A lesson in the theory of demand
Nice story. Never heard it, but doesn't mean it's not true. I think the simpler answer doesn't involve a taste for fairness, though I'm not denying that such a taste or moral predisposition exists. But the first mover has more to lose than the second mover. It's a Coasian world, right? So he will reason that if he is too greedy, the second mover will just refuse. The lower the amount that's at stake the "fairer" the first mover has to be to induce the second mover to accept. If I have $10 million in my hand, do I have to split evenly to get you to agree to my offer? $1 million would do quite nicely. In a purely self-interested world without fairness, the allocation might depend on the risk-aversion (if that's the right term for an unrepeatable interaction) of the first mover. It's not clear to me that there is a determinate solution even in that set up but the first mover clearly has the advantage.
Second comment. Alchian's attachment to the theory of demand results from his highly developed subjective theory of cost in which all cost is the highest foregone value. So cost is really just another term for demand.