Some parents urge their children to be the best in everything they do. They push them to be the best athlete, and the best scholar, and the best musician, and so on. Other parents urge their children to pursue whatever they are best at, whether it be athletics, academics or music. Some parents push their children to try their best. Still others try hard not to push their children to be the best, or even to try to be their best, because they worry about the psychological damage that such messages might cause. But most parents love their children, and however they raise them, they are trying the best they can on their behalf. After all, most parents genuinely want what is best for their children – they just have different conceptions of what that requires.

In seeking what is best for their children, most parents are implicitly buying into what has been the dominant view of individual rationality, at least in the West, since the time of the Greeks. On this conception, what it is to be rational is to act so as to maximise the overall quality of one’s life over the course of one’s life – that is, a rational agent chooses whatever options will make her life, as a whole, go as well as possible. The United States Army’s advertising slogan ‘Be All You Can Be’ taps into this picture of what it is for an individual agent to be rational.

Built into the standard conception of rationality are two fundamental assumptions. The first is that there is a best way for any life to be. The second is a more technical assumption – I’ll call it the Axiom of Transitivity for Better Than – which holds that for any three choices, if the first option is better than the second, and the second option is better than the third, then the first option must be better than the third.

The Axiom of Transitivity for Better Than generates a decision procedure for identifying the best of any finite set of options. Compare them two at a time. If the first is better, throw the second out. Then compare the third with the first. If the third is better, throw the first out. Proceed in this way, always choosing the best of each set of two alternatives. On this basis, if the Axiom of Transitivity for Better Than is true, we can determine the best of any finite set of n options, on the basis of n-1 pairwise comparisons.

Many people have challenged the first assumption in one of four ways. Some have pointed out that some options might be equally good, so there is no single best option. Others have suggested that some alternatives might be only roughly comparable, or on a par. On this view, two alternatives could be in the same ballpark, say the genius of Einstein or Mozart, or a legal career versus an academic career, without one being better than the other, or their being exactly equally good. Still others have suggested that in some rare cases two alternatives can be completely incomparable. And finally, some have noted that among an infinite number of possibilities, there might be no best one, just as there is no largest number in the infinite sequence 1, 2, 3, 4, …

Friends of the standard conception of rationality can readily amend their view to accommodate such worries. They can say that if two alternatives are equally good, genuinely incomparable or only roughly comparable, then there is no compelling reason to choose one rather than the other, so we can choose either, rationally. They can then add that we are finite beings, who usually must select from a finite set of options, and for all such cases an agent can rationally choose any option as long as there is no available option that is better than it. So, on this amended view, even if there might be no single best option, rationality can direct us to never choose a worse option over an available better option.

The second fundamental assumption has gone unchallenged for most of human history. Most philosophers, economists and others have assumed that the Axiom of Transitivity for Better Than must be true, in virtue of the meanings of the words better than, or as a matter of logic. Indeed, most have assumed that all comparative ‘-er than’ relations must be transitive. And most are. So, for example, if Ahmed is taller (or faster, or heavier) than Ilsa, and Ilsa is taller (or faster or heavier) than Quiping, then, indeed, Ahmed must be taller (or faster, or heavier) than Quiping.

But the second fundamental assumption about rationality could be deeply mistaken.

How tall Ahmed is – in absolute terms – just depends on internal facts about Ahmed. And those internal facts don’t change depending on whom Ahmed is compared with in terms of height. Ahmed is tall in comparison with mice, and short in comparison with giraffes, but the factors we consider in assessing someone’s height, and the way we appeal to those factors in determining whether one being is taller than another, never vary. This is why ‘taller than’ is a transitive relation. Since the factors that are relevant for comparing Ahmed with Ilsa in terms of height are the very same as the factors that are relevant for comparing Ilsa and Quiping in terms of height, and Ahmed with Quiping in terms of height, this ensures that if, in terms of those unchanging factors, Ahmed is taller than Ilsa, and Ilsa taller than Quiping, then Ahmed is taller than Quiping.

But ‘better than’ is different. Often the factors that are relevant for comparing one outcome with another to determine which of the two is better vary, depending on what alternatives are being compared. Thus, it can be that one outcome is better than a second, in terms of all the factors that are relevant for making that comparison, and a second outcome is better than a third in terms of all the factors that are relevant for making that comparison, and yet the first outcome might not be better than the third, because the factors that are relevant for determining which of those two outcomes is better might be different than the factors for comparing one or both of the other two sets of alternatives.

