# Are there mathematical models for multi-utility satisfaction?

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In several places I have seen reference to the law of diminishing returns with regard to satisfaction. A common toy example is eating cookies. The first cookie gives some amount of satisfaction, the second some smaller amount, and so on. It follows the satisfaction function $$S(n)$$ is a convex function of the number $$n$$ of cookies eaten.

This seems quite reasonable but I wonder how is it backed up experimentally? Are there studies where subjects are given various amounts of something and their satisfaction is somehow measured?

Three follow-up questions:

1. Has this been done using some objective measure of satisfaction rather than just asking the subject?

2. Are there any mathematical models for the shape of the convex function?

3. How do things change is there are several utilities? So $$S(n,x)$$ where $$n$$ is the number of cookies and $$x$$ describes the quality.