In my optimization class we were talking about maximin problems. A maximin problem is a problem that maximizes the lowest outcome (maximize f when f(x1,x2,…xn) = min{x1,x2,…xn}. Our professor made the claim that “… economics is focused on maximin problems or should be focused on”. I felt like this statement grossly understated my major and the range of economics. Even the optimization problems we face in economics have a wide variety such as profit optimization, game theoretical optimization, utility optimization, etc. not to mention the other sectors of economics that don’t include optimization. However, he does deserve some credit. Maximin problems are one way to solve a particular problems in economics.
When attempting to optimize it is nice to be all to confidently say choice x >= y, but comparing isn’t always this easy. An inequality can be difficult when x and y are not numerical values. Take the saying “…comparing apple and oranges” for example. The saying is conveying that the attributes of an apple(x), are in no way similar to an orange(y), making it impossible to say one is greater than the other. Believe it or not, economists have a solution to this problem; utility functions. Utility functions focus on one attribute; the sum benefit or cost of a choice or bundle. We are all familiar with utility functions; u(x) where x represents bundles or possible choices, and u represents the utility received for choosing x. Since u(x) usually produces one numerical value(the utility made from x decision) it is easy to compare choices (>,< or =) in the domain of u. This enables us to compare choices, and optimize our decision x. That was one hurdle economists have faced when comparing choices, but not the one that is uses maximin.
What if the outcomes are numerical, but has but as multiple sets of values. As an economist you’re trying to compare the point (2,1) and (4,0) by inequalities. One point is not strictly greater than the other. Similarly we can use our imagination to derive a function, a function that returns one value that can be compared to another singular value. One of the functions is a min{} function where f(x,y) = min{x,y}. This is a specific function that is used to describe specific situations. This function is also the maximin problem I was referring to. We are trying to maximize the minimum of the set (x,y). The decision mechanism the student was referring to was dictator games;
Thought Experiment: You are at dinner eating with your friends. You have finished eating but still have a lot of food left over, and your friends have made the mistake of getting off-campus meal plans, so they’re hungry. One’s a foodie and the other not so much, they only eat out of necessity. Who do you give your food to? And in what ratio? If you give all of your food to the foodie it will be appreciated and all you’ll soothe their hunger, but someone will go completely unfed. Even though the non-foodie won’t appreciate the food you can’t let them starve! That’s horrible! Even if the overall happiness between the two of them is highest when the foodie gets all the food. So you devise a maximin problem that says; I am only as happiest as the saddest person between them. This would feed the non-foodie only enough so that they are not starving and the foodie the rest, because they’ll appreciate it. (“The Division Problem” Planet Money)
This problem can be generalized to many tough problems in economics, but it cannot be generalized to all of economics…or can it? In economics we assume all resources are scarce. This means that we face allocating those resources to people in the best way possible. One way to compare distributions is maximin problems. Could economics be a huge, confusing, indescribable maximin problem? Another mathematician making accidental economic revelations, Nash would be proud.