1. Introduction
If evidence is available for ‘free’, it is presumably a good idea to pursue that evidence and take it into account when making decisions.Footnote 1 Good (Reference Good1967) proves as much for the case where Your degrees of belief are precise.Footnote 2 That is, Good proves that when Your beliefs or credence can be represented by a single probability function, free evidence cannot be detrimental. Good proves that, under standard conditions, Your expectation increases (or at least cannot decrease) if You pursue free evidence.Footnote 3
In the case in which Your credence is imprecise, however, no such nice result exists. Indeed, several authors have shown that, in such cases, You may effectively pay to avoid free evidence (Grünwald and Halpern Reference Grünwald and Halpern2004; Kadane et al. Reference Kadane, Schervish and Seidenfeld2008). These examples rely on the phenomenon of dilation (Seidenfeld and Wasserman Reference Seidenfeld and Wasserman1993; Pedersen and Wheeler Reference Pedersen and Wheeler2014). What happens in dilation is that conditionalizing on some evidence can cause an event whose prior probability was sharp to have an unsharp posterior probability. This ‘fuzzying’ of Your degrees of belief may lead to Your paying to avoid free evidence. This article investigates the puzzle suggested by this result.
Let us quickly set the scene. Many have argued, convincingly, that rationality permits imprecision: a rational agent may have incomplete preferences that are best represented by a set of probability- and utility-function pairs, rather than a single precise such pair.Footnote 4 But, rationality surely requires that an agent not pay to avoid free evidence. If all plausible decision theories for handling imprecision have the consequence that You may pay to avoid free evidence, we have a contradiction and must consider which of the premises should be given up.
The next section presents the apparent trilemma in more detail: we initially outline the setup for, and statement of, Good’s theorem and show, by example and in line with other authors, that his lesson about the invariable goodness of free evidence does not straightforwardly extend to the imprecise context. While not an original aspect of the article, we think it important to carefully explain Good’s reasoning about free evidence, and its prima facie extension to the imprecise realm, because this is a noteworthy and interesting issue for decision theory that deserves attention. We go on to clarify the premises of our proposed trilemma. This serves to highlight the main issue of the article: whether all plausible decision theories for handling imprecision do in fact fall afoul of free-evidence intuitions.
In order to discuss the commitments of different decision rules regarding free evidence, it is necessary to first elaborate, in section 3, how You should negotiate a dynamic- or sequential-decision problem (i.e., a decision problem that involves choices both now and in the future). We do not dispute the orthodox sophisticated-choice approach, but we show that it has not been adequately spelled out in the current literature, and the missing details leave room for controversy that is relevant to our free-evidence trilemma. So we effectively introduce and argue for our own version of sophisticated choice for the imprecise probabilist. In a sense, this middle section is the most important contribution of the article.
section 4 then examines an important, perhaps even the only plausible, decision rule for handling imprecision; this is the maximally permissive non-dominated-set (NDS) rule. The NDS rule does not preclude paying to avoid free evidence, but we prove that it at least never mandates such payment. section 5 comments on how this result fits into the broader decision theory literature on dynamic coherence. In particular, we note that previous work in the literature shows that no decision rule can do better than the NDS rule with respect to the pursuit of free evidence. section 6 summarizes our stance on the trilemma.
2. Good’s Theorem and Troubles for the Imprecise Probabilist
I. J. Good showed that it always pays in expectation for the precise probabilist to pursue free evidence. The proof can be found in Good (Reference Good1967). A key assumption is that You conform to standard Bayesian decision theory; that is, You are an expected utility maximizer and update Your beliefs in line with the rule of conditionalization.Footnote 5 Good’s result accords with intuitions about seeking evidence and experimentation being advantageous. Indeed David Miller (Reference Miller1994) notes that one of A. J. Ayer’s two main criteria for a successful account of scientific method is that the account should advise scientists to pursue free evidence. (The second criterion is that the account should advise the scientist to base inferences on all the evidence available.) Bayesian decision theory thus meets Ayer’s standards because it is not inconsistent with the two criteria (notwithstanding Miller’s arguments to the contrary). Carnap also discussed what he called the Principle of Total Evidence (PTE), which amounted to the claim that reasoners should not ignore available evidence when estimating a probability.Footnote 6
The problem set up for the proof is as follows: there is a partition of the state space H 1, …, Hr and some acts A 1, …, As that can have different payouts for the different events Hi. Your utility for an act in a particular event is U(Aj(Hi)). Your credences over the events are Pr(Hi). There is another partition of the state space E 1, …, Et. You are offered the chance to learn which Ek obtains. Pursuing the new evidence has no disutility associated with it, so the utilities of the basic outcomes in the decision problem are the same whether or not You learn. This is essentially the meaning of free evidence.Footnote 7 Good shows that, under these assumptions, opting for learning which Ek obtains has at least as high an expectation as not learning, and learning always has a higher expectation when it is possible that the new evidence may change Your choice of acts among A 1, …, As.
The idea behind the proof is illustrated in table 1.Footnote 8 Not learning effectively means choosing among the averages of the acts conditional on each Ek. Your expectation is thus the maximum value in the final “Average” column. Your expectation for learning, however, is the average of the maximums in each of the columns E 1, …, Et. In short, learning is always at least as good as not learning because the average of maximums is always at least as high as the maximum average.
