Abstract
Adam Elga (Philosophers’ Imprint, 10(5), 1–11, 2010) presents a diachronic puzzle to supporters of imprecise credences and argues that no acceptable decision rule for imprecise credences can deliver the intuitively correct result. Elga concludes that agents should not hold imprecise credences. In this paper, I argue for a two-part thesis. First, I show that Elga’s argument is incomplete: there is an acceptable decision rule that delivers the intuitive result. Next, I repair the argument by offering a more elaborate diachronic puzzle that is more difficult for imprecise Bayesians to avoid.
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Notes
Elga (2010) does not explicitly characterize Strict Rules using the schema above. Instead, he gestures towards this characterization with the midpoint rule. As Moss (ms., Appendix) argues, however, the midpoint rule endorses incoherent synchronic betting behavior.
And these actions remain permissible even if the payoff of the bet is slightly increased.
This rule is a synchronic version of the Caprice rule defended in Weatherson (ms. a).
See, most prominently, McClennen (1990).
Those familiar with the literature will note that the decision rule Planning above differs from two other diachronic decision rules discussed by Elga (2010): Narrowing and Sequence proposals. Narrowing proposals require agents to change their credal state after acting on the first bet. Such rules are antithetical to the evidentialist motivations of many imprecise Bayesians and are in violation of Separability. Sequence rules evaluate sequences of action for rationality directly; they include irreducible diachronic ought claims that take wide-scope over sequences of actions without issuing verdicts on the permissibility of individual actions. Sequence proposals raise a host of thorny questions. For my purposes here, I will take it for granted that such rules are off-the-table: a decision rule must issue in verdicts of the permissibility of an agent’s decision at a particular time.
After drafting this paper, I’ve discovered other authors working on related projects. Chandler (forthcoming), Sahlin and Weirich (2013) and Bradley and Steele (ms.) each argue that, using backwards induction, the popular decision rule \(\Upgamma\)-MaxiMin requires Elga’s agent to accept both bets. (The rule, however, has well-known problems of its own and, most importantly, is not a ‘permissive’ rule which is a property favored by most imprecise Bayesians.) In their independent manuscript, Bradley and Steele also allude briefly to a rule similar to the one I explore below.
∃x, y, .. such that (α i , x, y, ...) = s i \(\in \) A i .
There is some sequence of future permissible acts, (α p i+1 , \({{\upalpha}^{p}_{i+2}}\), ...) such that (α i , α p i+1 , \({{\upalpha}^{p}_{i+2}}\), ...) \(\notin\) A i . Note the appeal to future permissible acts is not circular, as there will be some last action in the sequence of acts.
For all sequences s i+1 of future permissible acts, \((\upalpha_{i}^{\prime})\bowtie s_{i+1} = s_{i} \in A_{i}.\)
Indeed, even accepting both bets is not what she objectively ought to have done—depending on whether p is true or false, she would have maximized her ends by accepting either Bet 1 or Bet 2 (but not both).
Thanks to Sarah Moss and Brett Topey for raising this objection.
Thanks to Tom Dougherty for raising this example and Miriam Schoenfield for pushing me on it.
See Hedden (ms., Sect. 4) for a discussion of this feature of diachronic Dutch Books. (As a ‘time-slice rationalist’, however, Hedden takes this observation to undercut support for diachronic Dutch Book arguments.)
This is a complaint levied against Resolute Choice theorists. See Kavka (1983) for an initial discussion.
A route may exist between Forward Looking and a sort of consequentialist justification. Although unacceptable sequences cannot always be avoided (see Sect. 5), the consequences for an agent that avoids sequence-dominated actions are typically better than an agent that does not, in that the former often avoids completing an unacceptable sequence. So, there may be a rule-consequentialist justification in the offing for a defender of Forward Looking. (Thanks to Alex Guerrero for discussion here. See also Buchak (ms., Chapter 6) for more on consequentialist justifications and dynamic choice theories.)
