Utility functions on indivisible goods

Alright kiddo, let's talk about utility functions on indivisible goods!

So imagine you have a big bag of candy, and you want to share it with your friends. But the thing is, you can't divide the pieces of candy - they're "indivisible." That means you can't break them up into smaller pieces.

Now, each of your friends might have different preferences for the kind of candy they want. Maybe one friend really likes chocolate, while another prefers gummy candies. This is where utility functions come in.

Utility functions are basically a way to measure how happy or satisfied someone is with a given situation. So in the case of candy sharing, your friends would have different utility functions that assign different levels of happiness to different kinds of candy.

Let's say your friend who loves chocolate has a utility function that gives them more happiness for each piece of chocolate they get. Meanwhile, your gummy-loving friend has a utility function that gives them more happiness for each piece of gummy candy they receive.

So when you divide up the candy among your friends, you want to make sure that each person gets the candy that will give them the most happiness (or highest utility). But since the candy is indivisible, you can't necessarily give everyone exactly what they want.

This is where things get a little more complicated, but bear with me. One way to determine the fairest distribution of the candy is to use an algorithm called the "Max-Min Fairness" algorithm. This algorithm basically tries to maximize the minimum utility of all your friends.

So let's say you have three friends - one who loves chocolate, one who loves gummy candy, and one who is pretty indifferent. You might give the chocolate lover 2 pieces of chocolate and 1 piece of gummy candy, the gummy lover 2 pieces of gummy candy and 1 piece of chocolate, and the indifferent friend 1 of each.

This distribution maximizes the minimum utility, because even though the indifferent friend doesn't really care what kind of candy they get, they still get something, and the chocolate and gummy lovers both get at least 2 pieces of their preferred candy.

So there you have it, kiddo - utility functions on indivisible goods! It's all about figuring out the best way to divide things up fairly when you can't break them into smaller pieces.