As with any other negotiation, how much is the house worth? And how much is it worth to you?
Further, how do you know what the other offer is? Could be the other offer was "full price" at $99,900. Or somewhat higher, but not $110,000. Be careful you're not negotiating against yourself: Your competitor bid $99,900, you bid $110,000, and now you're raising your bid.
So, figure out what you're really willing to pay. Make sure you know what the house is worth. Then submit a bid at the lower of those two amounts.
Hope that helps.