The balance and weight of reasons

Correspondence Nicholas Makins, King’s College London, London, UK. Email: ndmakins@gmail.com Abstract The aim of this paper is to provide a detailed characterisation of some ways in which our preferences reflect our reasons. I will argue that practical reasons can be characterised along two dimensions that influence our preferences: their balance and their weight. This is analogous to a similar characterisation of the way in which probabilities reflect the balance and weight of evidence in epistemology. In this paper, I will illustrate the distinction between the balance and weight of reasons, and show how this is crucial for an adequate account of preference and choice. The upshot is a more complete picture of a particular kind of decision, labelled by Isaac Levi and, more recently, Ruth Chang as ‘hard choices’. These are choices in which one option is better than another in some ways, the other is better than the first in some ways, but neither seems better overall. The distinction between the balance and weight of reasons presents a new way of understanding how hard choices vary by degree and what it is that makes them so hard.


| INTRODUCTION
A crucial part of practical deliberation involves weighing up our reasons for alternative courses of action. An agent evaluates the considerations that stand in favour of their different options and comes to some judgement about which is best, overall. In order to make a choice, one must settle where the balance of reasons lies. That is to say, whether or not, and the degree to which, our reasons favour one option over another. But this is not the only dimension of our reasons that influences preference and choice. An underappreciated aspect of practical deliberation is the overall weight of reasons that are at play in a decision. That is to say, the total amount that is at stake in a choice, or the gravity of the reasons that contribute to an agent's evaluation of their options. When new reasons come to light, two things may change: the degree to which one option is deemed preferable to the other, and the overall gravity of the reasons that are at play. This paper presents a philosophical exploration of this dimension of practical reason and the role it has to play in our understanding of preference and choice. In Section 2, I will explicate the distinction between the balance and weight of reasons and argue that the greater the weight of reasons underlying an evaluation, the more stable that evaluation will be when new reasons come to light. In Section 3, I will demonstrate how this account of the weight of reasons can fill some explanatory gaps in our understanding of 'hard choices'those in which one option seems better than another in some respects, the other seems better than the first in some respects, but neither seems better overalland the analysis of these choices in terms of options that are on a par. In particular, the weight of reasons plays a crucial role in the way in which hardness can vary by degree, while also explaining what it is about these choices that makes them genuinely hard.
First, a note of clarification on my use of the terms 'reason' and 'preference'. When I talk about reasons, I mean normative reasons. That is to say, those considerations that stand in favour of, or lend justificatory support to, the actions or attitudes of an agent. When I talk about preferences, I mean, all things considered, comparative evaluations of a variety of objects including options, outcomes, states of the world, or anything else that one can judge as more or less desirable. 1 The 'all things' here includes all of the normative reasons that a rational agent recognises as such. On this view reasons are, in some sense, prior to preferences and contribute to their formation. As will become clear in what follows, I think of the relation between preferences and reasons as similar to the relation between beliefs and evidence; rational preferences are based on, and sensitive to, our reasons in much the same way that rational beliefs are based on, and sensitive to, our evidence. However, the discussion of reasons that follows does not rely on any particular view about the source of reasons themselves, nor does it assume a position on the debate between internalism and externalism about normative reasons. In particular, although I see preferences as being responsive to reasons, those reasons in turn may depend on some conative attitude of the agent in question. Whatever the source of our normative reasons, the preferences of rational agents should be responsive to them and it is the way in which this is so that is my target.

| The distinction between balance and weight
Conventional accounts of preference characterise agents as either preferring one option A to another B, preferring B to A, or remaining indifferent between the two. However, this account misses some important features of the way in which agents may evaluate alternatives in a choice. Take indifference: one may be indifferent between A and B because there are no good reasons for either. For example, I have no reason to prefer the state of affairs in which the temperature in Winnipeg tomorrow is 25 C over the state of affairs in which the temperature in Winnipeg tomorrow is 27 C. I will not be anywhere near Winnipeg tomorrow and simply do not care about the weather there. With no reason to prefer one temperature to the other, I am indifferent. Alternatively, one may be indifferent between A and B because there are good reasons of equal strength in favour of both options. For example, Buridan's ass finds itself exactly halfway between two equally large and delicious piles of hay; with equally strong reasons in favour of each pile of hay, the ass is indifferent between the two. Lastly, one may be indifferent between A and B because there are good reasons of equal strength against both options. For example, when sailing home from the Trojan War through the Strait of Messina, Odysseus has 1 For a defence of this interpretation of preferences, see Hausman (2011) and Bradley (2017). For a critique, see Thoma (2019).
to plot a course either within reach of Scylla, a sea monster with six heads full of sharks' teeth, or Charybdis, a whirlpool that swallows into the deep whole ships and their crews. With equally strong reasons against each option, Odysseus is indifferent between the two, stuck between a rock and a hard place.