Table 1. Idea Behind the Proof
| E 1 | … | Et | Average | |
|---|---|---|---|---|
| A 1 | ∑Pr(Hi∣E 1)U(A 1(Hi)) | … | ∑Pr(Hi∣Et)U(A 1(Hi)) | ∑Pr(Hi)U(A 1(Hi)) |
| A 2 | ∑Pr(Hi∣E 1)U(A 2(Hi)) | … | ∑Pr(Hi∣Et)U(A 2(Hi)) | ∑Pr(Hi)U(A 2(Hi)) |
| ⋮ | ⋱ | ⋮ | ||
| As | ∑Pr(Hi∣E 1)U(As(Hi)) | … | ∑Pr(Hi∣Et)U(As(Hi)) | ∑Pr(Hi)U(As(Hi)) |
Consider the following simple example, depicted in figure 1. Note that the squares indicate choice nodes (places where You must make a choice) and the circles indicate chance nodes (uncertainty that is then resolved one way or another).Footnote 9 There are two urns labeled X and Y. You believe that urn X contains 10 black marbles and that urn Y contains 10 white marbles. One urn will be selected at random by the toss of a fair coin, and a marble drawn from it. You are offered a 2 to 1 bet on black: if black is drawn You end up with 3, and if white is drawn You lose your stake of 1. It should be obvious that learning (i.e., ‘down’ at choice point 0) has higher expectation in this case: if You were to learn that the draw is from X, You would bet on black and predict a win of 3, and if You were to learn that the draw is from Y, You would refrain from betting and receive 0. Thus, Your expectation would be 0.5 × 3 + 0.5 × 0 = 1.5, which is greater than 1, the expected utility of not learning (since if You chose up at node 0 You would choose to bet).
Figure 1. Simple decision problem.
It is worth noting that Good’s proof concerns the expected value of free evidence. It may be the case that You ‘get unlucky’, and Your expected utility in fact goes down upon receiving some evidence. For instance, You may learn that the draw is from Y in our example, giving an expectation of 0 (since if You learned Y You would choose not to bet), which is of course less than 1. This is a possible eventuality; nonetheless, the expected utility of learning free evidence (i.e., before a particular piece of evidence is actually received) is greater than or equal to not learning.
Consider a slightly different example: You believe that urn X definitely contains eight black and two white marbles and that urn Y contains two black and eight white. Again, learning which urn is drawn from has higher expectation, as You would tailor Your choice of Bet/Don’t Bet to the new evidence, as before. Here You would not be guaranteed the prize, by Your own lights, even if You learn that the draw is from urn X, but Your choice will accord with what You take to be the proportions of balls in the urn, whatever urn is revealed, and so overall You have a higher expectation.
There is a problem when we try to apply this to the imprecise case. That is, where Your belief is represented by a set of probability measures, 
, rather than just one measure. Call this set Your representor. Imagine a scenario similar to the last one. Imagine You believe that there are a total of 10 black and 10 white marbles distributed somehow among the urns X and Y. Each urn contains 10 marbles. An urn will be selected at random by flipping a fair coin, and a marble drawn from it. Using X and Y to refer to the propositions ‘the marble is drawn from urn X’ and ‘the marble is drawn from urn Y’, respectively, and using B and W to stand for the propositions ‘the marble drawn is black’ and ‘the marble drawn is white’, respectively, in the spirit of imprecise probabilism, what follows is a plausible characteristic of Your belief representor, 
: Pr(B∣X) = 1 − Pr(B∣Y) for all Pr ∈ 
. As such, Your credences before learning regarding the color of the marble drawn are as follows: 
(W) = {0.5} = 
(B). You believe that the number of white marbles and the number of black marbles are equal and that over the two urns their probabilities average out. It seems plausible that Your conditional credences are, however, imprecise: You have no information about how the marbles are distributed between the urns, and so Your representor, 
, plausibly includes the possibility that urn X certainly contains only white marbles and also the possibility that X certainly contains no white marbles, and everything in between. That is, 
(W∣X) = 
(B∣X) = 
(W∣Y) = 
(B∣Y) = {0, 1/10, …, 9/10, 1}, or, if convexivity were mandated (as per Levi Reference Levi1974, Reference Levi1980), Your representor would plausibly have 
(W∣X) = [0, 1], and likewise for the other conditional attitudes.Footnote 10 For reasons of notational convenience, we stick to using [0, 1] to represent the aforesaid imprecise conditional beliefs. So learning which urn is drawn from dilates Your probability for white from 0.5 to [0, 1], and likewise for black. Note that this analysis of the problem assumes the standard rule for updating imprecise beliefs, known as generalized conditioning, which basically amounts to pointwise conditioning: after learning X, say, {Pr(−∣X) : Pr ∈ 
} is Your new representor.Footnote 11
Recall that You are offered a 2 to 1 bet on black: if black is drawn You win 3, and if white is drawn You lose 1. Before learning, it is clear that this bet is advantageous and You should take the bet, rather than stick with the status quo. (As before, Your expectation for not learning, given the credences described above, is 1.) If You choose to learn, however, Your expectation for the bet will inevitably dilate from {1} to [−1, 3]. Whether the revealed urn is X or Y, the bet no longer has higher than or equivalent expected utility to the status quo: the two acts in each case become incommensurable. Pursuing free evidence is no longer straightforwardly better for You, even though it may lead to You changing Your choice of option.
Now imagine the same problem, except that choosing not to learn comes with a small penalty of δ (where δ > 0). This makes the problem more vivid, because now, if You choose not to learn, You are in fact paying to avoid learning. Figure 2 depicts this revised problem. You face a choice at the initial node (labeled 0) between learning and paying not to learn. let us assume Your credences are as before. Learning leads to evidence about the coin flip—whether the urn to be drawn from is X or Y. As per Good’s theorem, the final decision problem is constant, whatever happens, except that the option of not learning involves δ utility subtracted from all outcomes. If You choose not to learn, You effectively pay δ to avoid free evidence.Footnote 12
Figure 2. Problematic decision problem.
Imagine, for instance, that You conform to the so-called gamma-maximin rule, which holds that one should always choose the option that has the largest minimum expected utility.Footnote 13 You would reason as follows: if learning is chosen, there will inevitably be a choice between betting, with expected utility [−1, 3], and not betting, with expected utility 0. The latter has the greater minimum expected utility, so learning effectively leads to not taking the bet, which has 0 utility. (This process of working backward through a decision problem is referred to as sophisticated reasoning or backward induction. All of the examples here have involved sophisticated reasoning. We say more about this in the next section.) Not learning, however, will mean that the bet is chosen, and this has expected utility 1−δ from the initial vantage point. Assuming δ < 1, not learning is therefore preferable. You thus pay to avoid free evidence.
