Changed my mind about a job offer...how do I keep them happy and not burn any bridges?

After finishing my masters, I applied to two jobs 2 months ago and had invitations to interview from both. Both companies sent their invitations the same week.

So I interviewed with company A on Monday and was given an offer on Wednesday.

Then had an interview with company B on Thursday and was given a better offer on Friday.

I called company B the following Monday and told them I'd accept.

I called company A a few days later to formally reject the offer and stated what company B was offering. They counter offered a better deal than company B.

I have not signed with Company B, and I feel like I now want to sign with Company A.

But this is a very competitive market, and only these two companies will want my skill set. How do I tell Company B I changed my mind and will not accept their offer?

Any advice appreciated.

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