# Can the following statement be rationalized if it yields a choice function?

A person choose an alternative to maximize another person's suffering.

I thought we could define a sort of relation where the person suffers more from x than y. And if we can always do this, we can verify if it is complete and transitive.

Now, after we do this, how can we rationalize it? I thought that maximizing another person's suffering was similar to choosing the maximal element in a set. This would violate one of the conditions of rational choice (namely, we must always have the same preference when we consider the smaller set versus the bigger set).

How can we go along verifying the above statement?

There could be a problem with the definition of suffering... I would like to think of suffering as disutility (or decrease in utility) and I will follow on this definition. However, there might be a different conceptualization of suffering (50 shades of gray, where suffering actually increases utility etc.).

What concerns your question, if I want to maximize someone's suffering (disutility), thus minimize their utility, I need to know what their utility is. So let's start from it.

Consider the following scenario:

Consumer B has some utility function $$U_B(\boldsymbol{x_a}, \boldsymbol{x_b})$$, where $$\boldsymbol{x_a}$$ denotes actions of consumer A, while $$\boldsymbol{x_b}$$ denotes actions of consumer B himself. Consumer A wants B to suffer (therefore to have the least utility possible).

Minimizing something is the same as maximizing the opposite value. Therefore, we could say that

$$U_A(\boldsymbol{x_a}) = -U_B(\boldsymbol{x_a}, \boldsymbol{x_b})$$

Now the question remains, what do we know about the $$\boldsymbol{x_b}$$. Do we know they "play" sequentially and who starts? Do they play simultaneously etc... If B starts then the problem is reduced to searching $$\boldsymbol{x_a}$$ such that the utility is minimized, which would be the basic optimization problem. If A starts then he should know the reaction function of B. However, it would still be regular optimization problem.

Regarding your question, preferences of A are rational because we can describe them by utility function.