I may regret entering the discussion on this note, but I have seen @GuitarStv say several times over the years that slavery was a free market. I am astounded by the claim, and that nobody has repudiated this BS. If a transaction has three participants, and one of them (notably the one doing all the work) isn't free, by definition it is not a free market. 2/3≠1.
Agreed with your logic as derived from assumptions.
The problem though, is with your initial conditions. Specifically that there are three participants in the slave trade. There are only two participants - slave owner and slave vendor. The slave him/herself is the
commodity, not a participant. This is consistent with other commodities. Horses might be sold from one person to another, but the horse isn't considered a participant in the trade.
It is like saying that if two people burglarize my home, and freely agree in advance that one would get the TV and the other get any computers, that it was a free market because they freely agreed to the theft and the government didn't have a law against it. No, it was not a free market, because I did all the labor. It was theft. The government failure wasn't the lack of regulation of the free market, it was lack of illegalization of burglary. The government failure regarding slavery was not the lack of regulation of the free market, it was lack of illegalization of kidnapping, theft, assault, and a hundred other things that could be summed up in the single word "slavery." It was a failure that government didn't force everyone into a free market.
You appear to be using an unusual and non-standard definition of 'free-market'. The one that I'm using is:
"Free market, an unregulated system of economic exchange, in which taxes, quality controls, quotas, tariffs, and other forms of centralized economic interventions by government either do not exist or are minimal"
There's no mention about perfectly equal and equitable human rights there. It's purely a statement about government intervention in trade.
Kidnapping, theft, assault . . . these were all on the books as illegal actions against people while the slave trade was going on. The problem was that the unregulated market saw slaves as commodities. The United States required the 13th amendment to explicitly prevent this free trade.
The distinction is meaningful. Two salient reasons the South lost the war were first that Gen. Lee had the military mind of a child, and second that the Union had a much stronger industrial base. One of the reasons the south had a weak industrial base is that they did not have a free market. The slave owners and traders had little incentive to innovate or industrialize because they did not work. The slaves had no incentive to innovate because they did not profit and obviously industrialization was out of the question. Even had the South not lost the war, their lack of free market labor made it inevitable that they would have become weak and impoverished by comparison to the North. Within a few more decades the northern states would have become so strongly industrialized that the south would have gone from near equality to the modern equivalent of the "third world," and the economic system of slavery would have been the cause.
It's an interesting theory that the slave trade weakens a place economically through lack of innovation, and one that I suspect has some truth to it. It's entirely that the south's love of slavery would have led to their downfall at some point. There exists significant slave trade in many modern states today . . . including the UAE, India, and China. While I wouldn't call any of them meccas of innovation, they're also not economic weaklings because of the slavery they enjoy.
To summarize, by definition the slave trade was not a free market, because the slaves were not free. (I mean.. obviously... I really can't comprehend why this is not obvious.) Further, this is not just a semantic distinction.
Could you fully define what you believe a 'free market' is? It doesn't seem to be the commonly used definition - as human rights are not generally mentioned as part of a free market. This type of definition raises some interesting questions. For example, does oppression of gay and trans people prevent a market from being free in your eyes?