Jäger (2014): Rationalizable Signaling #
Jäger models pragmatic interpretation in a semantic game, a signaling game whose signals have
literal meanings and whose players may be unsure of each other's preferences. Starting from the
credulous receiver, each player in turn plays a best response to some belief that gives every
strategy the other might currently play positive weight, an unexpected signal being read as
true; the strategies that recur arbitrarily late are pragmatically rationalizable, the notion
Jäger and Ebert also develop, and they are rationalizable in the classical sense
(prs_rationalizable). Worked examples derive the scalar implicature of "some", Horn's division
of pragmatic labor, the context-dependent precision of round numbers and the breakdown of
communication under opposed interests, and contrast the model with Franke's iterated best
response.
Implementation notes #
- Beliefs are points of mathlib's
Convexity.StdSimplex; the paper's distributions over a setPof strategies, all of them or only those positive on all ofP, are the beliefs whose support is contained in, respectively equal to,P. - The iterated cautious response sequence is an iterate of
receiverStepand the pragmatically rationalizable strategies are its limit superior;mem_prsR_iffis the paper's definition, which printssforrin its receiver half. - A signal or an action that some alternative weakly dominates against every strategy the other
player might play is never a cautious response (
senderCR_subset,receiverCR_subset). This is the sound half of the note on computing the sequence, which goes on to compute cautious responses separately at each context and world. Since one belief serves all of them, that computation can overcount: with one context, receiversf ↦ a₁, f' ↦ a₂andf ↦ a₂, f' ↦ a₁, the sender wantinga₁inw₁anda₂inw₂, andf'costing0.2, sendingf'is optimal inw₁only if the first receiver has weight at most0.4and inw₂only if it has weight at least0.6. The stages displayed for Example 5 overcount in this way. With one context, a sender strategy that no mixture weakly dominates in ex-ante payoff is a cautious response (mem_senderCR_of_not_weaklyDominated), by Pearce's lemma. - The examples are set in
Fin n, thek-th world, signal or action of the paper beingk - 1; each example's docstring names its signals. - Section 3 asserts that under the right panel of Table 4 alone Sally would use "some but not
all" when not all is true and never use "some", so that the implicature needs the other
context. Under the definitions the unused "some" may be read as not-all after a belief revision,
and two rounds later it carries the implicature
(
NoImplicatureContext.some_still_implicates_not_all). Table 8 swaps Table 4's context labels. - The recurrence sets displayed for Example 2 are products, which by the paper's convention for
such tables contain every strategy; the sequence cycles through two stages and the pragmatically
rationalizable strategies are their union (
Opposed.prs_eq). - The stages displayed for Example 8 move away from the credulous ones, but the receiver's belief
about the sender's context has full support, so "100 m" stays read as
100 mand the sequence never moves; "exactly 100 m" is never used (Precision.exactly_never_used). The paper's verbal conclusions hold. - Table 14 labels the last sender set of its left panel
S₀; it isS₁. - Example 10's "some but not all" costs a parameter
cwith0 < c < 1, the paper saying only that the cost is small. Franke's iterated best response is his costless light system (Franke2011.receiverChain), which agrees with the paper's costly variant whenc < 1/2.
TODO #
- Examples 1, 3, 5 and 7, the uniform-prior variant of Example 6 in Section 6, and the notions of
credibility of Section 7. Example 7 contrasts the model with weak bidirectional optimality,
which the library has in
Pragmatics/Bidirectional.lean.
References #
Beliefs #
Recurrence in a periodic sequence of sets #
Semantic games #
A semantic game has contexts C, which carry each player's uncertainty about the other's
preferences, worlds W, signals F and actions A. It has a full-support prior over worlds, a
literal meaning for each signal, receiver utilities, and sender utilities that subtract a signal
cost from an outcome utility.
- prior : Convexity.StdSimplex ℝ W
The receiver's prior over worlds.
- prior_support : self.prior.weights.support = Finset.univ
Every world has positive prior probability.
- meaning : F → W → Prop
Whether signal
fis literally true at worldw. - vS : C → W → A → ℝ
The sender's utility of an outcome, by her context, the world and the receiver's action.
- cost : F → ℝ
The cost of sending a signal.
- uR : C → W → A → ℝ
The receiver's utility, by his context, the world and his action.
- meaningDecidable (f : F) : DecidablePred (self.meaning f)
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The sender's utility, the outcome utility less the signal's cost.
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The worlds at which a signal is literally true.
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The actions optimal for the receiver in context c when he updates the belief p with the
proposition φ (Definition 1).
