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Biased information and the exchange paradox

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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.

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Notes

  1. Most of the philosophical literature on the exchange paradox deals with probabilistic versions of the paradox that are more sophisticated than the simple form of the paradox just described. These more sophisticated versions of the paradox, which are engineered to avoid certain obvious objections to the argument for switching envelopes as formulated above, will be discussed in Sect. 2. I have opted to begin with a statement of the paradox in its simplest form so as not to burden the reader with unnecessary complications. The reader, however, should rest assured that the analysis offered in this paper will propose a solution to the exchange paradox in its most general form.

  2. See, e.g., McGrew et al. (1997), Clark and Shackel (2003) and Douven (2007).

  3. Douven (2007, p. 363; n. 3), for example, acknowledge that symmetry considerations cannot provide an explanation of why the agent ought not to switch after he has come to know the contents of his own envelope.

  4. This is at least true with respect to the more sophisticated, probabilistic versions of the paradox discussed in the philosophical literature (see, e.g., Sobel 1994; Norton 1998; Wagner 1999; Dietrich and List 2005). In the context of the simple form of the paradox, there are additional considerations that may be relevant for assessing the validity of the agent’s hypothetical deliberations. These issues will be discussed in more detail in Sect. 2 below.

  5. See, e.g., Sobel (1994), Norton (1998), Wagner (1999) and Kadane (2011, p. 278).

  6. By definition, probabilities are normalized, countably additive, real-valued functions. Since a probability function is normalized, it must assign to the disjunction of any countably infinite set of mutually exclusive and exhaustive events a probability of 1. Now if, on the one hand, each event in such a set is assigned a probability of 0, then, by countable additivity, the probability of their disjunction also equals 0. If, on the other hand, each event is assigned a non-zero probability, then again by countable additivity (and the Archimedean property of the real numbers), the probability of their disjunction is unbounded. In either case, we have a contradiction. If we wish to satisfy Eqs. (1) and (2), while, at the same time taking Pr to be a normalized function, there are two strategies that we could adopt: (i) assume that \({ Pr}(W=n)=0\), for all n, and that \({ Pr}\) is a finitely but not countably additive measure; or (ii) assume that \({ Pr}(W=n)\ne 0\), for all n, and that Pr takes values in some non-Archimedean field, such as that of the (extended) hyperreals. The former possibility is endorsed by de Finetti (2017, pp. 103–105), and the latter possibility is mentioned, though not discussed at any length, by Sobel (1994, pp. 73–74).

  7. Scott and Scott (1997, p. 35) for example, claim that “[t]he simple form of the paradox can be rejected... since the argument itself makes an assumption which entails a contradiction.” A similar point is made by Ian Jewitt, who writes: “As I see it, the monstrous hypothesis is the uniform density on the set of...integers. Accept this and you accept anything” (quoted in Nalebuff 1989, p. 176). To describe the simple form of the paradox as implying a contradiction is misleading, for the contradiction only arises once we accept the assumption that the agent’s prior state of belief can be represented by a probability function. This assumption, however, has quite dramatic implications. For instance, probabilities defined on infinite sets are not only non-uniform, they are radically non-uniform, since, for any arbitrarily small \(\varepsilon >0\), there must exist a finite set with probability greater than \(1-\varepsilon \). Thus, the agent can only assign probabilities to the possible contents of the two envelopes if he is willing to posit a finite set of values from which it is all but certain that the larger sum has been chosen, or, as de Finetti (2017, p. 103) puts it, if he is willing to treat an infinite set as ‘finite—up to trifles.’.

  8. We will return to this example in Sect. 6 below.

  9. More precisely, we refer to a paradoxical pair (p, f) as ‘proper’ if there exists a probability function Pr, which satisfies Eq. (1), and which induces the agent’s conditional assessments of likelihood via the equality \(p(n)={ Pr}(R=n+1|L=n)\). As noted above, the simple form of the paradox is not proper in this sense. Examples of proper paradoxical pairs can be found in Nalebuff (1989, pp.  177–178), Broome (1995, pp. 6–7) and Clark and Shackel (2000, pp. 423–424). These examples all use the same payoff function \(f(n)=2^n\). As a result, the conditional probability distribution p that is used in these examples is unnecessarily complex.

