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Synthese

, Volume 196, Issue 6, pp 2455–2485 | Cite as

Biased information and the exchange paradox

  • Anubav VasudevanEmail author
Article

Abstract

This paper presents a new solution to the well-known exchange paradox, or what is sometimes referred to as the two-envelope paradox. Many recent commentators have analyzed the paradox in terms of the agent’s biased concern for the contents of his own arbitrarily chosen envelope, claiming that such bias violates the manifest symmetry of the situation. Such analyses, however, fail to make clear exactly how the symmetry of the situation is violated by the agent’s hypothetical conclusion that he ought to switch envelopes on the assumption that his own envelope contains some specific amount of money. In this paper, I offer an explanation of this fact based on the idea that the agent’s deliberations are not only constrained by the epistemic symmetry reflected in the agent’s uniform ignorance as to the contents of the two envelopes, but also by a deeper methodological symmetry that manifests itself in the intuition that the two envelopes constitute equally legitimate sources of potential information. I interpret this intuition as implying that the agent’s final decision should reflect a point of rational equilibrium arrived at through an iterated process of counterfactual self-reflection, whereby the agent takes account of what he would have thought had he instead chosen the other envelope. I provide a formal model of this method of counterfactual self-reflection and show, in particular, that it cannot issue in the paradoxical conclusion that the agent ought to switch regardless of how much money is in his envelope. In this way, by correcting for the bias in the agent’s reasoning, the paradox is resolved.

Keywords

Exchange paradox Two-envelope paradox Bayesian decision theory 

References

  1. Aumann, R. J., Hart, S., & Perry, M. (2005). Conditioning and the sure-thing principle. In Discussion paper series (no. 393). Center for the Study of Rationality.Google Scholar
  2. Bar-Hillel, M., & Falk, R. (1982). Some teasers concerning conditional probabilities. Cognition, 11(2), 109–122.Google Scholar
  3. Bovens, L., & Ferreira, J. L. (2010). Monty Hall drives a wedge between Judy Benjamin and the sleeping beauty: A reply to Bovens. Analysis, 70(3), 473–481.Google Scholar
  4. Broome, J. (1995). The two-envelope paradox. Analysis, 55(1), 6–11.Google Scholar
  5. Christensen, R., & Utts, J. (1992). Bayesian resolution of the “exchange paradox”. The American Statistician, 46(4), 274–276.Google Scholar
  6. Clark, M., & Shackel, N. (2000). The two-envelope paradox. Mind, 109(435), 415–442.Google Scholar
  7. Clark, M., & Shackel, N. (2003). Decision theory, symmetry and causal structure: Reply to meacham and weisberg. Mind, 112(448), 691–701.Google Scholar
  8. de Finetti, B. (2017). Theory of probability: A critical introductory treatment. Hoboken: Wiley.Google Scholar
  9. Dietrich, F., & List, C. (2005). The two-envelope paradox: An axiomatic approach. Mind, 114(454), 239–248.Google Scholar
  10. Douven, I. (2007). A three-step solution to the two-envelope paradox. Logique et Analyse, 50(200), 359–365.Google Scholar
  11. Easwaran, K. (2008). Strong and weak expectations. Mind, 117(467), 633–641.Google Scholar
  12. Eckhardt, W. (2013). Paradoxes in probability theory. Berlin: Springer.Google Scholar
  13. Gaifman, H. (2013). The sure thing principle, dilations, and objective probabilities. Journal of Applied Logic, 11(4), 373–385.Google Scholar
  14. Jaynes, E. T. (1996). Probability theory: The logic of science. Louis, MO: Washington University in St. Louis.Google Scholar
  15. Kadane, J. B. (2011). Principles of uncertainty. Boca Raton: CRC Press.Google Scholar
  16. Kass, R. E., & Wasserman, L. (1996). The selection of prior distributions by formal rules. Journal of the American Statistical Association, 91(435), 1343–1370.Google Scholar
  17. Levi, I. (1987). The demons of decision. The Monist, 70(2), 193–211.Google Scholar
  18. Lewis, D. (1987). A subjectivist’s guide to objective chance. In Philosophical papers volume II. New York: Oxford University Press.Google Scholar
  19. McDonnell, M. D., & Abbott, D. (2009). Randomized switching in the two-envelope problem. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 465, 3309–3322.Google Scholar
  20. McDonnell, M. D., Grant, A. J., Land, I., Vellambi, B. N., Abbott, D., & Lever, K. (2011). Gain from the two-envelope problem via information asymmetry: On the suboptimality of randomized switching. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 467, 2825–2851.Google Scholar
  21. McGrew, T. J., Shier, D., & Silverstein, H. S. (1997). The two-envelope paradox resolved. Analysis, 57(1), 28–33.Google Scholar
  22. Meacham, C., & Weisberg, J. (2003). Clark and Shackel on the two-envelope paradox. Mind, 112(448), 685–689.Google Scholar
  23. Milgrom, P., & Stokey, N. (1982). Information, trade and common knowledge. Journal of Economic Theory, 26(1), 17–27.Google Scholar
  24. Nalebuff, B. (1989). Puzzles: The other person’s envelope is always greener. The Journal of Economic Perspectives, 3(1), 171–181.Google Scholar
  25. Norton, J. (1998). Where the sum of our expectation fails us: The exchange paradox. Pacific Philosophical Quarterly, 79(1), 34–58.Google Scholar
  26. Nover, H., & Hájek, A. (2004). Vexing expectations. Mind, 113(450), 237–249.Google Scholar
  27. Rubinstein, A., & Wolinsky, A. (1990). On the logic of “agreeing to disagree” type results. Journal of Economic Theory, 51(1), 184–193.Google Scholar
  28. Rudin, W. (1976). Principles of mathematical analysis. In International series in pure & applied mathematics. McGraw-Hill Publishing Co.Google Scholar
  29. Savage, L. J. (1972). The foundations of statistics. Mineola, New York: Dover Publications.Google Scholar
  30. Scott, A. D., & Scott, M. (1997). What’s in the two envelope paradox? Analysis, 57(1), 34–41.Google Scholar
  31. Selvin, S. (1975). On the monty hall problem. The American statistician, 29(3), 134–134.Google Scholar
  32. Shafer, G. (1985). Conditional probability. International Statistical Review = Revue internationale de statistique, 53(3), 261–275.Google Scholar
  33. Sobel, J. H. (1994). Two envelopes. Theory and Decision, 36(1), 69–96.Google Scholar
  34. Wagner, C. G. (1999). Misadventures in conditional expectation: The Two-Envelope problem. Erkenntnis, 51(2–3), 233–241.Google Scholar

Copyright information

© Springer Nature B.V. 2017

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of ChicagoChicagoUSA

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