263 listener). All proofs use rsa_predict.
Formalization Coverage #
- L1 predictions (§6): All 4 listener comparisons proved via
rsa_predict - S1 predictions (§6b): Speaker informativity preferences proved via
rsa_predict - Size principle (§6c): Eq. 2 demonstrated — S1 prefers smaller extensions
- Montague grounding (§5b): Feature semantics grounded in intersective predicate modification
- Structural properties (§5): Feature uniqueness/ambiguity proved by
rfl
The 4 qualitative findings from @cite{frank-goodman-2012}.
Each finding is an L1 comparison between two objects given a word. Findings 1–2 are the core pragmatic results (ambiguous words get narrowed). Findings 3–4 are baseline sanity checks (unique identifiers trivially select the referent).
- square_pragmatic_narrowing : Finding
Hearing "square" → blue_square over green_square. The core RSA insight: "green" uniquely identifies green_square, so a speaker saying "square" probably means blue_square. Evidence: model-human r = 0.99.
- blue_pragmatic_narrowing : Finding
Hearing "blue" → blue_square over blue_circle. Parallel inference: "circle" uniquely identifies blue_circle, so "blue" signals blue_square.
- green_unique_reference : Finding
Hearing "green" → green_square (unique identifier, no pragmatics needed).
- circle_unique_reference : Finding
Hearing "circle" → blue_circle (unique identifier, no pragmatics needed).
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The three objects in the classic reference game context (Figure 1).
Modeled as an enumeration rather than Color × Shape because the context has only 3 of 4 possible combinations — no green_circle.
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Characteristic function: ⟦feature⟧(object).
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- Phenomena.Reference.Studies.FrankGoodman2012.Feature.blue.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.blue_square = true
- Phenomena.Reference.Studies.FrankGoodman2012.Feature.blue.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.blue_circle = true
- Phenomena.Reference.Studies.FrankGoodman2012.Feature.green.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.green_square = true
- Phenomena.Reference.Studies.FrankGoodman2012.Feature.square.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.blue_square = true
- Phenomena.Reference.Studies.FrankGoodman2012.Feature.square.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.green_square = true
- Phenomena.Reference.Studies.FrankGoodman2012.Feature.circle.appliesTo Phenomena.Reference.Studies.FrankGoodman2012.Object.blue_circle = true
- f.appliesTo o = false
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@cite{frank-goodman-2012} reference game as RSA model.
SM Eq. S1–S4: ⟦w⟧(o) = 1 if w applies to o, 0 otherwise (Boolean semantics) L0(rₛ|w) = ⟦w⟧(rₛ) / Σ_o ⟦w⟧(o) (Eq. S3, literal listener) U(w; rₛ, C) = log L0(rₛ|w) − D(w) (Eq. S2, informativity − cost) P(w|rₛ, C) ∝ e^{α · U(w; rₛ, C)} (Eq. S1, soft-max speaker)
With D(w) = 0 and α = 1 (Luce choice rule).
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"green" uniquely identifies green_square.
"circle" uniquely identifies blue_circle.
"blue" is ambiguous between two objects.
"square" is ambiguous between two objects.
The feature semantics (Feature.appliesTo) is compositionally grounded
in Montague intersective predicate modification.
Each feature word denotes an intersective ⟨e,t⟩ predicate. Objects in the reference game context are uniquely characterized by predicate modification (conjunction) of their color and shape features:
⟦blue square⟧ = ⟦blue⟧ ⊓ₚ ⟦square⟧ = λx. blue(x) ∧ square(x)
This means the RSA meaning function cfg.meaning inherits its semantics from
Montague composition rather than being stipulated independently.
Each object is uniquely characterized by predicate modification of its color and shape features. Reference resolution in the game IS predicate modification: the listener identifies the referent by intersecting the denotations of the heard feature words.
The paper's key theoretical contribution is the size principle (@cite{tenenbaum-griffiths-2001}, Eq. 2): the speaker probability of an utterance is inversely proportional to its extension size |⟦u⟧|.
In RSA with α = 1 and belief-based scoring: S1_score(w, u) = L0(w|u)¹ = 1/|⟦u⟧| (for true utterances) S1_score(w, u) = 0 (for false utterances)
Features with smaller extensions are more informative and thus preferred by S1. The S1 predictions in §6b demonstrate this: unique features (size 1) always beat ambiguous features (size 2).
