Abstract
Consequentialists often assume rational monism: the thesis that options are always made rationally permissible by the maximization of the selfsame quantity. This essay argues that consequentialists should reject rational monism and instead accept rational pluralism: the thesis that, on different occasions, options are made rationally permissible by the maximization of different quantities. The essay then develops a systematic form of rational pluralism which, unlike its rivals, is capable of handling both the Newcomb problems that challenge evidential decision theory and the unstable problems that challenge causal decision theory.
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Notes
For reasons discussed in Sect. 5.3, it is the stable maximization, not the mere maximization, of a quantity that makes options rationally permissible.
See e.g. Rawls (1971).
See e.g. Buchak (2013).
This characterization assumes that we have set nonideal agents aside.
Other discussions of Newcomb problems and/or unstable problems include: Ahmed (2012, 2014a, b), Arntzenius (2008), Bales (2018), Bassett (2015), Briggs (2010), Eells (1982), Eells and Harper (1991), Gallow (2020), Gibbard and Harper (1978), Gustafsson (2011), Hare and Hedden (2016), Horgan (1981), Hunter and Richter (1978), Jeffrey (1983), Joyce (1999, 2012, 2018), Lewis (1981), Nozick (1969), Oddie and Menzies (1992), Rabinowicz (1988, 1989), Skyrms (1982, 1984, 1990), Spencer and Wells (2019) Stalnaker (1981), Wedgwood (2013), Weirich (1985, 1988, 2004), and Wells (2019).
For a defense of two-boxing, see Spencer and Wells (2019).
This example is from Spencer and Wells (2019: 34). It’s assumed that the agent is unable to randomize their choice.
This example is from Spencer and Wells (2019: 35).
The two relevant dependency hypotheses are \(k_W\), which says that the white box contains $100, and \(k_B\), which says that the black box contains $100, and: \(u(a_{LW}k_W) = 100 < 105 = u(a_{RW}k_W)\); \(u(a_{LW}k_B) = 0 < 5 = u(a_{RW}k_B)\); \(u(a_{LB}k_W) = 0 < 5 = u(a_{RB}k_W)\); and \(u(a_{LB}k_B) = 100 < 105 = u(a_{RB}k_B)\).
The proof draws on and is heavily indebted to Ahmed (2012).
There is a delicate question about when an option survives a restriction. I am inclined toward what J. Dmitri Gallow calls the simple view: that option \(a_n\) survives the elimination of some options just if some post-restriction option, \(a_m\), is such that, for any \(k \in K\), \(u(a_nk) = u(a_mk)\) and \(C(k|a_n) = C(k|a_m)\). One might wonder whether there are any decision problems that witness K-Selection and also permit restriction. I am convinced that there are. One (rather complicated) example is a variation on The Semi-Frustrater.
As before, there is a white box and a black box. One contains $100; the other contains $0. The agent gets the contents of the box they point to, plus $5 if they point right-handedly. It has not been settled whether the agent will have four options, being able to point to either box with either hand, or just two options, being forced to point either left-handedly or right-handedly to a selected box. The Semi-Frustrater has made two predictions. They predicted which box the agent would point to if choosing from four options, placing $100 in the box they predicted the agent would not point to, and they also predicted whether the agent would point left-handedly or right-handed if choosing from two options. If they predicted that the agent would point left-handedly if choosing from two options, then they flipped a fair coin, and selected the box that contains $100 if the coin landed heads. If they predicted that the agent would point right-handedly, then they selected the box that contains $0. The Semi-Frustrater’s predictions about whether an agent would point left-handedly or right-handedly to a selected box are 90%-reliable, whichever option is chosen. The agent knows all of this. Moreover, as the agent will learn if and only if they end up having two options, the white box has been selected.
If Q-monism is true, then, relative to the four-option decision problem, \(Q(a_{LW}) > Q(a_{RW})\). Eliminating the options that correspond to the black box leaves \(u(a_{RW}k)\), \(u(a_{LW}k)\), \(C(k|a_{RW})\), and \(C(k|a_{LW})\) the same, for every \(k \in K\), so the options corresponding to the white box survive the restriction. If Q is independent, then \(Q(a_{LW}) > Q(a_{RW})\) relative to the two-option decision problem. But K-Selection entails that \(a_{RW}\) is the only rationally permissible option relative to the two-option decision problem.
Cf. Sen (1970).
Cf. Jeffrey (1983).
We can also formulate V-ratificationism as a form of rational pluralism. The monistic and pluralistic formulations of V-ratificationism do not differ extensionally, but they differ metaethically—see Sect. 4.3.
If an option strictly K-dominates every other option, then it is the only ratifiable option.