These are all instances of indifference, but there is an important dissimilarity between the first caseconcerning my indifference about the weather in Winnipegand the other two. Whereas my indifference regarding the temperature in Winnipeg stems from my complete lack of interest in the alternative possibilities, Buridan's ass and Odysseus have a considerable amount at stake and compelling, but conflicting, reasons pulling them in opposing directions. This difference is not reflected in the balance of reasons, because, in terms of the overall comparative evaluation, they are all simply classified as indifferent. Rather, this difference is captured by the fact that the latter two are ambivalent, with strong reasons for mutually exclusive options.
A common metaphor for ambivalence sees competing reasons balancing on either side of a set of weighing scales. Reasons for one option accrue on one side, reasons for an alternative on the opposite side, and the weightier sum of reasons tips the balance in favour of the option to be pursued (Cullity, 2018;Lord & Maguire, 2016;Maguire & Snedegar, 2021). Different comparisons might differ from one another in terms of the balance of reasons, which is represented by the relative heights of the two sides of the scales. The more that reasons for one option outweigh those for an alternative, the further down that option is pushed. However, they may also differ in the total amount of weight on the scales. When new considerations come to light that are relevant to a choice, they may shift the balance in favour of one option or the other, but either way the overall weight of reasons will have increased. Although my attitude towards the weather in Winnipeg and the attitude of Buridan's ass towards the piles of hay are the same in terms of the balance of reasonswe are both indifferentthey differ in terms of the overall weight of reasons.
It should be noted that the weighing scales metaphor is only a crude model of practical reason and is not intended to capture every aspect of rational deliberation. It should not be taken to imply that our reasons always simply add their weights together, like inert pieces of cast iron, nor that they simply provide unqualified support for one option or an alternative. For example, Snedegar (2018) has argued that we should distinguish between reasons against one option and reasons for the alternatives, while Tucker (2022) has argued that a dual scale model of reasons is needed to adequately represent ethical theory. More complex setups such as these might be required for some applications of the scale metaphor. Nevertheless, the basic version of this model can be fruitful for representing certain aspects of the way in which we consider and compare our practical reasons. In particular, it allows us to get an intuitive grasp of the distinction between the balance and weight of reasons, and will be a helpful tool for exploring some of the explanatory and normative implications of this distinction.

| The weight of evidence and stability of credences
Recognising the role played by the weight of reasons, and its distinction from balance, has a number of important implications. The main point that I aim to demonstrate here is that the weight of our reasons affects the stability of our preferences. In particular, the greater the total weight of reasons on the scales, the less that a new reason will be able to change the balance. This idea mirrors a closely related point in epistemology, about the balance and weight of evidence. 2 Consider the following pair of cases, suggested by Popper (2002Popper ( [1959) to illustrate his famous 'paradox of ideal evidence': 2 See, for example, Peirce (1992), Keynes (1921), Carnap (1950), Jeffrey (1965), Skyrms (1977), Levi (1980Levi ( , 2011, Gärdenfors and Sahlin (1982), Joyce (1999) and Weatherson (2002).
Coin toss A I hand you a coin and tell you that it is fair. A quick look at the coin suggests no reason to think otherwise. What is your credence that the 6th toss of the coin will land heads? You might say ½.
Coin toss B I hand you a coin and tell you that it is fair. Again, a quick look at the coin suggests no reason to think otherwise. This time, however, you play with the coin for a while, flipping it repeatedly and keeping track of the results. Over 100 coin tosses, 50 land heads and 50 land tails. What is your credence that the 106th toss of the coin will land heads? You might say ½.
In some sense, you have said the same thing in response to both of these questions. In both cases, you have a credence of ½ in the proposition that the coin will land heads on some future toss. However, your credal state might well be thought to be quite different in the two cases. That difference lies in the weight of evidence that underlies your credences. In the first case, the only evidence you have is the appearance of the coin, my claim that it is fair, and your general knowledge of how coins tend to behave. In the second case, however, you have all of this evidence, plus that gained during the additional 100 coin tosses. This provides more evidence overall, but the balance of the evidence does not change. So, the balance of evidence is the same for both, but the weight of evidence is different. In the words of C.S. Peirce: […] we ought not to have the same feeling of belief in reference to all events of which the chance is even. In short, to express the proper state of our belief, not one number but two are requisite, the first depending on the Inferred probability, the second on the amount of knowledge on which that probability is based. (Peirce, 1992, p. 160) To see what difference the weight of evidence makes in practice, consider what happens to your credence in the 6th or 106th coin toss coming up heads as you observe tosses 1 to 5 or 101 to 105. Suppose that both sequences produce five consecutive heads. In the first case, your credence in the hypothesis that the 6th toss will land heads is going to increase quite substantially. The fact that all observed coin tosses have landed the same way up suggests that the coin is biased towards heads, which would increase the probability of heads on the next toss. In the second case, however, when you have already tossed the coin many times and seen an even distribution of heads and tails, a sequence of five heads in a row is not going to change your credence all that much. In fact, over a long run of coin tosses, we would expect to find just this sort of sequence of back-to-back heads every so often. Because the evidence in the first case is weightier, the associated credence is more stable. That is to say, a given piece of new evidence elicits a smaller change in one's credence.
