Objective and subjective value and rationality

The choices it is objectively rational for an agent to make depend upon what the pertinent facts are, regardless of whether the agent knows them, or could have known them, at the time of her decision. For example, it is objectively irrational for Jim to eat the poisoned apple, even if he had no way of knowing the apple was poisoned and had good reason to believe it was not. It is objectively rational for agents to perform the act with the greatest objective value. This is different from what it is subjectively rational for an agent to do. Subjective rationality depends on the agent's epistemic position. It is subjectively rational for Jim to eat the poisoned apple, given what he knew. It is subjectively rational for agents to perform the act with the greatest ‘subjective value’.

It is commonly held that it is subjectively rational to maximize expected value. Unfortunately, standard expected value theory is inconsistent with one of my claims in this article. Expected value theory requires that orderings of options in terms of their expected value are negatively transitive.Footnote ⁴ Negative transitivity requires that if x does not have greater expected value than y and y does not have greater expected value than z, then x does not have greater expected value than z. I will argue that the best account of rational decision theory allows that the ordering of options in terms of the value they have given our epistemic situation can be negatively intransitive. Therefore, expected value theory cannot be the correct decision theory. Indeed, it is perhaps not surprising that this is one of my conclusions because, in some form, the VP has itself been the main defence of negative transitivity and the other axioms of expected value theory, such as completeness and transitivity.Footnote ⁵ Since I reject expected value theory, I will talk about ‘subjective value’, which is not lumbered with a commitment to negative transitivity.Footnote ⁶ The term ‘subjective’ may mislead. It is not about the value of an option to an agent or as it is perceived by an agent. Subjective value is value simpliciter relativized to the epistemic position of the agent at the time.Footnote ⁷ Subjective value determines subjectively rational choice.

The ordering of options in terms of their subjective value can be negatively intransitive because the relation ‘neither subjectively better nor subjectively worse than’ is intransitive. However, it cannot be the case that ‘equal subjective value’ is an intransitive relation, because the semantics of ‘equality’ entail transitivity. So, I will have to introduce some novel terminology: I will say that if an option x is neither subjectively better nor subjectively worse than y, then x has ‘correlative subjective value’ to y.Footnote ⁸ This still allows that in some cases x and y could be equally subjectively good. In my framework, all options with equal subjective value have correlative subjective value, but not all options with correlative subjective value have equal subjective value.

Since our information is usually incomplete, actions that have the highest objective value do not always have the highest subjective value. By implication, the objectively rational act is not always the subjectively rational one.

It will not be clear what the target of the VP is unless we make it clear whether the argument is about the objective or subjective value of the options. I am going to look at two different versions of the VP. The first I will call the ‘Incomparability Value Pump’ (VPⁱ). The VPⁱ shows that Jim can get pumped even if he has perfect knowledge of the value of the options, provided options are related by intransitive objective value relations, such as ‘incomparable to’ or ‘on a par with’. I will call a betterness ordering with intransitive relations a ‘non-standard betterness ordering’. A betterness ordering with only the transitive relations ‘better than’ and ‘equally good’ is a ‘standard betterness ordering’.Footnote ⁹ The second version of the VP I will call the ‘Uncertainty Value Pump’ (VP^u). The VP^u shows that Jim can make subjectively rationally permissible choices and lose out in a value pump, provided that options can be ordered negatively intransitively in terms of their subjective value.

These different versions of the VP have, in some cases, not been adequately distinguished in the literature. Philosophers have sometimes talked about negatively intransitive preferences without considering whether these preferences are the product of incomparability or ignorance.Footnote ¹⁰ This is an important oversight. We can know which theory of value is affected by the VP only if we distinguish the aforementioned versions of it.

Foresight

Frederic Schick has argued that if Jim has foresight, he would see what is in store for him, reject the offer of A at t₂ and thus stop the pump.Footnote ¹¹ McClennan has rendered this more precise using backwards induction.Footnote ¹² With foresight, we can predict what we would do at a future choice node and act accordingly to avoid a bad result. (A ‘choice node’ is just a single point in time at which a choice is made in a choice series, e.g. t₁ and t₂ are both choice nodes.) However, Wlodek Rabinowicz has argued that even if we have foresight, we can lose out in a modified and more complex VP, provided we are persistently given new offers.Footnote ¹³

Mongin, Schick and McClennan attack the VP on the basis that it does not have the practical import many have thought.Footnote ¹⁴ The central point of this article, on the other hand, is that practical consequences are irrelevant to the assessment of an agent's rationality. So, I will assume that agents do not have foresight and that they do get pumped. We can then look at whether this matters for their rationality.

