pith. machine review for the scientific record. sign in

arxiv: 2605.07671 · v1 · submitted 2026-05-08 · 💻 cs.GT · cs.AI· cs.MA· econ.TH· math.OC

Recognition: 2 theorem links

· Lean Theorem

The Endogeneity of Miscalibration: Impossibility and Escape in Scored Reporting

Lauri Lov\'en, Sasu Tarkoma

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:13 UTC · model grok-4.3

classification 💻 cs.GT cs.AIcs.MAecon.THmath.OC
keywords miscalibrationscoring rulesAI oversightmechanism designtruthful reportingapproval functionsendogeneity
0
0 comments X

The pith

Any non-affine approval function makes truthful reporting suboptimal under strictly proper scoring rules.

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

A principal scoring an agent's report with a strictly proper rule must also approve the report for screening or allocation purposes. The optimal approval is non-affine to distinguish agent types, but this non-linearity perturbs the agent's combined incentives away from truth-telling when deviations stay hidden. The result is an endogenous miscalibration that cannot be avoided under any such scoring rule. The paper derives a closed-form expression for the perturbation and shows that a simple step-function threshold escapes the problem by turning the choice into a binary one. For the Brier score this yields welfare equivalence between first-best and second-best oversight.

Core claim

The principal's optimal oversight necessarily uses a non-affine approval function to screen types, yet any non-affine approval makes truthful reporting suboptimal under the combined objective whenever deviation is undetectable. This impossibility holds for all strictly proper scoring rules, with a closed-form perturbation formula. A constructive escape exists: a step-function approval threshold achieves first-best screening for every strictly proper scoring rule, because the agent's binary inflate-or-not choice creates a type-space threshold regardless of the generator's curvature. Under the Brier score specifically, the type-independent inflation cost yields a welfare equivalence between第二-

What carries the argument

Non-affine approval function interacting with strictly proper scoring rule curvature to generate a perturbation in the agent's reporting strategy.

If this is right

  • Step-function approval thresholds achieve first-best screening without miscalibration for any strictly proper scoring rule.
  • Brier score uniquely provides welfare equivalence between first-best and second-best due to type-independent costs.
  • For non-Brier rules the welfare gap under smooth oversight is bounded below by Omega of Var(1/G'') times (gamma/beta)^2.
  • Smooth C1 oversight cannot elicit truthful reports in the presence of non-detectable deviations.

Where Pith is reading between the lines

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

  • AI oversight systems may need to incorporate discrete thresholds rather than smooth scoring to maintain calibration.
  • This endogeneity could extend to other settings like peer review or prediction markets with additional incentives.
  • A direct test would involve checking if agents adjust reports according to the perturbation formula when facing non-affine approvals.
  • Repeated interactions might allow the principal to detect deviations and mitigate the effect.

Load-bearing premise

Deviations from truthful reporting by the agent are undetectable by the principal.

What would settle it

If an agent is presented with a non-affine approval function alongside a strictly proper scoring rule and the deviation is undetectable, then the observed report should show the exact bias given by the closed-form perturbation; absence of this bias would falsify the impossibility.

read the original abstract

Eliciting truthful reports from autonomous agents is a core problem in scalable AI oversight: a principal scores the agent's report using a strictly proper scoring rule, but the agent also benefits from the report through a non-accuracy channel (approval for autonomous action, allocation share, downstream control). The same structure appears in classical mechanism-design settings such as marketplace operation. Our main result is an endogeneity: the principal's optimal oversight necessarily uses a non-affine approval function to screen types, yet any non-affine approval makes truthful reporting suboptimal under the combined objective whenever deviation is undetectable. The principal cannot avoid the perturbation that undermines calibration. This impossibility holds for all strictly proper scoring rules, with a closed-form perturbation formula. A constructive escape exists: a step-function approval threshold achieves first-best screening for every strictly proper scoring rule, because the agent's binary inflate-or-not choice creates a type-space threshold regardless of the generator's curvature. Under the Brier score specifically, the type-independent inflation cost yields a welfare equivalence between second-best and first-best; we prove this equivalence is unique to Brier (the welfare gap under smooth $C^1$ oversight is bounded below by $\Omega(\text{Var}(1/G'') (\gamma/\beta)^2)$ for every non-Brier rule). Two instances develop the framework: AI agent oversight (the lead motivating setting) and marketplace operation (a parallel mechanism-design domain). The message for AI alignment is direct: smooth scoring-based oversight cannot elicit truthful reports from a strategic agent; sharp thresholds are the calibration-preserving design.