More carefully, I prove the following claim in the Appendix: Presented with a series B = (b 0, \(b_{0}^{\prime}\), b 1, \(b_{1}^{\prime}\)..., b n , \(b_{n}^{\prime}\)) of choices between bets on the truth of some proposition q over time, a miser (who is certain she is in conditions C) that follows Forward Looking will always enact a sequence of actions that is admissible at t = 0.
c(q) = 1 recommends opening Door 4; c(q) = 0 recommends opening Door 5; c(q) = .5 recommends opening Door 6
Although I don’t explore them here, there are plenty of other ways in which to complicate Annie’s decision problem so that the limited result of the Appendix does not apply. For instance: the bets could be on different propositions, Annie’s action with respect to past choice point could affect the utility associated with future bets, or the choices Annie makes in the past could affect which choices she faces in the future. As an instance of the last, consider an agent with two bets b 1 and b 2 that are incomparable in the sense that each maximizes expected utility according to some member of the agent’s representor. At t 1, the agent must choose to sweeten exactly one of b 1 or b 2 by adding a small bonus prize to the bet – although the bets remain incomparable after sweetening. At t 2, the agent chooses between the newly sweetened bet and the alternative unsweetened bet. Here it is difficult to see how forward coordination could prevent the agent from choosing an unsweetened bet. (Many thanks to Seamus Bradley for pointing out this example to me. A similar puzzle for imprecise preferences appears in Hedden (2013, Sect. 2.4)).
Weisberg (forthcoming) presents an argument to this end. In the case of precise agents, susceptibility to Dutch Books is thought to demonstrate that the agent is committed to regard as fair a series of bets that are clearly not fair. Because our credences are thought to fix our commitments of fairness of bets, the inconsistency in our commitments —brought to the fore by Dutch Book susceptibility—is emblematic of an inconsistency of our credences. Weisberg argues that, in the case of imprecise agents, this last step of this argument fails. Roughly, he argues that imprecise credences often underdetermine commitments to the fairness of bets—imprecise agents may be undecided as to whether or not a particular bet is fair. The choices of an imprecise agent, then, do not correspond with commitments to the fairness of bets.
Consider some contrived cases in which Annie is absolutely certain about her action with respect to Bet 2. If she is certain she will reject, it may seem strange to treat the case differently than a case in which she is presented with only the first bet. If she is certain that she will accept, it may seem unduly harsh to require Annie to accept the first bet. (Thanks to Carrie Ichikawa Jenkins and Jeff Russell for discussion here.)
For instance: one result of the rule is that the order in which the bets are presented to Annie affects which bets Annie must accept. Some have found this oddity objectionable (although I don’t). (Thanks to Joshua Schechter for discussion here.)
When I talk of θ i as a real number instead of as a probability function, I should be understood as referring to the pivot point at i (i.e. θ i evaluated for the relevant proposition q).
Fig. 2 
Pivot point of Bet i
References
Bradley, S., & Steele, K. (ms.). Subjective probabilities need not be sharp.
Buchak, L. (forthcoming). Risk and rationality. Oxford University Press.
Chandler, J. (forthcoming). Subjective probabilities need not be sharp. Erkenntnis.
Dorr, C. (2010). The eternal coin: A puzzle about self-locating conditional credence. Philosophical Perspectives, 24(1), 189–205.
Elga, A. (2010). Subjective probabilities should be sharp. Philosophers’ Imprint, 10(5), 1–11.
Hedden, B. (ms.). Time-slice rationality.
Hedden, B. (2013). Options and diachronic tragedy. Philosophy and Phenomenological Research. doi:10.1111/phpr.12048.
Ismael, J. (2012). Decision and the open future. In Adrian B. (Ed.), The future of the philosophy of time (pp. 149–168). Oxford: Oxford University Press.
Joyce, J. (2011). A defense of imprecise credences in inference and decision making. In T. S. Gendler & J. Hawthorne (Eds.), Oxford studies in epistemology (Vol. 4). Oxford: Oxford University Press.
Kavka, G. S. (1983). The toxin puzzle. Analysis, 43(1), 33–36.
McClennen, E. F. (1990). Rationality and dynamic choice. Cambridge: Cambridge University Press
Moss, S. (ms.). Credal dilemmas.
Sahlin, N.-E., & Weirich, P. (2013). Unsharp sharpness. Theoria. doi:10.1111/theo.12025.
Weatherson, B. (ms. a). Decision making with imprecise probabilities.
Weatherson, B. (ms. b). Lecture notes on game theory.
Weisberg, J. (forthcoming). You’ve come a long way, Bayesians. Journal of Philosophical Logic.