  10. Christensen and Utts (1992) provide a Bayesian solution to the simple form of the paradox based on a criticism of the agent’s prior probabilities. They write: “The paradox can be arrived at as the consequence of using an improper “noninformative” prior, and thus reaffirms the dangers of blindly using such priors” (Christensen and Utts 1992, p. 274). While the use of improper priors is not a standard part of Bayesian methodology, the appeal to such priors, in fact, has a long history in Bayesian statistics and many formal rules for prior selection, such as Jeffrey’s rule, yield improper distributions when applied in common statistical settings (see Kass and Wasserman 1996). For a Bayesian argument against the use of improper priors, see Jaynes (1996), ch. 15.

  11. The Bayesian principle of maximum expected utility is sometimes expressed in terms of expected value rather than expected gain. In these terms, the agent would be justified in switching envelopes provided the unconditional expected value that he assigns to the ticket in the right-hand envelope is strictly greater than that which he assigns to the ticket in the left-hand envelope. In finite contexts, the principle of maximizing expected value and that of maximizing expected gain are equivalent. However, in infinite contexts, the two may diverge (see n. 17 below).

  12. For a mathematically rigorous discussion of these issues, the reader is referred to any elementary textbook in analysis, e.g., Rudin (1976), ch. 3.

  13. It is sufficient to show that the terms of the series do not converge in absolute value. Suppose, for contradiction, that they do. Then:

    $$\begin{aligned} \sum _n {|{ Pr}(L=n-1,R=n)\times f(n)|}+{|{ Pr}(L=n,R=n-1)\times (-f(n))|} = \sum _n { Pr}(W=n)\times f(n) \end{aligned}$$

    is a convergent series. This implies that the sequence of terms in the series must converge to zero. But since (p, f) is paradoxical, it follows that \({ Pr}(W=n+1)f(n+1)>{ Pr}(W=n)f(n)\), for all n. Thus, regardless of how the terms are ordered, one will be able to find a subsequence of this sequence which is strictly increasing. Since all the terms are positive, this subsequence (and hence the sequence as a whole) does not converge to zero. Contradiction.

  14. Clark and Shackel (2000, p. 435) show that there exist proper paradoxes for which the agent’s average conditional expected gain has a finite positive value. The authors label such cases as ‘best’ paradoxes, implying that they are somehow more problematic than those paradoxical cases in which this average diverges to infinity. I agree with the point made by Meacham and Weisberg (2003) that no new difficulties are raised by these ‘best’ versions of the paradox, since the proposed error in the agent’s reasoning is not with the assumption that the weighted average of the agent’s conditional expected gains is positive, but rather with the assumption that the agent’s unconditional expected gain can be identified with this average.

  15. Of course, for the same reason, Bayesian decision theory does not imply that the agent ought to be rationally indifferent to the exchange, since the expected gain from switching envelopes is not 0. Thus, in order to justify the agent’s rational indifference, extra-Bayesian principles must be invoked. One such principle, based on the proposal made by Easwaran (2008), would require that a rational agent value switching envelopes at a price equal to the ‘weak’ expected gain of switching, i.e., the number to which a sequence of i.i.d. variables with the common distribution of the gain from switching envelopes converges in probability. This proposal, however, cannot supply a fully general explanation of why the agent ought to be indifferent to the exchange, for there are paradoxical scenarios in which the weak expected gain is not well-defined. This is the case, for example, with respect to the proper paradox described at the end of Sect. 2, since the relevant sequence of i.i.d. variables does not satisfy the weak law of large numbers. An alternative justification of the agent’s rational indifference is offered by Dietrich and List (2005), who posit as an axiom of decision theory an indifference principle which asserts that a rational agent ought to be indifferent between any two random variables which have the same distribution.

  16. In similar fashion, Kadane (2011, p. 278) writes: “In this case, the weak step is going from dominance (“whatever amount x is in your envelope, it is better to switch”) to the unconditional conclusion (“Therefore you don’t need to know x, it is better to switch”). That step is true if the expected utilities of the options are finite. However, here the expected utilities of both choices are infinite, and so the step is unjustified. Indeed, even though if you knew x it would be in your interest to switch envelopes, in the case where you do not know x, switching and not switching are equally good for you”.