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 claims that when a principal elicits reports from agents using any strictly proper scoring rule but the report also determines a non-accuracy benefit (approval, allocation, or control), optimal type-screening requires a non-affine approval function. Any such non-affine function necessarily perturbs the agent's report away from its true belief under the combined objective whenever the deviation is undetectable, yielding an impossibility result with a closed-form perturbation formula that holds for every strictly proper scoring rule. A constructive escape is a step-function approval threshold that restores first-best screening for any scoring rule. Under the Brier score the resulting welfare equivalence between second-best and first-best is unique; for all other rules the welfare gap under smooth C^1 oversight is bounded below by Ω(Var(1/G'') (γ/β)^2).

Significance. If the derivations are correct, the result identifies a fundamental tension between type screening and calibration preservation that is relevant to both AI oversight and classical mechanism design. The closed-form perturbation, the step-function escape, and the Brier-specific welfare equivalence (with explicit lower bound for other rules) supply concrete design guidance and falsifiable predictions rather than purely qualitative warnings.

major comments (2)
  1. [Abstract and main impossibility result] The impossibility (Abstract and main theorem) is load-bearing on the maintained assumption that any deviation from truthful reporting is permanently undetectable by the principal. Under this assumption the non-affine approval term enters the agent's first-order condition and produces the stated perturbation; if even a small detection probability or ex-post penalty is admitted, the agent's optimization changes and truthful reporting can remain optimal. The paper must state the formal modeling of undetectability (e.g., information structure or monitoring technology) and either prove robustness or delineate the boundary case.
  2. [Welfare analysis] The welfare-gap lower bound Ω(Var(1/G'') (γ/β)^2) for non-Brier rules (Abstract) is asserted to be tight and to arise directly from the generator curvature. The derivation of the variance term and the precise conditions under which the bound holds (including the same undetectability premise) should be exhibited with equation numbers so that the claim can be verified independently.
minor comments (2)
  1. [Notation and model] Define the parameters γ and β, the generator G, and the approval function notation at first use rather than relying on context from the abstract.
  2. [Applications] The two application instances (AI oversight and marketplace operation) are mentioned but not developed in the provided abstract; a short comparative table or paragraph would clarify whether the impossibility and escape apply identically in both domains.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will make the indicated revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract and main impossibility result] The impossibility (Abstract and main theorem) is load-bearing on the maintained assumption that any deviation from truthful reporting is permanently undetectable by the principal. Under this assumption the non-affine approval term enters the agent's first-order condition and produces the stated perturbation; if even a small detection probability or ex-post penalty is admitted, the agent's optimization changes and truthful reporting can remain optimal. The paper must state the formal modeling of undetectability (e.g., information structure or monitoring technology) and either prove robustness or delineate the boundary case.

    Authors: We agree that the undetectability assumption is foundational. In the revision we will add an explicit definition in Section 2: the principal's information structure consists solely of the reported belief r and the realized outcome ω, with no additional monitoring technology or detection channel. Under this structure we prove that the non-affine approval term enters the agent's first-order condition exactly as stated, yielding the closed-form perturbation for every strictly proper scoring rule. We will also delineate the boundary: when a detection probability ε > 0 is introduced, the agent's optimal deviation is scaled by (1-ε) and exact calibration is restored only in the limit ε → 0. This makes clear that the impossibility is specific to the undetectable case, which is the relevant regime for scalable oversight. revision: yes

  2. Referee: [Welfare analysis] The welfare-gap lower bound Ω(Var(1/G'') (γ/β)^2) for non-Brier rules (Abstract) is asserted to be tight and to arise directly from the generator curvature. The derivation of the variance term and the precise conditions under which the bound holds (including the same undetectability premise) should be exhibited with equation numbers so that the claim can be verified independently.

