In a free society, whoever is willing to pay more will be able to pay judges to rule in their favor.
In a free society you are not compelled to use a monopoly arbiter as you are with a state; given that you will have a choice, why would you choose someone that you know can be bribed when your opponent is willing and capable of doing so? You wouldn't. Such a business would have no customers. You aren't even forced to use an arbiter in a free society, although it would probably end up being the civilized thing to do, just as most people don't grow their own food but leave it to experts with appropriate knowledge and tools.
This fallacy stems from a fundamental misunderstanding of thinking of a free society as a state with extra stuff stapled to it, rather than looking at it from first principles and seeing how many state-caused problems just wouldn't exist any more. But their minds have so long been held back by narrow thinking—or worse, have a vested interest in it—that they find it hard to let go of the violence.
And of course people can and do bribe and influence judges in non-free societies, and without competition they are nowhere near as transparent (or technologically savvy—e.g., most of the time there's no reason disputants or witnesses would need to be in the same physical location as the "court") as those in a free society. (DBR)