Skip to main content
SpringerLink
Log in

A mechanism for the elicitation of second-order belief and subjective information structure

  • Research Article
  • Published:
Economic Theory Aims and scope Submit manuscript

Abstract

This paper describes a direct revelation mechanism for eliciting (a) decision makers’ range of subjective priors under Knightian uncertainty and their second-order introspective belief and (b) Bayesian decision makers’ range of posteriors and their subjective information structure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In robust Bayesian statistics, the elicitation of the set of priors is analogous to the elicitation of a single prior in Bayesian statistics (see Seidenfeld et al. 1995). Multi-prior models are part of the more general theory of imprecise probability, which allows for partial probability specifications. It is applicable when information is insufficient to identify a unique (prior) probability distribution (“it is useful for dealing with expert elicitation, because decision makers have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval.” [Wikipedia, Imprecise probability]).

  2. The term “second-order belief” is defined in the literature dealing with interactive decision making to describe one player’s belief of another player belief. It is also employed in the theory of decision making under uncertainty in the sense of its use here, namely to describe a decision maker’s belief over a set of priors (e.g., Seo 2009; Nascimento and Riella 2013; Giraud 2014).

  3. See also Galaabaatar and Karni (2013).

  4. Unlike the mechanism in this paper and the protocol of Chambers and Lambert, Prelec’s (2004) elicitation scheme is not designed for the elicitation of second-order beliefs and is, therefore, less pertinent for the problem under study.

  5. This is the case of a subjective expected utility maximizing decision maker whose preference relation on the set of acts is incomplete, but restricted to the subset of constant acts the preference relation is complete.

  6. Karni and Safra (2016) provide a detailed discussion of connection between their work and those of Dillenberger et al. (2014) and Lu (2016). They argue that, despite sharing some features, their work is different conceptually, methodologically, and structurally.

  7. The existence of \(\mu _{0},\) representing the decision maker’s second-order belief on the set of priors, is implied by the model of Karni and Safra (2016).

  8. The existence of \(\mu _{1}\), representing the subject’s subjective information structure, is implied by the models of Dillenberger et al. (2014) and Lu (2016).

  9. Kreps (1979) articulates this presumption as follows: “In many problems of individual choice, the choice is made in more than one stage. At early stages, the individual makes decisions which will constrain the choices that are feasible later. In effect, these early choices amount to choice of a subset of items from which subsequent choice will be made. This paper concerns choice among such opportunity sets, where the individual has a “desire for flexibility” which is “irrational” if the individual knows what his subsequent preferences will be” (Kreps 1979, p. 565). The focus of the discussion here is the subject’s subsequent beliefs rather than subsequent tastes.

  10. In Dillenberger et al. (2014) and Lu (2016) the delay is built in as the interim period in which the decision maker receives the information signal. The willingness to delay the choice is a manifestation of the value of the anticipated information.

  11. The same analysis pertains to Bayesian subjects with private information structures.

  12. It is easy to verify that the second-order condition is satisfied, so the first-order condition is necessary and sufficient for a maximum.

  13. Assume that \({{\mathcal {H}}}\) is endowed with the topology of weak convergence, and denote by \(\Sigma \) the Borel sigma algebra on \({{\mathcal {H}}}\).

  14. I assume that \({{\mathcal {F}}}\) together with the trace of \(\Sigma \) on \({{\mathcal {F}}\,}\) is the relevant measurable space.

  15. For a recent review, see Chambers and Lambert (2017).

  16. Note that \(\hat{F}\) is a second-order belief and is the same as \(\mu \) in this paper. Similarly, \(\hat{p}\) is a first-order belief and is the same as \( \pi \) in this paper. It is noteworthy, however, that the mechanism described in this paper does not require that the subject report his posterior belief.