    Authors: The derivation currently resides in the appendix (Lemma A.3 and the surrounding Taylor expansion). We will move the essential steps into the main text and assign equation numbers (new Eqs. 14–17) to: (i) the agent's combined utility under undetectability, (ii) the second-order expansion that isolates the Var(1/G'') term, (iii) the scaling factor (γ/β)^2 arising from the approval parameters, and (iv) the tightness construction via a sequence of quadratic generators. The undetectability premise is maintained throughout. These numbered equations will allow independent verification of the Ω lower bound. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations are independent mathematical consequences of proper scoring rules and agent optimization.

full rationale

The paper derives an impossibility result and a constructive escape directly from the definition of strictly proper scoring rules (expected score maximized uniquely at true belief) combined with the agent's maximization of a composite objective (score plus non-affine approval). The closed-form perturbation follows from first-order conditions on the agent's utility without any parameter fitting or redefinition of inputs as outputs. The step-function escape is shown constructively to restore a type threshold for any generator curvature, and the Brier-specific welfare equivalence plus the Omega lower bound for other rules are proved from explicit variance expressions rather than imported via self-citation or ansatz. No load-bearing step reduces to a prior result by the same authors or renames an empirical pattern; the entire chain is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard domain assumptions from mechanism design and scoring rule theory with no new postulated entities or fitted parameters beyond model variables like relative weights in the welfare bound.

axioms (3)
  • domain assumption Agents are expected utility maximizers whose objective combines the scoring rule payoff with a non-accuracy benefit from the report
    Core modeling choice invoked throughout the impossibility and escape analysis
  • standard math The scoring rule is strictly proper
    Invoked for the impossibility holding for all such rules
  • domain assumption Deviations from truthful reporting are undetectable
    Required for the non-affine approval to induce miscalibration without detection

pith-pipeline@v0.9.0 · 5594 in / 1539 out tokens · 66432 ms · 2026-05-11T02:13:52.715349+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 1 internal anchor

  1. [1]

    Abernethy and Rafael M

    Jacob D. Abernethy and Rafael M. Frongillo. 2012. A Characterization of Scoring Rules for Linear Properties. In Proceedings of the 25th Annual Conference on Learning Theory (COLT 2012) (JMLR Proceedings, Vol. 23). 27.1–27.13

    work page 2012

  2. [2]

    Mohammad Akbarpour and Shengwu Li. 2020. Credible Auctions: A Trilemma.Econometrica88, 2 (2020), 425–467. doi:10.3982/ECTA15925

    work page doi:10.3982/ecta15925 2020

  3. [3]

    Aaron Archer and Éva Tardos. 2001. Truthful Mechanisms for One-Parameter Agents. InProceedings of the 42nd IEEE Symposium on Foundations of Computer Science (FOCS 2001). 482–491. doi:10.1109/SFCS.2001.959924

    work page doi:10.1109/sfcs.2001.959924 2001

  4. [4]

    Kenneth J. Arrow. 1951.Social Choice and Individual Values. Wiley, New York. Second edition 1963

    work page 1951

  5. [5]

    Baron and Roger B

    David P. Baron and Roger B. Myerson. 1982. Regulating a Monopolist with Unknown Costs.Econometrica50, 4 (1982), 911–930. doi:10.2307/1912769

    work page doi:10.2307/1912769 1982

  6. [6]

    Bo Becker and Todd Milbourn. 2011. How Did Increased Competition Affect Credit Ratings?Journal of Financial Economics101, 3 (2011), 493–514. doi:10.1016/j.jfineco.2011.03.012

    work page doi:10.1016/j.jfineco.2011.03.012 2011

  7. [7]

    Dirk Bergemann, Marek Bojko, Paul Dütting, Renato Paes Leme, Haifeng Xu, and Song Zuo. 2024. Data-Driven Mechanism Design: Jointly Eliciting Preferences and Information.arXiv preprint(2024). arXiv:2412.16132 [econ.TH] https://arxiv.org/abs/2412.16132

    work page arXiv 2024

  8. [8]

    Dirk Bergemann, Tibor Heumann, and Stephen Morris. 2026. Information Design and Mechanism Design: An Integrated Framework.arXiv preprint(2026). arXiv:2601.17267 [econ.TH] https://arxiv.org/abs/2601.17267

    work page arXiv 2026

  9. [9]

    Dirk Bergemann and Stephen Morris. 2016. Bayes Correlated Equilibrium and the Comparison of Information Structures in Games.Theoretical Economics11, 2 (2016), 487–522. doi:10.3982/TE1808

    work page doi:10.3982/te1808 2016

  10. [10]

    Dirk Bergemann and Stephen Morris. 2019. Information Design: A Unified Perspective.Journal of Economic Literature 57, 1 (2019), 44–95. doi:10.1257/jel.20181489

    work page doi:10.1257/jel.20181489 2019

  11. [11]