By way of setting the stage for the remainder of the article, let us be explicit about the proposed trilemma:
P1. Paying to avoid free evidence is irrational.
P2. Incomplete preferences are not irrational and can be represented by sets of probabilities and utilities. In other words, imprecise probabilism is permissible.
P3. All plausible decision theories for handling imprecision sanction paying to avoid free evidence.
As mentioned earlier, P2 is defended elsewhere. We will revisit P1 later, but let us simply note, for now, that it is hard to conceive how ‘money’ spent on avoiding free evidence could be money well spent. Decision theory aside, any sensible person would surely ignore new free evidence that he or she might obtain, if it is really unhelpful, rather than pay to remain ignorant about it.
The initial focus of this section is the final premise, P3. What we have illustrated above is just that a very specific type of imprecise probabilist—one whose belief-updating rule is generalized conditioning and who chooses according to gamma-maximin—chooses making a payment over receiving free evidence. It remains to be seen, however, whether this is also a consequence of other decision theories for handling imprecision. Indeed, we now investigate whether there may be alternative generalizations of standard Bayesian decision theory that do not recommend this nonintuitive course of action.
We focus exclusively on alternative decision rules. That is, we do not here consider alternative rules for updating belief but rather assume generalized conditioning throughout. As mentioned above, this is arguably the most natural generalization of Bayesian conditioning for the imprecise context. Stated in full, the rule holds that Your posterior belief in some proposition H, after learning the proposition E, is just the set of conditional probabilities pertaining to all probability functions in Your representor for which the conditional is defined:
This is the rule we implicitly assumed for the example above, and it is subject to dilation. Of course, one might suggest that, given that dilation seems to be directly implicated in paying to avoid free evidence, we should explore alternative rules for updating imprecise beliefs that might preclude dilation and its attendant problems. Elsewhere we do explore this possibility (see Bradley and Steele Reference Bradley and Steele2014b), but this is not our focus in this article. Note just that we do not regard this to be a fruitful avenue for developing imprecise probabilism; dilation is not a good reason for departing from generalized conditioning.
So, as mentioned, we restrict our attention to decision rules and hold fixed generalized conditioning as the belief-update rule. The gamma-maximin rule assumed above is already criticized on other grounds (Seidenfeld Reference Seidenfeld2004), so there is hope that the bad results concerning dilation are specific to this rule. In what follows, our discussion of decision rules proceeds under the assumption that there is already an appropriate representation of Your beliefs and desires. In particular, here we assume that You have imprecise probabilistic beliefs, that is, Your belief representor is a set of probability distributions, 
, and yet, for simplicity, You have a precise utility function over basic outcomes.Footnote 14 The claims of this article pertain to this assumed setup—imprecise probabilities yet precise utility function—but we do not anticipate any problems in extending the results to the case of both imprecise probabilities and utilities. Before getting to the specifics of the various decision rules, however, it is important to first specify how, in general, one should negotiate a sequential-choice problem, since the question whether to pursue free evidence is a problem of this kind.
3. Sophisticated Choice for the Imprecise Probabilist
Recall that the problem we set up in the previous section involved several ingredients: first, a particular update rule; second, a particular decision rule; and third, a particular approach to sequential choice. In the example, we had generalized conditioning, gamma-maximin, and sophisticated choice. We take the first and third of these for granted throughout. We will see, in a moment, an alternative to gamma-maximin that fares better in the kind of problems we are looking at. But first, we want to spell out a little more carefully what sophisticated choice amounts to in the current setting. For precise probabilities, sophisticated choice is the orthodoxy,Footnote 15 but for imprecise probabilities, it has not been fully fleshed out what sophistication means. We take the following discussion to flesh out the correct version of sophisticated choice.
3.1. Sophisticated Choice, User-Friendly Version
Recall the sequential-choice reasoning in the previous section. You look at the terminal nodes and work out which act maximizes minimum expectation with respect to Your representor. This done, You now treat those terminal nodes as if the choice were fixed. From the earlier node, You treat the earlier choices as if they were choices that determinately led to the gamma-maximin-best act. You effectively act as if You know Your future self will choose that way.
Let us recast the imprecise decision problem we discussed earlier as a group decision problem in which there is a common utility for the agents in the group.Footnote 16 Let us also state, in accordance with the suggested decision rule, that the group chooses by evaluating an act by the lowest expected value assigned to it by some committee member (recall fig. 2). Each agent knows that if the group ends up at node 1, the group choice will be to bet since every member agrees that the expected value of betting at node 1 is 1 − δ. Each agent (at node 0) also knows that if the group chooses to learn and ends up at node 2, the group will choose not to learn, since there is a committee member who assigns expected value −1 to betting: the agent who thinks all the black marbles are in urn Y (call this agent Mr. White). Likewise, at node 3 the group will choose not to bet since there is an agent who evaluates betting at −1: the agent who thinks all the black marbles are in urn X (call this agent Mrs. Black). So at node 0, any agent evaluates choosing to learn as a 50:50 gamble between two refusals to bet, resulting in expected value of 0. Thus, every committee member agrees at node 0 that paying not to learn is better, given what they believe about how the committee will choose at later nodes. This is the problem of free evidence for the imprecise probabilist recast as a problem for group belief.
But consider the following alternative reasoning. Note that each committee member is a precise probabilist, and as such, Good’s theorem applies: each committee member thinks that learning increases expectation. So each committee member agrees that choosing to learn is uniquely permissible. Since any plausible decision rule should respect unanimity of this kind, there is no problem of free evidence for the imprecise probabilist.