Extension size: number of objects a feature applies to.
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Unique features have extension size 1; ambiguous features have size 2.
All RSA levels derive from cfg:
cfg.L0 u w— L0 posterior P(w|u)cfg.S1 w u— S1 policy P(u|w)cfg.L1 u w— L1 posterior P(w|u)
The pragmatic inferences arise because S1 prefers informative utterances: a speaker wanting green_square says "green" (unique), so "square" signals blue_square. Similarly, a speaker wanting blue_circle says "circle", so "blue" signals blue_square.
Core RSA result: L1("square") assigns higher probability to blue_square than green_square. A speaker wanting green_square would say "green" (uniquely identifying). Saying "square" signals blue_square.
L1("blue") assigns higher probability to blue_square than blue_circle. A speaker wanting blue_circle would say "circle" (uniquely identifying). Saying "blue" signals blue_square.
L1("green") assigns higher probability to green_square (the only object "green" applies to).
L1("circle") assigns higher probability to blue_circle (the only object "circle" applies to).
The pragmatic listener predictions above (§6) derive from speaker behavior: S1 prefers informative utterances — those that uniquely identify the target. These S1 comparisons are the mechanism behind pragmatic narrowing.
S1 prefers "green" over "square" for green_square. "green" uniquely identifies green_square; "square" is ambiguous.
S1 prefers "circle" over "blue" for blue_circle. "circle" uniquely identifies blue_circle; "blue" is ambiguous.
S1 is indifferent between "blue" and "square" for blue_square. Both features share with exactly one other object, so L0(blue_sq|blue) = 1/2 = L0(blue_sq|square), hence equal S1 scores.
The size principle in action: S1 prefers features with smaller extensions. Unique features (extensionSize = 1) dominate ambiguous ones (extensionSize = 2). This is Eq. 2 of the paper: P_S(u|w) ∝ 1/|⟦u⟧|.
Map each empirical finding to the RSA model prediction that accounts for it.
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The RSA model accounts for all 4 qualitative findings from @cite{frank-goodman-2012}.
@cite{frank-goodman-2012} tested RSA predictions across 7 distinct context types varying feature overlap between a target object and 2 distractors. The contexts are characterized by how many feature dimensions each distractor shares with the target (Table 1, Figure 2).
Context 2/2b is the canonical blue_square / blue_circle / green_square example
from §2–7 above. The parameterized system below generalizes all 7 contexts via a
single mkRefGame constructor: a ContextSpec of 4 booleans determines the
full applicability matrix, and mkRefGame builds the RSAConfig from it.
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Feature values in a 2-dimensional reference game. dim1_a / dim2_a are the target's values on each dimension. b-variants are d1-specific; c-variants are d2-specific.
- dim1_a : Dim2Feature
- dim1_b : Dim2Feature
- dim1_c : Dim2Feature
- dim2_a : Dim2Feature
- dim2_b : Dim2Feature
- dim2_c : Dim2Feature
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Context specification: 4 booleans encoding distractor overlap with target. Each boolean indicates whether a distractor shares the target's value on a given dimension.
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Derive the boolean applicability matrix from a context spec. Target features (dim1_a, dim2_a) always apply to target. A distractor gets the target's feature if the spec says it shares that dimension; otherwise it gets its own unique feature (b for d1, c for d2).
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- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim1_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.target = true
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim2_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.target = true
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim1_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d1 = ctx.d1_dim1
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim1_b Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d1 = !ctx.d1_dim1
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim2_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d1 = ctx.d1_dim2
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim2_b Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d1 = !ctx.d1_dim2
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim1_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d2 = ctx.d2_dim1
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim1_c Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d2 = !ctx.d2_dim1
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim2_a Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d2 = ctx.d2_dim2
- ctx.applies Phenomena.Reference.Studies.FrankGoodman2012.Dim2Feature.dim2_c Phenomena.Reference.Studies.FrankGoodman2012.Obj3.d2 = !ctx.d2_dim2
- ctx.applies x✝¹ x✝ = false
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Build a reference game from a boolean applicability matrix. All contexts share: SM Eq. S1 (e^{α·log L0}), α = 1, uniform priors, D(w) = 0.