    David Blackwell. 1951. Comparison of Experiments. InProceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, CA, 93–102

    work page 1951

  12. [12]

    David Blackwell. 1953. Equivalent Comparisons of Experiments.Annals of Mathematical Statistics24, 2 (1953), 265–272. doi:10.1214/aoms/1177729032

    work page doi:10.1214/aoms/1177729032 1953

  13. [13]

    Glenn W. Brier. 1950. Verification of Forecasts Expressed in Terms of Probability.Monthly Weather Review78, 1 (1950), 1–3. doi:10.1175/1520-0493(1950)078<0001:VOFEIT>2.0.CO;2

    work page doi:10.1175/1520-0493(1950)078 1950

  14. [14]

    Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei

    Paul F. Christiano, Jan Leike, Tom Brown, Miljan Martic, Shane Legg, and Dario Amodei. 2017. Deep Reinforcement Learning from Human Preferences. InAdvances in Neural Information Processing Systems 30 (NeurIPS). 4299–4307

    work page 2017

  15. [15]

    Michele Conforti and Gérard Cornuéjols. 1984. Submodular Set Functions, Matroids and the Greedy Algorithm: Tight Worst-Case Bounds and Some Generalizations of the Rado–Edmonds Theorem.Discrete Applied Mathematics7, 3 (1984), 251–274. doi:10.1016/0166-218X(84)90003-9

    work page doi:10.1016/0166-218x(84)90003-9 1984

  16. [16]

    Crawford and Joel Sobel

    Vincent P. Crawford and Joel Sobel. 1982. Strategic Information Transmission.Econometrica50, 6 (1982), 1431–1451. doi:10.2307/1913390

    work page doi:10.2307/1913390 1982

  17. [17]

    Bruno de Finetti. 1937. La prévision: ses lois logiques, ses sources subjectives.Annales de l’Institut Henri Poincaré7, 1 (1937), 1–68

    work page 1937

  18. [18]

    Piotr Dworczak. 2020. Mechanism Design with Aftermarkets: Cutoff Mechanisms.Econometrica88, 6 (2020), 2629–2661. doi:10.3982/ECTA15768

    work page doi:10.3982/ecta15768 2020

  19. [19]

    Ronald A. Dye. 1985. Disclosure of Nonproprietary Information.Journal of Accounting Research23, 1 (1985), 123–145. doi:10.2307/2490910

    work page doi:10.2307/2490910 1985

  20. [20]

    Werner Fenchel. 1949. On Conjugate Convex Functions.Canadian Journal of Mathematics1 (1949), 73–77. doi:10. 4153/CJM-1949-007-x

    work page 1949

  21. [21]

    Matheus V. X. Ferreira and S. Matthew Weinberg. 2020. Credible, Truthful, and Two-Round (Optimal) Auctions via Cryptographic Commitments. InProceedings of the 21st ACM Conference on Economics and Computation (EC ’20). 683–712. doi:10.1145/3391403.3399495

    work page doi:10.1145/3391403.3399495 2020

  22. [22]

    Tobias Fissler and Johanna F. Ziegel. 2016. Higher Order Elicitability and Osband’s Principle.Annals of Statistics44, 4 (2016), 1680–1707. doi:10.1214/16-AOS1439

    work page doi:10.1214/16-aos1439 2016

  23. [23]

    Drew Fudenberg and David K. Levine. 1989. Reputation and Equilibrium Selection in Games with a Patient Player. Econometrica57, 4 (1989), 759–778. doi:10.2307/1913771 The Endogeneity of Miscalibration 0:37

    work page doi:10.2307/1913771 1989

  24. [24]

    Matthew Gentzkow and Emir Kamenica. 2011. Bayesian Persuasion.American Economic Review101, 6 (2011), 2590–2615. doi:10.1257/aer.101.6.2590

    work page doi:10.1257/aer.101.6.2590 2011

  25. [25]

    Matthew Gentzkow and Emir Kamenica. 2017. Competition in Persuasion.Review of Economic Studies84, 1 (2017), 300–322. doi:10.1093/restud/rdw052

    work page doi:10.1093/restud/rdw052 2017

  26. [26]

