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arxiv: 1907.09115 · v1 · pith:M2AHPROCnew · submitted 2019-07-22 · 💻 cs.GT · cs.AI

Measuring Belief and Risk Attitude

Sven Neth (University of California , Berkeley) This is my paper

Pith reviewed 2026-05-24 18:02 UTC · model grok-4.3

classification 💻 cs.GT cs.AI
keywords risk-weighted expected utilitysubjective probabilitybelief measurementrisk attitudeRamsey methoddecision theoryobservable preferencesBuchak
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The pith

Observable preferences reveal both risk attitudes and subjective probabilities for risk-weighted expected utility maximizers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Ramsey's 1926 approach, which extracted subjective probabilities from preferences under the assumption of expected utility maximization, to agents who instead maximize risk-weighted expected utility as defined by Buchak. It first shows how to recover the agent's risk-weighting function directly from choices over lotteries or bets, then uses that function to back out the agent's beliefs about event probabilities. A sympathetic reader would care because many observed decisions violate the independence axiom of standard expected utility theory, so a method that works without that axiom broadens the range of agents whose beliefs can be measured from behavior alone.

Core claim

Assuming an agent is a risk-weighted expected utility maximizer, their risk attitude can be measured from observable preferences over acts, and this measurement can then be used to determine the agent's subjective probabilities over states, extending the spirit of Ramsey's original proposal to a strictly wider class of agents.

What carries the argument

Risk-weighted expected utility representation, in which a risk function distorts the cumulative probabilities attached to ranked outcomes before they enter the expectation.

If this is right

  • Risk attitudes become measurable without assuming the independence axiom of expected utility.
  • Subjective probabilities can be recovered for agents whose choices violate sure-thing or independence principles.
  • Belief-elicitation procedures apply directly to a larger set of decision makers in economic and psychological experiments.
  • The same preference data can separate the contribution of risk attitude from the contribution of belief.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be used to compare risk functions across different populations or contexts once the representation is assumed.
  • In settings where agents are modeled computationally, the procedure supplies a route from choice data to separate estimates of belief and risk attitude.
  • If the risk function turns out to be stable across domains, the approach would support portable measurement of beliefs in field data.

Load-bearing premise

The agent is a risk-weighted expected utility maximizer.

What would settle it

A set of observed preferences that cannot be represented by any risk function and utility function under the risk-weighted expected utility formula, or a case where the derived probabilities produce inconsistent predictions when tested against new choices.

read the original abstract

Ramsey (1926) sketches a proposal for measuring the subjective probabilities of an agent by their observable preferences, assuming that the agent is an expected utility maximizer. I show how to extend the spirit of Ramsey's method to a strictly wider class of agents: risk-weighted expected utility maximizers (Buchak 2013). In particular, I show how we can measure the risk attitudes of an agent by their observable preferences, assuming that the agent is a risk-weighted expected utility maximizer. Further, we can leverage this method to measure the subjective probabilities of a risk-weighted expected utility maximizer.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends Ramsey's (1926) preference-based elicitation of subjective probabilities—originally for expected-utility maximizers—to agents whose preferences satisfy Buchak's (2013) risk-weighted expected utility (RWEU) representation. It first recovers the agent's risk function r(·) directly from observable preferences under the maintained RWEU hypothesis, then uses the recovered r to elicit the subjective probability measure P.

Significance. If the derivation holds, the result meaningfully widens the class of agents for whom both risk attitudes and beliefs can be measured from choice data without introducing auxiliary functional-form assumptions or free parameters beyond the RWEU axioms themselves. This is a clean theoretical contribution to the literature on revealed-preference elicitation.

major comments (2)
  1. [§4] §4, the step that recovers r from indifferences between compound lotteries: the argument assumes that the relevant acts can be ranked in a way that isolates r without circular reference to P, but the manuscript does not explicitly verify that the constructed ranking is independent of the unknown P for all admissible r (including non-strictly-increasing cases).
  2. [Theorem 2] Theorem 2 (elicitation of P): the uniqueness claim for P appears to rest on the already-recovered r being fixed; it would be useful to see an explicit statement that the procedure remains well-defined when the agent's r is only identified up to the equivalence class permitted by the RWEU representation theorem.
minor comments (2)
  1. [Abstract / §1] The abstract and introduction use 'risk attitudes' and 'risk function' interchangeably; a brief clarification of the distinction (especially for readers unfamiliar with Buchak) would help.
  2. [§2] Notation for the weighting function r is introduced in §2 but not contrasted with the more common 'probability weighting' terminology in prospect theory; a one-sentence remark would reduce potential confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the contribution, and the constructive suggestions for improving clarity. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4, the step that recovers r from indifferences between compound lotteries: the argument assumes that the relevant acts can be ranked in a way that isolates r without circular reference to P, but the manuscript does not explicitly verify that the constructed ranking is independent of the unknown P for all admissible r (including non-strictly-increasing cases).

