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Randomization and dynamic consistency

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Abstract

Raiffa (Q J Econ 75:690–694, 1961) has suggested that ambiguity aversion will cause a strict preference for randomization. We show that dynamic consistency implies that individuals will be indifferent to ex ante randomizations. On the other hand, it is possible for a dynamically consistent ambiguity averse preference relation to exhibit a strict preference for some ex post randomizations. We argue that our analysis throws some light on the recent debate on the status of the smooth model of ambiguity This debate rests on whether the randomizations implicit in the set-up are viewed as being resolved before or after the (ambiguous) uncertainty.

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Notes

  1. Numerous experiments have confirmed this observation; Camerer and Weber (1992), Trautmann and van de Kuilen (2015).

  2. Raiffa (1961) considers the original composition of the Ellsberg urn with 30 red balls and 60 balls which were either black or yellow. Unlike the standard 3-colour Ellsberg urn, in the modified example, slightly less than one-third of the balls are red.

  3. See also the discussion in Ghirardato (1997).

  4. See in particular Wakker (2010) Sections 4.9 and 10.7.

  5. Anscombe and Aumann (1963, p. 201), Assumption 2 (Reversal of order of compound lotteries).

  6. The ex post randomizations in this framework may be identified with a subset of the horserace-lottery acts in the AA framework. In particular, the ex post randomization \(f_{D}g\) may be identified with the horserace-lottery act H where for each s in S, \(H\left( s\right) =\mu \left( D\right) \delta _{f(S)}+\left( 1-\mu \left( D\right) \right) \delta _{g(S)}\) and \(\delta _{x}\) is the degenerate lottery that yields the outcome x with probability 1. That is, an ex post randomization can be naturally identified with a horserace-lottery act in which the second stage is a binary lottery.

  7. This can be seen as a weak version of the AA reversal of order assumption, which only applies to acts which are independent of the S-states.

  8. This assumption is similar in spirit to the device independence assumption in Eichberger and Kelsey (1996).

  9. If \(a\in A\left( {\varOmega }\right) \) then for given r, s, \(a\left( r,s,\cdot \right) \) is an act whose outcome only depends on the ex post randomizing device and hence may be described by a distribution function \(F_{a\left( r,s,\cdot \right) }\).

  10. Since we use a specific filtration our axioms are not symmetric between the ex post and ex ante randomizations.

  11. In this sense it may also be viewed as a weaker form of the “coherence” condition of Skiadas (1997), expression (6) p. 353.

  12. This may be viewed as a special case of ‘issue preference’, as in Ergin and Gul (2009), or ‘source dependence’, as in Chew and Sagi (2008).

  13. This can be made compatible with our assumption that the R-space is a continuum by identifying \(r_{1}\) with \(\left[ 0,\frac{1}{2}\right] \subseteq R\), and \(r_{2}\) with \(\left( \frac{1}{2},1\right] \subseteq R\).

  14. One may find this result less compelling in a normative sense than a strict preference which violates dynamic consistency. If preferences satisfy an appropriate continuity property, however, then we can construct a violation of dynamic consistency which only involves strict preferences. The argument in the introduction provides an outline of the proof.

  15. Bade (2011) makes a related point. She shows that, in a strategic setting, an ambiguous randomizing device would not help a player in a game. In her model the equilibria with ambiguous randomizations coincide with conventional Nash equilibria.

  16. The indifference \(\left( x_{C}y\right) E\left( y_{C}x\right) \thicksim \left( x_{C}y\right) E\left( x_{C}y\right) \) arises because \(\mu \left( C\right) =1/2\) implies that \(\left( x_{C}y\right) \) and \(\left( y_{C}x\right) \) give the same outcomes with the same probabilities. Intuitively they should be indifferent conditional on \(E^{c}\). Indifference is implied by Assumption 2.1 which says objective randomizations with the same probabilities are indifferent and if they have the same conditional probabilities they are indifferent in the corresponding conditional preferences.

  17. In the quotation, the notation of events and acts has been adapted to the one used in this paper.

  18. As in the previous quotation, we have adapted the notation to the one used in this paper.

  19. There is also an earlier literature, see for instance, Brewer (1963) or Fellner (1963).

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Authors and Affiliations

  1. Department of Economics, Universität Heidelberg, Heidelberg, Germany

    Jürgen Eichberger

  2. Research School of Economics, Australian National University, Canberra, Australia

    Simon Grant

  3. School of Economics, University of Queensland, Brisbane, Australia

    Simon Grant

  4. Department of Economics, University of Exeter, Exeter, EX4 4PU, UK

    David Kelsey

Authors
  1. Jürgen Eichberger
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  2. Simon Grant
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  3. David Kelsey
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Correspondence to David Kelsey.

Additional information

Research supported by a Leverhulme Research Fellowship. We would like to thank the referee and editor of this journal, Peter Klibanoff, Tigran Melkonyan, Zvi Safra, Marciano Siniscalchi, Peter Wakker and seminar audiences at the Universities of Bristol, Cergy-Pontoise, Exeter, LSE and RUD 2013 for helpful comments.

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Eichberger, J., Grant, S. & Kelsey, D. Randomization and dynamic consistency. Econ Theory 62, 547–566 (2016). https://doi.org/10.1007/s00199-015-0913-8

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