Source: https://www.quora.com/How-does-Ubers-surge-pricing-algorithm-work
On a very basic level, surge pricing is a direct function of the supply-demand curve. When available cars/drivers (supply) are scarce relative to the number of Uber requests by potential passengers (demand), Uber begins to raise a multiplier (2x, 3x, etc) in order to shift the curves and match supply with demand. The idea being that while at price point A, off-line drivers will not feel it is worth going after customers, at a higher price point B, they will feel the opportunity is worth it and will come online and thereby add supply to better match demand. Finding this “tipping point” to add the marginal driver into the available pool is the goal of the algorithm and presumably will scale until enough drivers are available and online to meet demand.

If you are asking about the exact specifics of the algorithm, I am obviously not qualified to answer this as I do not work for Uber. But my suspicion is the core algorithm is not outrageously complex and particularly groundbreaking in sophistication given it is a simple supply-demand function at the end of the day. If there are 10x as many Uber requests as Uber cars available, my guess is there is a gradual step-function type scaling mechanism until that imbalance of requests-to-drivers is alleviated and then vice versa as too many drivers come online enticed by the surge pricing structure. 

But what I surmise is extremely sophisticated and impressive (especially given the caliber of the Uber team and based on what I’ve seen on their amazing/dedicated data science blog as a humble reader), is the calibration of this scaling algorithm. If I had to guess (and based on what I’ve experienced anecdotally as a user of the app in different cities), I would bet the algorithm is custom tailored and calibrated to each location as price elasticities almost certainly vary across different cities depending on a huge multitude of variables: income, distance/sprawl, traffic patterns, car ownership, etc. With the massive troves of user data that Uber probably has collected, they most likely have tweaked the algos for each city to adjust for these varying sensitivities to surge pricing. Throw in some machine learning and incredibly rich data and you’ve got yourself an incredible, constantly-evolving algorithm.