The battle for robo-taxi dominance is heating up.

Alphabet's robo-taxi unit, Waymo, stoked the flames Tuesday by announcing plans to start commercial service in Germany next year.

Both stocks were down in midmorning trading. Alphabet fell 1.4% to $338.62. Tesla lost 1.3% to $345.83.

Waymo operates in several U.S. cities-San Francisco, Los Angeles, and Dallas are examples-with plans for more, including Las Vegas, Tampa, and Detroit. It is testing vehicles in London and Tokyo as well.

Tesla launched its robo-taxi service last summer in Austin, Texas. Today, the service is in a handful of cities with small fleets.

Last week, both Tesla and Waymo received approval to operate robo-taxis in Nevada-5,000 and 1,000 respectively.

Tesla is also working to get its Full-Self Driving driver assistance software approved in Europe. The Netherlands gave the OK earlier this year. More than a million drivers in the U.S. use the software.

FSD approval is probably the first step before Tesla tries to extend its robo-taxi business across the Atlantic.

Waymo and Tesla are the heavyweights in robo-taxis. They have the history, AI technology, and operating businesses. There are several others vying for robo-taxi business, including Amazon.com's Zoox. China is also a hotbed. Pony.ai, DiDi, and WeRide offer autonomous rides there.

Winning market share is worth billions, even trillions in market value.

Morgan Stanley analyst Andrew Percoco values Tesla's FSD and robo-taxi business at roughly $1 trillion, six times the value ascribed to Tesla's EV business.