References

  • Anscombe, F.J., Aumann, R.J.: A definition of subjective probability. Ann. Math. Stat. 34, 199–205 (1963)

    Article  Google Scholar 

  • Bewley, T.F.: Knightian decision theory: part I. Decis. Econ. Finance 25, 79–110 (2002)

    Article  Google Scholar 

  • Brier, G.W.: Verification of forecasts expressed in terms of probabilities. Mon. Weather Rev. 78, 1–3 (1950)

    Article  Google Scholar 

  • Chambers, C.P., Lambert, N.S.: Dynamically eliciting unobservable information. In: Proceedings of the 15th ACM Conference on Electronic Commerce, pp. 987–988 (2014)

  • Chambers, C.P., Lambert, N.S.: Introduction to dynamic belief elicitation. ACM SIGecom Exch. 14, 80–83 (2015)

    Article  Google Scholar 

  • Chambers, C.P., Lambert, N.S.: Dynamic belief elicitation. Unpublished manuscript (2017)

  • Dillenberger, D., Lleras, J.S., Sadowski, P., Takeoka, N.: A theory of subjective learning. J. Econ. Theory 153, 287–312 (2014)

    Article  Google Scholar 

  • Galaabaatar, T., Karni, E.: Subjective expected utility with incomplete preferences. Econometrica 81, 255–284 (2013)

    Article  Google Scholar 

  • Giraud, R.: Second order beliefs models of choice under omprecise risk. Theor. Econ. 9, 779–816 (2014)

    Article  Google Scholar 

  • Good, I.J.: Rational decisions. J. R. Stat. Soc. Ser. B (Methodol.) 14, 107–114 (1952)

    Google Scholar 

  • Grether, D.M.: Financial incentive effects and individual decision making. Cal Tech, Social Science Working Paper 401 (1981)

  • Kadane, J.B., Winkler, R.L.: Separating probability elicitation from utility. J. Am. Stat. Assoc. 83, 357–363 (1988)

    Article  Google Scholar 

  • Karni, E.: A mechanism design for probability elicitation. Econometrica 77, 603–606 (2009)

    Article  Google Scholar 

  • Karni, E., Safra, Z.: A theory of stochastic choice under uncertainty. J. Math. Econ. 63, 164–173 (2016)

    Article  Google Scholar 

  • Kreps, D.M.: A representation theorem for ‘preference for flexibility’. Econometrica 47, 565–576 (1979)

    Article  Google Scholar 

  • Lu, J.: Random choice and private information. Econometrica 84, 1983–2027 (2016)

    Article  Google Scholar 

  • Nascimento, L., Riella, G.: Second order ambiguous beliefs. Econ. Theory 52, 1005–1037 (2013). https://doi.org/10.1007/s00199-011-0675-x

    Article  Google Scholar 

  • Prelec, D.: A Bayesian truth serum for subjective data. Science 306, 462–466 (2004)

    Article  Google Scholar 

  • Qu, X.: A mechanism for eliciting a probability distribution. Econ. Lett. 115, 399–400 (2012)

    Article  Google Scholar 

  • Roth, A.E., Malouf, M.W.K.: Game-theoretic models and the role of information in bargaining. Psychol. Rev. 86, 574–594 (1979)

    Article  Google Scholar 

  • Savage, L.J.: Elicitation of personal probabilities and expectations. J. Am. Stat. Assoc. 66, 783–801 (1971)

    Article  Google Scholar 

  • Seidenfeld, T., Schervish, M.J., Kadane, J.B.: A representation of partially ordered preferences. Ann. Stat. 23, 2168–2217 (1995)

    Article  Google Scholar 

  • Seo, K.: Ambiguity and second-order belief. Econometrica 77, 1575–1605 (2009)

    Article  Google Scholar 

Download references

Acknowledgements

I am grateful to Zvi Safra, Andrei Savochkin, Jay Lu, Christopher Chambers, Nicholas Lambert and an anonymous referee for their useful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Economics, Johns Hopkins University, Baltimore, MD, 21218, USA

    Edi Karni

  2. Warwick Business School, University of Warwick, Coventry, UK

    Edi Karni

Authors
  1. Edi Karni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Edi Karni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karni, E. A mechanism for the elicitation of second-order belief and subjective information structure. Econ Theory 69, 217–232 (2020). https://doi.org/10.1007/s00199-018-1162-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00199-018-1162-4

Keywords

JEL classification

Search

Navigation