  17. In formulating the Bayesian solution this way, I have tacitly attributed to the Bayesian the following principle of rational decision-making:

    1. (EG)

      For any two real-valued random variables X and Y, a rational agent ought to prefer X to Y if the expected gain that comes from choosing X rather than Y, i.e., \(E(X-Y)\), converges absolutely to some strictly positive value.

    One might, however, view the following principle as a more intuitive formulation of the Bayesian principle of maximum expected utility:

    1. (EV)

      For any two real-valued random variables X and Y, a rational agent ought to prefer X to Y if both the expected value of X (i.e., E(X)) and the expected value of Y (i.e., E(Y)) converge absolutely, and \(E(X)>E(Y)\).

    The principle (EG) implies (EV), but not vice versa. One reason that a Bayesian might prefer the more restrictive (EG) to the less restrictive (EV) is that (EG) can justify certain intuitions about dominance reasoning that cannot be explained by (EV). For example, only (EG) can explain why it is preferable to play the St. Petersburg’s game for free to playing it for \(\$1\), and why it is preferable to play the Altadena game to playing the Pasadena game (see Nover and Hájek 2004).

  18. The sure-thing principle was first introduced by Savage (1972). In commenting on the principle, Savage remarked that “except possibly for the assumption of simple ordering, I know of no other extralogical principle governing decisions that finds such ready acceptance” (Savage 1972, p. 21). Nevertheless, for various reasons, Savage did not include the sure-thing principle as an axiom of his theory, but instead thought it preferable to regard the principle as “a loose one that suggests certain formal postulates.” For a discussion of how the sure-thing principle can be formalized in a theory of rational preferences, see Gaifman (2013).

  19. One might be concerned that by modifying the scenario in this way, we have altered the problem in some crucial respect. After all, it can sometimes be the case that merely being told that at some later point in time one will come to know the value of a certain unknown quantity, can itself provide one with a reason to act. Suppose, for example, that you are deciding whether or not to purchase a certain mystery novel when you are informed that your talkative and rather indiscreet friend, whom you know to have just finished reading the novel, will be paying you a visit this evening. Anticipating that by the end of the visit, you will know how the mystery is resolved, you decide not to purchase the book. While this may be a perfectly reasonable thing to do, this is only because the enjoyment that you will derive from reading the novel depends not just on how the book ends, but also on whether or not you know how the book ends prior to reading it. In the context of the exchange paradox, this is not the case since it is assumed that all the agent cares about is money, so that apart from its potential monetary implications, no value is attached to his coming to know the number of his ticket. For a discussion of apparent violations of the sure thing principle that are owing to the value attached to knowing a certain quantity, see Aumann et al. (2005).

  20. Many philosophers argue for a sharp distinction between synchronic constraints on an agent’s conditional reasoning, and diachronic constraints on that agent’s reasoning after he has come to learn that a certain condition has been satisfied (see, e.g., Levi 1987). Most, however, would be willing to acknowledge that if an agent is committed to the claim that, conditional on a given assumption, he ought to prefer one act to another, then he ought to prefer that act to the other, were he to come to know that the assumption is true, provided that he has not acquired any additional relevant information.

  21. Dietrich and List (2005) argue that the intuitive force for the sure-thing principle derives from the ‘state-wise’ dominance principle according to which, if one act is to be preferred to the other in every possible state of the world, then it is to be preferred to the other unconditionally. The crucial difference between this state-wise dominance principle and the sure-thing principle is that the latter only requires that the preference obtain for each event in a partition of the space of possible states of the world. If the partition is not maximally refined, then these events may be realized by states of the world which are such that, in those states, the preference would be reversed. Dietrich and List (2005) claim that in these cases the sure-thing principle loses its intuitive force. On this point, I disagree. The intuitive force of the sure-thing principle, in my view, derives from the irrationality of the ritual act of refraining from adopting the preferences one knows one would form on the basis of a certain piece of information unless and until one has that information in hand. This sort of intuition can be used to support not just a state-wise but an event-wise reading of the sure-thing principle.