Note that in this pair of cases, the balance of evidence for some propositions might differ. For example, in the first case, as I have suggested, the balance of evidence suggests that the coin is likely to be biased, but in the second case it does not. Nonetheless, the cases allow us to see that there are some propositions for which the balance of evidence is the same, but the weight is different, and this will manifest as greater stability in the credence produced by the weightier body of evidence.

| The weight of reasons and stability of preferences
What I aim to demonstrate here is that something similar should be thought to happen when it is the weight of our practical reasons rather than our evidenceor reason for beliefthat differs. That is to say, when making comparative evaluations of alternatives in a choice, the greater the weight of reasons being considered, the less that a new reason will be able to tip the scales in either direction. 3 To make this point clear, consider an agent's preferences over various sums of money. Suppose that this agent is an expected utility maximiser and their preferences can be represented by a cardinal utility function that increases linearly with money. In other words, the only reasons underlying their preferences concern money, and every £1 contributes equally strong reasons to every other. There is a sense in which the strength of their preference for £10 over £5 is greater than that for £1000 over £800. After all, £10 is twice as good as £5, but £1000 is only 1.25 times as good as £800. This is a difference in the balance of reasons; the scales are tipped further to one side when £10 and £5 are being compared than when £1000 and £800 are. This difference will manifest in the agent's choices between lotteries with the relevant sums of money as possible pay-offs. For example, consider the agent's preferences over the pairs of lotteries described in Table 1, in which the outcome depends on whether a red ball or a yellow ball is pulled out of an urn.
The greater degree of preference for £10 over £5 than for £1000 over £800 means that the agent will take Lottery II over Lottery I at a lower probability of winning than that at which they will take Lottery III over Lottery IV. If offered a choice between Lottery I and Lottery II, the agent would be indifferent when the probability of a red ball p(Red) = 0.5, prefer Lottery I when p(Red) > 0.5, and prefer Lottery II when p(Red) < 0.5. However, if offered a choice between Lottery III and Lottery IV, the agent would be indifferent when p(Red) = 0.8, prefer Lottery III when p(Red) > 0.8, and prefer Lottery IV when p(Red) < 0.8. This difference in the probability at which they are indifferent is a result of the difference in the strength of preference.
However, the weight of reasons underlying the preference for £1000 over £800 is greater than that for £10 over £5. Simply put, there is more money on the table. This means that, even though the latter preference is stronger in the way described above, the former preference relation is more stable: it would require larger changes to the quantities in order to be reversed. The amount by which the outcome of picking a red ball after choosing Lottery II would have to be improved in order to make it preferable to the outcome of picking a red ball after choosing Lottery I is >£5. The amount by which the outcome of picking a red ball after choosing Lottery IV would have to be improved in order to make it preferable to the outcome of picking a red ball after choosing Lottery III is >£200. In other words, when weightier reasons are at stake, a fixed change to the possible outcomes will elicit a smaller change in a rational agent's T A B L E 1 Two pairs of lotteries.

THE BALANCE AND WEIGHT OF REASONS 5
preferences. 4 This also means that, when neither of two options is deemed better than the other, how strong any new reasons will have to be in order to break the tie will depend, in part, on the weight of reasons already at stake. We will return to this point in Section 3. The effect of the amount at stake on the stability of one's preferences can also be seen in more complex, realistic decisions. Consider another pair of choices.
Moving house I am moving house and must decide where to live. I have narrowed down the options to two: remain in London, where I have always lived, or move to Otley, West Yorkshire. In London I have family and friends, and access to the outstanding arts and culture of a global metropolis. Otley, on the other hand, presents the opportunity to live within touching distance of the Yorkshire Dales and the excitement of getting to know somewhere new. On balance, I prefer Otley.
Dessert I must choose whether to have vanilla or chocolate ice cream after dinner this evening. The only reason I have for either is the amount of gustatory pleasure I would derive from each. Although I like ice cream, there simply aren't very weighty reasons on the scales here. On balance, I prefer vanilla.
