Advertisement

Synthese

, Volume 145, Issue 1, pp 89–109 | Cite as

Trying to Resolve the Two-Envelope Problem

  • Casper J. Albers
  • Barteld P. Kooi
  • Willem Schaafsma
Article

Abstract

After explaining the well-known two-envelope ‘paradox’ by indicating the fallacy involved, we consider the two-envelope ‘problem’ of evaluating the ‘factual’ information provided to us in the form of the value contained by the envelope chosen first. We try to provide a synthesis of contributions from economy, psychology, logic, probability theory (in the form of Bayesian statistics), mathematical statistics (in the form of a decision-theoretic approach) and game theory. We conclude that the two-envelope problem does not allow a satisfactory solution. An interpretation is made for statistical science at large.

Keywords

Probability Theory Game Theory Mathematical Statistic Statistical Science Satisfactory Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Broome, J. 1995The Two-Envelope ParadoxAnalysis.55611Google Scholar
  2. Chase, J. 2002The Non-Probabilistic Two Envelope ParadoxAnalysis.62157160CrossRefGoogle Scholar
  3. Christiansen, R., Utts, J. 1992Bayesian Resolution of the “Exchange Paradox”The American Statistician.46274276Google Scholar
  4. Clark, M., Shackel, N. 2000The Two-Envelope ParadoxMind.109415442CrossRefGoogle Scholar
  5. Ferguson, T. 1967Mathematical Statistics, a Decision-Theoretic ApproachAcademic PressNew YorkGoogle Scholar
  6. Gardner, M. 1982Aha! Gotcha: Paradoxes to Puzzle and DelightW.H. FreemanNew YorkGoogle Scholar
  7. Gellius, A. 1946The Attic Nights of Aulus Gellius (Noctes Atticae)Harvard University PressCambridge, MAtranslated by J. C RolfeGoogle Scholar
  8. Heijenoort, J.~v. (1967). Logical Paradoxes’ in P. Edwards (ed.), The Encyclopedia of Philosophy, Vol. 5, Macmillan Company & The Free Press, 45–51Google Scholar
  9. Jackson, F., Menzies, P., Oppy, G. 1994‘The Two Envelope “Paradox”Analysis544345MathSciNetGoogle Scholar
  10. Kraitchik, M. 1943Mathematical RecreationsGeorge Allen & UnwinLondonGoogle Scholar
  11. Linzer, E. 1994The Two Envelope ParadoxThe American Mathematical Monthly.101417419Google Scholar
  12. McGrew, T., Shier, D., Silverstein, H. 1997The Two-Envelope Paradox ResolvedAnalysis.572833CrossRefGoogle Scholar
  13. Nalebuff, B. 1988Cider in Your Ear, Continuing Dilemma, the Last Shall Be the First, and MoreJournal of Economic Perspectives.2149156Google Scholar
  14. Nalebuff, B. 1989The Other Person’s Envelope is Always GreenerJournal of Economic Perspectives.3171181Google Scholar
  15. Neumann, J., Morgenstern, O. 1944Theory of Games and Economic BehaviourPrinceton University Press.Princeton, NJ.Google Scholar
  16. Nikaidô, H. 1953On a Minimax Theorem and its Applications to Functional AnalysisJournal of the Mathematical Society of Japan.58694Google Scholar
  17. Nikaidô, H. 1959On a Method of Proof for the Minimax TheoremProceedings of the American Mathematical Association.10205212Google Scholar
  18. Pearson, K. 1920The Fundamental Problem of Practical StatisticsBiometrika.13116Google Scholar
  19. Savage, L. 1951The Theory of Statistical DecisionJournal of the American Statistical Association.465567Google Scholar
  20. Schaafsma, W. 1971The Neyman-Pearson Theory for Testing Statistical HypothesesStatistica Neerlandica.25127Google Scholar
  21. Smullyan, R. 1997The Riddle of Scheherazade, and Other Amazing Puzzles, Ancient and ModernKnopfNew York.Google Scholar
  22. Stewart, I. 2000Mathematical Recreations: Paradox LostScientific American.68889Google Scholar
  23. Wald, A. 1947Book Review of (Neumann and Morgenstern 1944) reprinted fromThe Review of Economic Statistics.244752Google Scholar
  24. Wald, A. 1964Statistical Decision FunctionsJohn WileyNew YorkGoogle Scholar
  25. Zabell, S. 1988aLoss and Gain: The Exchange Paradox’Bernardo, J. M.DeGroot, M. H.Lindley, D. V.Smith, A. F. M. eds. Bayesian Statistics, 3. Proceedings of the Third Valencia International MeetingClarendon PressOxford233236Google Scholar
  26. Zabell, S. 1988bSymmetry and its DiscontentsSkyrms, B.Harper, W. L. eds. Causation, Chance and CrecedenceKluwer Academic PublishersDordrechtGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Casper J. Albers
    • 1
  • Barteld P. Kooi
    • 1
  • Willem Schaafsma
    • 1
  1. 1.Mathematics & Natural ScienceUniversity of GroningenGroningenThe Netherlands

Personalised recommendations