As others have stated, Adidas started the modern 'arms race' with the Boost foam. But even that claim in somewhat untrue. The high tech arms race started years prior to that, and there were other game changers along the way.
Fila is an interesting one. They dropped their carbon-fiber marathon shoe in 1996. That shoe was briefly THE shoe; through 2002 they were winning loads of majors and setting course records. (Tergat ran a 59"06 in them in Lisbon in 2000; this world record was not ratified due to elevation drop). Even in 2002 the Fila was the 'fastest' production shoe in existence, having been worn by the rabbit Korir when Tergat out kicked him by one second in Berlin for a new world record of 2:04.55. (Tergat's Nikes were prototypes).
All of this success did not translate to sales. In 2003 Fila was sold to an American hedge fund, and their performance division defunded.
Adidas then took over with its light Adizero line, which dominated for a number of years. This was the shoe Geb and others set their world records in. This shoe came as a result of intense collaboration with PHD candidates; the tech they developed was released to the world in 2008. Another PHD collaborator, Geng Luo, preferred Nike over Adidas; he moved to the US (Oregon, to be exact) and won a graduate school competition Nike had sponsored. Nike hired him soon after. Lou and his team spent three years working on carbon fiber plates; this tech was not new, but the ZoomX foam they created was. Enter the 2016 Olympic Olympics, and you know the rest of the story.
Historically, shoemakers seem to have a 5 or so year run at the top before new tech comes along. There's no irony here, only an ever-present dedication to make the world's greatest shoe.