In virtue of the weightier reasons involved, my preference in the choice of where to live will be more stable than in the choice of which dessert to have. That is, one of the options would have to be improved by a greater degree for my preference ordering to change. For example, although I prefer vanilla to chocolate, I would prefer chocolate plus £100 to vanilla and no monetary bonus. This is not the case in the choice between London and Otley; adding £100 to the option of living in London will not reverse my preference for Otley. In fact, to change that preference ordering would require a far more substantial change to the reasons for the two options. And this seems to be exactly how things should be! Evaluating these alternatives involves a comparison between weighty reasons. It feels like an important, high-stakes decision, with substantial considerations in favour of each option. Therefore, it would seem absurd for one's preferences regarding this important life choice to be so easily reversed by a comparatively small monetary sweetener. Thus, the analogy with the weight of evidence holds: when considering a great deal of evidence to reach a certain credence, that credence is more stable in the face of new evidence, and when there are strong reasons at stake when making comparative evaluation between choice options, this evaluation will be more stable in the face of new reasons.

| Potential objections
There are a number of ways that one might be tempted to explain the difference between these preferences without reference to the weight of reasons. First, one may claim that money simply does not feature as a reason for me in the decision of where to live, so financial incentives cannot tip the balance. It may be tempting to think that important personal matters like this should depend on considerations more profound and important than a bit of extra cash. This line of thought, however, does not hold up to much scrutiny. Even if we were to stipulate in the case 4 Another way of putting this is to say that a new reason will have different relative weight, depending on the weight of those reasons already under consideration: the greater the weight of existing reasons, the less a new reason weighs relative to them, so the less difference it makes to the overall comparative evaluation. that a significant reason to prefer Otley over London was the far lower cost of living, this would not force us to accept that adding £100 to the option of living in London would reverse my preference for Otley. Financial concerns are often a pivotal factor for even highly personal choices, but this does not mean that such choices can always be swayed by any given monetary improvement to one of the options.
Alternatively, one might think that the qualitative difference between the options makes it difficult to assess the strength of additional reasons, so the overall change needs to be bigger. Because London and Otley are qualitatively very different, it is hard to compare the reasons for living in each with enough precision for something like an additional £100 to make a difference, but chocolate and vanilla ice cream are sufficiently similar for the choice to be sensitive to this additional reason. While it is true that qualitative difference makes evaluative comparisons less precise, as I will discuss in more detail in Section 3, this does not fully account for the asymmetry between these cases. Suppose, instead, I were to choose between moving to Otley and moving to Ilkley, another market town between the city of Leeds and the Yorkshire Dales. Life in these two towns is fairly similar, but I have a preference for Otley over Ilkley, because, among other things, it has better pubs. £100 still would not be enough to swing this decision, even though the options are qualitatively so similar. However, if I were instead choosing between vanilla ice cream and a cigarette after dinner, I might clearly prefer the ice cream even though they are qualitatively very different, but this preference could be reversed by adding a £100 bonus to one of the options. In other words, I would prefer the ice cream to the cigarette, but would prefer the cigarette plus £100 to the ice cream. So, even when the options are more qualitatively similar, the effect of additional reasons is smaller if the overall weight of reasons is greater.
Lastly, it may appear as though the difference between these cases can simply be captured in the balance of reasons, with no need to complicate matters with the distinction between balance and weight. Perhaps the degree to which my reasons favour Otley over London is greater than that for vanilla over chocolate. This is not right either. Suppose, instead, we were to compare my preference for Otley over London to my preference for vanilla over banana ice cream. Although I in fact prefer Otley to London, the reasons for each are fairly evenly balanced and it feels like something of a close call. My preference for vanilla over banana, on the other hand, is clear cut, as I do not enjoy banana ice cream anywhere near as much as vanilla. Even then, while my preference for Otley over London could not be switched by £100, my preference for vanilla over banana could. £100 in the context of the weak reasons that settle ice cream choice has far more sway than £100 in a decision as important as where to live. Even when the balance of reasons is more uneven in one comparison than another, the comparison with the greater overall weight of reasons will be less sensitive to additional reasons.
The difficulty with applying this idea to preferences is that standard accounts of preference do not provide rich enough information to quantify the degree of preference for one option over another, nor the absolute value of an option. The only information taken to be captured by preferences is an ordering of alternatives and, at most, information about the ratios of value difference between different options. For example, we can say that the difference between A and B is twice the difference between C and D, but we cannot say that A is twice as good as B, nor that A has an absolute value of x and B has an absolute value of y. This is what is captured by the familiar idea in decision theory that the cardinal utility functions used to represent preference orderings are unique only up to positive affine transformation. Talking in terms of the balance and weight of reasons, on the other hand, gives an intuitive picture of the idea. We have, at the very least, an intuitive grasp of what it means to say that the balance of reasons in the choice of dessert is similar to that in the choice of where to live, but that the overall weight of reasons is far greater in the latter choice.
