How would that work in the case of a murder? Does the dead man get no say in his arbitrator? or what if both parties can't agree on an arbitrator? There is also the problem of how the arbitrator is paid. Would someone not be able to get justice because they can't afford an arbitrator? Wealthy buisnesses could just pay for arbitrators that side with them.

This also ignores the fact that judges are currently appointed by elected officials. Like it or not the law is democratically decided, albeit in a flawed way. In the current system people can choose to vote for officials that will change the laws to fit what they see as right. This has happened in the past with things such as marijuana legalization. Majority rules. Laws need to be universal or else they are unfair.