  22. Suppose that the exchange paradox is viewed as a game between two players, each of whom is arbitrarily assigned one of the two envelopes. The fact that the agents have reason to swap envelopes may be seen as presenting a counterexample to the “no-trade” theorem, first established by Milgrom and Stokey (1982). This theorem asserts that no trade is possible between ideally rational agents who possess common knowledge that resources have been distributed pareto optimally. As was the case with respect to the sure-thing principle, a Bayesian might respond to the paradox by claiming that the no-trade theorem, understood as a corollary to the principle of maximum expected utility, is only valid in finite settings where it can be assumed that the agent’s preferences will be preserved under disjoint unions (cf. Rubinstein and Wolinsky 1990). A similar counter to the Bayesian could then be made by asserting that the no-trade theorem reflects a more fundamental intuition about rational behavior than what is captured by the restricted finitary version of the principle that can be established within the Bayesian framework.

  23. This way of justifying the agent’s decision to switch is discussed in Clark and Shackel (2003). In citing the long-run returns that the agent acquires from switching, I do not mean to suggest that the Bayesian principle of maximum expected utility derives its ultimate justification from considerations of long-run strategic efficiency. Such consideration can, however, help to sharpen our intuitions and to highlight the structural assumptions that underlie the application of Bayesian methods.

  24. This is reflected in the fact that the standard dutch book argument for the rule of Bayesian conditionalization interprets an agent’s conditional probabilities as determining what that agent counts as fair odds on a proposed bet which will be called off in case the conditioning event does not occur (see, e.g., Lewis 1987; Levi 1987).

  25. For a forceful and compelling presentation of this point, see Shafer (1985). Shafer argues that the Bayesian theory of subjective probability only has normative significance against the background of a well-defined ‘protocol’, or set of rules which determine, at every stage of the game, the information and options available to the agent. He emphasizes this point by indicating how the standard dutch book derivation for the rule of conditionalization presupposes a specific protocol, observing that this result “makes clear how thoroughly the Bayesian theory assimilates practical problems to the picture of games of chance, and forces us to ask how this assimilation can be justified” (Shafer 1985, p. 262).

  26. Any number of similar examples could be employed to equal effect in the following discussion. For instance, the principle of total evidence has been used to disambiguate, in similar fashion, Feller’s puzzle of the two boys (Bar-Hillel and Falk 1982), the Monty Hall problem (Selvin 1975), and the Sleeping Beauty Problem (Bovens and Ferreira 2010), to name just a few.

  27. In order to arrive at this conclusion, it need not be the case that the passerby chooses which of the two cards to observe on the basis of a fair coin toss. It is enough to justify the agent’s decision to accept the bet to assume that the passerby’s choice of which card to observe is statistically independent of the color of the chosen card.

  28. Biased information is the complementary notion of what is referred to by Shafer (1985) as ‘exact evidence’. This is evidence which, in the context of a particular game of chance, describes nothing more or less than that the agent is at a particular stage of the game. As Shafer acknowledges, not all information need be exact, and to treat a given piece of information as such is simply to “draw an analogy with the evidence we would have for a particular outcome in a particular game of chance” (Shafer 1985, p. 268). Shafer does not further discuss the issue of how we ought to act on the basis of inexact, or biased information. Biased information may also be usefully compared with ‘inadmissible’ information, first introduced by Lewis (1987). Inadmissible information is information which can effect a divergence between our subjective credence in a given proposition, and our belief in the objective chance of the event described by that proposition. Similarly, biased information is information which can effect a divergence between our preference for a certain action and our belief as to the long-run optimality of the strategy implemented by that action.

  29. Compare this point with the following remark by Shafer (1985): “We do not always have protocols for new information in practical problems. This does not mean that we must despair of subjective probability judgment in such problems. But it does mean that the logic of subjective judgment is not as clear and powerful in these problems as we might like. In particular, it means that it is not normative to use the Bayesian theory in such problems. Alongside the Bayesian theory, which draws an imperfect analogy between practical problems and a game of chance, we should consider other theories, which rely on analogies to other canonical examples” (Shafer 1985, p. 262).

