They decide it in response to supply and demand. Would you really expect them to raise prices when there is less demand and more supply and lower them when the opposite?

Which is what allows agents to set their prices. If prices are too high, some riders exit or more drivers enter which causes prices to decline. If prices are too low, some drivers exit or more riders enter which causes prices to increase. Is this not the ordinary mechanism of market pricing?

I'm not an economist or a laissez-faire capitalist, but I don't think that's the ordinary mechanism of market pricing. My understanding is that the individual actors on the market independently price their goods in competition with one another. In this case Uber is the sole authority in setting the price of rides, while in a market this would be more distributed. To quote Wikipedia on the free market:

"In a free market, the laws and forces of supply and demand are free from any intervention by a government or other authority and from all forms of economic privilege, monopolies and artificial scarcities."[1]

[1] https://en.wikipedia.org/wiki/Free_market

What Uber is in this context is basically an intermediary, like a retailer that sits between a manufacturer and a consumer, or a general contractor that sits between a subcontractor and a property owner.

What you have then isn't a transaction between the driver and the rider, it's two transactions, one between the driver and Uber and another between Uber and the rider. Each transaction is negotiated. Uber changes their offers to each party based on supply and demand, and if either party doesn't like it they can hold out for a better offer or go use Lyft. And since Uber does actually adjust pricing based on supply and demand, the better offer actually comes when enough people hold out.

The way you negotiate with them is by delaying your purchase/sale until they meet your price. It's a functioning market.