The ‘no foresight’ assumption has a bearing on whether my versions of the VP are concerned with subjectively rational choice or objectively rational choice. Foresight is a feature of Jim's epistemic position and so has no bearing on what he is objectively rationally permitted to do. Since I am leaving it open that Jim's lack of foresight bears on what he is rationally permitted to do, my two versions of the VP are about subjective, not objective, rationality. If foresight would not protect Jim from loss, then my discussion of the VPⁱ applies by implication to objectively rational choice as well. Even though neither version mentions objectively rational choice, I can still draw conclusions about the objective value of options by assuming that Jim has perfect epistemic access to the value of the options. If Jim has perfect knowledge of the value of the options, then the objective value of the options equals the subjective value of the options. For instance, if Jim has perfect knowledge of the objective value of A+ and B and these options have equal objective value, then A+ and B have correlative subjective value. Therefore, by assuming that Jim has perfect knowledge of the objective value of the options, one can infer from claims about objective value to claims about subjectively rational choice and vice versa.

Premise 1ⁱ: choice between incomparable options

‘x >^o y’ denotes that x is objectively better than y. ‘x =^o y’ denotes that x and y are objectively equally good. ‘x ~^o y’ denotes that x is objectively on a par with y or objectively incomparable to y. ‘P_t(x)’ denotes that Jim is rationally permitted to choose option x at time t. ‘EA’ denotes that Jim has perfect epistemic access to the value of the options over the course of the series of choices. Premise 1 of the VPⁱ goes as follows:

1. 1ⁱ. In a pairwise choice only between options x and y at t and where the agent lacks foresight of future trades that will be offered:

   1. (a) If ( (EA) ∧ (x > ^o y) ), then P_t (x) ∧ ¬P_t(y).

   2. (b) If ( (EA) ∧ (x =^o y) ), then P_t(x) ∧ P_t(y).

   3. (c) If ( (EA) ∧ (x ~^o y) ), then P_t(x) ∧ P_t(y).

1ⁱ infers from objective value to rational permissibility. The subjective value of options equals the objective value of the options when the agent has perfect knowledge of the value of the options. For instance, if EA and x > ^o y, then x is subjectively better than y. Thus, it is subjectively rational to choose x. The premise is still only about subjectively rational permissibility because Jim lacks foresight. If the agent has foresight of future trades, 1ⁱ(a), (b) and (c) might all be false. What he is required to do if he has foresight depends on what is the correct ‘choice strategy’.Footnote ¹⁶ At t₁, given the available evidence, Jim has no reason to believe he will be offered future trades, so the value of x and y is all that should concern him.

Premise 2ⁱ: incomparability

Premise 2 of the VPⁱ is about the ranking of three options in terms of their objective value.

1. 2ⁱ. For three options, A+, A and B from t_n to t_n+m for all m ≥ 1, if a choice is made at each choice node in the series:

   1. (a) A ~^o B

   2. (b) A⁺ ~^o B

   3. (c) A⁺ >^o A

A is objectively incomparable with B; A+ is objectively incomparable with B; and A+ is objectively better than A. Negative intransitivity is possible because of the intransitivity of incomparability and parity. 2ⁱ states that the value of the options stays constant over time. This is because, as I will show in section VIII, some philosophers argue that the objective value of the options can alter over time by virtue of the fact that an agent chose them earlier in the series.

Premise 3ⁱ: practical loss

Premise 3 of both versions of the VP makes a claim about what choices agents are permitted to make at different points in time. This is the incomparability version.

1. 3ⁱ. If (EA ∧ (P_tn(x) ) ∧ (y <^o x) ), then ¬P_tn+m(y) for all m ≥ 1.

In words, 3ⁱ says that where agents have perfect epistemic access to the value of options, they are not permitted to end up with an option that is objectively worse than one they could have had earlier in the series. We are now in a position to prove the following theorem:

Theorem – Premises 1ⁱ, 2ⁱ and 3ⁱ are logically inconsistent.