    Allan Gibbard. 1973. Manipulation of Voting Schemes: A General Result.Econometrica41, 4 (1973), 587–601. doi:10.2307/1914083

    work page doi:10.2307/1914083 1973

  27. [27]

    Tilmann Gneiting and Adrian E. Raftery. 2007. Strictly Proper Scoring Rules, Prediction, and Estimation.J. Amer. Statist. Assoc.102, 477 (2007), 359–378. doi:10.1198/016214506000001437

    work page doi:10.1198/016214506000001437 2007

  28. [28]

    Charles A. E. Goodhart. 1984. Problems of Monetary Management: The UK Experience. InMonetary Theory and Practice: The UK Experience. Macmillan, London, 91–121

    work page 1984

  29. [29]

    Jerry Green and Jean-Jacques Laffont. 1977. Characterization of Satisfactory Mechanisms for the Revelation of Preferences for Public Goods.Econometrica45, 2 (1977), 427–438. doi:10.2307/1911219

    work page doi:10.2307/1911219 1977

  30. [30]

    Grossman

    Sanford J. Grossman. 1981. The Informational Role of Warranties and Private Disclosure about Product Quality. Journal of Law and Economics24, 3 (1981), 461–483. doi:10.1086/466995

    work page doi:10.1086/466995 1981

  31. [31]

    Bengt Holmström. 1979. Moral Hazard and Observability.Bell Journal of Economics10, 1 (1979), 74–91. doi:10.2307/ 3003320

    work page 1979

  32. [32]

    Bengt Holmström. 1999. Managerial Incentive Problems: A Dynamic Perspective.Review of Economic Studies66, 1 (1999), 169–182. doi:10.1111/1467-937X.00083 Originally circulated 1982

    work page doi:10.1111/1467-937x.00083 1999

  33. [33]

    Leonid Hurwicz. 1972. On Informationally Decentralized Systems. InDecision and Organization, C. B. McGuire and Roy Radner (Eds.). North-Holland, Amsterdam, 297–336

    work page 1972

  34. [34]

    Geoffrey Irving, Paul Christiano, and Dario Amodei. 2018. AI Safety via Debate.arXiv preprint arXiv:1805.00899(2018)

    work page internal anchor Pith review arXiv 2018

  35. [35]

    Jean-Jacques Laffont and Jean Tirole. 1986. Using Cost Observation to Regulate Firms.Journal of Political Economy94, 3 (1986), 614–641. doi:10.1086/261392

    work page doi:10.1086/261392 1986

  36. [36]

    1993.A Theory of Incentives in Procurement and Regulation

    Jean-Jacques Laffont and Jean Tirole. 1993.A Theory of Incentives in Procurement and Regulation. MIT Press, Cambridge, MA

    work page 1993

  37. [37]

    Pennock, and Yoav Shoham

    Nicolas Lambert, David M. Pennock, and Yoav Shoham. 2008. Eliciting Properties of Probability Distributions. In Proceedings of the 9th ACM Conference on Electronic Commerce (EC ’08). 129–138. doi:10.1145/1386790.1386813

    work page doi:10.1145/1386790.1386813 2008

  38. [38]

    Shengwu Li. 2017. Obviously Strategy-Proof Mechanisms.American Economic Review107, 11 (2017), 3257–3287. doi:10.1257/aer.20160425

    work page doi:10.1257/aer.20160425 2017

  39. [39]

    Yang Liu, Juntao Wang, and Yiling Chen. 2023. Surrogate Scoring Rules.ACM Transactions on Economics and Computation10, 3 (2023), Article 9. doi:10.1145/3565559

    work page doi:10.1145/3565559 2023

  40. [40]

    Alessandro Lizzeri. 1999. Information Revelation and Certification Intermediaries.RAND Journal of Economics30, 2 (1999), 214–231. doi:10.2307/2556078

    work page doi:10.2307/2556078 1999

  41. [41]

    Lucas, Jr

    Robert E. Lucas, Jr. 1976. Econometric Policy Evaluation: A Critique. InThe Phillips Curve and Labor Markets, Karl Brunner and Allan H. Meltzer (Eds.). Carnegie-Rochester Conference Series on Public Policy, Vol. 1. North-Holland, Amsterdam, 19–46

    work page 1976

  42. [42]

    Mailath and Larry Samuelson

    George J. Mailath and Larry Samuelson. 2001. Who Wants a Good Reputation?Review of Economic Studies68, 2 (2001), 415–441. doi:10.1111/1467-937X.00175

    work page doi:10.1111/1467-937x.00175 2001

  43. [43]