Here is a real-world example. In the US, many favour a policy of affirmative action of the following form. They believe that some preference should be given for hiring African Americans for certain positions over whites. The justification of their view lies in the peculiar historical relation between whites and African Americans in the US, including the history of slavery, lynching, Jim Crow laws, and so on. Notice, on this view, there would not be reason to give preference to Mexican Americans over whites, nor to African Americans over Mexican Americans; since, to put it crudely, Mexican Americans were not enslaved by whites, nor were African Americans enslaved by Mexican Americans.

On this view, there could be three candidates: Mr White, Mr Mexican American, and Mr African American, such that all things considered – that is, taking account of all of the factors that were relevant for making each comparison – we might judge that hiring Mr White was better than hiring Mr Mexican American, and that hiring Mr Mexican American was better than hiring Mr African American, and yet that hiring Mr African American was nonetheless better than hiring Mr White. Here, we have a violation of the Axiom of Transitivity of Better Than, and the reason for that violation, as given above, is that the factors that are relevant for considering the desirability of hiring Mr White are, at least in part, different depending on whether the alternative is hiring Mr Mexican American, or Mr African American.

Asked how they avoid this kind of dilemma, the WHO member succinctly replied: ‘We fudge!’

Here is a second example. Like many national and international health organisations, the World Health Organization (WHO) often adopts a cost-effectiveness approach in choosing between alternative health policies. Basically, this involves trying to get as much ‘bang for their buck’ in spending their scarce resources, and this requires making certain trade-offs between quality and quantity. So, for example, if they could cure a very bad disease that affected a relatively small number of people, or a slightly less bad disease that affected twice as many people, they would do the latter. However, this kind of trade-off, between quality and quantity seems plausible only for some cases. If the gap in disease severity were large enough, the WHO would no longer worry about the less severe disease, no matter how many people it afflicted. So, the WHO is not in the business of preventing short, mild itches or headaches, no matter how many people might be afflicted by such maladies.

But this combination of plausible views is incompatible with the Axiom of Transitivity of Better Than, since there could be a spectrum of maladies such that the first was very severe, but affected only a few thousand people, the second was slightly less severe, but affected far more people, the third was slightly less severe than the second, but affected even more people, and so on, until the last merely involved very short, mild itches, but they affected most of the Earth’s population. Using the cost-effectiveness reasoning that seems plausible for such comparisons, curing the second malady might be judged better than curing the first, curing the third better than curing the second, curing the fourth better than curing the third, and so on. Together with the Axiom of Transitivity for Better Than, these pairwise judgments imply that it would be better to cure the last malady than the first. But almost no one believes that. Curing the first malady seems clearly better than curing the last. Here, the factors that seem relevant for comparing the maladies that are adjacent to each other along the spectrum of maladies – which licenses the kind of trade-offs between quality and quantity described – are different from the factors that seem relevant for comparing the maladies at the opposite ends of the spectrum, where trade-offs between quality and quantity no longer seem permissible.

Notice, in this case, there is no best option for the WHO to choose. Worse, whatever option they choose, there is another available option that seems clearly better. Asked how they avoid this kind of dilemma, one member of the WHO tasked with actually making such decisions succinctly replied, as I recall: ‘We fudge!’ He was acutely aware that the WHO followed certain kinds of reasoning for certain choices, but that this reasoning inevitably led to a conclusion they couldn’t – and shouldn’t – accept.

It is striking how often people purchase an item, and immediately regret having bought that item, rather than an alternative item that they might have bought. This common reaction is normally thought to reflect some kind of psychological shortcoming in the agent. And no doubt often it does. But there is another possible explanation for buyer’s remorse being so ubiquitous. It might be that we often face a series of options for which the Axiom of Transitivity for Better Than fails. In those cases, there will be a series of options where the first will be better than the second, the second better than the third, the third better than the fourth, and so on, but the last option will be better than the first.

In that case, we would have an example of what the economists call a cycle, and a rational agent will have good reason to prefer the first option to the second (because it is, after all, better), and good reason to prefer the third option to the second, and so on, but the rational agent will also have good reason to prefer the first option to the last. In that case, there is no best option. Worse, it is guaranteed that whatever option one chooses, there will have been another available option that was better. In that case, it will be natural for people to suffer buyer’s remorse, as they will come home, compare the purchase they made with another one they might have made, and decide, correctly, that they chose the worse outcome. The problem is that if they were confronting a cycle of intransitive alternatives, such buyer’s remorse will be inevitable.