Equations
- g.optimalActions c φ p = Finset.univ.argmax fun (a : A) => ∑ w ∈ φ, p.weights w * g.uR c w a
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Best responses and cautious responses #
A sender strategy maps a context and a world to a signal, a receiver strategy a context and a signal to an action. A player's belief combines a belief about the other's strategy with one about the other's context.
The receiver's best responses to a belief about the sender's strategy and context maximize expected utility under the prior in each of his contexts (Definition 2).
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The sender's best responses to a belief about the receiver's strategy and context maximize expected utility in each context and world (Definition 2).
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The sender's cautious responses to a set of receiver strategies, the best responses to a belief whose support is exactly that set, with every receiver context possible (Definition 3).
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The receiver's cautious responses to a set of sender strategies (Definition 3).
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The iterated cautious response sequence #
A signal is unexpected under a set of sender strategies if none of them ever sends it.
Equations
- Jaeger2014.SemanticGame.Unexpected S f = (f ∉ ⋃ s ∈ S, Set.range (Function.uncurry s))
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The credulous receiver strategies, optimal under the prior updated with each signal's literal meaning.
Equations
- g.credulous = {r : C → F → A | ∀ (c : C) (f : F), r c f ∈ g.optimalActions c (g.extension f) g.prior}
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One step of the sequence keeps the receiver's cautious responses to the sender's cautious
responses to R that read every unexpected signal as literally true under some full-support
revised belief.
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The receiver stages of the iterated cautious response sequence (Definition 4).
Equations
- g.icrR n = g.receiverStep^[n] g.credulous
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The sender stages, the cautious responses to the receiver stages (Definition 4).
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The pragmatically rationalizable sender strategies, those recurring arbitrarily late in the sequence (Definition 5).
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The pragmatically rationalizable receiver strategies (Definition 5).
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Once the sequence cycles, the pragmatically rationalizable receiver strategies are the union of the stages of one period.
Once the sequence cycles, the pragmatically rationalizable sender strategies are the union of the stages of one period.
Rationalizability #
A strategy pair is rationalizable if it lies in a pair of sets each of whose members is a best response to some belief supported inside the other set (Definition 6, after Osborne).
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Pragmatically rationalizable strategy pairs are rationalizable (Theorem 1). The sequence runs on the finitely many sets of strategies, so it is eventually periodic, and the recurrence sets themselves witness rationalizability, since a recurring strategy responds cautiously to a late stage of the other player, which lies inside the other recurrence set.
Best responses, pointwise #
Definition 2 maximizes over whole strategies, but the objectives separate: the sender's at a context and world depends only on the signal sent there, and the receiver's is a sum over signals.
The sender's expected utility of a signal in a context and world.
Equations
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The receiver's expected utility of an action on a signal, counting the occasions on which the signal is sent.
Equations
- g.receiverEU σ q c f a = ∑ s : C → W → F, σ.weights s * ∑ c' : C, q.weights c' * ∑ w : W, g.prior.weights w * if s c' w = f then g.uR c w a else 0
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Dominance #
A signal that an alternative weakly dominates against every receiver strategy and context the sender considers possible, strictly against one, earns less under every full-support belief, so it is never a cautious response; symmetrically for the receiver's actions, compared at the worlds where the signal is sent.
A signal outside T c w that some signal weakly dominates against R, strictly against one
member, is never a cautious response there.
A strategy whose every choice weakly dominates against a nonempty set responds cautiously to it.
The ex-ante payoff of a sender strategy against a receiver strategy in a context is the sender's utility at each world, weighted by the prior.
Equations
- g.senderPayoff c s r = ∑ w : W, g.prior.weights w * g.uS c w (s c w) (r c (s c w))
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With one context, a sender strategy that no mixture of strategies weakly dominates in ex-ante
payoff against a nonempty set of receiver strategies responds cautiously to it, the converse of
senderCR_subset. Pearce's lemma for weak dominance gives a belief of full support on the set
under which the strategy is optimal, and the objectives separate across worlds.
An action outside T c f that some action weakly dominates at the worlds where f is sent,
strictly at one, is never a cautious response there.
A receiver strategy whose every action weakly dominates where its signal is sent responds cautiously to a nonempty set of sender strategies.
When the signals of T c w weakly dominate every signal against R and each other signal is
strictly beaten by one of them, the cautious responses to R choose from T.
When the actions of T c f weakly dominate every action where f is sent and each other
action is strictly beaten by one of them, the cautious responses to S choose from T.