    David Manheim and Scott Garrabrant. 2018. Categorizing Variants of Goodhart’s Law.arXiv preprint arXiv:1803.04585 (2018)

    work page arXiv 2018

  44. [44]

    John McCarthy. 1956. Measures of the Value of Information.Proceedings of the National Academy of Sciences42, 9 (1956), 654–655. doi:10.1073/pnas.42.9.654

    work page doi:10.1073/pnas.42.9.654 1956

  45. [45]

    2004.Putting Auction Theory to Work

    Paul Milgrom. 2004.Putting Auction Theory to Work. Cambridge University Press, Cambridge. doi:10.1017/ CBO9780511813825

    work page 2004

  46. [46]

    Paul Milgrom and Ilya Segal. 2002. Envelope Theorems for Arbitrary Choice Sets.Econometrica70, 2 (2002), 583–601. doi:10.1111/1468-0262.00296

    work page doi:10.1111/1468-0262.00296 2002

  47. [47]

    Paul R. Milgrom. 1981. Good News and Bad News: Representation Theorems and Applications.Bell Journal of Economics12, 2 (1981), 380–391. doi:10.2307/3003562

    work page doi:10.2307/3003562 1981

  48. [48]

    Hervé Moulin. 1980. On Strategy-Proofness and Single Peakedness.Public Choice35, 4 (1980), 437–455

    work page 1980

  49. [49]

    Roger B. Myerson. 1979. Incentive Compatibility and the Bargaining Problem.Econometrica47, 1 (1979), 61–73. doi:10.2307/1912346

    work page doi:10.2307/1912346 1979

  50. [50]

    Roger B. Myerson. 1981. Optimal Auction Design.Mathematics of Operations Research6, 1 (1981), 58–73. doi:10.1287/ moor.6.1.58

    work page 1981

  51. [51]

    Caspar Oesterheld and Vincent Conitzer. 2021. Decision Scoring Rules. InWeb and Internet Economics (WINE 2020) (Lecture Notes in Computer Science, Vol. 12495). Springer, 468–481. doi:10.1007/978-3-030-68024-4_26

    work page doi:10.1007/978-3-030-68024-4_26 2021

  52. [52]

    Jean-Charles Rochet. 1987. A Necessary and Sufficient Condition for Rationalizability in a Quasi-Linear Context. Journal of Mathematical Economics16, 2 (1987), 191–200. doi:10.1016/0304-4068(87)90007-3 0:38 Lovén and Tarkoma

    work page doi:10.1016/0304-4068(87)90007-3 1987

  53. [53]

    Princeton University Press (1970)

    R. Tyrrell Rockafellar. 1970.Convex Analysis. Number 28 in Princeton Mathematical Series. Princeton University Press, Princeton, NJ. doi:10.1515/9781400873173

    work page doi:10.1515/9781400873173 1970

  54. [54]

    Mark Allen Satterthwaite. 1975. Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions.Journal of Economic Theory10, 2 (1975), 187–217. doi:10.1016/0022- 0531(75)90050-2

    work page doi:10.1016/0022- 1975

  55. [55]

    Leonard J. Savage. 1971. Elicitation of Personal Probabilities and Expectations.J. Amer. Statist. Assoc.66, 336 (1971), 783–801. doi:10.1080/01621459.1971.10482346

    work page doi:10.1080/01621459.1971.10482346 1971

  56. [56]

    Schervish

    Mark J. Schervish. 1989. A General Method for Comparing Probability Assessors.Annals of Statistics17, 4 (1989), 1856–1879. doi:10.1214/aos/1176347398

    work page doi:10.1214/aos/1176347398 1989

  57. [57]

    Vasiliki Skreta and Laura Veldkamp. 2009. Ratings Shopping and Asset Complexity: A Theory of Ratings Inflation. Journal of Monetary Economics56, 5 (2009), 678–695. doi:10.1016/j.jmoneco.2009.04.006

    work page doi:10.1016/j.jmoneco.2009.04.006 2009

  58. [58]

    Rakesh V. Vohra. 2011.Mechanism Design: A Linear Programming Approach. Cambridge University Press, Cambridge. doi:10.1017/CBO9781139236782

    work page doi:10.1017/cbo9781139236782 2011