Proof – Assume that Jim has perfect epistemic access to the value of the options. Suppose for reductio that 2ⁱ is true. Let t₁ and t₂ be two points in time and suppose that Jim is offered a choice between A⁺ and B at t₁, and between B and A t₂. Premise 1ⁱ implies that P_t1(B) in the choice between A⁺ and B, since A⁺ ~^o B. Now consider the choice made at t₂. 1ⁱ implies that P_t2(A), since A ~^o B. However, since P_t1(A⁺) and A+ >^o A, premise 3ⁱ implies that ¬P_t2(A).

One of the premises of the argument must be false. We need to be careful to separate the issue of what the premises logically imply from the issue of whether any of the premises provide any reason to believe that the other premises are false. I will argue that 3ⁱ does not provide reason to believe that 1ⁱ and/or 2ⁱ are false. However, I also believe that 3ⁱ is true and, along with 1ⁱ, implies that 2ⁱ is false.

Premise 1^u: choice under uncertainty

Premise 1 of the VP^u relates subjective value and rational permissibility. ‘x >^s y’ denotes that x is subjectively better than y. ‘x #^s y’ denotes that x and y have correlative subjective value.

1. 1^u. In a pairwise choice only between options x and y at time t and where the agent lacks foresight of future trades that will be offered:

   1. (a) If x > ^s y, then P_t(x) ∧ ¬P_t(y).

   2. (b) If x #^s y, then P_t(x) ∧ P_t(y).

Premise 2^u:

Premise 2^u is about the rankings of options in terms of their subjective value.

1. 2^u. For three options, A+, A and B at t_n and t_n+m for all m≥1, if a choice is made at each choice node in the series:

   1. (a) A #^s B

   2. (b) A⁺ #^s B

   3. (c) A⁺ >^s A

The ranking of these options in terms of their subjective value is negatively intransitive. This is a product of the intransitivity of correlative subjective value.

Premise 3 and the remainder of the VP^u

Premise 3 of the VP^u makes a claim about which choices agents are rationally permitted to make at different points in time:

1. 3^u. If (P_tn(x) ∧ (y <^s x) ), then ¬P_tn+m(y) for all m≥1.

3^u says we are not permitted to end up with an option which is subjectively worse than one we could have had at an earlier point in the series. We are now in a position to prove the following theorem:

Theorem – Premises 1^u, 2^u and 3^u are logically inconsistent.

Proof – Suppose for reductio that 2^u is true. Let t₁ and t₂ be two points in time and suppose that the agent is offered a choice between A⁺ and B at t₁, and between B and A t₂. Premise 1^u implies that P_t1(B) in the choice between A⁺ and B, since A⁺ #^s B. Now consider the choice made at t₂. 1^u implies that P_t2(A), since A #^s B. However, since P_t1(A⁺) and A+ >^s A, 3^u implies that ¬P_t2(A).

1^u, 2^u and 3^u are inconsistent. One of them must be false. My position on the VP^u is different from my position on the VPⁱ. As before, I argue that 3^u does not provide reason to believe that 1^u and/or 2^u are false. However, this time, I argue that there is good reason to believe that 2^u can be true.

A note on preferences

Before I critique these arguments, a note on preferences. The VP has often been presented as an argument about the rationality of preferences. The preference version of the argument can be inferred from both of these arguments. This is because the subjective value of the options determines the preferences it is subjectively rational to have; and the objective value of the options determines the preferences it is objectively rational to have.Footnote ¹⁷ So, for example, the preference version of the VP^u would say that Jim ought to be indifferent between A+ and B because they have correlative subjective value. So, he may choose either. He chooses B. Then, he is offered to swap A for B. He ought to be indifferent between A and B, so he may choose either. He chooses A. All we need for the preference version of the Value Pump to get going is for preferences to be negatively intransitive, which is what rationality requires if options are ranked as specified by 2^u and 2ⁱ.Footnote ¹⁸ If the VP is the wrong way to determine the subjective or objective value of options, then it is the wrong way to determine the rationality of preferences.

The intransitivity of correlative subjective value

I will argue that 2^u is true and that an examination of the argument which shows this provides some initial confirmation for my abductive argument that the VP^u does not provide us with reasons to reject 1^u or 2^u.

