Docs / Studies / Studies.HawkinsEtAl2025
Hawkins, R. D., Tsvilodub, P., Bergey, C. A., Goodman, N. D. & Franke, M. (2025). Philosophical Transactions of the Royal Society B: Biological Sciences, 380(1932)
The PRIOR-PQ model of Hawkins et al. 2025 explains the answers to polar questions by the questioner's decision problem. A base respondent gives a random true and safe response, one that resolves the question for anyone who knows it. The questioner asks the question whose answers most raise the value of its decision problem, and a pragmatic respondent infers the decision problem from the question and trades informativity against relevance to it.
RSA.listeners of
the base respondent and of the questioner, so a response no world produces leaves the zero
measure; the questioner and the pragmatic respondent are Kernel.softmax kernels.q entails not knowing whether r",
under which its safe example "Yes, we take Visa" to "Do you take Visa or Mastercard?" would be
unsafe. Safe takes the reading "not knowing r" that the examples force, and applies to a
mention apart from the polar answer it accompanies ("No, but we take American Express").R0 by the truth-only R0' in the questioner's update as the respondent
models it, since under R0 the update is undefined at unsafe responses. The display primes
only the informativity term; the file uses R0' for relevance too.A response r is safe for the polar question q when not knowing the answer to q entails
not knowing r, that is, when every belief state that knows r resolves q.
Used by safe_iff_mem_polar, instDecidableSafeOfFintypeOfDecidablePredMemSet, Model.trueSafe, polar_exhaustive_safe, mention_safe_iff
A response is safe exactly when it resolves the polar question, since resolution is downward closed.
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A PRIOR-PQ model gives the propositions that questions and responses denote, the questioner's decision problems, and the cost of responses.
The proposition a polar question asks about.
The proposition a response asserts.
The utility function of a decision problem.
The questioner's prior over worlds under a decision problem.
The production cost of a response.
Used by Model.trueSafe, instDecidablePredMemSetTrueSafeOfFintypeOfQuestionOfResponse, Model.R0, Model.R0', Model.R0_apply, Model.R0'_apply, instIsFiniteKernelR0, instIsFiniteKernelR0', Model.R0_apply_singleton_ne_zero_iff, Model.R0'_apply_singleton_ne_zero_iff, Model.R0_real_singleton, Model.isMarkovKernel_R0, Model.listener_R0'_apply_singleton_ne_zero_iff, Model.listener_R0_real_lt_iff, Model.questionScore, Model.questionScore_eq_valueOfInformation, Model.questioner, instIsFiniteKernelQuestioner, instIsMarkovKernelQuestionerOfNonempty, Model.questioner_apply_singleton_ne_zero, Model.goalPosterior, instIsFiniteKernelGoalPosterior, Model.goalPosterior_real_lt_iff, Model.respondentScore, Model.respondentScore_of_beta_eq_one, Model.respondent, Model.respondent_apply_singleton_ne_zero_of_beta_eq_one, Model.respondent_truthful, cards
The parameters of PRIOR-PQ are the rationalities of the policy, the questioner and the
respondent, the weight of cost, and the weight β of action relevance against informativity.
The rationality αℵ of the policy.
The rationality αQ of the questioner.
The rationality αR of the respondent.
The weight w_c of response cost.
The weight β of action relevance.
Used by Model.questionScore, Model.questionScore_eq_valueOfInformation, Model.questioner, instIsFiniteKernelQuestioner, instIsMarkovKernelQuestionerOfNonempty, Model.questioner_apply_singleton_ne_zero, Model.goalPosterior, instIsFiniteKernelGoalPosterior, Model.goalPosterior_real_lt_iff, Model.respondentScore, Model.respondentScore_of_beta_eq_one, Model.respondent, Model.respondent_apply_singleton_ne_zero_of_beta_eq_one, Model.respondent_truthful
m.trueSafe w q is the set of responses true at w and safe for q.
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Used by instDecidablePredMemSetTrueSafeOfFintypeOfQuestionOfResponse, Model.R0, Model.R0_apply, Model.R0_apply_singleton_ne_zero_iff, Model.R0_real_singleton, Model.isMarkovKernel_R0, Model.listener_R0_real_lt_iff, trueSafe_nonempty
The definition is too large to print here; read it in the source.
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The base respondent (2.1) answers q at w uniformly among the true and safe responses.
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Used by Model.R0_apply, instIsFiniteKernelR0, Model.R0_apply_singleton_ne_zero_iff, Model.R0_real_singleton, Model.isMarkovKernel_R0, Model.listener_R0_real_lt_iff, Model.questionScore, Model.questionScore_eq_valueOfInformation, instIsMarkovKernelFinsetCardRespR0ActGoalCards
The truth-only base respondent R0' (§2c) answers uniformly among the true responses.
Built from
Used by Model.R0'_apply, instIsFiniteKernelR0', Model.R0'_apply_singleton_ne_zero_iff, Model.listener_R0'_apply_singleton_ne_zero_iff, Model.respondentScore_of_beta_eq_one, yes_value_eq_exhaustive, generalYes_posterior_ne_zero
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Between two worlds of equal prior where a response is true and safe, the questioner favours the one with fewer true and safe alternatives (§2b).
