I have the following problem:
Imagine that N kids want to buy candy from a candy shop, each i-th kid has $D_{i}$ dollars, they give all the money to one of the kids to go and buy, at the store there are M options of candies, each k-th candy option has $S_{k}$ of them available in stock, but also every kid "i" like (or in other words "values") each candy "k" differently, as $U_{ik}$ dollars.
The kid objective is to configure and buy the candy in such a manner that the money spent by the i-th kid approaches the total value of candy it receives, without surpassing the limit in stock for each candy type. And also when it's not possible to give then exactly what they spent, buy the candy in such a way that is fair.
How do i begin to solve this? I'm really lost where to search for references.
I don't know what "fair" means to you.
There's the Shapley Value ( https://en.wikipedia.org/wiki/Shapley_value ). It is an axiomatic approach that captures ideas about fairness, then shows there is a unique way to divide the surplus among the players.
There's the core https://en.wikipedia.org/wiki/Core_(game_theory). The idea here is that any kid could refuse to participate and go to the store themselves, or even convince some of the other kids to join them and deviate from the group. In simple models, there is a result called ``equal treatment in the core'', where kids with similar amounts of money and similar tastes get the same outcome. This is another way to formalize the idea of fairness as coming from bargaining power.
You could also look at the Nash bargaining solution.
If the question is just about maximizing candy purchases subject to wealth constraint and then dividing up the spoils, you could just use Pareto optimality. Put a welfare weight on each child, solve the utility maximization problem, then trace out the Pareto frontier as the weights vary. Everything on the frontier is efficient, and you can pick whatever outcome seems most fair to you.