[the Sun King]

There is one thing that has troubled me with libertarianism:

Should a child or the child's parents be responsible for the child's actions? Let's say in education for example: should the child or the child's parents be the one to decide on what school the child attends to? Since if there is no government-enforced public education in a libertarian society.

If the parents were to decide doesn't that violate the child's physical integrity? But if the child is to decide doesn't that lead to poor decisions since child cannot be mature enough to make those decisions that would affect his or her entire life?

Also what age should the child be responsible for his or her own actions? In other words when does one become an fully responsible adult? Or will there be no concept of adulthood at all?

[Goldberk]

There is one thing that has troubled me with libertarianism:

Only one thing?

Let's say in education for example: should the child or the child's parents be the one to decide on what school the child attends to? Since if there is no government-enforced public education in a libertarian society.

This is an interesting question in any society? The philosophical question should be around what rights and individual has to chose an action that effects not only themselves but others, and what rights others have to influence an individuals choice that effects them. The particular question should be a consideration of capacity, at what age (if ever) can an individual make a rational choice, and further at what age does one become an individual.

[Voluntarism]

Should a child or the child's parents be responsible for the child's actions?

There is obviously indeterminacy associated with children and their actions. In essence, as a child grows up they acquire understanding and respect for the rights of others until they become a full adult. To the extent that they can deliberately rather than inadvertently violate someone else's rights as a result of their actions is therefore highly dependent on the situation and the child's development. Any punishment needs to reflect this.

With specific reference to your question, parents should not have any different legally enforceable obligation associated with the actions of their child compared to the actions of anyone else in society. Except in specific circumstances, this means "No, parents should not be legally responsible for a child's actions".

In my opinion however, I would say that most people in most societies would say that parents have a moral obligation to compensating victims of certain actions of their children - particularly actions that occurred while the child should have been supervised. For example, if my 4-year old daughter damaged a book or toy we borrowed from a friend, there would be a moral obligation to at least offer to replace it or with an item of similar value.

Let's say in education for example: should the child or the child's parents be the one to decide on what school the child attends to?

Sensible parents obviously listen to the wishes of their children. Whilst the child is dependent on the financial support of their parents however, the decision ultimately rests with the parents.

Also what age should the child be responsible for his or her own actions? In other words when does one become an fully responsible adult? Or will there be no concept of adulthood at all?

There are many aspects to people being mature enough to be a fully responsible adult. Kids mature in different aspects in various ways, but the underlying biological consistently point to certain fixed ages like 16 or 18 being "good enough" milestones for determining that the majority of children should have matured enough. Most parents would 'set free' their children at the ages they deemed appropriate. One couple I know let their 10 and 12 year old boys set up and operate their own go-kart business on weekends as they deemed them capable enough to do so responsibly while another waited couple until their boy was 14 before they let them do a newspaper delivery job.

[Eran]

The Rothbardian thesis is that while children, like all humans, are self-owners, parents effectively act as trustees, as the child is, at least initially, unable to act as a self-owner.

Children, according to Rothbard, can claim their independence by simply moving out of their parents' house. While they live under the protection (figuratively) of their parents, children are subject to whatever (reasonable) demands their parents make. As soon as the child is able and willing to leave, he can claim his independence. That might happen at 14 for some people, and at 23 for others.