Persuasive non-practical arguments show that correlative subjective value is intransitive and therefore that orderings of options in terms of their subjective value can be negatively intransitive.Footnote ²² Suppose Sally has to choose between going to an Indian restaurant and a Chinese restaurant.Footnote ²³ Both restaurants only offer a two-course set menu for £18 and these menus vary from week to week. Sally has been to both restaurants before and she has no reason to think that either will be objectively better, simply because she has not yet eaten the particular meals on offer that evening. (This allows there to be a truth of the matter about which meal would be objectively better.) Given Sally's epistemic situation, the restaurants have correlative subjective value, so she should be indifferent between them. To bracket the issue of the normative significance of past choices or intentions, assume that she does not make a choice. Now suppose Sally finds out that the Chinese has put up prices. Their set menu now costs £18.01. This change seems too insignificant to make it the case that Sally rationally ought to choose the Indian restaurant. Sally should be indifferent between the Indian and the slightly worse Chinese, but prefer the original Chinese to the worse Chinese. Therefore, the restaurants are ordered negatively intransitively in terms of their subjective value.

Examples like these are legion in recent discussions of comparability. Indeed, there is widespread agreement among philosophers who have considered the Small Improvements Argument (SIA) on the possibility of negatively intransitive orderings of options in terms of their subjective value. The argument above is a variant of an SIA. It can plausibly be explained by Sally's epistemic deficit. But many philosophers argue that the SIA shows that sometimes the trichotomy of objective value relations ‘better than’, ‘worse than’ and ‘as good as’ does not apply between two options. Incomparabilists, like Broome and Raz, and proponents of a tetrachotomy of value relations, like Chang, argue that it is possible for options to be negatively intransitively ordered in terms of their objective value. This implies it is possible for options to be ordered negatively intransitively in terms of their subjective value, because subjective value equals objective value if an agent has perfect knowledge of the value of the options. Indeed, if it were true that the VP^u showed that options could not be ranked negatively intransitively in terms of their subjective value and if, as seems plausible, some SIAs exploit borderline cases of vague predicates, then the VP^u would have shown by implication that supervaluationism and all non-epistemicist theories of vagueness are false.Footnote ²⁴ No theorist of vagueness has yet thought that the VP^u even bears on the truth of his or her theory. It might do, but significant further argument is required to show that it does.

The epistemicist theory of vagueness might be thought to offer a way out of this. If some SIAs exploit borderline cases of vague predicates and epistemicism is true, then there seems to be room for an account which says that the betterness ordering is standard.Footnote ²⁵ It might be thought that this implies that the epistemicist at least must accept that options cannot be ordered negatively intransitively in terms of their subjective value. Since epistemicism is a respectable theory of vagueness, a respectable theory rules out the possibility of negatively intransitive orderings in terms of their subjective value. This is not correct. The epistemicist diagnosis of small improvements cases is that proponents of non-standard betterness orderings confuse our ignorance of a ranking with the lack of a ranking. That is, they confuse negatively intransitive orderings of options in terms of their objective value with negatively intransitive orderings of options in terms of their subjective value. It is this diagnosis which allows epistemicists to account for the phenomena in small improvements cases without granting that the betterness ordering is non-standard. Thus, negatively intransitive orderings of options in terms of their subjective value are accepted across the spectrum among those who have considered the SIA: from incomparabilists to proponents of a tetrachotomy of value relations to epistemicist comparabilists, like myself.

Initial confirmation of the abductive argument

My argument up until now serves two functions. First, it establishes that correlative subjective value is intransitive. If the other parts of 1^u and 2^u can be defended, then 3^u must be false. Second, an analysis of the argument for intransitive correlative subjective value provides initial support for my abductive argument. Practical arguments seem to be intuitively irrelevant to the assessment of arguments like the one above for correlative subjective value. The case for intransitive correlative subjective value is so clear that if it can be shown that it commits us to incurring practical costs, then we must accept those costs. Similarly, the claim in 2^u(c) that ‘A+ >^s A’ is just a brute fact to which we must adjust our theories of rational decision. We could avoid practical loss by denying that conjunct of 2^u, but there are good non-practical arguments against doing so. The same holds for arguments for the other conjuncts of 2^u. Any change to that premise recommended solely on practical grounds looks ad hoc. The best explanation for this is that the VP^u does not provide us with reasons to change premise 2.