  30. Interestingly enough, it turns out that there do exist certain variations of this game in which it can be shown that the most efficient strategy is not that which instructs the agent to switch regardless of the number of the observed ticket. See, e.g., McDonnell et al. (2011). These analyses measure the efficiency of a strategy in terms of its weak expected gain, as defined in Easwaran (2008). The sub-optimality of always switching in certain paradoxical settings has been confirmed by computer simulations (McDonnell and Abbott 2009; Eckhardt 2013, ch. 8).

  31. This is an important feature of the exchange paradox that distinguishes it from other examples in which one must reason on the basis of biased information. In this particular case, the biased information that one receives is relevant for assessing what information one would have received from a different, but equally legitimate source.

  32. This marks one difference between the exchange paradox and the example of the contradictory forecasts described at the end of Sect. 4. Because, in this case, the agent is engaged in what is effectively a zero-sum game against his counterfactual self, it makes sense to treat his counterfactual expected gains as his own expected losses. With respect to the example of the contradictory forecasts, however, this approach obviously does not make sense. So, in the context of that example, some other model would be needed to formalize the instruction to ‘take into account’ what one would have thought had one received the report of the other forecaster. I thank an anonymous referee of an earlier version of this paper for bringing this point to my attention.

  33. More specifically, if \(p(n)=1/2\) and \(f(n)=c^{n}\,(c>1)\), then \(L_{m}(n)=k^{m}L_{0}(n)\), where \(k=1-\left( \frac{c^{2}+1}{2c}\right) \). Thus, if \(c<2+\sqrt{3}\), \(L(n)=0\), for all n, otherwise L(n) is everywhere undefined. This marks a critical point between convergent and divergent oscillation. If we wish to ensure convergence in a case in which f grows at a rate faster than \(c^n\), we could do so by increasing, in corresponding measure, the coefficient of damping. In the context of the model this would amount to assigning a smaller relative weight to one’s counterfactual expectations when correcting for the bias in one’s initial judgments.

  34. Since this specific example is merely illustrative, we omit the proof of this claim.

  35. It may seem like a very strange result that the agent should only switch were he to suppose that his envelope contains an odd-numbered ticket. I cannot give any intuitive justification for this result. It is perhaps worth noting, however, that the prior probability distribution which underlies this result is itself asymmetrical in terms of the parity of the winning ticket. Specifically, this distribution has a unimodal form which peaks at the even value 0. This shape of the distribution is what gives rise to the strange effect, for the typical pattern of damped oscillation proceeds as normal until the point at which the agent is either forced to entertain either the possibility that the right-hand envelope contains ticket number 0; or the possibility that, had be been informed of contents the right-hand envelope, then he would have judged it possible that the left-hand envelope contains ticket number 0; or the possibility that, had be been informed of the contents of the right-hand envelope, he would have judged it possible that, had he been informed of the contents of the left-hand envelope, he would have judged it possible that the right-hand envelope contains ticket number 0, etc. At this point, the agent’s expected gain starts to diverge in one direction or the other.

  36. In general, if the number of the winning ticket has a lower bound of zero, then if the agent assumes that his own envelope contains ticket number n, he will only be able to revise his expectations in accordance with the above methodology n times before the assumption that both agent’s have equally legitimate expectations can no longer be sustained. This limitation of the method, in fact, serves to highlight a rather interesting point, namely, that while it is often the case that two agents have common knowledge that their respective information was obtained from equally legitimate sources, it is almost never the case that they have common knowledge that the conclusions that they form on the basis of this information are equally legitimate.

References

  • 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.

  • Bar-Hillel, M., & Falk, R. (1982). Some teasers concerning conditional probabilities. Cognition, 11(2), 109–122.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Broome, J. (1995). The two-envelope paradox. Analysis, 55(1), 6–11.

    Article  Google Scholar 

  • Christensen, R., & Utts, J. (1992). Bayesian resolution of the “exchange paradox”. The American Statistician, 46(4), 274–276.

    Google Scholar 

  • Clark, M., & Shackel, N. (2000). The two-envelope paradox. Mind, 109(435), 415–442.

    Article  Google Scholar 

  • Clark, M., & Shackel, N. (2003). Decision theory, symmetry and causal structure: Reply to meacham and weisberg. Mind, 112(448), 691–701.