One dependent monism on offer is defended by Wedgwood (2013). Wedgwood defends B-monism, where \(B(a) = \sum _K ( C(k|a)(u(ak) - \frac{\sum _A u(ak)}{\#A}))\). Three Shells is a counterexample to B-monism. In Three Shells, no matter what the agent’s credences are:
$$\begin{aligned}&B(a_A) = \sum _K \left( C(k|a_A)\left( u(ak) - \frac{\sum _A u(ak)}{\#A}\right) \right) \approx (1)\left( 5 - \frac{5}{3}\right) + (0)\left( 0 - \frac{19}{3}\right) + (0)\left( 0 - \frac{19}{3}\right) = \frac{10}{3};\\&B(a_B) = \sum _K \left( C(k|a_B)\left( u(ak) - \frac{\sum _A u(ak)}{\#A}\right) \right) \approx (0)\left( 0 - \frac{5}{3}\right) + (1)\left( 9 - \frac{19}{3}\right) + (0)\left( 10 - \frac{19}{3}\right) = \frac{8}{3} \end{aligned}$$; and
$$\begin{aligned} B(a_C) = \sum _K \left( C(k|a_C)\left( u(ak) - \frac{\sum _A u(ak)}{\#A}\right) \right) \approx (0)\left( 0 - \frac{5}{3}\right) + (0)\left( 10 - \frac{19}{3}\right) + (1)\left( 9 - \frac{19}{3}\right) = \frac{8}{3}. \end{aligned}$$Gallow (2020) defends a different form of dependent monism. I have argued elsewhere that Gallow's preferred form of dependent monism also admits of counterexamples.
The most sustained argument against optimism is Briggs’ (2010). Briggs argues that any adequate decision theory must verify two principles—a Pareto principle and a self-sovereignty principle—and then proves that no decision theory can verify both. I think that an adequate decision theory must falsify both: the Pareto principle is refuted by The Semi-Frustrater, and the self-sovereignty principle is refuted by Three Shells.
Some representative quotations:
For an alternative way of trying to answer the problem of consequentialists credentials, see Hammond (1988).
For an orthogonal critique of using representation theorems to answer the problem of consequentialist credentials, see Meacham and Weisberg (2011).
The representation theorem at best establishes a claim of coextensionality, so those with reductive ambitions must combine the representation theorem with some additional principles. Representation theorems do not play any role in my preferred way of solving the problem of consequentialist credentials, so I won’t hazard a guess at what the additional principles might be.
Familiar representation theorems have been monistic; they aim to arrive at a representing quantity. But as a helpful reviewer points out, one could, in principle, attempt to prove a pluralistic representation theorem, which purports to show that rational agents act as if they maximize one quantity in one sort of circumstance and act as if they maximize another quantity in another sort of circumstance. It would be interesting to explore the prospects of pluralistic representation theorems, but I shall not do so here.
For example, I have not yet been able to find any (remotely plausible) way of d-scoring quantities that (a) entails that V weakly D-dominates U and (b) does not entail that the d-score of V sometimes exceeds the d-score of actual value. A method of d-scoring quantities cannot be adequate unless in ensures that the d-score of actual value is never exceeded, so this amounts to an outstanding challenge to V-enthusiasts.
Proof: Let \(\alpha\) be the quantity maximized by exactly the actual value maximizing options at every \(\langle w,d \rangle\). For any quantity Q and any \(\langle w,d \rangle\), \(@(\alpha ,w,d) \ge (Q,w,d)\), since the average actual value of the options that maximize actual value at \(\langle w,d \rangle\) cannot be less than the average actual value of the options that maximize Q relative to \(\langle w,d \rangle\). Hence, for any d, \(S(\alpha ,d) \ge S(Q,d)\).
If d is some particular decision problem, then \(Q_1\) is d-coincident with \(Q_2\) if, for each \(\langle w, d \rangle\), \(Max(Q_1,w,d) = Max(Q_2,w,d)\). For any quantity Q and any decision d, there are uncountably many quantities that are d-coincident with Q, and every quantity that is d-coincident with Q has the same d-score as Q does.
Or anyway, much of what sets them apart is score. We may also want to impose some formal conditions, like continuity.
A similar conception of guidance is defended in Spencer and Wells (2019: 38–40).
Condition (2) is akin to, but not quite equivalent to, a principle that Hare (2011: 196) calls “Reasons are not Self-Undermining.”
See Spencer and Wells (2019: 38–40).
Note an important distinction here. What condition (2) requires is that it be true and certain relative to \(C^a\) that a maximizes the quantity relative to \(C^a\), not that it be true and certain relative to \(C^a\) that a maximizes the quantity relative to C. Thanks to Arif Ahmed for discussion here.