One possible response to the abductive argument is that the revisions only seem ad hoc and that we cannot infer from this that the VP^u does not provide us with reasons since the revisions might seem ad hoc, but not be ad hoc. My reply is that it is hard to see why we would be incapable of intuitively recognizing the reasons the VP^u provides, even after close examination of the arguments. It is hard to see why we would always have the intuition that a particular change is not recommended for good reasons, even though there are good reasons. This does not deductively prove that the VP^u does not provide good reasons, but by far the best explanation of it is that the VP^u does not provide us with good reasons.

Many proponents of the VP implicitly accept that practical considerations should give way in the face of good non-practical arguments. For example, Martin Peterson rules out the denial of the principle of irrelevant alternatives as a solution to the VP, calling it ‘metaphysically odd’, and rules out Chang's claim that there are different senses of rational permissibility calling it ‘ad hoc’.Footnote ²⁶ He infers from this and the VP that options cannot be incomparable or on a par. But what if denying the possibility of incomparability or parity is just as metaphysically odd or ad hoc as denying the principle of irrelevant alternatives? If metaphysical oddness or ad hocness is sufficient reason to reject a proposed solution to the VP, then it should be sufficient reason to reject other proposed solutions to the VP. Peterson neglects this possibility and so does not consider the possibility that the VP does not provide us with reasons. When Peterson says that Chang's solution to the VP is ad hoc, he means that it is ad hoc for non-practical reasons. This is instructive. First, if he thought that the denial of parity were ad hoc on a non-practical basis, then, by consistency, he would have to accept that the VP is unsound. Second, the fact that the revision seems ad hoc provides confirmation for my abductive argument that the VP is impotent.

My arguments here are, if sound, bad news for expected value theory. First, I have provided some initial confirmation for my abductive argument that the VP^u does not provide reason to reject either premise 1^u or premise 2^u. Since the VP has been the main argument offered for the axioms of expected value theory, this is troubling. Second, there are very strong and widely accepted arguments which imply that options can be ordered negatively intransitively in terms of their subjective value. This implies that expected value theory is incorrect.Footnote ²⁷

Past choices add value

It might be argued that choosing an option at one choice node adds value to that option at subsequent choice nodes. One might endorse what I call the ‘Past Choices’ principle:

Past Choices (PC): For some option x, in a choice only between x and y, if an agent chooses x rather than y at t₁, then x has more value (for that agent) at t_1+n for all n≥1.

‘For the agent’ is in brackets because on some possible defences of PC, the option only has additional value at subsequent choice nodes relative to the choosing agent in particular, but not for other agents. Ruth Chang could be interpreted to be endorsing PC in the following passage:

The ‘then’ in the last sentence is like ‘therefore’: Chang takes the fact that PC allows people to avoid getting pumped to be a sufficient reason to change a theory of rational decision. She says that having chosen y at t₁, we have ‘most reason’ to choose y over x-minus at t₂. Elsewhere, she says that there is an isomorphism between reasons and value: ‘If an alternative has some value, then there will be a corresponding reason, and if there is a reason to choose an alternative, then there will be a corresponding value it bears.’Footnote ³¹ If we have more reason to choose y than x-minus at t₂ and there is an isomorphism between reasons and value, then y must have more value than x at t₂, i.e. y must be better than x at t₂. Thus, Chang can plausibly be interpreted as endorsing PC.Footnote ³² PC implies that 2ⁱ is false because it implies that if B is chosen at t₁, then it is better than, as opposed to incomparable to or on a par with, A at t₂. PC entails that Jim can rationally avoid getting pumped because he is not rationally permitted to choose A at t₂.Footnote ³³

The problems with this argument provide further confirmation for my abductive argument. Non-practical considerations count against PC, and the practical downsides of the denial of PC seem to be irrelevant to the assessment of whether it is true or not.