How one defines a precise measure on the balance and weight of reasons will depend on the specific view one adopts about the nature of reasons themselves. Whether you think that a reason to φ is evidence that φ-ing is right (Whiting, 2018) or that one ought to φ (Kearns & Star, 2008, 2009Silverstein, 2017;Thomson, 2008), a premise in good reasoning that leads one to φ (Hieronymi, 2005;McHugh & Way, 2016;Setiya, 2014;Way, 2017), an explanation of why it is good or right to φ (Broome, 2013;Finlay, 2014;Hyman 2015;Raz, 1999;Toulmin, 1950), a primitive notion (Parfit, 2001;Scanlon, 1998) or something else entirely, one will need to provide a different account of how to define and measure the balance and weight of reasons. However, the motivation for distinguishing these two dimensions of practical reason does not depend on any particular conception of reasons. In light of this, the ability to capture balance and weight as distinct variables may be considered a desideratum of any full account of reasons. Whether any particular account is better able to accommodate this distinction than others remains an open question for future work. However one defines a measure on the weight of reasons, it is my contention that this feature of practical deliberation has important explanatory implications for a particular class of decisions known as 'hard choices'. There are a number of well-known proposals for how such choices should be analysed and what implications they have for practical rationality. In what follows, I will focus on one such proposal: that hard choices are those between options that are 'on a par'. I will suggest that this account leaves open some important gaps in our understanding of the kind of decision at stake, but that these gaps can be filled once we recognise the role of the weight of reasons. In combination, the ideas of parity and the weight of reasons provide the conceptual tools to explain hard choices, and lay the foundation for how agents can deliberate and act in this context.

| Incomparability and parity
There are many ways in which a decision may be difficult. For example, it might be cognitively demanding, emotionally painful or require significant willpower. Some choices may be difficult because they are sensitive to considerations about which the decision-maker is ignorant or highly uncertain. However, there is a distinct class of decisions that have become known as hard choices, following Levi's (1986) book of the same name. These are choices in which one option seems better than another in some ways, the other seems better than the first in some ways, but neither seems better than the other, all things considered. 5 The different 'ways' in which the options are better than one another concern different types of value that they exhibit, or different kinds of reason that support them. 6 This presents a puzzle for our understanding of the ways in which alternatives can relate to one another.
When making quantitative comparisons between two objects, say A and B, it is natural to think that either A is greater than B, B is greater than A, or else they are equal. This most obviously applies for simple physical quantities: either I am taller than you, you are taller than me, or we must be the same height. The same is also true of more complex, abstract quantities, such as the price of a good, the mortality rate of a disease or the number of words in a language. When evaluating options in a decision, however, things are not always so simple. Sometimes it seems that neither of two alternatives is better than the other, but nor are they equally good. A classic example of this, suggested by Raz (1986), is that of a person leaving school and trying to choose what career to pursue. They have an excellent academic record and a place at a prestigious law school, but are also a talented musician and stand a good chance of making it as a professional clarinettist. So they ask themself, which is better: a successful career as a lawyer, or a successful career as a clarinettist? The legal career is better than the career as a clarinettist in 5 For simplicity, I will stick to decisions with two options, but in reality, of course, individuals may face hard choices between any number of possible alternatives. 6 For a related discussion of the structure of value and the structure of practical reasons, see Chang (2016). terms of salary, job security and social impact. Becoming a clarinettist, on the other hand, is better in terms of creative expression and aesthetic appeal. It might well be said that it is neither true that the law career is better than the musical career, nor that the musical career is better than the law career. In this case, however, that does not necessarily mean that they are considered equal. For if that were the case, then any small improvement to either would settle the choice. But we could add £100 to the hypothetical salary of the law career and it would still not seem as though this option was better than the alternative. The extra money would improve this option, but it would not break the tie, so the options must not be precisely equally good.