  • de Finetti, B. (2017). Theory of probability: A critical introductory treatment. Hoboken: Wiley.

    Book  Google Scholar 

  • Dietrich, F., & List, C. (2005). The two-envelope paradox: An axiomatic approach. Mind, 114(454), 239–248.

    Article  Google Scholar 

  • Douven, I. (2007). A three-step solution to the two-envelope paradox. Logique et Analyse, 50(200), 359–365.

    Google Scholar 

  • Easwaran, K. (2008). Strong and weak expectations. Mind, 117(467), 633–641.

    Article  Google Scholar 

  • Eckhardt, W. (2013). Paradoxes in probability theory. Berlin: Springer.

    Book  Google Scholar 

  • Gaifman, H. (2013). The sure thing principle, dilations, and objective probabilities. Journal of Applied Logic, 11(4), 373–385.

    Article  Google Scholar 

  • Jaynes, E. T. (1996). Probability theory: The logic of science. Louis, MO: Washington University in St. Louis.

    Google Scholar 

  • Kadane, J. B. (2011). Principles of uncertainty. Boca Raton: CRC Press.

    Book  Google Scholar 

  • Kass, R. E., & Wasserman, L. (1996). The selection of prior distributions by formal rules. Journal of the American Statistical Association, 91(435), 1343–1370.

    Article  Google Scholar 

  • Levi, I. (1987). The demons of decision. The Monist, 70(2), 193–211.

    Article  Google Scholar 

  • Lewis, D. (1987). A subjectivist’s guide to objective chance. In Philosophical papers volume II. New York: Oxford University Press.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • McGrew, T. J., Shier, D., & Silverstein, H. S. (1997). The two-envelope paradox resolved. Analysis, 57(1), 28–33.

    Article  Google Scholar 

  • Meacham, C., & Weisberg, J. (2003). Clark and Shackel on the two-envelope paradox. Mind, 112(448), 685–689.

    Article  Google Scholar 

  • Milgrom, P., & Stokey, N. (1982). Information, trade and common knowledge. Journal of Economic Theory, 26(1), 17–27.

    Article  Google Scholar 

  • Nalebuff, B. (1989). Puzzles: The other person’s envelope is always greener. The Journal of Economic Perspectives, 3(1), 171–181.

    Article  Google Scholar 

  • Norton, J. (1998). Where the sum of our expectation fails us: The exchange paradox. Pacific Philosophical Quarterly, 79(1), 34–58.

    Article  Google Scholar 

  • Nover, H., & Hájek, A. (2004). Vexing expectations. Mind, 113(450), 237–249.

    Article  Google Scholar 

  • Rubinstein, A., & Wolinsky, A. (1990). On the logic of “agreeing to disagree” type results. Journal of Economic Theory, 51(1), 184–193.

    Article  Google Scholar 

  • Rudin, W. (1976). Principles of mathematical analysis. In International series in pure & applied mathematics. McGraw-Hill Publishing Co.

  • Savage, L. J. (1972). The foundations of statistics. Mineola, New York: Dover Publications.

    Google Scholar 

  • Scott, A. D., & Scott, M. (1997). What’s in the two envelope paradox? Analysis, 57(1), 34–41.

    Article  Google Scholar 

  • Selvin, S. (1975). On the monty hall problem. The American statistician, 29(3), 134–134.

    Article  Google Scholar 

  • Shafer, G. (1985). Conditional probability. International Statistical Review = Revue internationale de statistique, 53(3), 261–275.

    Google Scholar 

  • Sobel, J. H. (1994). Two envelopes. Theory and Decision, 36(1), 69–96.

    Article  Google Scholar 

  • Wagner, C. G. (1999). Misadventures in conditional expectation: The Two-Envelope problem. Erkenntnis, 51(2–3), 233–241.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Philosophy, University of Chicago, 1115 E. 58th Street, Chicago, IL, 60637, USA

    Anubav Vasudevan

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Correspondence to Anubav Vasudevan.

Appendix

Appendix

In this appendix, we prove the following claim: If \(p(n)=1/2\) and f(n) is a polynomial function of order k, then \(L_{m}(n)=0,\) for all n and for all \(m>k\).