Built from , ,
The policy (2.2) of a decision problem under beliefs π is a softmax over expected utility
with rationality αℵ.
Used by value
The value V(D) of a decision problem is the expected utility of following its policy.
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Used by value_eq_of_ae, Model.questionScore, Model.questionScore_eq_valueOfInformation, Model.respondentScore_of_beta_eq_one, yes_value_eq_exhaustive
The expected value to the questioner of asking q under the decision problem d (2.3) is
the value of d updated (2.4) by the base respondent's answer, less the weighted cost.
Built from , , ,
Used by Model.questionScore_eq_valueOfInformation, Model.questioner
The question score is the value of information of the question under the policy value, plus the value of the prior, less the expected cost.
Built from , , , ,
The questioner (2.3) soft-maximizes the question score with rationality αQ.
Built from , ,
Used by instIsFiniteKernelQuestioner, instIsMarkovKernelQuestionerOfNonempty, Model.questioner_apply_singleton_ne_zero, Model.goalPosterior, Model.goalPosterior_real_lt_iff
Built from , ,
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The respondent's belief about the decision problem after hearing q is that of a listener
to the questioner at the prior πD, π(D ∣ q) ∝ Q(q ∣ D) π(D).
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Used by instIsFiniteKernelGoalPosterior, Model.goalPosterior_real_lt_iff, Model.respondentScore_of_beta_eq_one
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Between decision problems of equal prior, the respondent comes to favour the one under which the question was the likelier, so a question signals the goal.
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The pragmatic respondent's score for answering q with r (2.5) is the expectation over the
inferred decision problem of informativity, weighted by 1 - β, and action relevance, weighted
by β, less the weighted cost. Both terms read the questioner's belief after the truth-only
update, and informativity is its negative divergence from the respondent's belief πW. The
expected divergence is a lower integral, so a response at infinite divergence scores ⊥.
The definition is too large to print here; read it in the source.
Built from ,
Used by Model.respondentScore_of_beta_eq_one, Model.respondent
Built from , , , , ,
The pragmatic respondent (2.5) soft-maximizes its score with rationality αR.
Built from , ,
Used by Model.respondent_apply_singleton_ne_zero_of_beta_eq_one, Model.respondent_truthful
Action relevance alone does not enforce truthfulness (§2c), since at β = 1 the respondent
gives every response, false ones included, positive probability.
Built from , ,
When β < 1, a respondent never gives a response false at a world it considers possible (§2c).
Built from , ,
A card is one of the three credit cards of §3a.
Derives DecidableEq, Fintype, Repr.
Used by Resp, instMeasurableSpaceFinsetCard, instDiscreteMeasurableSpaceFinsetCard, Goal, cardUtility, Resp.denote, instDecidablePredFinsetCardMemSetDenote, cards, instDecidablePredFinsetCardMemSetQuestionRespActGoalCards, instDecidablePredFinsetCardMemSetResponseRespActGoalCards, polar_exhaustive_safe, mention_safe_iff, trueSafe_nonempty, instIsMarkovKernelFinsetCardRespR0ActGoalCards, yes_value_eq_exhaustive
The questioner either stays or goes.
Derives DecidableEq, Fintype, Repr, Inhabited.
Used by cardUtility, cards
A response is a polar answer to whether any card of S is accepted, the mention of some
accepted cards, or the exhaustive list of the accepted cards.
Derives DecidableEq, Fintype.
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Used by instMeasurableSpaceResp, Resp.denote, instDecidablePredFinsetCardMemSetDenote, cards, instDecidablePredFinsetCardMemSetResponseRespActGoalCards
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The decision problem U1 turns on whether any of the questioner's cards C is accepted,
and U2 on whether any card is accepted.
Derives DecidableEq, Fintype.
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Used by cardUtility, cards, instIsProbabilityMeasureFinsetCardPriorRespActGoalCards
The utility of §3a is 5 for going when a relevant card is accepted or staying when none is, and 0 otherwise.
Built from , ,
r.denote is the set of worlds at which the response r is true.
Built from ,
The definition is too large to print here; read it in the source.
Built from , ,
In the credit-card model, a question asks whether any card of a set is accepted, worlds are the sets of accepted cards, priors are uniform over the eight worlds, and a response costs the cards it mentions.
The definition is too large to print here; read it in the source.
Built from , , , ,
Used by instDecidablePredFinsetCardMemSetQuestionRespActGoalCards, instDecidablePredFinsetCardMemSetResponseRespActGoalCards, instIsProbabilityMeasureFinsetCardPriorRespActGoalCards, polar_exhaustive_safe, mention_safe_iff, trueSafe_nonempty, instIsMarkovKernelFinsetCardRespR0ActGoalCards, yes_value_eq_exhaustive, generalYes_posterior_ne_zero
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Mentioning accepted cards is safe iff one of them was asked about ((1), (2), Examples.ex1,
Examples.ex2).
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In (3) (Examples.ex3), for a questioner holding the card asked about, yes has the same
decision value as the exhaustive list.
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In (5) (Examples.ex5), after yes to the general question a world accepting only MasterCard
keeps positive posterior, but not after yes about American Express.
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