Question about common property and the settlement of disputes over commonly owned property?

You can't resolve this ethical problem because, by design, it is too abstracted from reality. Luckily communist politics are not based on methodological individualism and abstract reasoning, but instead an analysis of the concrete, material conditions of existence that we are rooted in.

In real life:
- a community would not consist of two people.
- it is relations of power, and not reason, that would determine the outcome in this scenario.

Issues like the above could be settled by a vote on the proposals, since communities in the real world generally don't consist of an A and a B.

This Ancap 'argument' is childishly oversimplified and circular.

One party announcing a property right does not solve the problem at all.

In the example, how would 'A' claiming ownership of the river through the act of damming it be an end to the conflict? Why would 'B' have to take this as the end of the matter? If ownership equals control then 'B' could just retake ownership by demolishing the dam.

In private property societies, "ownership" and "control" are definitely not the same thing. For example, in a public company, the shareholders own the company, but the CEO controls it.

For some real-world examples of how common property has been managed see here:
https://blogs.lse.ac.uk/lsereviewofbooks/2012/06/17/elinor-ostroms-work-on-governing-the-commons-an-appreciation/

The ancap makes the argument that this scenario effectively shows how collective/common ownership is an oxymoron: ownership of something is defined as having control over it; if A wins, and the dam is built, it shows that B does not have control, and ownership, over the river. Likewise, if B wins, and the dam isn’t built, it therefore goes that A does not have control, and therefore ownership, over the river. Either you have ownership, or you don’t. There is no in between.

A(sshole) can still use the river, as it's common property.... It's like saying that A should be entitled to start mining on public land, since not allowing A to do so would mean that the public or B "wins" and can continue using it. The entire argument is utterly ridiculous and has nothing to do with reality, as mentioned above.

Suppose we take a river that is collectively owned collectively in a community consisting of two people: A & B. Both are equally affected by the river, and likewise are equally affected by any change in the river.

A wants a dam to be built in the river, and B doesn’t. Neither is willing to change their position, as there are only two, diametrically opposed outcomes to the conflict: either the dam is built, or it is not. No compromise via consensus is possible. Who wins the conflict?

We also don't have to engage in abstract philosophical scenarios, as factory owners commodifying and damming up public rivers is something that actually happened during the industrialization of the US and other countries. In New England, the lower classes had often relied on rivers like the Merrimack as an important food source. Merchants/capitalists then came along and constructed dams and canals to power their blast furnaces and textile mills located along these rivers, disrupting the ecology of the rivers and consequently the lower classes' subsistence practices. Here's the environmental historian Ted Steinberg describing this commodification of nature:

Steinberg wrote: Long before water became a commodity for powering New England's factories, before the dams and canals produced energy, farmers relied on rivers and streams to provide food for the family economy. (58)

Steinberg wrote: The mills along the lower Merrimack incorporated the natural wealth available in the countryside into their designs for production and in so doing produced more than just cloth. They generated a chain of ecological and social consequences that spilled out beyond the factories, affecting places and people more than 100 miles away in a completely different state. Nothing better demonstrates the ways in which industrialization led to a major rationalization and reallocation of natural resources, enriching some at the expensive of others. (61)

Presumably the person who posed this scenario assumed that he belongs to the upper classes (the Boston Brahmins in the case of New England) and would not have his subsistence practices impacted by the construction of such dams. However, I think that he is actually just a confused worker who has come under the influence of ancap propaganda. I also have a feeling that he would react a bit differently if he were an actual farmer or member of the lower classes in early nineteenth-century New England.

The Ancap uses this to further his argument that exclusive property ownership is superior because it was adopted as a conflict avoiding norm.

"conflict avoiding norm"! It's also worth noting that, in my example based on actual history, farmers and others vigorously resisted the damming up of rivers by merchants and capitalists in New England:

Steinberg wrote: This grand scheme for efficiently controlling the entire river amounted to a massive redistribution of water wealth from New Hampshire to Massachusetts. Some considered what the Boston Associates had done to be tantamount to theft, and they rose up to defy them. In 1859, a group protested by trying to destroy a dam in Lake Village, New Hampshire, a key structure in the Boston Associates' designs for water. (60)

Steinberg wrote: Others eschewed violence and turned to the law to redress their grievances. What they found was that the law too was changing, evolving in a direction that conceived of water as a tool for bolstering the new industrial order. (60)

I also hardly need to mention the class conflict within the textile mills and other factories that were powered by these dams and canals.

The argument is also a bit confusing as it explains nothing about this "third party" (let's call them C for capitalists) or how their exclusive ownership over natural resources is supposed to make conflict between A and B, or between them and C, disappear. It was also assumed in the argument that there were only two people, so I'm not quite sure where this C character fits into things. As far as history is concerned, the people under the banner of private property were often the same ones damming up rivers in order to power their factories. To put it in mathematical terms C(apitalists) = A(ssholes). Their control over these natural resources, as I mentioned above, hardly led to any reduction in conflict.

Of course, the "AnCap" would have difficulty explaining how someone could "own" a river by "mixing" their labour with it -- unless they accept someone just saying "I own this" (which they claim to do).

Then there is the question of a company of two people who own it together -- what happens if they disagree over what to do with it? Is our "AnCap" going to ban all ownership by more than a single person?

It is funny to see how an "AnCap" always seem to forget that capitalism is generally marked by group ownership -- genuine individual ownership of resources is rare (i.e., resources which are worked on by one person). It is almost like they don't understand what capitalism is...

Just to note, as an aside, communism, i.e. the common ownership of the means of production, distribution and exchange, is the immediate attainment of human development but is not its ultimate form.