Lemma 1

If \(p(n)=1/2\) and f(n) is a polynomial of order k, then

$$\begin{aligned} L_{m}(n)=\sum _{k=0}^{2m+1}\left( {\begin{array}{c}2m+1\\ k\end{array}}\right) \frac{(-1)^{k+m+1}f(n+k-m)}{2^{m+1}} \end{aligned}$$

Proof

The proof proceeds by induction on m. If \(m=0\), the theorem is easily verified. Suppose the theorem holds for all numbers less than m (\(m\ge 1\)). Then:

$$\begin{aligned} L_{m}(n)= & {} L_{m-1}(n)-\frac{1}{2}\left( L_{m-1}(n-1)+L_{m-1}(n+1)\right) \\= & {} \frac{1}{2^{m}}\left( \sum _{k=0}^{2m-1}\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) (-1)^{k+m}f(n+k-m+1)\right) \\&-\frac{1}{2^{m+1}}\left( \sum _{k=0}^{2m-1}\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) (-1)^{k+m} f(n+k-m)\right) \\&-\frac{1}{2^{m+1}}\left( \sum _{k=0}^{2m-1}\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) (-1)^{k+m} f(n+k-m+2)\right) \\= & {} \frac{1}{2^{m+1}}\left\{ 2\left( \sum _{k=1}^{2m} \left( {\begin{array}{c}2m-1\\ k-1\end{array}}\right) (-1)^{k+m+1}f(n+k-m)\right) \right\} \\&+\frac{1}{2^{m+1}}\left( \sum _{k=0}^{2m-1}\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) (-1)^{k+m+1} f(n+k-m)\right) \\&+\frac{1}{2^{m+1}}\left( \sum _{k=2}^{2m+1}\left( {\begin{array}{c}2m-1\\ k-2\end{array}}\right) (-1)^{k+m+1}f(n+k-m)\right) \end{aligned}$$

Regrouping the terms according to the argument of f, we have:

$$\begin{aligned} L_{m}(n)= & {} \frac{1}{2^{m+1}}\left\{ (-1)^{m+1}f(n-m)+(-1)^{m} (2m+1)f(n-m+1)\right\} \\&+\frac{1}{2^{m+1}}\left\{ (-1)^{m+1}(2m+1)f(n+m)+(-1)^{m}f(n+m+1)\right\} \\&+\frac{1}{2^{m+1}}\left\{ \sum _{k=2}^{2m-1}(-1)^{k+m+1}f(n+k-m)\left[ 2\left( {\begin{array}{c}2m-1\\ k-1\end{array}}\right) \right. \right. \\&\left. \left. +\,\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) +\left( {\begin{array}{c}2m-1\\ k-2\end{array}}\right) \right] \right\} \end{aligned}$$

From the recursive definition of the binomial coefficient:

$$\begin{aligned} 2\left( {\begin{array}{c}2m-1\\ k-1\end{array}}\right) +\left( {\begin{array}{c}2m-1\\ k\end{array}}\right) +\left( {\begin{array}{c}2m-1\\ k-2\end{array}}\right)= & {} \left[ \left( {\begin{array}{c}2m-1\\ k\end{array}}\right) +\left( {\begin{array}{c}2m-1\\ k-1\end{array}}\right) \right] \\&+\left[ \left( {\begin{array}{c}2m-1\\ k-1\end{array}}\right) +\left( {\begin{array}{c}2m-1\\ k-2\end{array}}\right) \right] \\= & {} \left( {\begin{array}{c}2m\\ k\end{array}}\right) +\left( {\begin{array}{c}2m\\ k-1\end{array}}\right) \\= & {} \left( {\begin{array}{c}2m+1\\ k\end{array}}\right) \end{aligned}$$

Substituting back into the above, we obtain the desired result. \(\square \)

The theorem follows from Lemma 1 and the fact that for any polynomial P(n) of order \(<k\):

$$\begin{aligned} \sum _{j=0}^{k}(-1)^{j}\left( {\begin{array}{c}k\\ j\end{array}}\right) P(j)=0. \end{aligned}$$

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Vasudevan, A. Biased information and the exchange paradox. Synthese 196, 2455–2485 (2019). https://doi.org/10.1007/s11229-017-1550-5

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