It's a race to either profitability or the death of the opposition, at which point subsidies disappear and prices rise, so does Ubers cut of the fares.
once you knock out the current infrastructure it is hard for some one to reenter and compete. And in some locations they may lose but on the whole they would maintain 85%+ market share and set the prices. Local firms would know they are operating almost with permission of e.g. Uber, because Uber (or whoever wins) could set local prices lower to put the firm out of business again. This time however the subsidy would be by higher prices across the rest of their market.