The hierarchy of arbitrage: Revisions

Revisional arbitrage as a method to extend reliability

One arbiter can manage only a restricted number of clients. How much, is an open question, but however efficient, the number will be hardly greater than 10000, thus, very small compared with the number of people on Earth. Let's name this number N_arbiter.

Assume we manage a list of reliable arbiters. If this list is supposed to contains some nontrivial information, it's length is restricted: We have to find some nontrivial information about these arbiters, which needs some time, and our time is restricted. Let's denote the number of poeple on our list N_list, which is hardly greater than 1000. Then, the maximal number of people we can rely on because they have reliable arbiters is N_arbiter×N_list. This number is much smaller than the number of people on Earth, thus, the chance that the average stranger will be somebody we can cooperate with is minimal.

But there is a way to get around this limit: There may be arbiters of arbiters - revisional instances. If we trust such a revisional instance, we need not trust the first instance. All we need is that the rules are sufficiently strong so that we loose only if or the person we want to cooperate with, or the arbiter, or the revisional arbiter violates its rules. In the first two cases, they will appear on the black list for violating their rules. In the last case, we have found by personal experience that our trust was unjustified. So we remove the revisional arbiter from our list of reliable persons.

With the second instance, which, we can assume, can handle the same number N_arbiter of arbiters, the upper limit of people we can cooperate with increases to N_arbiter×N_arbiter×N_list, which is much larger. And, of course, we can iterate this game if necessary. With a third instance, the upper limit becomes N_arbiter×N_arbiter×N_arbiter×N_list, which for N_arbiter=1000 already allows to cover the whole population of Earth.

Let's name this solution of the problem the hierarchical one. What is the difference between this network solution and a classical court hierarchy with lower and upper courts, and a highest court on the top whose decisions are final?

• The first difference is that there is no instance like the state which assigns a given arbiter some higher status. It is, instead, a subdivision of labor which leads to such a hierarchy: Some arbiters decide to specialize themself on second or third order arbitration cases. It depends on their professional quality if they become accepted as revisional arbiters by the lower instances or not.
• The revisional arbiter may declare himself as a revisional arbiter. But if some algorithm for reliability evaluation accepts this declaration as reasonable is far away from being clear: The algorithm may ignore revisional arbiters with insufficiently strong rules. It may ignore revisional arbiters with a too small number of clients. Thus, it depends on the professional success of the arbiter if he becomes acknowledges by such reliability evaluation programs.
• There is no central instance which prescribes criteria for accepting revisional arbiters. Different reliability evaluation programs may chose very different criteria, thus, their evaluation of the network may give very different hierarchies, even to differently structured hierachies (one, say, with two hierarchy levels, another one with three, while a third one may allow some arbiters to be part of different levels). In this sense, there simply is no objective hierarchy.
• There is no "highest court". Every arbiter has his own rules, every arbiter has some other arbiter which decides if the arbiter has followed its rules.

Conceptual problems with revisional arbitrage

If every arbiter has some arbiter who decides if he has followed his own rules, doesn't this lead to infinite loops? What if somebody never accepts the arbitrage, and goes to the next arbiter, up to infinity?

Then, the arbiter may and will (as any person) accept different arbiters. What if two arbiters have different opinions about a particular case? A and B are in conflict. Arbiter C decides in favour of A. B goes to arbiter D for revision against C's decision, and D decides in favour of B. But now A goes to the other arbiter E of C for another revision. And E decides in favour of A. And so on. Who wins the case?

One simple solution of these problems is that the revisional instance does not change the decision of the arbiter. This is only something which can be done by the arbiter himself, and only in some circumstances described by his rules. Instead, the arbiter has to pay compensation to the side which has been treated unjustly. This solution gives security to the winner of the case (he will not loose anything by future decisions of revisional courts). And the initial loser gains what he wants if he wins the revision - he will be compensated for the injustice.

This solution may be not ideal in a case where hatred is involved, say, if one side wants to see the perpetrator in prison much more than getting some money. But for most of the civil cases this solution is sufficient. Of course, it has side effects: It increases the costs of arbitrage, because arbitrage becomes a risky job, and the arbiter has to be paid for accepting this risk. So, it is not unreasonable to search for better solutions.

But it seems quite obvious that the revisional arbiter cannot change the decision himself. He can judge only that the arbiter has violated his own rules, and, therefore, has to be fined. But this fine can depend on the future behaviour of the arbiter: If he changes the wrong decision, therefore partially undoing the harm created by his wrong decision, the fine may be reduced. If, and how often, the arbiter can change his decision depends on the rules of the particular arbiter. It seems reasonable to impose some upper limit.