I still find it hard to believe that I may be the first person to make tort and criminal restitution into a mathematical formula that encourages both sides to be as honest and fair as possible in claiming what the most just and equitable amount of restitution should be. Since I am so far unsuccessful in finding this formula in other writers, I will call this the Hobbsian Just Restitution Formula until someone can show me someone who derived it before I did, originally around 1999 though not published till now.
In any case where a wrong is claimed to have been done, whether by intent or negligence, there is the temptation for the alleged perpetrator to seek paying less restitution than is just, and for the victim to seek receiving more restitution than is just. This may not always be so, but a justice system should have a way to minimize this tendency. This formula is a way that can minimize the temptation for each party to have the scales of justice lean in their favor instead of truly balancing.
Take the restitution claimed appropriate by the perpetrator to be Rx, and by the victim to by Ry. Now if they are unable to resolve this alone, they can take this to a third party to act as an arbitrator and judge, whose decision is Rj. Assume for now that Rj is not beyond the bounds of Rx to Ry. If so, then Rx d Rj d Ry. Now here is the important formula to determine who pays the cost of the arbitrator:
Ct (Ry ' Rj) / (Ry ' Rx) = Cy
and Ct (Rj ' Rx) / (Ry ' Rx) = Cx
where Ct is the total cost of arbitration and Cy and Cx are the costs of arbitration for the victim and perpetrator to pay respectively.
By this method, if the victim has requested restitution equal to the arbitrated decision, then he will pay for none of the costs of arbitration. Likewise the perpetrator could avoid paying arbitration costs if he had chosen what the arbiter decides. In the event that the judgment falls in between, then they each will have to pay part of the cost in direct proportion to how much they desired more than the arbitrated decision of an equitable amount.
By this method, each party would be financially encouraged to avoid excessive claims to minimize the arbitration costs to be paid. If they can come to a point where the spread of desired restitution is less than estimated arbitration costs, then they could recognize it might be cheaper to resolve the issue without a professional paid arbitrator. The more complex the issue, the more arbitration will likely cost, and the incentive to avoid paying for arbitration also increases.
It is possible that the arbitrator could decide that appropriate restitution is beyond the Rx to Ry borders. Rules can be decided before hand if the boundary points would be limiting or not. It is also possible that the parties do not let the arbitrator know the boundary points until after the decision is reached, if one or both feel that such might unfairly influence the result. They could then just be intended for use to determine who pays arbitration costs. However, it would still be necessary for the parties to know each other's desired restitution value. It would make no sense to keep it secret from the counter party, go to arbitration, and then discover that you would have agreed to an amount without an arbitrator.
What about lawyers? Different arbitrators might allow or disallow them. In this system, they would rarely if ever be necessary, and most often increase the expected cost for each side that uses one. An arbitrator should be unbiased, and should help both sides explain as well as understand the strength of each argument given by each side. An arbitrator's goal is not only to make a just decision, but also to explain and convince both sides that the decision is just. This is not the goal of lawyers.
What about non-monetary compensation? This is a significant possibility, and a further complication that could add to the cost of arbitration. If a proposed compensation is to be labor of the perpetrator to the victim, then as the victim would act as employer, then the repayment wage rate would have to be agreed upon as any other wage rate. The arbitrator could help decide this, as well as look for third parties who would accept labor and then pay the labor wage to the victim.
This system does not require the alleged perpetrator to agree that any wrong was done. He can even claim that since no wrong was done, he is owed compensation for the counter party wasting his time. The formula would still encourage him to not overestimate this counter claim.
Additionally, despite modern practice, there is no reason to think this system would not also be best for criminal cases. Punitive damages could be part of restitution paid to the victim, not as fines paid to the state. When victims do not receive restitution through a criminal justice system, there is an increase in both extremes of letting crimes go unreported and unpunished, due to lack of incentive, and of victims groups advocating punishments that do more harm to the perpetrator than would be done by fair restitution. (See Bruce Benson's books The Enterprise of Liberty and To Serve and Protect for historical examples of restitution-based free market courts and their successes, as well as the failures created when states monopolized criminal justice systems.)
Religious groups have usually realized that to submit to states in the area of justice is to make the religion a pawn of the state. Jews have always had a Beit Din to resolve problems religious, civil and sometimes even criminal. Islam has Sharia courts for the same purpose. Christians are required to do the same, per the Apostle Paul in 1 Corinthians 6. (However, I don't know of Christian groups that actually have established even informal courts or respected arbitrators for this purpose, other than religious bodies intended to decide purely religious dogmatic conflicts.)
There are many possible complications, but none of them really diminish the value of this formula. What if one party refuses to go to any arbitrator? What if some arbitrators are corrupt? What about appeals? These and many more are all legitimate questions. To answer them all would make this essay a textbook. However, other writers have already insightfully answered these questions, including Murray Rothbard's The Ethics of Liberty, Power and Market, Bruce Benson's The Enterprise of Liberty, Morris and Linda Tannehill's The Market for Liberty, and David Friedman's The Machinery of Freedom.
These authors (and more like Anthony de Jasay and J.C. Lester) provide quite different methodologies, but reach similar conclusions about the efficiency of polycentric law and arbitration compared to authoritarian, conquest-based law.
Could this be the kernel of truth to the concept of justice? Is justice something that must be created and defined by a government whose origins in conquest of weaker parties precede and supercede its concept of justice (sovereign immunity)? Or is justice a principle where conflicting parties resolve disputes without allegiance to any party above, beyond, superceding, or immune to justice?
Given that people tend to be utility maximizers, judicial systems will be tools to maximize the utility of those who control them. In a state, its laws are written to maximize the utility of the ruling coalitions as opposed to the whole 'people,' the subjects, or any other partition of it. If any social group seeks to be classless and judicially independent, it cannot form a state where any weaker party is subject to law made by the stronger. Nor can any part of the group be immune to the principles of justice.
There could be many social groups that could make themselves judicially sovereign-independent. Religious groups mentioned already have such rules, and might intend to be an example to outsiders in being independent from states. It is possible to have multiple overlapping social judicial groups, but this would create hierarchical limitations, as individuals would need priorities as to which social group (with its justice) to place primary association. For example, a conflict arises between parties who are in the same religious judicial group and the same business judicial group. Both groups could not require all members to seek justice for internal conflicts only through that group's arbitration system. Or if these groups do, and have differing memberships, then individuals who joined both would have to violate one of the groups' primacy agreements if such a conflict arose.
The reason that justice without sovereign immunity would naturally exist in cohesive social groups is that individuals obtain significant utility from their social group, and ostracism by one's social group could provide a persuasive influence to those who would consider refusing to accept an arbitrated decision. Several millennia of diasporic Jewish social groups all over the world provide plenty of historical evidence for this non-state judicial system to be sufficient to maintain internal peace and order, especially relative to a state justice system. Social groups then compete non-geographically in the market for members, based on many factors, justice likely being one.