    Authors: We appreciate the referee highlighting the need for explicit verification. The construction in §4 uses compound lotteries with objective probabilities that are known and fixed, so the ranking of these acts depends only on the values of r at those fixed points and is independent of the unknown subjective P by construction. To make this fully rigorous and to cover all admissible r (including any non-strictly-increasing cases permitted by the maintained axioms), we will add a short lemma or remark in the revised §4 that explicitly confirms the independence from P. revision: yes

  2. Referee: [Theorem 2] Theorem 2 (elicitation of P): the uniqueness claim for P appears to rest on the already-recovered r being fixed; it would be useful to see an explicit statement that the procedure remains well-defined when the agent's r is only identified up to the equivalence class permitted by the RWEU representation theorem.

    Authors: We agree that an explicit statement would improve the presentation. The RWEU representation theorem identifies the risk function r only up to the equivalence class consistent with the axioms (typically preserving the ordering of weighted sums). Our elicitation procedure for P in Theorem 2 is invariant under this equivalence, because any two equivalent r functions yield the same ranking of the relevant acts once P is recovered. We will add a clarifying remark immediately after the statement of Theorem 2 noting that the procedure remains well-defined for any representative of the equivalence class. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation extends Ramsey's elicitation procedure to agents satisfying the RWEU representation theorem from Buchak (2013), an external citation with no author overlap. The method first recovers the risk function r from observable preferences under the maintained RWEU hypothesis, then recovers the subjective probability measure P. Both steps are presented as direct consequences of the representation theorem and the observable preference data; no parameter is fitted to a subset and then relabeled as a prediction, no self-citation chain is load-bearing, and the RWEU assumption is explicitly stated rather than smuggled in. The construction therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that agents maximize risk-weighted expected utility; no free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The agent is a risk-weighted expected utility maximizer as defined in Buchak (2013).
    This is explicitly required to extend Ramsey's measurement procedure to the wider class.

pith-pipeline@v0.9.0 · 5613 in / 1065 out tokens · 22452 ms · 2026-05-24T18:02:59.173404+00:00 · methodology

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Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Springer, doi:10.1007/978-1-4939-2712-8

    Stephen Abbott (2001): Understanding Analysis. Springer, doi:10.1007/978-1-4939-2712-8

    work page doi:10.1007/978-1-4939-2712-8 2001

  2. [2]

    Econometrica 21(4), pp

    Maurice Allais (1953): Le Comportement de l’Homme Rationnel devant le Risque: Crit ique des Postulats et Axiomes de l’Ecole Americaine . Econometrica 21(4), pp. 503–546, doi:10.2307/1907921

    work page doi:10.2307/1907921 1953

  3. [3]

    Scale-Free Networks: Complex Webs in Nature and Technology

    Jos´ e Luis Berm´ udez (2011): Decision Theory and Rationality . Oxford University Press, doi:10.1093/acprof:oso/9780199548026.001.0001

    work page doi:10.1093/acprof:oso/9780199548026.001.0001 2011

  4. [4]

    Wiley Series in Probability and Mathematical Statistic, Wiley

    Patrick Billingsley (1995): Probability and Measure, 3 edition. Wiley Series in Probability and Mathematical Statistic, Wiley. 364 Measuring Belief and Risk Attitude

    work page 1995

  5. [5]

    Dialectica 58(4), pp

    Richard Bradley (2004): Ramsey’s Representation Theorem . Dialectica 58(4), pp. 483–97, doi:10.1111/j.1746-8361.2004.tb00320.x

    work page doi:10.1111/j.1746-8361.2004.tb00320.x 2004

  6. [6]

    Scale-Free Networks: Complex Webs in Nature and Technology

    Lara Buchak (2013): Risk and Rationality . Oxford University Press, doi:10.1093/acprof:oso/9780199672165.001.0001

    work page doi:10.1093/acprof:oso/9780199672165.001.0001 2013

  7. [7]

    In Christopher Hitchcock & Alan Hajek, edi- tors: Oxford Handbook of Probability and Philosophy , Oxford University Press, pp

    Lara Buchak (2016): Decision Theory . In Christopher Hitchcock & Alan Hajek, edi- tors: Oxford Handbook of Probability and Philosophy , Oxford University Press, pp. 789–814, doi:10.1093/oxfordhb/9780199607617.013.40

    work page doi:10.1093/oxfordhb/9780199607617.013.40 2016

  8. [8]

    Ethics 127(3), pp

    Lara Buchak (2017): T aking Risks Behind the V eil of Ignorance . Ethics 127(3), pp. 610–44, doi:10.1086/690070

    work page doi:10.1086/690070 2017

  9. [9]

    Dialectica 27(1), pp

    Donald Davidson (1973): Radical Interpretation . Dialectica 27(1), pp. 314–28, doi:10.1111/j.1746-8361.1973.tb00623.x

    work page doi:10.1111/j.1746-8361.1973.tb00623.x 1973

  10. [10]

    Annales de l’Institut Henri Poincar´ e17, pp

    Bruno De Finetti (1937): La Pr ´evision: Ses Lois Logiques, Ses Sources Subjectives . Annales de l’Institut Henri Poincar´ e17, pp. 1–68

    work page 1937

  11. [11]

