Ask HN: How to approach two competing job offers - is bidding war an option?

I studied Computer Science, and I recently graduated as a bachelor. I went on to apply to two major Silicon Valley companies, let's call them A and B, and aced the interviews.

I got an offer from A, which I would have happily accepted had I not had another company still contemplating their offer. Now B contacted me, not yet ready to give an offer, but they mentioned that their offer would likely be significantly larger if they would be able to see the offer from A in writing.

I got my offer from A both verbally and in informal writing to my e-mail. I find it clear that if I asked them for the offer in writing now, they would certainly know what's happening (given that I've kept them waiting for some time now). I told this to B already previously, they understood, but it would certainly benefit me if I had it in writing now.

How should this game be played in your opinion? I actually prefer A, and if B's offer were roughly the same size, I would be very happy to take A. However, I am wondering whether I am a wussy if I play it safe now, and take no action, and should I instead try to get some competition between these two. There's also a small chance that A is trying to lowball me with their offer, since I might be too humble analyzing my own value. All this leads me to think that I might just want to get the offer in writing, not caring what they think about it, but I am very very open to other ideas.

Also, I know that I should probably never try to bluff, and that's my intention, too - I'll never try to inflate my offer if I am not really willing to take the competing one. These both are great companies, and B can become better in my mind if their offer triumphs on the financial side.