[Bur19] — Signalling Games, Sociolinguistic Variation, and #
the Construction of Style
Linguistics and Philosophy 42: 419–450.
Overview #
Social Meaning Games (SMGs) model how sociolinguistic variant choice conveys social information. A speaker's use of -ing vs -in' induces listener inferences about persona traits (competent, friendly, etc.). The framework combines [Lew69]'s signalling games with RSA-style Bayesian reasoning to derive both style shifting (intra-speaker variation across contexts) and social stratification (inter-speaker variation across classes) from the same principles.
Architecture #
The meaning function is grounded in the Eckert–Montague lift from
EckertMontague.emMeaningMI: each variant's Eckert field (a set of
indexed properties) is lifted to persona compatibility via intersection
semantics. The grounding theorem ingMeaning_eq_emMeaningMI verifies
that the study's meaning function matches the theory-layer derivation.
Each context is a belief-based RSA over personae (worlds) and variants
(utterances): the speaker Sk is the softmax of L₀⁶ (α = 6) and the
listener L1k the Bayesian posterior (PMF.posterior) against the
context-specific persona prior, which also weights L₀ (Burnett eq. 11).
Predictions are exact PMF-value comparisons.
Key predictions #
- Per-persona variant preference: cool-guy prefers -in' ~69%
- Style shifting: casual→careful flips the cool-guy's preference
- Stern-leader exclusion: -in' is incompatible with stern leader
- Listener interpretation: Rice/Pelosi/Bush /t/ release predictions
- Bulletproofing: strong prior overwhelms variant effects (Bush)
- Cross-reference: model predictions close to [Lab12] data
Social properties (Burnett example (5)). Two bipolar dimensions: competence (competent/incompetent) and warmth (friendly/aloof).
- competent : PersonaTrait
- incompetent : PersonaTrait
- friendly : PersonaTrait
- aloof : PersonaTrait
Instances For
Equations
- Burnett2019.instDecidableEqPersonaTrait x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- Burnett2019.instReprPersonaTrait = { reprPrec := Burnett2019.instReprPersonaTrait.repr }
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- Burnett2019.instDecidableEqPersona x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- Burnett2019.instReprPersona = { reprPrec := Burnett2019.instReprPersona.repr }
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- Burnett2019.instReprPersona.repr Burnett2019.Persona.doofus prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Burnett2019.Persona.doofus")).group prec✝
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Eckert fields (Burnett example (10)):
- [-ing] = {competent, aloof}
- [-in'] = {incompetent, friendly}
The meaning function is derived via the Montagovian Individual / intersection semantics (Burnett footnote 14, Table 1): persona p is compatible with variant v iff p shares at least one property with v's Eckert field.
The property space for Burnett's simplified example.
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Persona membership as a Finset.
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- Burnett2019.Persona.coolGuy.toFinset = {Burnett2019.PersonaTrait.competent, Burnett2019.PersonaTrait.friendly}
- Burnett2019.Persona.sternLeader.toFinset = {Burnett2019.PersonaTrait.competent, Burnett2019.PersonaTrait.aloof}
- Burnett2019.Persona.doofus.toFinset = {Burnett2019.PersonaTrait.incompetent, Burnett2019.PersonaTrait.friendly}
- Burnett2019.Persona.asshole.toFinset = {Burnett2019.PersonaTrait.incompetent, Burnett2019.PersonaTrait.aloof}
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Eckert fields for (ING) (Burnett example (10)).
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The ING grounded field: both Eckert fields are consistent.
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- Burnett2019.ingField = { indexedProperties := Burnett2019.ingEckertField, indexed_consistent := Burnett2019.ingField._proof_1 }
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Meaning via the EM intersection lift: persona p is compatible with variant v iff p shares ≥1 property with v's Eckert field.