The conclusion Raz draws from this example is that the two are incomparable. The reason that neither is better than the other, but nor are they equally good as career options, is that they are just too different to be compared. According to this view, hard choices are those in which the options are incomparable, and two options are incomparable if and only if it is neither true that one is better than the other, nor true that they are of equal value. This definition reveals a tacit assumption that better than, worse than, and equal to are the only comparative relations that can hold between options. However, a number of philosophers have claimed that there are other ways in which options may be related. Prominent among these is Ruth Chang, who has argued that there are situations in which neither option is better or worse than the other, nor are they equally good, but they are comparable, since they are 'on a par' (Chang, 2002(Chang, , 2012(Chang, , 2017. The idea is that options may be in the same overall neighbourhood of value, in some respect, but be so qualitatively different to one another that they are not related by any of the three conventional comparative relations. Two options that are on a par both exhibit some particular type of value, but in very different ways. They are in the same overall neighbourhood of value, because the comparison between them does not favour either one over the other. Therefore, neither is better, nor worse, than the other. Nonetheless, they are not equally good, because, in light of their qualitative dissimilarity, there is an evaluative difference between them. So, they are comparable, despite neither being better than, worse than, nor equal to, the other. They are on a par. The idea of being in the same overall neighbourhood of value is somewhat vague, and I will suggest a more precise definition in Section 3.2, but this general overview should suffice for present purposes. 7 The concept of parity can be elucidated by the very case that Raz used to illustrate incomparability. Neither the career as a lawyer, nor the career as a clarinettist is better than the other in terms of goodness as a career. And this tie cannot be broken by adding £100 to the salary of the law career, so they are not equally good. However, this does not mean that there is no possible tie-breaker. If we were to increase the salary of the law career in the comparison by ten times, this might well make this option decisively better than the alternative. Or, if we stipulated that the law career was guaranteed to result in redundancy and subsequent unemployment midcareer, this too might settle the matter. The reasons in favour of becoming a clarinettist are more closely balanced against those for the legal career in the original example than they are for these alternative versions of the legal career that break the tie. Clearly, then, we can say something about the comparison between the original options. Since they both exhibit a degree of goodness as a career, but neither is better than the other, they are in the same overall neighbourhood of value. But the significant qualitative differences between them mean that they are not exactly equal. Rather, they are on a par. Choices between options that are on a par are hard because, although neither option is better than the other, this does not mean that there is nothing at stake, as would be the case if they were of exactly the same value. In contrast, when choosing between options that are precisely 7 A third way of understanding hard choices is in terms of indeterminacy. The idea is that comparative evaluations are vague, so there are some cases in which it is neither true nor false that one option is better than the other. For arguments in favour of this view, see Broome (1997Broome ( , 2000 and Constantinescu (2012). For criticism of this view, see Chang (2002) and Carlson (2004Carlson ( , 2013. Chang (2017) also explains why the hardness of choices cannot be accounted for solely by a margin for error that is brought about by our ignorance or uncertainty.
THE BALANCE AND WEIGHT OF REASONS equally balanced, such as which one of multiple copies of the same book to pick from a shelf, neither option is better than the other, but this choice is not hard, because it does not really matter which copy one picks; the choice can be made in a completely arbitrary, thoughtless way. The qualitative difference between the value of alternatives that are on a par means that the choice matters, but the reasons on the table fail to settle which of the options is to be pursued. This, so the reasoning goes, is what makes 'hard choices' hard.

| Degrees of hardness
The concept of parity appears to provide a compelling explanation of hard choices. As it stands, however, this account has a substantial weakness, in that it treats as equivalent choices that differ in important ways. For Chang, parity, and therefore hardness, seems to be binary. Choices are either hard, if the options are on a par, or not otherwise. But this ignores some important differences between various choices in which the options are on a par. Consider the example that Chang uses throughout her (2017) paper 'Hard choices': a choice between apple pie and lemon sorbet for dessert. While apple pie is deliciously warm and comforting, and lemon sorbet is deliciously tart and refreshing, neither seems tastier overall. Chang shows how this is a choice between options that are on a par: neither is better than the other, and nor are they equal, but they are comparable, since they are in the same overall neighbourhood of value. In this case, we can test that they aren't precisely equal, because a dollop of whipped cream would improve the apple pie, but still might not break the tie with the lemon sorbet, because of the significant qualitative difference between them. To borrow a phrase from Caspar Hare, the choice is (quite literally) 'insensitive to mild sweetening' (Hare, 2010, p. 237). This analysis puts the choice of dessert on an equal footing with other hard choices, such as Raz's example of the choice between a career as a lawyer and a career as a clarinettist. But this does not seem right. For one thing, the choice between apple pie and lemon sorbet is just not very hard at all. It is not the sort of choice that people tend to agonise over or worry about very much, and certainly not to the degree that seems proportionate in the decisions at life's major junctures. We often settle choices like the one Chang considers in an arbitrary way, without any real trouble. If someone were to be stuck when picking what to order in a restaurant and left the choice to the toss of a coin, we would not be dismayed by their disregard for the import of the matter in the way that we might were someone to adopt that approach to choices such as whether to pursue a particular career, marry a partner or conceive a child. Yet Chang's account of parity provides no clear way to distinguish between these choices. There seems to be a gradable dimension of hard choices that is missing from the analysis in terms of parity. Moreover, if the choice of career is a harder choice than the choice of dessert, but both choices meet the criteria for parity, then parity alone cannot explain why the former is so much harder than the latter. Given that some choices between options that are on a par are much harder than others, this account fails both to meet the need for a graded notion of hardness, as well as to provide an explanation of what it is that makes the truly hard choices so hard.