A signal true at a single world is read, under any full-support belief, as that world's best action.
Examples #
The data rows annotated with a signal and a world of a model, as pairs of the model's indices, looked up by the paper's names.
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The literal meanings of the paper's three signals over two worlds, f₁ (0) true at w₁
(0), f₂ (1) at w₂ (1) and f₁₂ (2) at both.
Equations
- Jaeger2014.literalMeaning f w = (f = 2 ∨ ↑f = ↑w)
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Equations
- Jaeger2014.instDecidablePredFinOfNatNatLiteralMeaning f x✝ = id inferInstance
The uniform prior over two worlds.
Equations
- Jaeger2014.uniform = Convexity.StdSimplex.duple 0 1 Jaeger2014.uniform._proof_1 Jaeger2014.uniform._proof_1 Jaeger2014.uniform._proof_2
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The scalar games of Section 3 and Example 4 #
In the scalar games of Section 3 the speaker says "all" (0), "some but not all" (1, which
costs 2) or "some" (2), in the world where all is true (0) or the other (1), and the
hearer acts on either world or hedges (2), which pays hedge c in context c. Both players
have the same payoffs.
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Example 2: opposed interests #
Example 2 pairs the opposed utilities of Table 6 with Example 1's signals f₁, f₂, f₁₂.
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With opposed interests (Example 2) the sequence cycles with period two. The pragmatically
rationalizable senders are the liar and the truth-teller, and the receivers are those reading f₁
and f₂ differently, f₁₂ either way.
Section 3: the no-implicature context alone #
The right panel of Table 4 on its own, where the hedge pays 6.
Equations
- Jaeger2014.NoImplicatureContext.game = Jaeger2014.scalarGame fun (x : Unit) => 6
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Contrary to the claim on p. 683, the no-implicature utilities alone yield the scalar implicature. The pragmatically rationalizable sender uses "some" exactly when not all is true and never uses "some but not all".
Example 4: the scalar implicature across two contexts #
Example 4 has two contexts, the hedge paying 6 in 0 and 9 in 1 (Table 8).
Equations
- Jaeger2014.Scalar.game = Jaeger2014.scalarGame fun (c : Fin 2) => if c = 0 then 6 else 9
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In Example 4, whatever the context, the pragmatically rationalizable sender uses "some" exactly when not all is true and never the costlier "some but not all", and every pragmatically rationalizable receiver reads "some" as "not all".
Every pragmatically rationalizable receiver reads the paper's "Some boys came in." as "not all", in either context.
Example 6: Horn's division of pragmatic labor #
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Horn's division of pragmatic labor (Example 6). The pragmatically rationalizable strategies are the convention on which the cheap form marks the frequent world and the costly form the rare one.
Every pragmatically rationalizable receiver reads the two sentences about stopping the car as the paper reports, the plain one as the regular way and the periphrastic one as an abnormal way.
Example 8: precision of number words #
In Example 8 the distance is 100 m in w₁ (0) and 101 m in w₂ (1). The signals are
"100 m" (0), "exactly 100 m" (1) and "101 m" (2), the last two costing 9/2, and each
action suits one world, paying 10 when precision matters (context 0) and 4 otherwise
(Table 12).
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In Example 8 the sequence never leaves the credulous receiver. The pragmatically rationalizable sender says "101 m" only when precision matters and otherwise "100 m", and the receiver reads every signal literally.
The round number is pragmatically ambiguous, precise when precision matters and vague when it does not.
Contrary to the stages displayed for Example 8, "exactly 100 m" is never used.
Example 10 and Section 6: cautious against best response #
In Example 10 "some" (0) is true in both worlds, "all" (1) in the world 0 where all is
true and "some but not all" (2) in the other world 1.
Equations
- Jaeger2014.SomeAll.meaning f w = (f = 0 ∨ ↑f = ↑w + 1)
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Equations
- Jaeger2014.SomeAll.instDecidablePredFinOfNatNatMeaning f x✝ = id inferInstance
Example 10's game (Table 13), with "some but not all" costing c.
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The denotations of the signals, which are the literal receiver of the iterated best response of [Fra11].
Equations
- Jaeger2014.SomeAll.denotation f = Finset.filter (Jaeger2014.SomeAll.meaning f) Finset.univ
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Under iterated best response the literal receiver is already a fixed point, in which no world sends "some" and "some" keeps both readings.
Under iterated cautious response, in Example 10, the pragmatically rationalizable receiver reads "some" as "not all", and the sender uses "some" when not all is true.