If the Meta-Frustrater is perfect, then in both examples:
\(V(a_{RW}) = V(a_{RB}) = (105)(0.1) + (5)(0.9) = 15;\) and
\(V(a_{LW}) = V(a_{LB}) = (100)(0.5) + (0)(0.5) = 50.\)
The U-values are sensitive to the agent’s credences over A. If, for example, \(C(a_{RW}) = C(a_{RB}) = C(a_{LW}) = C(a_{RW}) = 0.25\), then, in both examples:
\(U(a_{RW}) = U(a_{RB}) = (105)(0.5) + (5)(0.5) = 55;\) and
\(U(a_{LW}) = U(a_{LB}) = (100)(0.5) + (0)(0.5) = 50.\)
Cf. Lewis (1981).
An option together with the laws and past, insofar as those are beyond the agent’s control, do not always suffice to determine the utility of the world; for sometimes we must also specify parts of the present or future that are beyond the agent’s control. (For example, one can construct a variant of The Frustrater where the Frustrater’s prediction is not causally affected by the agent’s choice but occurs after the agent’s choice. The utility of a world depends on the Frustrater’s prediction, but an option together with the past and laws, insofar as those are beyond the agent’s control, might not settle what the Frustrater predicted, and thus might not settle the utility of the world.) In such cases, we should identify K, not with \(\{lh_1, lh_2, ..., lh_n \}\), but instead with \(\{lhf_1, lhf_2, ..., lhf_m \}\), where each lhf specifies the laws, past, present, and future, insofar as those are beyond the agent’s control. If we identify K with \(\{lhf_1, lhf_2, ..., lhf_m \}\), we gradually coarsen in the same way, removing time-slices by their temporal order, producing ever shorter initial segments, and then finally removing the laws, themselves. Thanks to a helpful reviewer for pressing me on this point.
One alternative I find attractive is purely modal. Each fact is assigned some counterfactual fixity, à la Kment (2014), and we gradually coarsen by progressively removing the facts with the least counterfactual fixity. This purely modal characterization of K is harder to work with, but probably superior.
Assuming that the agent is certain that the prediction was made instantaneously j units prior to the time of decision makes the metaethical transition sudden. For every \(U^i \prec U^j\), \(U^i(a_A) + U^i(a_B) = 100\). And, for every \(U^k \succeq U^j\), \(U^k(a_A) + U^k(a_B) \approx 0\). If we drop the assumption that the agent is certain that the prediction was made j units prior to the time of decision, the metaethical transition might instead be gradual. If the decrease is gradual, then emphasizing stable maximization might be important. In a version of The Frustrater in which the agent is uncertain when the prediction was made, it may be the case that the least member of U that is stably maximized, say, \(U^j\), is maximized both by, say, \(a_A\) and \(a_E\). This sort of co-maximization would not make \(a_A\) rationally permissible, however, because \(a_A\) will not stably maximize \(U^j\). In fact, neither \(a_A\), nor \(a_B\) stably maximize any member of U. If \(a_A\) maximizes some \(U^j\), then \(\sum C(k^j|a_A)u(a_Ak^j) < \sum C(k^j|a_a)u(a_Bk^j)\), since the agent then will regard \(a_A\) as evidence in favor of \(a_B\)-friendly \(k^j\)s. But the co-maximization would make \(a_E\) rationally permissible, since \(a_E\) stably maximizes the least member of U that is stably maximized, whatever that proves to be.
There is one added complication. As Bernhard Salow pointed out to me, according to U-pluralism as formulated, it is essential that the Meta-Frustrater makes his prediction before the minions do. If the minions make their prediction first, then the options that stably maximize the least member of U that is stably maximized will be the left-handed options. I am not sure whether this prediction is wrong. (Flipping the temporal order makes my intuitions less clear.) But when I am inclined to think that flipping the temporal order makes no normative difference, I am inclined, not to abandon U-pluralism, but to adopt an alternative conception of K. See note 44.
Much work on bounded rationality is similarly animated by a constrained optimization conception of rationality. See e.g. Bossaerts and Murawski (2017), Gigerenzer (2008), Griffiths et al. (2015), Griffiths and Tenenbaum (2006), Halpern et al. (2014), Icard (2018), Lorkowski and Kreinovich (2018), Paul and Quiggin (2018), Russell and Subramanian (1995), Simon (1956, 1957, 1983), Vul et al. (2014), and Weirich (1988, 2004).