What has gone wrong? We have two ways of reasoning about the decision problem that You or Your credal committee face, and they seem to give conflicting conclusions. To see what goes wrong with the latter kind of reasoning, we need to take a detour into backward induction, through Greek myths, game theory, and group decision making.
On its way back from Troy, Odysseus’s ship must sail past the sirens. Circe has warned Odysseus that the sirens’ song will drive him and his crew mad and cause them to crash their ship on the treacherous rocks thereabouts. Wily Odysseus orders his crew to stop their ears with beeswax; Odysseus, however, is curious to hear what the sirens’ song sounds like. Odysseus has the option of ordering his crew to tie him to the mast or to be left unhindered. Possible courses of action once within range of the sirens are for Odysseus to stay on board or to dive into the sea and drown. Odysseus would rather not be tied to the mast but prefers that to drowning. He knows that the sirens will cause him to jump overboard if he remains unhindered. So he opts to have himself tied to the mast, in order to stop his jumping overboard at a later time.
Let us be a little clearer about Odysseus’s thinking. There are two relevant times at which Odysseus must make a choice: he must decide (now) whether to be tied to the mast and later whether to jump into the sea and drown. If he is tied to the mast now, he will not be able to jump overboard. Also, if he is not tied to the mast, the sirens will drive him mad and make him jump overboard. In a sense, Odysseus’s most preferred option is to not be tied to the mast but then to remain on the boat when in range of the sirens. The problem is that he knows now that his future self will not stay on board if he is unhindered. Odysseus is essentially treating his future self as an agent he does not have full control over and treating his decision problem as a kind of game: his current self versus his future self. Given what he knows about his future self’s preferences (as influenced by the sirens’ song), his choice (now) should be that which gets him the best outcome on the assumption that his future self will act in accordance with his future preferences. So he should tie himself to the mast now so that his future self does not have the option to jump into the sea. This is an example of backward induction: a reasoning strategy that is ubiquitous in game theory.
To reiterate: Odysseus’s most preferred option would be to not be tied to the mast but then to refrain from jumping overboard.Footnote 17 He does not, however, act on this preference, because he is sophisticated enough to take into account facts about his future choices that are not under his current control. This makes Odysseus a sophisticated chooser as opposed to a naive chooser who would act on his current preference for not being tied up and ignore the inevitable future bad consequences of such actions. Sophisticated choice is clearly better than naive choice: it simply amounts to not ignoring pertinent facts about Your future choices.
Let us go back to the second line of reasoning discussed earlier that seemed to conclude that all the committee members agreed that learning was best. Consider the two extreme committee members Mrs. Black and Mr. White who think that all the black (white) marbles are in urn X. Mrs. Black and Mr. White make up a committee that must collectively make decisions on whether to learn and how to bet. Mrs. Black thinks that learning increases expectation because, if X is true, betting is the right option and, if Y is true, not betting is the right option. Mr. White thinks that learning increases expectation because, if X is true, not betting is the right option and, if Y is true, betting is the right option. Now consider things from Mrs. Black’s point of view. She is reasoning at node 0 about which option she prefers. She knows that if the committee arrives at node 1, she and Mr. White will disagree about what to do, and she is not sure how such disagreement will be resolved. So at node 0, Mrs. Black cannot discount the possibility that at node 1 the committee will choose not to bet (despite that being a bad option according to Mrs. Black). So what Mrs. Black and Mr. White agree on is not that the group decision should be to choose to learn but merely that if each of them individually were in control of the future choice, learning would be the right choice. That is, each committee member only believes learning is the right choice because of what the postlearning choices would be if that committee member were in charge. Given each committee member’s uncertainty about how future choices will in fact be resolved, it is not clear that the committee will make the ‘correct’ choice from that member’s perspective. And thus, the members do not agree that learning is necessarily the right choice, given what they know about the committee’s future choices. And indeed, if the committee members knew that the committee resolved disagreements by using gamma-maximin, they would agree that not learning is the better group decision.
3.2. Modeling Assumptions
This section serves two purposes: we introduce some formalism that we will need later, and we also make clear some assumptions we make in what follows. First, we will be using the language of choice functions that take as inputs the available options and output the set of choice-worthy or admissible options. For the precise probability case, the standard choice function is the function that takes a set of options and outputs the set of options that maximize expected utility.
In accordance with our sophisticated choice approach, Your current options are the acts available to You at the current time and not paths through the decision tree or plans of how to act at the current node and all future nodes. Further, from Your current perspective, You take how You anticipate You will act at future nodes to be fixed: to be part of the state space of Your current decision.
Let Oi be the options available at node i. Let us consider the simple case of the precise example in figure 1. So, for i = 1, 2, 3, Oi is the set Bet, Don’t Bet. And O0—the options at node 0—is Learn, Don’t Learn. The choice function summarizes the options that would be chosen at the node, so for node 1, C(O1) = Bet. And likewise C(O2) = Bet, and C(O3) = Don’t Bet. These facts about future choices should be encoded in the state space of Your decision problem at node 0.Footnote 18 As such, this problem should be summarized as per table 2. The events X and Y have probability 0.5 each. Note that the outcomes in this table are determined by the outcomes that the future choices will lead to. Some of these outcomes are lotteries. The lottery 〈3, B; −1, W〉 should be read as follows: ‘win 3 if B, lose 1 if W’. By assumption (recall fig. 1), the lotteries in the X column reduce to a sure outcome of 3, since Pr(B∣X) = 1, and the lottery in the Y column reduces to a sure outcome of −1, since Pr(B∣Y) = 0.Footnote 19 Now we again apply the choice rule to this decision problem, and thus determine C(O0). As noted earlier, learning has a higher expected utility (EU = 1.5) than not learning (EU = 1).
Table 2. Problem in Figure 1
| C(O1) = Bet; C(O2) = Bet; C(O3) = Don’t Bet | ||
|---|---|---|
| X | Y | |
| Don’t learn | 〈3, B; −1, W〉 | 〈3, B; −1, W〉 |
| Learn | 〈3, B; −1, W〉 | 0 |
Consider a further example. Table 3 represents Your node 0 decision problem as depicted in figure 2, with the assumption that the gamma-maximin decision rule will be applied at any future choice nodes.