    Mind 126(501), pp

    Edward Elliott (2017): Ramsey Without Ethical Neutrality: A New Representation Th eorem. Mind 126(501), pp. 1–51, doi:10.1093/mind/fzv180

    work page doi:10.1093/mind/fzv180 2017

  12. [12]

    Fishburn (1981): Subjective Expected Utility: A Review of Normative Theories

    Peter C. Fishburn (1981): Subjective Expected Utility: A Review of Normative Theories. Theory and Decision 13(2), pp. 139–99, doi:10.1007/BF00134215

    work page doi:10.1007/bf00134215 1981

  13. [13]

    University of Chicago Press, doi:10.2307/2183 328

    Richard Jeffrey (1990): The Logic of Decision, 2 edition. University of Chicago Press, doi:10.2307/2183 328

    work page doi:10.2307/2183 1990

  14. [14]

    Wakker (2003): Preference F oundations for Nonexpected Utility: A Generalized and Simplified T echnique

    V eronika K¨ obberling & Peter P . Wakker (2003): Preference F oundations for Nonexpected Utility: A Generalized and Simplified T echnique . Mathematics of Operations Research 28(3), pp. 395–423, doi:10.1287/moor.28.3.395.16390

    work page doi:10.1287/moor.28.3.395.16390 2003

  15. [15]

    Koopman (1940): The Bases of Probability

    Bernard O. Koopman (1940): The Bases of Probability . American Mathematical Society 46, pp. 763–74, doi:10.1090/S0002-9904-1940-07294-5

    work page doi:10.1090/s0002-9904-1940-07294-5 1940

  16. [16]

    Synthese 27(3-4), pp

    David Lewis (1974): Radical Interpretation. Synthese 27(3-4), pp. 331–44, doi:10.1007/BF00484599

    work page doi:10.1007/bf00484599 1974

  17. [17]

    Machina (1987): Choice under Uncertainty: Problems Solved and Unsolved

    Mark J. Machina (1987): Choice under Uncertainty: Problems Solved and Unsolved . Journal of Economic Perspectives 1(1), pp. 121–54, doi:10.1257/jep.1.1.121

    work page doi:10.1257/jep.1.1.121 1987

  18. [18]

    Machina & David Schmeidler (1992): A More Robust Definition of Subjective Probability

    Mark J. Machina & David Schmeidler (1992): A More Robust Definition of Subjective Probability . Econo- metrica 60(4), pp. 745–80, doi:10.2307/2951565

    work page doi:10.2307/2951565 1992

  19. [19]

    Journal of Philosophy 104(5), pp

    Samir Okasha (2007): Rational Choice, Risk Aversion, And Evolution . Journal of Philosophy 104(5), pp. 217–35, doi:10.5840/jphil2007104523

    work page doi:10.5840/jphil2007104523 2007

  20. [20]

    Journal of Economic Psychology 24(1), pp

    Adam Oliver (2003): A quantitative and qualitative test of the Allais paradox us ing health outcomes. Journal of Economic Psychology 24(1), pp. 35–48, doi:10.1016/S0167-4870(02)00153-8

    work page doi:10.1016/s0167-4870(02)00153-8 2003

  21. [21]

    Giovanni Parmigiani & Lurdes Y . T. Inoue (2009): Decision Theory: Principles and Approaches . Wiley Series in Probability and Mathematical Statistic, Wiley, d oi:10.1002/9780470746684

    work page doi:10.1002/9780470746684 2009

  22. [22]

    Journal of Economic Behavior and Organization 3(1), pp

    John Quiggin (1983): A Theory of Anticipated Utility . Journal of Economic Behavior and Organization 3(1), pp. 323–43, doi:10.1016/0167-2681(82)90008-7

    work page doi:10.1016/0167-2681(82)90008-7 1983

  23. [23]

    Econometrica 68(5), pp

    Matthew Rabin (2000): Risk aversion and expected-utility theory: A calibration t heorem. Econometrica 68(5), pp. 1281–92, doi:10.1111/1468-0262.00158

    work page doi:10.1111/1468-0262.00158 2000

  24. [24]

    F. P . Ramsey (1926): Truth and Probability. In R.B. Braithwaite, editor: The Foundations of Mathematics and other Logical Essays , 1999 electronic edition edition, Harcourt, pp. 156–98, do i:10.4324/9781315887814

    work page doi:10.4324/9781315887814 1926

  25. [25]

    Journal of the American Statistical Association 66(336), pp

    Leonard Savage (1971): Elicitation of Personal Probabilities and Expectations . Journal of the American Statistical Association 66(336), pp. 783–801, doi:10.2307/2284229

    work page doi:10.2307/2284229 1971

  26. [26]

    Savage (1954): The F oundations of Statistics

    Leonard J. Savage (1954): The F oundations of Statistics. Wiley Publications in Statistics

    work page 1954

  27. [27]

    Sierpi´ nski (1922): Sur les fonctions d’ensemble additives et continues

    W . Sierpi´ nski (1922): Sur les fonctions d’ensemble additives et continues . Fund. Math. 3(1), pp. 240–46, doi:10.4064/fm-3-1-240-246

    work page doi:10.4064/fm-3-1-240-246 1922