Equations
- Burnett2019.ingMeaning Eckert2008.INGVariant.velar Burnett2019.Persona.coolGuy = true
- Burnett2019.ingMeaning Eckert2008.INGVariant.velar Burnett2019.Persona.sternLeader = true
- Burnett2019.ingMeaning Eckert2008.INGVariant.velar Burnett2019.Persona.asshole = true
- Burnett2019.ingMeaning Eckert2008.INGVariant.velar Burnett2019.Persona.doofus = false
- Burnett2019.ingMeaning Eckert2008.INGVariant.apical Burnett2019.Persona.coolGuy = true
- Burnett2019.ingMeaning Eckert2008.INGVariant.apical Burnett2019.Persona.sternLeader = false
- Burnett2019.ingMeaning Eckert2008.INGVariant.apical Burnett2019.Persona.asshole = true
- Burnett2019.ingMeaning Eckert2008.INGVariant.apical Burnett2019.Persona.doofus = true
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Grounding theorem: the inline meaning function equals the
theory-layer emMeaningMI applied to the ING Eckert fields.
-ing is compatible with 3 personae (Table 1: excludes doofus).
-in' is compatible with 3 personae (Table 1: excludes stern leader).
Each social context is a belief-based RSA over personae (worlds) and variants (utterances), α = 6 ([Bur19] p. 435):
L₀(p | v) ∝ ⟦v⟧(p) · π(p) (Bayesian literal listener, `L0k`)
S₁(v | p) ∝ L₀(p | v)⁶ (`Sk`)
L₁(p | v) ∝ π(p) · S₁(v | p) (`L1k`, `PMF.posterior`)
The persona prior enters L₀ directly (Burnett eq. 11): the "naive listener"
weights compatible personae by π, not uniformly — this context-dependence
drives style shifting. meaningE carries that prior-weighted meaning; the
speaker is the softmax of L₀⁶ and the listener the Bayesian posterior
against the same prior π (which sums to 1 for every context).
Prior-weighted literal meaning ⟦v⟧(p) · π(p), lifted to ℝ≥0∞.
Equations
- Burnett2019.meaningE π v p = if Burnett2019.ingMeaning v p = true then ENNReal.ofReal ↑(π p) else 0
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Denominator value D_v = Σ_p ⟦v⟧(p)·π(p), assembled from the four
per-persona meaning values (ENNReal.ofReal of π p or 0).
Prior-weighted literal listener L₀(· | v) : PMF Persona.
Equations
- Burnett2019.L0k π hπ v = RSA.L0OfMeaning (Burnett2019.meaningE π) v ⋯ ⋯
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L0 value: L₀(p | v) = ⟦v⟧(p)·π(p) / D_v, an exact ENNReal.ofReal.
Speaker S₁(· | p) : PMF INGVariant, softmax of L₀⁶ (α = 6).
Equations
- Burnett2019.Sk π hπ p = PMF.normalize (fun (v : Eckert2008.INGVariant) => (Burnett2019.L0k π hπ v) p ^ 6) ⋯ ⋯
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Speaker value: S₁(v | p) = L₀(p|v)⁶ / Σ_v' L₀(p|v')⁶.
Speaker Z value: Σ_v L₀(p|v)⁶, assembled from the two L0_vals.
The persona prior sums to 1 (every context is a distribution).
The listener's persona prior π : PMF Persona.
Equations
- Burnett2019.priorK π h1 = PMF.ofFintype (fun (p : Burnett2019.Persona) => ENNReal.ofReal ↑(π p)) h1
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Pragmatic listener L₁(· | v) : PMF Persona, the Bayesian posterior.
Equations
- Burnett2019.L1k π hπ h1 v = PMF.posterior (Burnett2019.Sk π hπ) (Burnett2019.priorK π h1) v ⋯
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Listener reduction (<): L₁ prefers p₂ iff the prior-weighted
speaker scores do; the marginal cancels.
Casual-context prior (Burnett Table 2): voters at the barbecue think Obama is aloof (personae with aloof get more weight).
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Speaker at the casual context.
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Careful-context prior (Burnett Table 5): journalists think Obama is incompetent (incompetent personae get more weight).
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Speaker at the careful context.
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Rice: uniform prior — unfamiliar politician (Burnett Table 10).
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- Burnett2019.ricePrior x✝ = 1 / 4
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Listener for the Rice item (uniform prior).
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Pelosi: listeners believe she is inarticulate (Burnett Table 13).