Table 3. Problem in Figure 2 Using Gamma-Maximin
| C(O1) = Bet; C(O2) = Don’t Bet; C(O3) = Don’t Bet | ||
|---|---|---|
| X | Y | |
| Pay not to learn | 〈3 − δ, B; −1 − δ, W〉 | 〈3 − δ, B; −1 − δ, W〉 |
| Learn | 0 | 0 |
Both of the above examples obscure an important fact about the general case of imprecise sophisticated choice, that is, that a choice function may fail to determine a unique admissible act. Let us go through the choice problem again, but using a different choice rule. Let us imagine You use the permissive NDS decision rule: all Your choice function does is rule out options that are expectation dominated. We discuss this rule in more detail in the next section. You are at node 0 reasoning about how Your future selves will behave. Your node 1 self is easy: the acts available—Bet, Don’t bet—each have precise utility 1 − δ and −δ, respectively. So Your node 1 self will choose to bet. Now consider Your node 2 self. Your beliefs at node 2 have dilated, and with them, Your expectations. The expectations for Bet and Don’t bet are [−1, 3] and 0, and neither act expectation dominates the other. So from node 0, You do not know how Your node 2 self will act. The same reasoning goes for node 3. We can thus represent Your node 0 decision problem as per table 4. Note that choice function C can return a set of acts.
Table 4. Problem in Figure 2 for Non-dominated-Set Rule
| C(O1) = Bet; C(O2) = {Bet, Don’t Bet} = C(O3) | ||
|---|---|---|
| X | Y | |
| Pay not to learn | 〈3 − δ, B; −1 − δ, W〉 | 〈3 − δ, B; −1 − δ, W〉 |
| Learn | {0, 〈3, B; −1, W〉} | {0, 〈3, B; −1, W〉} |
So how should Your node 0 self reason about these unknown future choices? How should learning be valued at node 0? The choice to learn amounts to a 50:50 chance of one or another future decision problem, neither of which You know how You will solve. The next section presents our answer to this question. The presentation may seem complicated, but we maintain that the complications are a feature of imprecise decision theory when it comes to sequential choice. As mentioned, this is one aim of our article: to make explicit how imprecise decision rules should be characterized in the sequential-choice setting. While others (notably Seidenfeld Reference Seidenfeld2004; Kadane et al. Reference Kadane, Schervish and Seidenfeld2008) have considered how key rules fare in particular sequential-choice problems, with similar findings to our own, no one has provided a clear account of what it means to do backward induction when numerous incommensurable options are choice-worthy at future nodes.Footnote 20
4. The Non-Dominated-Set Rule
The latter part of the discussion in the previous section alluded to an imprecise decision rule that is less ‘opinionated’ than the gamma-maximin rule. Roughly, if members of Your credal committee disagree in their preference between options, then Your overall preference between these options is simply indeterminate, and there is no saying which will be chosen in a decision between them. This is effectively the NDS rule, otherwise known as Sen-Walley maximality. We will define it in more precise terms shortly. The NDS rule is the most permissive decision rule on the table: it would be a serious flaw of any decision rule that it sanctioned performing some action when a dominating action was available (i.e., if the choice rule made permissible something outside of the NDS choice set). This makes NDS interesting for two reasons. First, it is a very plausible decision rule, as it does not contrive a preference between incommensurable options where there is none, so to speak. Second, being the most permissive rule, NDS is pivotal in our discussion. If even the NDS rule made paying to avoid free information obligatory, then this would be a killer blow to imprecise probabilism, since any other less permissive rule would inherit this flaw.
So how the NDS rule fares in ‘free evidence’ situations is more important than how gamma-maximin fares in those situations. In what follows, we first give a formal definition of the NDS rule (sec. 4.1), and then we prove a result for the rule that relates to the third premise of our trilemma (sec. 4.2). The result is not surprising given the previous work mentioned above, but no one has yet given a treatment as general as ours or as explicit with respect to sequential-choice reasoning.Footnote 21 We provide such a general treatment and articulate the correct albeit controversial approach to sophisticated choice.
4.1. Formal Definition
Formally, the NDS rule can be defined in terms of its ‘choice set’ (i.e., the set of admissible options that it returns) as follows:
This needs some unpacking. The meaning of Aj <EU Ak that accords best with the current literature is as follows: option Aj has lower expected utility than option Ak according to all probability distributions in Your representor. In such a case, Aj is strongly dominated by Ak and is thus not choice-worthy. This definition of <EU will not quite do, however, given the complications that can arise for the imprecise probabilist making sequential decisions that we canvassed at the close of the previous section.
In order to spell out an adequate definition of <EU and thus the NDS rule for sequential contexts, it helps to return to our example of the imprecise probabilist facing the decision problem in figure 2. In accordance with backward induction, we determine what will be chosen at the later nodes: it is clear enough that C(O2) = C(O3) = {Bet, Don’t Bet} by the NDS rule, since neither betting nor not betting EU-dominates the other.Footnote 22 Indeed, thinking in terms of our credal committee, Mrs. Black and Mr. White disagree on what choices at nodes 2 and 3 maximize expected utility. As before, C(O1) = Bet, since all committee members agree that this option at node 1 has higher expected utility. The upshot is that the problem facing the NDS agent (the credal committee) at the initial node is in fact the problem represented in table 4. Everyone in the committee sees, at node 0, that at nodes 2 and 3 there will be disagreement among the committee members, and either betting or not betting could be chosen at those locations. So the choice set for node 2 contains both the Bet and Don’t Bet options, likewise for node 3. Thus, the outcome of opting to learn—which is a mixture of the outcomes of nodes 2 and 3—amounts to a mixture of two sets of two future acts.
