There is already plenty enough competition in driving people for money that it's hanging right there on the It's-not-worth-doing-this line.
Competition drives down costs, but only asymptotically to the cost of production. Double the competition, you get maybe a few cents lower prices, if it's already close to the cost of production.
But you know what doesn't move asymptotically? Incentive to cheat. As this news story illustrates: When you live on the margin, when everyone is barely getting by, those who cheat even a little will have a tremendous advantage. It's the same reason there's so much doping in Tour de France: all the legitimate ways of getting even a tiny edge on your competition have already been exhausted. And the cost of policing, of keeping people honest, quickly eclipses the tiny savings from getting ever so slightly closer to the asymptote.
So your auction scheme would make things worse, I guarantee it. I don't know exactly what way they would find to cheat, but they'd find one. If not they wouldn't be driving for long.
If glibertarians got their way on such things, you'd certainly get cheap taxis, but the only way taxi drivers could make a living was by occasionally selling one of their passengers to the Mafia's kidney harvesting operation.