Burnett (2019): Signalling games, sociolinguistic variation and the construction of style #
This file formalizes Burnett's social meaning games. A speaker choosing between -ing and -in' conveys something about the persona they are constructing, and the paper takes that inference to be the equilibrium of a signalling game between a rational speaker and listener: each variant is compatible with the personae sharing a property with its indexical field, the listener infers a persona from the variant against a prior, and the speaker chooses the variant that best conveys the persona they are after. Two kinds of variation come out of the one model. A speaker holding a persona fixed changes variants as the context changes the listener's prior, style shifting, here Labov's Obama who prefers -in' at a barbecue and -ing with the journalists; and a listener holding the variant fixed infers different personae from different speakers, so that a strongly stereotyped speaker such as the paper's Bush conveys nothing by the choice.
The property space is the example's two dimensions, competence and warmth, and the four
personae of example (6) are its maximal consistent sets (personae_eq). The meaning of a
variant is the lift of its indexical field to persona compatibility, and every prediction is an
instance of the game's speaker preferring the more informative variant (prefers_iff).
Implementation notes #
The contexts are priors over personae given as integer weights, read off the paper's tables as the text describes them, more mass on the aloof personae at the barbecue and on the incompetent ones with the journalists. The speaker is the softmax of the literal listener at the paper's rationality 6 with no costs, and the pragmatic listener the Bayesian posterior against the same prior; no normaliser is evaluated, preferences being compared with the pipeline's lemmas, which cancel them. The persona selection of Section 4.3, a softmax over the speaker's values of the personae and the marginal rate of a variant, is not represented.
References #
Personae and variants #
The social properties of example (5), two poles of competence and two of warmth.
- competent : PersonaTrait
- incompetent : PersonaTrait
- friendly : PersonaTrait
- aloof : PersonaTrait
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Equations
- Burnett2019.instDecidableEqPersonaTrait x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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- One or more equations did not get rendered due to their size.
The dimension a trait is a pole of.
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- Burnett2019.PersonaTrait.competent.dimension = SocialMeaning.Dimension.competence
- Burnett2019.PersonaTrait.incompetent.dimension = SocialMeaning.Dimension.competence
- Burnett2019.PersonaTrait.friendly.dimension = SocialMeaning.Dimension.warmth
- Burnett2019.PersonaTrait.aloof.dimension = SocialMeaning.Dimension.warmth
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A persona of the example is a maximal consistent set of its properties, the two poles of each dimension being incompatible.
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The cool guy, competent and friendly.
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The stern leader, competent and aloof.
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The doofus, incompetent and friendly.
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The arrogant asshole, incompetent and aloof.
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The four personae of example (6) are the maximal consistent sets of the property space.
The indexical fields of example (10) have -ing index competence and aloofness and -in' incompetence and friendliness.
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The grounded field of (ING), both fields being consistent.
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- Burnett2019.ingField = { indexes := Burnett2019.ingEckertField, isIndepSet := Burnett2019.ingField._proof_1 }
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The model #
The literal listener conditions the persona prior on the personae the variant meets, the speaker is its softmax at the paper's rationality 6 with no costs, and the pragmatic listener inverts the speaker against the same prior.
The literal listener conditions the prior on the variant's Eckert–Montague field.
Equations
- Burnett2019.L0 prior = RSA.listener prior (RSA.truthful fun (v : Eckert2008.INGVariant) => ↑(Burnett2019.ingField.personae v))
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The speaker, the softmax of the literal listener at rationality 6 (p. 435), without costs.
Equations
- Burnett2019.S1 prior = RSA.speaker 6 (fun (x : Eckert2008.INGVariant) => 0) (⇑(Burnett2019.L0 prior)) MeasureTheory.Measure.dirac
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The pragmatic listener inverts the speaker against the prior.
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- Burnett2019.L1 prior = RSA.listener prior ⇑(Burnett2019.S1 prior)
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The extensions differ in one persona each #
-ing meets every persona but the doofus and -in' every persona but the stern leader, so the two extensions share the cool guy and the asshole and differ exactly in those two. That is the whole of what the speaker's choice turns on.
The persona each variant rules out, the doofus for -ing and the stern leader for -in'.
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The extensions share the cool guy and the asshole, so one outweighs the other exactly when the persona it keeps, the other's excluded one, outweighs the persona it rules out.
For a persona either variant can convey, the speaker prefers the variant that rules out more prior mass, the more informative one. Since the two variants differ only in ruling out the doofus and the stern leader, the choice is settled by which of those two the context finds likelier.
The contexts #
A context is a prior over personae, given as integer weights read off the paper's tables as the text describes them.
At the barbecue the voters take Obama to be aloof (Table 2), so the aloof personae carry more mass.
Equations
- Burnett2019.casualWeight π = if Burnett2019.PersonaTrait.aloof ∈ ↑π then 3 else 2
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With the journalists he is taken to be incompetent (Table 5).
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- Burnett2019.carefulWeight π = if Burnett2019.PersonaTrait.incompetent ∈ ↑π then 3 else 2
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Rice is unfamiliar, so the listener's beliefs are uniform (Table 10).
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- Burnett2019.riceWeight x✝ = 1
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Pelosi is taken to be inarticulate (Table 13).
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- Burnett2019.pelosiWeight π = if Burnett2019.PersonaTrait.incompetent ∈ ↑π then 9 else 1
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Bush is taken to be inarticulate and aloof, almost to certainty (Table 15).
Equations
- Burnett2019.bushWeight π = if Burnett2019.PersonaTrait.incompetent ∈ ↑π ∧ Burnett2019.PersonaTrait.aloof ∈ ↑π then 97 else 1
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Variant choice #
Every prediction about the speaker is an instance of prefers_iff.
At the barbecue the stern leader outweighs the doofus, so -in', which rules the stern leader out, is the more informative variant, and a persona either variant can convey is conveyed by it. That is Obama's cool guy, predicted to use -in' about 69% of the time (p. 435).
The asshole, also conveyable either way, goes the same way at the barbecue.
Style shifting arises because with the journalists the doofus outweighs the stern leader instead, so -ing is now the more informative variant and the same cool guy prefers it. Neither the speaker nor the meaning has changed, only the context's prior, and with it which variant rules more out.
Bulletproofing arises because Bush's listeners are almost certain he is inarticulate and aloof, and the two personae the variants distinguish carry the same small weight, so neither variant rules out more than the other, the speaker is indifferent, and variant choice conveys nothing at all (pp. 444–445).
The same holds of Rice, whose listeners have no prior beliefs to speak of.
The predicted direction is the observed one, the cool guy taking -in' at the barbecue and -ing with the journalists while Obama's rate of -in' falls from the casual through the careful to the formal style ([Lab12]).
Interpretation #
What the listener does with a variant is the posterior over personae. A persona only one variant can convey produces it with certainty, while a persona either can convey splits its production between them, so the exclusive persona wins the posterior whenever the prior does not favour the other. That is the shape of the paper's interpretation results: a released /t/ points at the stern leader, a flapped one at the doofus.
The stern leader can only be conveyed by -ing and the doofus only by -in', so each is produced with certainty by the persona it is exclusive to.
A variant gives no posterior mass to a persona it cannot convey, so hearing -ing rules out the doofus and hearing -in' rules out the stern leader, whatever the listener believed beforehand.
With no prior beliefs the literal listener spreads its mass evenly over the three personae the variant can convey, the game-theoretic literal listener of Definition 4.1.