So our NDS agent faces the decision problem depicted in table 4: the complicated case involving outcomes with sets of acts and thus sets of utilities. For each probability distribution Pri ∈ 
(or each committee member i), the evaluation of an act Aj (here learning) may thus amount to a set of expected utilities.
Let us now continue with unpacking the statement Aj <EU Ak:
where EUi(Aj) is the expected utility of act Aj according to probability function Pri. It can be a set of values if act Aj is not the terminal node of the decision tree, because of Your uncertainty about Your post-Aj choices. Of course, the problem now shifts to defining <P. All that has been stipulated about <P is that, in the case in which both EUi(Aj) and EUi(Ak) are singletons, <P amounts to the simple ‘is less than’ relation for numbers. Thus, when Aj and Ak lead to outcomes with precise expected utilities for each Pri, Aj <EU Ak has its standard meaning. Given that the standard NDS rule appeals to a strictly greater than relation, it makes sense to define <P as follows:
The relation <P is sometimes known as the relation of interval dominance. This makes sense because it holds exactly when one utility interval is entirely above another. Note that it does behave appropriately in the precise limit.Footnote 23
We can now finish working through the decision problem in figure 2 using our fully worked out NDS rule. We have already established, via backward induction, that the decision problem at the initial node is given in table 4. The question is whether learning EU-dominates not learning, or vice versa. Consider just one of the extreme probability distributions—that of committee member Mrs. Black—Pr1, where Pr1(B∣X) = 1 and Pr1(B∣Y) = 0. For this probability distribution alone, the lotteries in table 4 will thus be evaluated as per table 5. Note that the lottery outcome in the X column is evaluated as 3 (or 3 − δ), since Pr1(B∣X) = 1. Thus, the lottery outcome in the Y column is evaluated as −1 (or −1 − δ).
Table 5. Problem in Figure 2 for the Non-dominated-Set Rule, for Pr1 (Mrs. Black)
| C(O1) = Bet; C(O2) = {Bet, Don’t Bet} = C(O3) | ||
|---|---|---|
| X | Y | |
| Pay not to learn | 3 − δ | −1 − δ |
| Learn | {0, 3} | {0, −1} |
To work out Mrs. Black’s evaluation of the acts in table 5 (i.e., learning vs. paying not to learn), we need to generalize the idea of expected value to sets of utilities. An informal characterisation of expected value is ‘the sum of the probability-weighted utilities’. So it is the probability of the state times the utility of the act in that state, summed over the states. To evaluate the act ‘Learn’ in table 5, we need a ‘sum of probability-weighted utility sets’. This means characterizing what it means to weight a set of utilities by a probability and then what it means to add sets of probability-weighted utility together. In other words, we need to evaluate
We will need to give an appropriate gloss on what × and + mean in this context. We suggest that p × A = {pa : a ∈ A}. That is, the multiplication is done to each element of A. As for A + B, we take this to mean {a + b : a ∈ A, b ∈ B}. This is the full set of possible sums of elements of A and B.Footnote 24 This proposal for how to treat sums and products of sets of values can be motivated in the same way we motivated the NDS rule: this is the most permissive plausible proposal for how to deal with arithmetic operations on sets of values. So this approach is pivotal in the same sense that the NDS rule is. Returning to the example, this gives
Paying not to learn has precise utility {1 − δ}. Thus, it is not the case that EU1(Learn) <P EU1(Pay not to learn), nor vice versa.Footnote 25 Accordingly, just by looking at Mrs. Black’s evaluation of the options (let alone the evaluations of her fellow committee members), we see that neither learning nor paying not to learn EU-dominates the other. So at the initial node, both options are admissible by the NDS rule; that is, C({Learn, Pay not to learn}) = {Learn, Pay not to learn}. The NDS rule therefore sanctions paying not to learn, but in this case, at least, the rule does not require or mandate paying for ignorance.Footnote 26 This is an improvement on the gamma-maximin rule.
4.2. A Positive Result
While we have seen that paying not to learn may be an admissible option, according to the NDS rule, there is some good news: for this pivotal imprecise decision rule, learning free evidence is always itself an admissible option. That is, it is never the case that not learning or else paying not to learn is uniquely admissible or obligatory, when learning free evidence is also an available option.
The structure of the proof follows our working of the decision problem above, but the idea here is to show that this analysis generalizes to all decision scenarios of the following form: Your beliefs may be imprecise; that is, Your belief representor, 
, may consist of more than one probability distribution. (Your utility function over basic outcomes is nonetheless precise.) You face a decision as to whether to pursue free evidence, sensu Good (Reference Good1967). As before, it is a question whether to make a choice between a set of acts now or else make the choice between the same set of acts (with basic outcomes unchanged) after learning some evidence. We assume, for simplicity, that the decision to be made now or after learning is not itself a sequential decision problem. For rhetorical reasons, the option of not learning is adjusted to include a payment of some small δ. So either You can learn and then make a decision or You can pay to not learn and make the decision from a state of ignorance.
This is the claim to be proven, with reference to the general decision problem just described:
For the imprecise agent abiding by the NDS rule, learning free evidence is always an admissible option. That is, not learning or else paying not to learn free evidence is never uniquely admissible, when learning is also an available option.
We start by summarizing Good’s result. Let us say that we have acts whose outcomes depend on which of the Hms is true. Call the acts the Ajs. We are trying to work out whether it pays in expectation to learn which Ek is true. The Hms partition the state space, as do the Eks. The expectation for not learning is
That is, You know You will pick the act that maximizes Your utility after having chosen not to learn. Using the fact that Pr(Hm) = ∑kPr(Ek)Pr(Hm∣Ek) and rearranging the order of summation, we have that this is equal to
Now consider learning some Ek. Expectation after learning Ek is
So expectation for learning is the probability-weighted sum of these expectations:
We can move the Pr(Ek) inside the summation so this is equal to
These expressions for learning (3) and not learning (2) differ only in the order of the summation and maximization. So all we need to do to prove the theorem is show that, for any function f(j, k), we have
Then if we consider f(j, k) = ∑mPr(Ek)Pr(Hm∣Ek)U(Aj(Hm)), we have the result we wanted. So let us consider the j that maximizes ∑kf(j, k), call it j 0. Clearly maxjf(j, k) ≥ f(j 0, k) for any k. Thus,
That, in brief, is Good’s theorem.Footnote 27
Returning to our own decision scenario: let O denote the set of acts at the initial node: Learn and Don’t learn/Pay not to learn. Let A denote the set of acts for the decision problem that needs to be solved: Bet/Don’t bet (i.e., the acts that are the rows in table 1). The possible evidence forms a partition E: urn X or urn Y.