This paper has been long in the making. I can’t remember all of the people who have helped it along, but they include: Arif Ahmed, Sara Aronowitz, Adam Bales, D. Black, R. A. Briggs, Tyler Brooke-Wilson, David Builes, Nilanjan Das, Kevin Dorst, Kenny Easwaran, Branden Fitelson, J. Dmitri Gallow, Ned Hall, Caspar Hare, Brian Hedden, Wes Holliday, Michele Odisseas Impagnatiello, Boris Kment, Daniel Muñoz, L. A. Paul, Agustín Rayo, Bernhard Salow, Haley Schilling, Miriam Schoenfield, Ginger Schultheis, Robert Stalnaker, P. Quinn White, members of the Rutgers Formal Epistemology and Decision Theory Reading Group (2015), attendees of the Cambridge Decision Theory Workshop (2016), members of the Northeastern Philosophy Reading Group (2017), attendees of the Ranch Workshop (2019) and especially my commentator there, Josh Dever, a helpful reviewer at another journal, and an extraordinarily helpful reviewer at this journal. A special thanks to Marshall Louis Reaves for help with the computer simulation, and an extra special thanks to Ian Wells, who was my co-equal partner as these ideas started taking shape.
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Appendix: Proof of Supervenient Optimality
Appendix: Proof of Supervenient Optimality
The proof of Supervenient Optimality has two parts. First, we show that U weakly D-dominates any supervenient quantity that diverges from U. Then we show that any quantity that is distinct from U, but does not diverge from U, violates a plausible continuity constraint.
If \(Max(Q,w,d) \not \subset Max(U,w,d)\) for any point \(\langle w,d \rangle\), I will say that there is a point of divergence between Q and U. If Q is supervenient and there is a point of divergence between Q and U, then U weakly D-dominates Q. After all, suppose that \(\langle w,d \rangle\) is a point of divergence between Q and U. Since Q and U are both supervenient, the d-score of Q is the average of the U-values of the options in Max(Q, w, d), and the d-score of U is the average of the U-values of the options in Max(U, w, d). Hence, since at least one member of Max(Q, w, d) fails to maximize U relative to d, the average of the U-values of the options in Max(Q, w, d) is strictly less than the average of the U-values of the options in Max(U, w, d). Hence, \(S(Q,d) < S(U,d)\). Moreover, if Q is supervenient, then, for any d, \(S(Q,d) \le S(U,d)\). So it follows that U weakly D-dominates Q. And given my assumption that the ordinal rankings of quantities respect relations of weak D-domination, it follows that U scores higher than does Q.
The supervenient quantities that are distinct from U, but score as highly as U, are subset quantities: quantities that are always maximized by U-maximizing options, but not always maximized by every U-maximizing option. (Think, for example, about the quantity that corresponds to being the leftmost U-maximizing option.) But subset quantities violate an intuitively plausible continuity constraint. If U is a utility function, and \(u(w)=x\), then let \(u^{w,\epsilon }\) and \(u^{w,-\epsilon }\) be utility functions that are exactly like U, except that \(u^{w,\epsilon }(w) = x+ \epsilon\) and \(u^{w,-\epsilon }(w) = x- \epsilon\). If \(d = \langle C,u,A,K \rangle\), then let \(d^{w,\epsilon } = \langle C,u^{w,\epsilon },A,K \rangle\) and let \(d^{w,-\epsilon } = \langle C,u^{w,-\epsilon },A,K \rangle\). The relevant continuity constraint then can be stated as follows:
Utility Continuity. If \(a \notin Max(Q,w,d)\), then, for any world \(w_i\) there is some \(\epsilon\) such that \(a \notin Max(Q,w,d^{w_i,\epsilon })\) and \(a \notin Max(Q,w,d^{w_i,-\epsilon })\).
In effect, Utility Continuity says that small changes to utilities assigned to any particular world should precipitate only small changes in the values that a quantity assigns to options.
To see that every subset quantity violates Utility Continuity, suppose that Q is a subset quantity, and suppose that a is among the options that maximize U at \(\langle w,d \rangle\), but not among the options that maximize Q at \(\langle w,d \rangle\). There will then be some a-world, \(w_i\), to which the credence function in d assigns nonzero probability, which is such that, increasing its utility, while keeping the utility of every other world the same, increases the U-value of a, but does not increase the U-value of any other option in A. So, for any \(\epsilon\), a uniquely maximizes U at \(\langle w,d^{w_i,\epsilon } \rangle\). Since Q is a subset quantity, a also uniquely maximizes Q at \(\langle w,d^{w_i,\epsilon } \rangle\). But that shows that Q violates Utility Continuity.
Thus, Supervenient Optimality holds: U is the highest-scoring supervenient quantity that satisfies Utility Continuity.
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Spencer, J. Rational monism and rational pluralism. Philos Stud 178, 1769–1800 (2021). https://doi.org/10.1007/s11098-020-01509-9
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DOI: https://doi.org/10.1007/s11098-020-01509-9