I did not, but only because in my case the monetary damage was not sufficient to pursue it. I would have liked to have seen those responsible be prosecuted for it, though.

If the monetary damage was high enough vis-a-vis the ability of the public system to deliver me restitution for my losses, I'd very quickly be hiring private detectives yes.

How do you imagine the outcome would have been different in a hypothetical world with an entirely privatized justice system?

I was in this situation, I lost roughly 2k£. I tried to go to court for 8k£ in small claims which is what the law allowed for as a compensation for the crime.

This guy got arrested by the police, he was released and then run away to Dubai where he's from. He came back after a month and opened a company under a false name and started doing the same scam. I knew his address, I reported everything to the police and the police dropped the case without doing anything.

I'm already paying 25+k£ in taxes every year. If I didn't have to spend that and there would be no police or laws, private companies (aka burly guys with guns) would be needed to ensure citizens safety, in exchange for a fee. Different private companies would need to interact with each other to settle down cases, eventually agreeing on a set of laws to compensate people subjected to crimes.

In this hypothetical world, I would have been happy to pay for my protection agency to go and fetch this individual and bring him to a private court where my protection agency and his protection agency could debate whether the crime happened or not.

It was a blatant crime so there was no reason I would lose and that guy should have been forced to return my money (according to laws agreed on by contracts between our private protection agencies) or work in jail until I'm repaid.

How does this hypothetical world not end up with people that can afford to hire the more powerful protection agency be favored in those settlements?

Not that I'm happy with the current system, but I can't really see how a privatized one would not end up with an even bigger class divide where the laws are only enforced against people that are on the same level as you or below.

Because private court systems would have a reputation as well. You can also have 3rd party companies auditing and reviewing the system. If a court system is notably unfair, it will lose business.

I don't see why corrupting a private judge would be easier than corrupting a public one.

The private one at least could default if they had no more business due to their unfairness.

You realize you have been treated the same as somebody that 2k was all his belongings and now walked without cash to the private security firm :))))

For what history says about such firms, look up Pinkerton agents and London's policing before the current system.

The difference is that I'm paying quite a bit for the government.

An equivalent would be paying the expensive private security firm subscriptions for 10 years and then expecting service when something bad happens.

In this hypothetical world you’d be paying way more in protection money than you are currently paying in taxes to the local mafia gang who would have long murdered all the other private protection agencies.

The local mafia gang would be just one of the private protection agencies with armed forces.

There is definitely a competition between them; but is violence or negotiation the most business effective way of dealing with it?

Warfare is expensive, you risk men lives and weapons aren't cheap.

If there is violence on the street or if protection money is too high, no-one would want to live there and there would be less people to sustain your army.

I think the incentives are aligned to solve disputes in a pacific way.

The UK used to have private prosecution. If you're interested in this topic, there's a chapter on it in the book Legal Systems Very Different From Our Own.