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Each context is defined by its overlap pattern: d1_dim1 × d1_dim2 × d2_dim1 × d2_dim2.
| Context | d1 dim1 | d1 dim2 | d2 dim1 | d2 dim2 | Shared features |
|---|---|---|---|---|---|
| 1/1 | ✗ | ✗ | ✗ | ✗ | 0 |
| 1/2 | ✗ | ✗ | ✗ | ✓ | 1 |
| 1/3 | ✗ | ✓ | ✗ | ✓ | 2 (both dim2) |
| 2/2a | ✓ | ✓ | ✗ | ✗ | 2 (both d1) |
| 2/2b | ✓ | ✗ | ✗ | ✓ | 2 (cross) |
| 2/3 | ✓ | ✓ | ✗ | ✓ | 3 |
| 3/3 | ✓ | ✓ | ✓ | ✓ | 4 |
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Each context type produces characteristic L1 predictions. The key contrasts:
Pragmatic narrowing (context 2/2b): A shared feature identifies the target because the distractor's S1 prefers a different unique feature. This requires asymmetric alternative options — the core RSA insight.
Symmetric non-narrowing (context 1/2): A shared feature does NOT narrow when both objects sharing it have equally good unique alternatives. S1 avoids the shared feature symmetrically for both, so L1 is uniform on it.
Symmetric indistinguishability (contexts 2/2a, 2/3, 3/3): When two objects share all features, no utterance can distinguish them — even pragmatic reasoning cannot break the symmetry.
Uniform informativity (context 1/3): When all objects share a feature and each has a unique alternative, S1 avoids the shared feature equally for all objects, making L1 uniform on that feature.
1/1: No feature overlap — all features uniquely identify their object. dim1_a uniquely picks out the target.
1/2: d2 shares dim2 with target, but pragmatic narrowing does NOT distinguish them via dim2_a. Both target and d2 have symmetric "escape routes" (unique dim1 features), so S1 avoids dim2_a equally for both. Contrast with 2/2b where asymmetric sharing enables narrowing.
1/2: The unique dim1 feature still identifies the target.
1/3: Both distractors share dim2. Each object has a unique dim1 feature, so S1 avoids dim2_a equally for all → L1("dim2_a") is uniform.
1/3: Unique dim1 features still work despite dim2 overlap.
2/2a: d1 shares both dimensions with target — they are featurally indistinguishable. No utterance can separate them.
2/2b: Pragmatic narrowing — the canonical FG2012 result. d1 shares
dim1, d2 shares dim2. S1(d1) prefers dim2_b (unique), so
L1("dim1_a") → target over d1. Matches §6 square_pragmatic_narrowing
with different types (Dim2Feature/Obj3 vs Feature/Object).
2/3: d1 shares both dims (indistinguishable from target). dim1_a cannot separate target from d1.
3/3: All objects share all features — completely indistinguishable. L1 is uniform; no utterance provides any information.
RSA's S1 score decomposes as:
S1(w|u) ∝ exp(α · (log L0(w|u) − D(w)))
where log L0 corresponds to Q1 (informativeness) and D(w) to Q2
(brevity). This model uses D(w) = 0 for all words, enforcing Q1 only.
The size principle — S1 prefers features with smaller extensions —
IS the Q1 sub-maxim: smaller extension = more informative = higher
log L0. With no cost term, there is no Q2 pressure.
This places @cite{frank-goodman-2012} in what @cite{dale-reiter-1995} call the "No Brevity" regime: Q1 is enforced, Q2 is not. The reference game is a probabilistic version of the REG (Referring Expression Generation) problem that D&R solve with the Incremental Algorithm — both produce a referring expression that identifies a target among distractors, but RSA does it via probabilistic inference rather than deterministic attribute selection.
RSA S1 implements a decomposition of Grice's Quantity maxim into two independent components: Q1 (informativeness, via log L0) and Q2 (brevity, via utterance cost D(w)). This model sets D(w) = 0, so only Q1 is active — the size principle (§6c) IS Q1 maximization. The independence of Q1 and Q2 is what allows selective enforcement: this model enforces Q1 while relaxing Q2 entirely.