Acknowledgments
This paper has benefited greatly from conversations and comments from many friends and teachers. Thanks to Harjit Bhogal, Rachel Briggs, Lara Buchak, Tom Dougherty, James Joyce, Jason Konek, Leon Leontyev, Miquel Miralbes Del Pino, Miriam Schoenfield, Daniel Singer, Eric Swanson, and Brett Topey. Special thanks to Seamus Bradley, Dmitri Gallow, Carrie Ichikawa Jenkins, Joshua Schechter, Sarah Moss, and Brian Weatherson. I am also grateful to audiences at the 2012 Australasian Association of Philosophy Conference, the 2012 Brown University Shapiro Graduate Philosophy Conference, the University of Pennsylvania and, especially, to participants of the Fifth Formal Epistemology Festival and the 2013 Bellingham Summer Philosophy Conference.
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Appendix: proof
Appendix: proof
In this Appendix, I show that Forward Looking cannot be subjected to simple diachronic puzzles. More carefully, we show:
Main Claim:
Presented with a series B = (b 0, \(b_{0}^{\prime}\), b 1, \(b_{1}^{\prime}\)..., b n , \(b_{n}^{\prime}\)) of choices between bets on the truth of some proposition q over time, a miser (who is certain she is in conditions C) that follows Forward Looking will always enact a sequence of actions that is admissible at t = 0.
Proof
Suppose the agent’s representor is c r(q) = [λ, η]. Call the probability functions that are members of the agent’s representor her ‘credal members’. Note that each choice at i between b i or \(b_{1}^{\prime}\) is associated with a ‘pivot point’ that bifurcates the unit interval such that one bet b i maximizes expected utility according to all the credal members that assign probability to q greater than that pivot point. The alternative bet maximizes expected utility according to all members that assign probability less than that pivot point. Either action maximizes expected utility according to the member that assigns probability equal to that pivot point (Fig. 2). Let θ i denote this member of the agent’s representor that assigns probability equal to the pivot point at i. Footnote 22
Either (a) the pivot point for no choice points are in the agent’s representor or (b) there is some last choice point, l such that the pivot point for that bet is in the agent’s representor \(\theta_l \in [\lambda , \eta ] \). If (a) the result is trivial: all members of the agent’s representor recommend the same action for each choice point, so that sequence of actions will be enacted and is admissible. If (b), we show the following by induction:
Lemma:
Any action that is permissible maximizes expected utility according to θ l .
Base Case
For any choice point i ≥ l, any action that is permissible maximizes expected utility according to θ l .
With respect to choice l, both actions maximize expected utility according to θ l . With respect to choices i after l, by assumption θ i \(\notin\) [λ, η]. Thus, for each of these choices, there is a unique permissible action and the sequence of those actions maximizes expected utility according to all members of the agent’s representor. Thus, that sequence maximizes expected utility according to θ l .
Induction Step
Suppose for choices i > m, any action that is permissible maximizes expected utility according to θ l . Then, for choice m any action that is permissible maximizes expected utility according to θ l .
There are three cases: either (i) θ m \(\notin \) [λ, η] (ii) θ m = θ l or (iii) θ m \(\in \) [λ, η] and θ m ≠ θ l . Cases (i) and (ii) are trivial. If θ m \(\notin\) [λ, η] then all members of the agent’s representor recommend the same action with respect to choice m, so that unique permissible action is recommended according to θ l . If θ m = θ l then both actions with respect to choice m maximize expected utility according to θ m and thus both actions maximize expected utility according to θ l .
In case (iii), some members of the agent’s representor recommend the same action that maximizes expected utility according to θ l ; some recommend the action that does not maximize expected utility according to θ l (Fig. 3). However, by the induction hypothesis, the action that maximizes expected utility according to θ l guarantees the agent an admissible sequence of actions (because all of the future actions will maximize expected utility according to θ l \(\in \) [λ, η]). And the alternative action can be combined with one of the permissible actions at choice l to form an inadmissible sequence. Thus the action that maximizes expected utility according to θ l sequence-dominates the alternative action.
Case (iii)
This proves the Lemma. The Main Claim follows immediately: all permissible actions maximize expected utility according to θ l , thus the sequence of actions that is enacted is admissible.
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Sud, R. A forward looking decision rule for imprecise credences. Philos Stud 167, 119–139 (2014). https://doi.org/10.1007/s11098-013-0235-2
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DOI: https://doi.org/10.1007/s11098-013-0235-2