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Listener for the Pelosi item.
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Bush: listeners almost certain he is {inarticulate, aloof} (Burnett Table 15).
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Listener for the Bush item.
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Denominators, literal-listener, and speaker values #
Bush bulletproofing: exact posterior values #
Cool-guy at the barbecue prefers -in' over -ing (~69% vs ~31%). Burnett (p. 435): "we predict that Obama will use -in' around 69% of the time [...] which is close to what Labov found" (72%).
Stern leader only uses -ing: -in' is incompatible (Table 1). This predicts ~0% -in' in formal contexts where Obama constructs the stern leader.
The doofus only uses -in': -ing is incompatible (Table 1).
In the careful context, the cool-guy now prefers -ing over -in'. The prior shift reverses the informativity ranking.
The /t/ release variable has the same mathematical structure as (ING). Relabeling: articulate↔competent, inarticulate↔incompetent (same friendly/aloof). Variants: released [tʰ]↔-ing, flapped [ɾ]↔-in'. The Eckert fields are structurally identical (Burnett example (19)): [tʰ] = {articulate, aloof}, [ɾ] = {inarticulate, friendly}.
We reuse the same types and meaning function, since the math is isomorphic. The personae reinterpret as: coolGuy ↔ {articulate, friendly}, sternLeader ↔ {articulate, aloof}, doofus ↔ {inarticulate, friendly}, asshole ↔ {inarticulate, aloof}.
The asshole prefers -in' in the casual context (both variants are compatible, but -in' is more informative given the prior).
Rice: released /t/ triggers {articulate, aloof} = stern leader (Burnett Table 11). With uniform prior, the exclusive variant (only -ing compatible) gets double the L1 weight.
Rice: flapped /t/ triggers {inarticulate, friendly} = doofus (Burnett Table 11). Symmetric to the released case.
Pelosi: released /t/ predominantly triggers {inarticulate, aloof} — the strong prior that she is inarticulate overwhelms the released /t/ association with articulateness (Burnett Table 14).
Pelosi: flapped /t/ triggers {inarticulate, friendly} (Table 14).
Bush "bulletproofing" (Burnett p. 444, Table 16): the prior is so extreme that variant choice has no practical effect. Both released and flapped /t/ yield >90% {inarticulate, aloof}.
Cross-reference: the SMG model's qualitative predictions match the directional pattern observed in [Lab12]'s data on Obama's (ING) rates. The model predicts the cool-guy persona prefers -in' in casual context and -ing in careful context; the data shows Obama's -in' rate decreasing monotonically from casual (72%) through careful (33%) to formal (3%).
Burnett's Social Meaning Game (SMG): a signalling game in which a speaker's variant choice conveys social information about their persona. The SMG reuses [Fra11]'s IBR machinery — the naive listener, strategic speaker, and uncovering listener are all instances of IBR reasoning applied to a social-meaning interpretation game.
The key design choice: toInterpGame converts any SMG into Franke's
InterpGame, so SMG agents reuse the existing IBR iteration machinery.
The grounding theorem naiveListener_eq_L0 verifies that this reuse
is semantically correct: the SMG L₀ definition produces the same
results as running Franke's L₀ on the converted game.
A Social Meaning Game (Burnett Def. 4.1): a signalling game where variant choice conveys social information.
P: persona types (social categories the listener is trying to infer)V: variant types (linguistic forms the speaker chooses)prior: probability distribution over personaemeaning: whether a variant is compatible with a persona (derived from the EM field:vmeanstiff the EM lift ofvincludes personat)socialEval: the speaker's utility μ(t, v) — how much personatvalues being associated with variantv
- prior : P → ℚ
Prior probability over personae.
Prior is non-negative.
- meaning : V → P → Bool
Semantic meaning: is variant
vcompatible with personat?
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Convert a Social Meaning Game to Franke's interpretation game.
This is the key architectural bridge: SMG analysis reuses the existing IBR machinery from [Fra11] rather than reimplementing iterated best response.
