(I cannot rule out that all of what I'm about to say has already been said, and already been refuted, somewhere in the literature.)

Rawls' difference principle states (someone correct me if I've got it wrong) that a deviation from perfect equality is just if, and only, if it would improve the position of the least well-off people in society.

Rawls' argument for the difference principle is as follows (again, please correct me if I've got it wrong). A deviation from perfect equality is just if, and only if, it would be chosen by a rational individual faced with a veil of ignorance about her own qualities (Premise 1). Such an individual would only want to deviate from perfect equality if doing so would improve the position of the least well-off people in society (Premise 2). Therefore, she would only choose to deviate from perfect equality if doing so would improve the position of the least well-off people in society (Conclusion).

The argument is clearly valid, so any criticisms must be directed toward its premises. Premise 1 says that a social institution is just if and only if it would be chosen by a rational individual behind a veil of ignorance. And Premise 2 says that a rational individual behind a veil of ignorance would maximise the position of the least well-off people in society. Conditional on accepting that justice is the right criterion for deciding what sort of institutions we should have, Premise 1 is fairly unobjectionable. Premise 2, on the other hand, is problematic.

Premise 2 is tantamount to the claim that it is rational to play a minimax strategy when faced with a veil of ignorance. It therefore implies that there is such a thing as a rational preference. Rationality is defined as doing whatever best satisfies one's preferences, conditional on one's preferences; it is a property of decision-making or behaviour, not a property of preferences. Arguably then, the difference principle is predictated on a category error.

The difficulty of maintaining that it is rational to minimax behind a veil of ignorance becomes clearer when we consider other possible strategies. Indeed, an obvious alternative strategy is to maximise expected utility. (I explain the difference between minimaxing and maximising expected utility at the end of the post.) Now, I am not arguing that maximising expected utility is more rational than minimaxing. Rather, I am pointing out that the difference principle rests on the claim that minimaxing is somehow more rational than maximising expected utility (or playing any other conceivable strategy).

A different interpretation of Rawls' difference principle is as follows. In referring to what a rational individual would do, Rawls did not mean to claim that there is such a thing as a rational preference. Rather, he was simply describing what a typical person would do. In other words, he was making a testable claim about what we should expect people to do when faced faced with a veil of ignorance.

Again, this interpretation is problematic. First, the population is almost certainly heterogeneous with respect to preference for equality. There are probably some people who would minimax; some who would maximise expected utility; and some who would play a different strategy altogether. Second, the difference principle no longer necessarily implies that deviations from perfect equality are only just if they improve the position of the least well-off people in society. Rather, it implies that deviations from perfect equality are just so long as the typical person tends to choose them. Indeed, if empirical enquiry reveals that people tend to maximise expected utility, say, then very large deviations from perfect equality may turn out to be just.

**The difference between minimaxing and maximising expected utility**

The diagram above depicts a veil of ignorance thought experiment (the numbers are completely hypothetical, and were chosen purely for the sake of exposition). Scenario 1 corresponds to perfect equality; the least well-off experience just as much utility as the most well-off. Scenario 2 corresponds to a situation where the least well-off experience 2 units of utility, the middle experience 3 units of utility, and the most well-off experience 4 units of utility. And scenario 3 corresponds to a situation where the least well-off experience 1 unit of utility, the middle experience 4 units of utility, and the most well-off experience 7 units of utility.

According to Rawls' difference principle, a rational individual behind a veil of ignorance should play the minimax strategy: she should maximise her minimum gain. The minimum gain from Scenario 1 is 1; the minimum gain from Scenario 2 is 2; and the minimum gain from Scenario 3 is 1. Therfore, she should choose Scenario 2.

However, an alternative strategy would be for an individual to maximise expected utility: he could maximise his average gain. The average gain from Scenario 1 is 1; the average gain from Scenario 2 is 3; and the average gain from Scenario 3 is 4. Therefore, if an individual were maximising his expected utility, he should choose Scenario 3.