The classic bait-and-switch works as follows. A company advertises one item for a very low price. A customer goes to purchase the item, which is no longer available in the store (perhaps it never was). The customer is then offered another, fancier, more expensive item to buy. Having gone out to buy an item of that sort, he might be reluctant to go home empty-handed, and could end up going home buying an item much fancier, but also much more expensive than he originally intended to. Having done so, he might feel, and most people would agree, that the customer has been cheated by an unscrupulous merchandiser.

A variation of the bait-and-switch can be almost as effective. A store advertises an item – say, a car – at a very good price. When the customer arrives, the car might be available at the price advertised. But, the salesman points out that the car is inferior in many ways to another car with a host of additional options, that don’t cost too much more. The customer might agree that the additional options are worth the extra price, and decide to buy the fancier car. At that point, the customer could be shown a third car, which has even more options for not too much more than the second car. The customer might again agree that the additional options are worth the extra price, and decide to buy the even fancier car. In the end, the customer could drive home with a much fancier car than he really needed or wanted, and he might feel like a fool for having fallen for the salesman’s various enticements. Having arrived home, with his brand-new car, he might decide that he would have been better off with the much cheaper, stripped-down model that he originally went in to buy. And he might be right.

If I’m right, attempts to identify the best possible life for one’s child might be doomed to failure

This phenomenon is extremely common. When it occurs, the customer often feels like he has been stupid and/or bamboozled. He might explain his behaviour in terms of akrasia – weakness of the will – he just couldn’t help himself when offered all the extra bells and whistles. Economists might offer an even more harsh assessment. In accordance with the second fundamental assumption underlying the standard conception of rationality, the Axiom of Transitivity for Better Than, such a person is clearly irrational, since he prefers the second car to the first, and the third to the second, even though, as his final judgment revealed, he also prefers the first car to the third. And virtually everyone agrees that salespeople who engage in such activities are unscrupulous, taking advantage of human weakness and/or irrationality.

Yet, as we have seen, another possibility looms. The Axiom of Transitivity for Better Than might not be true. The car-buyer might not have been suffering from weakness of the will or irrationality, and the salesperson might not have been particularly unscrupulous. It could be that the second car really was better than the first, in terms of all the factors relevant for making that comparison, and the third car really was better than the second, in terms of all the factors relevant for making that comparison, and yet the first car was better than the third, in terms of all of the factors relevant for making that comparison. In that case, the salesperson can hardly be faulted for pointing out to the customer that there was second car, available for purchase, that was an even better option than the first, and a third that was an even better option than the second. And the customer, in turn, might have simply been responding to good reasons, in deciding, correctly, that he would be better off buying the second car, rather than the first, and the third rather than the second. But, of course, if the Axiom of Transitivity for Better Than isn’t true, then that could explain why, despite all that, the customer could have ended up with an outcome that was worse than another available outcome he might have had instead, namely, purchasing the first car.

Return to the parent who wants what is best for her child. Perhaps, at some point, she might consider urging her child to become an elementary-school teacher, which is a noble profession, and has many advantages. But if the child is talented, she might think that the life of a college professor would be even better, as it would also involve teaching, but perhaps bring more intellectual stimulation, better pay and greater respect. However, compared with the life of a professor, she might decide the life of a solicitor would be better still. It, too, would have intellectual stimulation, and perhaps even greater pay, with fewer years of schooling required, and much better job prospects. Yet, compared with the life of a solicitor, that of an investment banker might seem even better, with a host of social and economic benefits that few solicitors could hope to attain. Despite all that, at the end of the day, the life of an elementary-school teacher might seem better than that of the investment banker, involving less stress, more time with one’s family, the ability to live somewhere other than in a major financial centre, and so on.

Faced with such a cycle of choices, most would urge that we think carefully, and determine which life would actually be best. Supposedly, this would be the one that involves the best overall combination of benefits and burdens. But if I’m right, attempts to identify the best possible life for one’s child might be doomed to failure. Unfortunately, given the complexity of the normative world in which we live, it could well be the case that, for a given set of possible lives, the first is better than the second, which in turn is better than the third, which in turn is better than the fourth, and so on, and yet the first might not be better than the last.

This is a puzzling and disturbing possibility. It can leave parents at a loss as to what to recommend to their child, since for any option they might recommend, there is another option that would be better. Learning to live with that fact, if it is a fact, will require a massive rethinking of what it is to be a rational agent, and to make good choices.