This presents a problem for an analysis of hard choices based on the parity relation, but this problem is surmountable. Once we properly understand the balance and weight of reasons, we can fill in the remaining gaps in our understanding of hard choices. In order to see this, however, it is important first to add one final detail to the concept of parity: not only can two options be on a par, but there is a range of values over which variations on one option are on a par with a fixed alternative. This point is illustrated by thinking about the variations on the legal career considered above. When compared to the career as a clarinettist, there are some versions of the legal career that are better and some that are worse, but there are multiple variations on this option that fall within a zone of parity with respect to the alternative. Broome (1997) has suggested an illuminating way of thinking about the structure of betterness orderings in this way. Although Broome was illustrating his claim that there are zones of indeterminacy in the betterness relation, the same point can be made about zones of parity instead (Chang, 2002). The idea works as follows: suppose we are comparing two alternatives: a career as a lawyer and a career as a clarinettist. We hold a standard version of the career as a clarinettist, C, fixed. We imagine a series of n variations on the legal career, L 1 -L n , and rank them from best to worst. We can then proceed through this series of variations on the legal career, comparing each in turn to the alternative, C. This will divide L 1 -L n into three zones: those that C is worse than, those that C is better than, and those that C is neither better nor worse than (see Figure 1). If this central zone contains a single item, then that item and C are precisely equally good, since any change to that item for the better or worse will change the comparative relation to either better than or worse than, thereby breaking the tie. If, however, there are multiple items in this zone, then they will be neither better nor worse than the alternative, but nor will they be equally good either. Instead, they are on a par. Both the original legal career and the legal career with an extra £100 fall within this zone of parity, while the legal career with a salary that is ten times greater and the legal career that will lead to redundancy and unemployment both fall outside of this range, as one is deemed to be better than the alternative, and the other worse.
With these ideas in mind, we are now in a position to provide a more precise definition of parity, by differentiating it from other comparisons in which neither option is better than the othertiesin terms of the way in which the tie might be broken. First, when neither is better than the other, but any improvement to one option relative to the other would be enough to break the tie, then they are equally good. For example, consider an individual choosing which bureau de change at which to buy some foreign currency and evaluating the alternatives solely on the basis of the exchange rate on offer. For two bureaus that offer the same exchange rate, neither is better than the other; they are tied. This tie is in fact a case of equality, since any improvement to the exchange rate offered by either would be enough to break the tie and settle the choice. Second, when neither of two options is better than another and no conceivable improvement to either could break the tie, then they are completely incomparable. For example, it does not make sense to say that a particular bureau de change is better or worse than a career in law, and there is no improvement to the exchange rate or salary on offer that could settle the comparison. The objects are simply not comparable. 8 Lastly, there are those cases in which it is not the case that any improvement to either option would break the tie, but there is This sort of comparison that is not well-defined is referred to by Chang as non-comparability.

THE BALANCE AND WEIGHT OF REASONS
some conceivable improvement that would. In the comparison between musical or legal careers, it is not the case that any improvement would break the tie, since adding £100 to the salary in the legal career does not decisively make it the better option, but there are some improvements that could break the tie, such as much larger changes to the salary or career prospects. These are cases of parity, which we can now define as follows: two options are on a par with respect to one another if (i) neither option is better than the other; (ii) at least one option could be made better than the other by some conceivable improvements; and (iii) it is not the case that one option would be made better than the other by any improvement. This classification of the different ways in which a comparative evaluation may result in a tie indicates that the differences between them are really just a matter of degree. Precise equality and complete incomparability are simply the extremities on a spectrum of tied pairs of options, varying in terms of the strength of tie-breaker required to settle the matter. This immediately suggests a way of making sense of the gradable nature of hard choices that I claimed was missing from Chang's account in terms of parity: the hardness of a choice between options that are on a par depends on how different things would have to be in order for the choice to be settled. That is to say, the degree to which one of the options would have to be improved in order to break the tie between them. In the final sub-section, I will demonstrate how the degree of hardness, thought of in this way, depends on two orthogonal factorsthe degree of qualitative difference and the weight of reasonsand how these, rather than parity alone, are crucial for an adequate explanation of hard choices.

| Weight of reasons, qualitative difference and hardness
The degree of hardness in a choice depends on the amount by which one of the options would have to be improved in order for one to be strictly better than the other. When any improvement to either option would be enough to break the tie, the options are equally good. When no conceivable improvement to either could break the tie, the options are completely incomparable. Between these two extremes lie all pairs of options that are on a par. But what determines how much an option would have to be changed in order to break the tie between options that are on a par? This depends on two factors.