The mapping:
- States = Personae (what the listener tries to infer)
- Messages = Variants (what the speaker chooses)
- meaning = SMG meaning (EM field compatibility)
- prior = SMG prior over personae
Equations
- smg.toInterpGame = { State := P, Message := V, meaning := smg.meaning, prior := smg.prior, stateFintype := inst✝³, messageFintype := inst✝², stateDecEq := inst✝¹, messageDecEq := inst✝ }
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The naive listener (Burnett Def. 4.2): L₀(t | v) = 1/|⟦v⟧| if ⟦v⟧(t), 0 otherwise.
This is Franke's literal L₀ — uniform over compatible types, NOT
Bayesian conditioning on the prior. The prior is passed through to
toInterpGame but Franke's HearerStrategy.literal ignores it,
distributing probability uniformly over trueStates.
The Bayesian L₀ (L₀(t | v) ∝ Pr(t) · ⟦v⟧(t)) is what Burnett's
RSA model uses (eq. 11). That prior-weighted version lives in the
meaningE / L0k pipeline of §3, not here.
Equations
- Burnett2019.naiveListener smg v t = (RSA.IBR.L0 smg.toInterpGame).respond v t
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Grounding theorem: The SMG naive listener IS Franke's L₀ applied to the converted game. True by construction.
The strategic speaker (simplified): S₁(v | t) ∝ μ(t, v) · ⟦v⟧(t).
This normalizes raw social evaluation scores over compatible variants,
producing a closed-form rational speaker. This is a simplification of
Burnett's Def. 4.3 / eq. (13), which uses soft-max over log-L₀:
P_S(v | π) ∝ exp(α · ln(L₀(π | v))). The full RSA formulation with
belief-based S₁ scoring lives in the Sk speaker of §3, not here.
Unlike Franke's best-response speaker (which maximizes hearer success), the SMG speaker maximizes social utility: a persona chooses variants that make the listener more likely to infer a desirable persona.
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The uncovering listener (Burnett Def. 4.4): L₁(t | v) ∝ Pr(t) · S₁(v | t).
The listener uses Bayes' rule to infer the speaker's persona from the observed variant choice, using the strategic speaker's production probabilities as the likelihood.
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Construct a Social Meaning Game from a grounded field, prior, and social evaluation function.
The meaning function is derived from the EM field: variant v
is compatible with a persona set p iff v's indexed properties
are a subset of p's properties.
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The SMG's naiveListener is Franke's literal L₀ — uniform over
compatible personae. This captures the exclusion structure (which
personae are ruled out by each variant) but not the prior-weighted
refinement. The quantitative predictions (69% -in' for cool guy,
style shifting) use the belief-based speaker Sk of §3 (Burnett eq. 13:
P_S(m|π) ∝ L₀(π|m)^α), which incorporates the context-specific prior
into the meaning function to recover Bayesian conditioning. We
construct an SMG from the study's types and prove structural
properties.
Obama's social value function μ at the barbecue ([Bur19], Table 6, p. 438).
Cool guy ({competent, friendly}) is most valued (μ = 2); asshole ({incompetent, aloof}) is least (μ = 0). The μ function reflects what the speaker (Obama) most wants the listener to infer about him in this context.
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The (ING) Social Meaning Game for the casual context ([Bur19], Def. 4.1 + Table 2 + Table 6).
This connects the study's types to the §8 game structure,
exercising SocialMeaningGame, naiveListener, and
toInterpGame.
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The SMG meaning is grounded in the Eckert–Montague intersection
lift — connecting the game structure to the compositional
semantics layer via ingMeaning_eq_emMeaningMI.
The naive listener excludes stern leader after hearing -in' (incompatible: stern leader = {competent, aloof} shares no property with [-in'] = {incompetent, friendly}).
The naive listener excludes doofus after hearing -ing (incompatible: doofus = {incompetent, friendly} shares no property with [-ing] = {competent, aloof}).
The naive listener assigns equal probability (1/3) to all compatible personae. Franke's literal L₀ is uniform over ⟦v⟧: since 3 personae are compatible with each variant, each gets 1/3.
This is the structural content of the meaning function: each variant partitions personae into compatible (1/3) and incompatible (0).