The Uber price didn't change, even after instructing the driver to take a different route. It should have been cheaper as the route I suggested was shorter and quicker.

A victim of upfront pricing. but don't despair!

Your city must offer Upfront Pricing (more details here).   This is when you agree to a price, before the trip.  Uber's claim is that this promotes transparency and reduces any confusion after the trip.

The problem about upfront pricing is that it's based on the route Uber specifies at the beginning of the trip.  This means that you pay for the expected duration and distance (and tolls/fees) that their algorithm has determined.  If you, a passenger, come up with another route that's shorter, they will still charge you the same price.  

This seems unfair, especially because Uber does not have to honor upfront pricing if the trip unexpectedly takes longer. (See the controversial article here.)

Here are what you can do:

• During the trip, here's a cool trick. Change your destination, by telling your driver or updating the final destination in the Uber app. This voids the upfront pricing and switches over the fare to a duration and distance based calculation.  This is taking advanage of the upfront pricing's rules!
• After the trip, contact Uber and claim that the fare was too expensive for the route traveled: https://help.uber.com/h/0487f360-dc56-4904-b5c9-9d3f04810fa9

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Uber Driver, CEO of TaxiFareFinder and Unleashed, LLC