How do you foster competition between teams to increase innovation?

With team inspiration, coordination and inner strength 

We try to find some common objective performance metric and gamify it for rewards. 

One way of using competition to drive innovation is to create sub teams and have them play out opposing scenarios, or different combinations of options and strategic choices available and see who can get to the same targets (and how!). Usually a synthesis of the approaches followed by the sub teams gives the least risky and most likely optimal solution to the entire problem!  

Sense of competition must also include a culture of camaraderie so that when the project is over and new teams are formed there is still a positive working relationship between all the people.  One of the ways that I do this is to allow the winning team to pick their reward ( within reason) but it also should include all the teams.  I.e.  the winning team can choose a pizza party that includes all the people working on the project.

One way that has worked for me is to ensure that the competing teams have strong engagement with a shared strategic or tactical goal, but differing philosophies or technical approaches that they are committed to proving, and then harnessing each team's enthusiasm to prove their correctness via prototyping their solutions.  Insisting on a probative prototype from each team will help drive innovation.

Alternately, I have also challenged competing teams to take a common/shared technology and implement said technology in *different* ways/use cases, again, driven by their engagement with the area(s) of interest.  This is sort of the inverse function of my prior example.

Both approaches rely on team members' drive and engagement with technology to inspire them to do new and better things.