First, the size of the zone of parity within which the options lie. The wider the range of values over which variations on one option are on a par with respect to a fixed alternative, the more that option will have to be changed in order to break the tie. Options that are on a par can be compared, since they are in the same overall neighbourhood of value, but this comparison is, in some sense, imprecise (Chang, 2016;Parfit, 2016). The less precise the comparison, the wider the range of values over which variations on the options are on a par. This, in turn, depends on the degree of qualitative difference between the two. Why should we think that the degree of precision depends on the degree of qualitative difference between the objects being compared? This claim is simply an extension of the idea that it is the qualitative differences between alternatives that makes comparisons between them imprecise in the first place. As Derek Parfit says, When two things are qualitatively very different, these differences would often make it impossible either that one of these things is better than the other by some precise amount, or that both things are precisely equally good. (Parfit, 2016, p. 113) If qualitative difference gives rise to imprecision in the comparison between items, it is natural to think that greater qualitative difference will result in a greater degree of imprecision, reflected in the size of the zone of parity. 9 9 The matter of specifying precisely what it means for there to be greater qualitative difference between items being compared remains a question for future research, but I hope that the reader will find the idea intuitively clear in the cases discussed. Second, the amount by which one of the options would have to be improved in order for a tie to be broken depends on the stability of the evaluation of each alternative in question. That is to say, the degree to which one's evaluation of some object changes in response to new reasons that stand for or against it. And this, as I argued in Section 2, depends on the overall weight of reasons. The weightier one's reasons when evaluating alternatives, the less sensitive this evaluation will be in response to new reasons, so the stronger those new reasons will have to be in order to break a tie.
In summary, if you are considering one option that falls within the zone of parity with respect to another, how easy it is to break this tie depends on the extent of this zone and how sensitive the evaluation is to additional reasons. How far it has to go and how easy it is to move. And these factors, in turn, depend on the degree of qualitative difference and the overall weight of reasons, respectively. Thinking about hard choices in terms of how different things would have to be in order for the choice to be settled not only provides an account of how hardness can vary by degree, but also opens the door to some further explanation of why it is that the choices at the harder end of the scale are so troublesome. The qualitative difference and weight of reasons that determine the degree to which one option would have to be improved to be considered better than another, when taken in combination, explain the hardness of these choices.
When an individual is ambivalent, with strong reasons for mutually exclusive alternatives, they cannot avoid choosing an option that they have pro tanto reason not to choose; some of their reasons will always be left unmet. The ambivalent agent cannot avoid some cause for regret, some consideration in light of which they have done the wrong thing, no matter what they choose. Moreover, the weightier one's reasons, the stronger the residual reasons that cannot be met and the more one may feel torn, or divided. 10 'In conflict situations', says Raz, 'our best efforts still leave us short' (Raz, 2011, p. 187). And in situations of greater conflict, our best efforts leave us even shorter.
When faced with such a choice, but the balance of reasons favours one option over the alternative, we can take consolation in the fact that we are acting as our reasons, taken together, dictate. Although we must still leave some reasons unmet, we can feel reassured that our choice is justified, safe in the knowledge that it is what any rational person would have done. However, when our reasons conflict and neither option is better supported by reasons than the other, no such consolation is forthcoming. One must make a choice that one has pro tanto reason not to make, but one cannot say why that set of reasons, rather than the other, is left unmet. This is captured by Chang's suggestion that in these choices 'our reasons run out' (Chang, 2017, p. 2). Part of what makes hard choices hard is that we must leave some reasons unmet without justification. The weight of reasons determines the degree to which the options would have to be different in order for the choice to be settled and the strength of the reasons that are left unmet.
The weightier one's reasons for competing alternatives, and the greater degree of qualitative difference between them, the stronger the residual reasons that cannot satisfactorily be met and the further one is from being able to rationally justify this choice. It is this idea that leads Frankfurt (2004, p. 95), in his meditation on ambivalence and wholeheartedness, to characterise ambivalence as 'a disease of the mind', echoing Saint Augustine before him. Although, strictly speaking, Frankfurt's focus is volitional ambivalence, rather than ambivalence in one's reasons, much of what can be said of the former applies equally well to the latter: The opportunity to act in accordance with his own inclination is a doubtful asset for an individual whose will is so divided that he is moved both to decide for a certain alterative and to decide against it. Neither of the alternatives can satisfy him, since each entails frustration of the other. (Frankfurt, 1999, p. 102) 10 See Makins (2022) for a detailed discussion of this form of ambivalence and how it may be understood as a non-cognitivist variety of moral uncertainty, when the conflicting reasons concern moral considerations.
When neither of two alternatives is better supported by reasons than the other, then being able to act in accordance with one's reasons is, likewise, a 'doubtful asset'. Either way, one will have to leave the reasons for one of these options unmet and there is no justification for failing to satisfy one set of reasons rather than the other. The hardness of a choice, thought of in terms of the degree to which things would have to be different for the choice to be settled by the balance of reasons, is directly related to the strength of the reasons that are ultimately left unmet.

| CONCLUSION
In summary, I have argued for the following claims: (1) a meaningful distinction can be drawn between the balance and weight of reasons in practical deliberation; (2) the greater the weight of reasons underlying one's preferences, the more stable those preferences will be in response to new reasons; (3a) the weight of reasons, along with the degree of qualitative difference between alternatives, provides an account of how hard choicesthose between options that are on a parvary by degree; and (3b) these factors also explain what makes those choices at the hard end of this spectrum so challenging.