Now consider the plight of the imprecise agent employing our refined version of the NDS rule, as outlined in section 4.1. Assume Your representor, 
, consists of probability functions (i.e., committee members) Pr1, Pr2, Pr3, … . Take any one of these probability functions Pri. Quite simply, for this committee member i, we know from Good’s theorem itself that, if this committee member were a one-person committee, learning is at least as good as not learning. That is not necessarily the case, given our assumption of imprecise probabilism—there may be other members in the committee. Nonetheless, the acts in A that committee member i deems maximal at each of the respective future choice nodes associated with learning, corresponding to each Ek that may be learned, will feature in the choice sets for these nodes (since, by assumption, the NDS rule is applied at these nodes too). Thus, when we evaluate learning versus not learning for any committee member i, the probability-weighted average of utility sets for learning, as discussed above, will be a set of utilities that includes a utility that is guaranteed, by Good’s theorem, to be at least as good as the utility of not learning. So for any committee member i, not learning does not (strictly interval-) dominate learning. Therefore, not learning (and thus paying not to learn) does not EU-dominate learning, and so learning is always admissible, for our fully worked out version of the NDS rule. The basic idea is that if an act is maximal for some agent, it cannot be dominated.
For clarity, we restate this proof sketch now more formally. For any probability function Pri, each possible evidence Ek ∈ E is associated with an act with maximum precise expected utility among the set of available acts A. Call this act aik. (Assume, with no loss of generality, that there is a unique such act with maximum precise expected utility for each Pri and each Ek.)
There is also an act in A that has maximum expected utility, in the unconditional sense (before learning), according to Pri. Again, assume, with no loss of generality, that there is a unique such act, which we label 
.
We know, by Good’s theorem, that
Here Ui is the utility function that is paired with probability function Pri.
The right-hand term above amounts to the value of not learning for this probability function Pri in the representor, that is, EUi(Don’t learn).
The left-most term does not amount to the value of learning with respect to Pri, however, because we have not taken into account sophisticated choice. The value of learning for each Pri depends on what would be the sets of admissible acts if each respective Ek was learned (the Ck(A)). For each Ek, these choice sets will at least include the union of all the acts that have maximum expected utility, conditional on Ek, for the probability distributions Pri in
:
The set of expected utilities that Pri assigns to this choice set under the assumption that Ek obtains is then
So the value of learning according to the probability function Pri is
As noted in section 3, we take the above expression to equate to the set of all combinations of probability-weighted sums of utilities, given the utility sets associated with each Ck(A). Therefore, one of the utility values in the set EUi(Learn) is ∑kPri(Ek) × Ui(aik). That is, the expected utility of choosing according to the expectations of Pri for each k.
Recall that, by Good’s theorem, this expression is greater than or equal to the value of not learning, for the probability function in question. Thus, given our definition of <P, it is not the case that
Therefore, it is not the case that
That is, learning is always admissible (given a choice of learning and not learning) by our refined NDS rule.Footnote 28
4.3. Return to the Trilemma
If the only plausible decision rule was the NDS rule, the trilemma stated in section 2 would stand. We have seen that this rule sanctions paying to avoid free evidence, so P3 is true. And P1 and P2 are also apparently true, but the three together yield a contradiction. The positive result proved above, however, suggests a possible resolution: this is to deny P1 in favor of a weaker premise, P1′, stating:
P1′ Paying to avoid free evidence, or even avoiding free evidence, should never be uniquely admissible, when pursuing free evidence is also an available option.
Note that in the precise case, we can satisfy the stronger principle that learning free evidence before making a decision has to be at least as good as not learning. This means the precise agent never pays to avoid free evidence. The above principle amounts to the weaker claim that learning cannot be worse. Since the precise Bayesian has complete preferences, these two principles amount to the same thing in the precise case. In the imprecise case, however, they are importantly different. We have seen that the refined NDS rule satisfies P1′, but the gamma-maximin rule does not.
5. Broader Considerations of Dynamic Coherence
Could we do better than the NDS rule? Should we be looking for a decision rule that never even permits paying to avoid free evidence? We think the answer is no. There is a sense in which decision rules like NDS cannot be improved on.Footnote 29
If one wants to advocate some form of nonprobabilistic epistemology, and if one wants to link this up with decision theory, then some axiom of the standard expected utility representation theorems must be denied. The plausible candidates for denial are the ordering postulate and the independence postulate. The NDS rule violates ordering, while the gamma-maximin rule upholds ordering but violates independence.
