How low should I make my initial offer? Is there a way to guess what the seller will accept?

I suggest using a buyer's agent to help you determine what the market value for the home you are considering is, how long that home has been on the market, and whether the seller has already dropped the price and by how much. Then, I suggest offering 80%. You can always negotiate up, but once you set your low bar you aren't going to get any lower. Also, remember when the price isn't very negotiable, other parts of the contract may be, such as closing costs, appliances, fees, etc.