If I had my choice of perfect worlds, there would be no government in it. People purchase protection from a private company of their choice. This company, in turn, subscribes to a system of laws which is privately written. Independent judges interpret the law fairly, or they don't get the business next time. Some legal systems will come into conflict, which will be resolved by a payment in one direction or another. The price one pays for a legal system determines the amount of conflict one bears. Poor people obviously get a cheap one which doesn't allow for much conflict. But it does cover them against the essentials -- no murder, no theft. In the end they get more justice by buying it in an efficient market than what they're currently getting through government -- arguably less than zero.
There's a lot of reasons to expect that this would result in better laws.
Obviously, in my perfect world, some parents still hurt their own children, so there must be provision in people's laws to protect children against their parents as well as others. And equally obviously, some people will seek to employ inordinate violence against their attackers, so there must be provision in people's laws against that as well. How do I know that these provisions will be there? Because people will purchase a subscription to a legal system without knowing whether it will be used for or against them. So, they will shop carefully. Will they make mistakes even though I'm supposing a perfect world? Sure they will. But the mistakes come from their own choice of legal system, and they have the power to correct that.
None of us has the power to correct our governmental legal system single-handedly. A private legal system, on the other hand, would quickly triangulate on what the majority desire for justice. The legal systems that gave out the justice that most people wanted would be cheapest, which would tend to bring in people from the sidelines to the same majority legal system. In practice most legal systems would be very similar to each other, and would be very close to optimal. This should be contrasted with the current governmental systems, which tend to produce laws optimized for special interests.
If you want an example of how this might work, take an extremely difficult example -- abortion. Clearly many people want the freedom to abort their babies. Many people also think that's murder. They would each choose legal systems that allowed or disallowed abortion. How, then, would these legal systems work? How could you both allow and prohibit abortion?
Let's follow an example. A woman gets pregnant and decides to carry the baby to term. Fine. She's not disobeying anybody's laws. Let's say that she decides to have an abortion. Obviously she hasn't chosen the anti-abortion legal system. She would contract with a doctor for an abortion. However, the pro-choice legal system has been paid to include a term that says that an anti-abortion protection firm will be informed of such a contract. Maybe the woman doesn't like this very much, but she chose that system, and besides it made her legal system cheaper. Now the anti-abortion firm knows that she's serious about getting an abortion. They offer to buy her all the medical care she needs to deliver the baby, and will find parents willing to adopt the baby.
Now here's where it gets tricky. Exactly what happens depends on exactly how many people are in favor of abortion choice and how much they're willing to spend to get their way. Let's say that the anti-abortion people are in the minority. They won't have the resources to help every women, but they'll have it for some, as many as possible. So, even though they're in the minority, they'll get their way more often than they do now.
Let's suppose it's the other way around -- that the pro-choicers are in the minority. It's likely that their legal system will be more expensive, because it includes the choice of abortion. It also requires them to seek counselling before getting an abortion. It also imposes a mandatory 7-day waiting period. Both of these were purchased by the anti-abortion majority, who have large resources at their disposal. A license to have an abortion might cost some serious amount of dollars.
Do poor people get screwed by a private market for law? Yes, absolutely, no question about it. If they weren't screwed, you'd have a hard time calling them "poor", or saying that "poor" was a bad thing to be. The harder question to answer is whether they are screwed more or less under a system of governmental legislated law as under a system of purchased private market law. At least under private law, somebody can purchase a subscription and donate it to them. If you think nobody would do that, you must first take your magic wand and wave away the existance of the Carnegie libraries.
In this manner, through a market for law, you have people purchasing, not voting for, law systems. To the extent that they purchase non-controversial, majority law, it's cheap. If they want to do something most people disagree with, it costs them money, and not many people can afford to do it. Contrast this with the current system where every man has a vote regardless of how strong he feels about the subject, and every decision is decided regardless of how many people feel strongly about it.