Luckily when it comes to regular cake cutting there may be a host involved that just wants to give his guests something nice. And the guests hopefully just enjoy the cake instead of being overly sensitive about the fairness of the divisions.
But rather as a stand-in for any resource that people want to figure out a fair, equitable way to distribute.
So you should read this as a guide to how different countries might be willing to fairly divide a contested piece of land, when different parts of the land have different qualities (e.g. suitability for farming, value in mining) and the different countries value those things differently (e.g. one is good at building mines, the other prefers to protect the environment).
There are a lot of conflicts in the world over resources that need to be divvied up in some way, where everyone really is looking out for the self-interest of their state or nation or community.
There's maaany possible justificatons for contesting previously non-contested resources. Historic or otherwise, justified or not. And those reasons change over time, change of leadership, economic or climate conditions, or even technological progress (resource previously considered uninteresting suddenly becomes valuable - and thus, often contested). Fairly (?) dividing whatever is contested, doesn't solve that problem.
One fix is to just accept that changing circumstances enhance or degrade whatever piece of the pie you got.
Or to keep 'moving the knives' continously. (Re-)distribution possibly enforced by military power. Oh wait... that is how it works!
Only straw-man arguments in economics use self interest to literally mean like “hedonic self interest” or layman’s self interest.
Economics actually does the opposite, the “self interest” is basically “whatever people want”. What do people want? They want to maximize their utility! What is utility? Whatever people want!