A bit of history:
Before any contract was awarded on JSF - in 1996 - there was a plan to fund two engines all the way through the program. Because P&W already had the F119 in EMD, and the JSF engine was expected to build on F119, it was a leader-follower arrangement whereby P&W supported the X-plane program and the development aircraft, and GE/RR designed a new-centerline engine on a later schedule, but would be ready for full-rate production.
See this:
http://www.defense.gov/transcripts/t...nscriptid=3814
In 1996 a lot of people remembered how the Great Engine War went - the F100 failed repeatedly and P&W somehow could not get it fixed until the GE F110 was real and snapping at their heels. But after that the AF found it had two good engines. The Navy, too, hated the F-14's TF30 and was not happy about a single-P&W-engine JSF.
I'm not sure that GE ever really pitched an engine for any of the three X-plane designs. McD/NG/BAE's whole point was that their engine could use a stock F119; for either of the other designs the existence of the F119 gave P&W an advantage. So there was no real competition there either.
But much of the story had been forgotten ten years later, so P&W's fairy tale - that the F136 had been a pork-barrel project from the outset - was sold quite successfully.