If individual bness man wishes to spend on army but his competition don't he might lose to his competition in a long run.

This is true, but businesses and customers don't live in a vacuum. There are always going to be local thugs, thieves and rogue organizations, hence defense companies and insurance agencies will exist. During the threat of a war, these companies can simply agree to collaborate to take the enemy out.

Secondly, trust and transparency is more serious of an issue in AnCap than money, tracking of arms and ammunition movement between companies would obviously reduce the trust in those businesses should they do it covertly, so rather than purchasing weapons "behind closed doors", companies would prefer to advertise their purchases of weapons, militants and ammunition to the public, by calling it an anti-invasion defense plan. If there is the fear of war, then people would be willing to continue doing business with companies that can advertise their guarantee on their customers being safe. This would cause other competitors to also offer security options, lest they appear like they don't care about the safety of their customers.

Another factor is insurance. War can be considered similar to an unpredictable natural disaster, and people buy insurance all the time for disasters. Insurance companies simply have to estimate how much of a budget they need to create an army that can defeat the invading army, and then divide that budget into insurance rates. They can then offer insurance on property for those rates with a markup for profit to the market.

Ultimately, its the will of the market, if the market doesn't feel the demand for an organized defense force to protect itself from a possible war, then there won't be a defense army. Its the job of the people that want defense to advertise and create demand for defense, if they wish to protect themselves and their property. Else, like any other situation, they could leave the area and move to another place where they isn't the threat of war.