What is a system where the government decides what is everyone's fair share, and what are some problems with that system

The correct answer is - socialist/communist government.

The socialist or communist governments have a policy where they tend to make all the people in the country equal by wealth, property, opportunities in life, social status.

In practice that has been done in the manner of taking land from one person that has a lot and give to the people that do not have land or have very little, giving them all the same exact amount of land. Or in the work places, the people that had higher wages got deduction, while the people that had smaller wages got an increase, thus all of them to take the same amount.

The biggest problem in this type of policies is that the people do not get what they deserve. Some may work better and more productive, some will be leaching of them and not contribute a lot, but at the end of the day they will get the same wages. Also, some people have worked very hard for generations in order to be able to have more land and gradually increase, and than they got striped of it without any compensation and their land was given to people that did not contributed at all to the gain of that land.