“One definition is that fairness = equal shares, the other that fairness = compensation in proportion to contribution.”
In either case, the word “fairness” is subjective. Who decides how much a person should be paid for his goods or his work ? If it’s set forcibly by a government authority or a thug with a baseball bat, no sane person outside of the government would consider that “fair”. As for the “free market”, if ten people are stranded on a desert island and one of them has the only remaining can of beans, the highest bidder probably wouldn’t consider the price he had to pay very fair.
A government that takes it upon themselves to legislate fairness must necessarily be ready to forcibly take from one constituency to give to another. By definition, fair or not fair, that’s not a free country.
Or, freedom isn’t fair.
In such a situation, there is no market, because the item in question is quite literally beyond price.