PC is subject to a variation of Michael Bratman's ‘bootstrapping objection’.Footnote ³⁴ Suppose that Jim faces a choice between A and B at t₁ and chooses B. Given PC, at t₂ B is better than A, for Jim. Note that Jim's choice of B does not make just a small improvement to B. B and A are not on an unstable knife-edge in terms of value: they are in a zone of parity, not at a single point of equal value. So, a minute improvement to B cannot be enough to make it better than A; the improvement must be reasonably large. Now, there must be some improved version of A, A*, which for Jim is on a par with B at t₂. If Jim is offered B and A*, he may permissibly choose B. He does so. This adds value to B for Jim, again. We can iterate this cycle of choices indefinitely. Merely by choosing B over and over again, B becomes a source of enormous value for Jim. And yet it is still humble old B; all that's happened is that Jim has chosen it over and over again. Imagine B is a career as a lawyer. PC implies that just by choosing the same career as a lawyer over and over again, it is better for Jim than a career as a clarinettist + £1 billion. This is true even though it is obvious that he should choose the career as a clarinettist. We should not accept this implication of PC. Therefore, we should not accept PC.Footnote ³⁵

There are good non-practical reasons to reject PC. The mere fact that it saves people from getting pumped does not provide reason to accept it: it simply seems ad hoc. The fact that it does provides further confirmation for my abductive argument. Moreover, from the practical point of view, PC is very desirable. If it were true, there would be potentially enormous practical benefits. If practical arguments provide us with reasons, then it seems as though the stronger the practical advantages of a theory if it were true, the stronger reason we would have to believe it to be true. And yet we have no reason at all to accept PC or other false theories which would have large practical benefits if they were true. The best explanation of this is that no practical arguments provide us with reasons. Chang herself evidently does not take the VPⁱ to be good reason to reject the possibility of parity because of what she takes to be the good non-practical arguments in favour of parity. But she does not consider the non-practical defects of PC, and so she does not consider the possibility that the VP does not give us reason to accept these defects, just as, in her view, it does not give us reason to reject parity.

Rationality and value

The second way to make past choices have normative significance would be to deny 1ⁱ(c):Footnote ³⁶

1. 1ⁱ. In a pairwise choice only between options x and y at t and where the agent lacks foresight of future trades that will be offered:

   1. (a) If ( (EA) ∧ (x > ^o y) ), then P_t (x) ∧ ¬P_t(y).

   2. (b) If ( (EA) ∧ (x =^o y) ), then P_t(x) ∧ P_t(y).

   3. ( c) If ( (EA) ∧ (x ~^o y) ), then P_t(x) ∧ P_t(y).

Chang could be interpreted to be arguing that by virtue of our choice of B at t₁ rationality constrains us from picking A at t₂, even though A and B are incomparable or on a par at t₂. Once again, Chang's proposal looks ad hoc in spite of its practical benefits. This completes the cumulative case for my claim that the VP does not provide us with reasons to reject premises 1 and 2.

In her ‘Parity, Internal Value and Choice’, Chang sets out the VP and then says:

As before, the ‘then’ in the first sentence is like ‘therefore’: Chang takes the VP to be the sole reason justifying the claims she goes on to make. First, she suggests that even though A and B are on a par, Jim is rationally required to choose B. Second, she suggests that there are different senses of ‘rational permissibility’ and that there is a sense in which Jim may rationally choose either A or B at t₂ and there is a sense in which he may not.

Chang does not offer any good non-practical justification for the claim that Jim is rationally forbidden from choosing A at t₂; all she has said is that this allows Jim to avoid getting pumped. And yet her claim seems to demand non-practical support, just as rejecting parity demands non-practical support. We need a non-practical explanation of why it is that sometimes agents are permitted to choose either of two options which are on a par, and in some cases they are not. A justification of why it is that we are allowed to choose two options which are on a par would go something like this: ‘it is rational to maximize value and since the options are on a par, neither option is better, so we maximize value whichever option we choose’. The problem is that we need to be given an explanation of why this rationale does not apply to the choice between A and B at t₂. The only reason Chang gives is that this allows agents to avoid getting pumped. In spite of that, the difference seems to be ad hoc.Footnote ³⁸ This completes my abductive argument. It might be true that there are good non-practical reasons to accept Chang's alteration to premise 1, but if there are, then these non-practical arguments would be doing the work, not the VP.

Chang's second claim is that there are different senses of rational permissibility. It is worth quoting Chang again:

Here, Chang confuses the claim that there are different senses of permissibility with the claim that there are different conditions in which it is permissible to choose either option. Her point is not that there is ‘a sense in which’ Jim may permissibly choose A at t₂. It is that in a choice between B and A, Jim may permissibly choose A, had he not already chosen B earlier in the choice series.Footnote ⁴⁰ Therefore, we may safely put Chang's claim to one side in discussions of the VP.