An ancap does not attempt to predict how the market will solve problems. If individuals could do that governments might work. The ancap can only speculate. For example, if a "foreign entity" violates a contract and does not make amends, insurance would cover the loss. The insurance company would then report it to a ratings agency, and the violator of the contract would find doing business much more difficult. This is only speculation of course. The market may find an even more effective solution.
Here's a wonderful talk by Robert P. Murphy, entitled "The Market for Security". It addresses many of the concerns of statists.
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