Any denial of independence leads to a pragmatic kind of dynamic incoherence: effectively refusing free money. Indeed, the dynamic incoherence can be seen as a sort of ‘information aversion’ or paying to avoid free evidence (Seidenfeld Reference Seidenfeld1988; Wakker Reference Wakker1988; Al-Najjar and Weinstein Reference Al-Najjar and Weinstein2009). This is easily shown via the decision problem in figure 3. Let A, B, C stand for lotteries and ApC stand for the mixed lottery ‘A if Z, C otherwise’ where Z has probability p. Let ≻ be Your strict preference—that is, ≻ is asymmetric and transitive. Independence is the principle that states that A ≻ B if and only if ApC ≻ BpC for all C. A violation of independence is thus a set of lotteries A, B, C where A ≻ B but it is not the case that ApC ≻ BpC. Assume a strong violation of independence (i.e., one involving a preference reversal): BpC ≻ ApC. A sophisticated agent with the above independence-violating preferences reasons about the decision in figure 3 as follows (Seidenfeld Reference Seidenfeld1988; Steele Reference Steele2010): You start by looking at the ‘terminal’ decisions: those decisions with no further choices after them. In this case, those are the choices at nodes 1 and 2. In each case, the arrow to the right highlights the hypothetically chosen option. Both these choices can be read off the preferences: A ≻ B, so A is chosen at node 1, but BpC ≻ ApC, so BpC is chosen at node 2. Now consider the choice from node 0. Since You know that Your node 1 choice will be A, the ‘up’ option of node 0 is effectively the gamble ApC. The ‘down’ option of node 0 is a choice between the two gambles ApC and BpC. Since You prefer BpC to ApC, You prefer to go down at node 0. Since the preference is strict, there is some small amount of money, δ, that You would pay to go down if offered this decision problem. Recall that p is the probability of event Z. Note that choosing up is effectively choosing to learn whether Z is true before making Your choice. Since You prefer down to up, You would pay to avoid learning whether X has occurred. The only thing needed to get this example going was a strong violation of independence.Footnote 30
Figure 3. Decision tree that yields information aversion (after Seidenfeld Reference Seidenfeld1988).
We advocate, instead, dropping the ordering postulate: the induced preference can be incomplete. As we have seen, such decision rules also sometimes suffer from issues with information aversion. However, they merely permit—never mandate—paying to avoid free evidence. This is an advantage over independence-violating decision rules. See also Bradley and Steele (Reference Bradley and Steele2014a) for further discussion of permission and obligation in the context of sure loss and sequential choice.
6. Concluding Remarks
Let us return to the trilemma that we began with:
P1. Paying to avoid free evidence is irrational.
P2. Incomplete preferences are not irrational and can be represented by sets of probabilities and utilities. In other words, imprecise probabilism is permissible.
P3. All plausible decision theories for handling imprecision sanction paying to avoid free evidence.
Of course, one could always deny P2 and claim that imprecise probabilism is evidently irrational. But we think there are compelling reasons for beliefs being imprecise, and so we consider this ‘way out’ to be a last resort (but see Al-Najjar and Weinstein Reference Al-Najjar and Weinstein2009 for a different opinion).
Alternatively, one could deny that ‘free evidence’ has any natural meaning. This could be reasonable: ‘free’ only makes sense with respect to some notion of value, and notions of value seem intimately tied to theories of decision. One might say that evidence is free by the lights of some decision theory just in case You would not pay to avoid this evidence if You were abiding by the theory. In terms of our trilemma, this move denies P3—that decision theories for handling imprecision allow paying to avoid free evidence—by claiming that the evidence is not free, by definition. Of course, the spin-off of this move is that P1 is rendered vacuous. A resolution of this sort might be appealing to those who already regard free evidence a merely technical term in the precise context.
We do not consider this a reasonable response, since Good’s definition of free evidence seems intuitive and applicable to a wide class of decision theories. Recall that evidence is free by the standards of Good’s proof if learning this evidence does not otherwise change Your decision problem. That is, You update Your beliefs according to conditionalization, but the available options A 1, …, As and the utilities of the outcomes associated with these options remain the same, whatever evidence is learned.Footnote 31
One might object to P1 on the grounds that there are cases of intuitively free evidence that You might reasonably want to avoid. For instance, imagine You are trying to decide between reading Murder on the Orient Express and going for walk, and then someone offers to tell You who killed Mr. Ratchett before You make the decision. Common sense tells us that it may be disadvantageous for You to accept this evidence, even if it comes with no explicit charge, which seems strangely at odds with Good’s theorem.
Closer inspection of the above case reveals that the evidence, while not incurring a monetary or other material cost, is not free in the Good sense because it may change the outcomes of the reading option: reading the novel when You know who did it is not the same act as reading the novel when You are one step behind Poirot and his little gray cells. That is to say that there is intrinsic value to Your current state of ignorance with respect to the identity of the murderer: you enjoy your reading of the novel more while not knowing who did it. Kadane et al. (Reference Kadane, Schervish and Seidenfeld2008) discuss a similar case (example 12). States of belief having intrinsic value offer well-known exceptions to various Bayesian theorems, for example, Dutch book arguments for additivity or conditionalization.
What we have done in this article is examine premises P1 and P3, with their meanings taken at face value. We have shown that something stronger than P3 in fact holds—it is not just the ‘extant’ decision theories that permit paying to avoid evidence but in fact any plausible decision theory for handling imprecision permits paying to avoid free evidence in some cases. On the positive side, however, we have refined the consequence of ‘paying to avoid evidence’. Certain decision theories—those akin to the NDS ruleFootnote 32—do indeed permit paying to avoid free evidence in ‘problem cases’ that apparently all involve dilation, but these theories never have avoiding free evidence as the uniquely admissible option when learning is available.
So, if we are willing to revise premise P1 in favor of P1′, then there is no longer a contradiction. Note that P3 can also be revised to P3′ so that it is more informative:
P1′ Paying to avoid free evidence, or even avoiding free evidence, should never be uniquely admissible, when pursuing free evidence is also an available option.
P2. Incomplete preferences are not irrational and can be represented by sets of probabilities and utilities. In other words, imprecise probabilism is permissible.
P3′ For a certain class of decision rules akin to the Sen-Walley maximality rule, learning free evidence is always admissible, but paying to avoid free evidence may sometimes be admissible too. There are no plausible decision rules that fare better than this, in the sense of never permitting paying to avoid free evidence.
This is, in our opinion, the best weakening or ‘way out’ of the original trilemma. It gives us the conclusion that the NDS rule is as good as it gets for the imprecise probabilist when it comes to negotiating free evidence.
