Remember that bargaining in the souks is expected.
It is not really possible to give an accurate indication of how much to start the bargaining at in relation to the initial asking price, but a general idea would be to aim for approximately 50% off. Prices are set on a daily, even, hourly basis, depending on how much has been sold on a given day (or period of hours), while also reflecting the vendor's personal estimation of the potential client.
The souks are often a good reflection of the basic economic principles of supply and demand, particularly with regard to the demand side. If a lot of products have been sold by a particular merchant he/she will raise the price, and may refuse to sell any more products for the rest of that day (or for days) unless the price is much higher than usual. If there are many tourists around prices go higher and bargaining even small amounts off the asking price becomes quite difficult.
In addition, the seller will generally inspect the client, whose dress and possessions (particularly if the potential client sports an expensive Swiss watch, camera, etc) are usually the main indication of how high the price may be set above the usual. However, the potential client's attitude is also taken into consideration.
Taking all this and other factors into account (such as the time of day, day of the week, season, etc.), initial prices may be up to 50 times or more in excess of normal prices, especially for more expensive items, such as carpets.
Carpets, however, are a very specialized item and it is necessary to have at least a cursory understanding of production techniques and qualities. If possible an ability to distinguish between hand-made and machine-made carpets, hand-dyes, and the like is helpful to avoid being utterly duped.
Bargaining is an enjoyable experience for most vendors and they prefer clients that don't appear hurried and are willing to take the time to negotiate. It is most often actually necessary to give reasons why you believe the price should be lower. The reasons you might give are limited only by your imagination and often lead to some very entertaining discussions.
Common reasons may include: the price of the item elsewhere, the item not being exactly what you are after, the fact that you have purchased other items from the stall/store, that you have built a rapport with the vendor after discussing football and so forth. On the other hand, if there is little movement in the price after some time, the best advice is to begin leaving, this often has the result of kick-starting the bidding anew, and if not, it is likely that the merchant is actually unwilling to go further below a given price, however absurd.
It is also important to show a genuine interest for the workmanship of the product for sale, no matter how disinterested you may actually be in what you are buying. This does not, however, mean that you should appear over-enthusiastic, as this will encourage the vendor to hold his or her price. Rather, it is important to project a critical appreciation for each article/object. Any defects are either unacceptable or a further opportunity to bargain the price down.
You should take caution to never begin bidding for unwanted items or to give the vendor a price you are unwilling or unable (with cash on hand) to pay. Try to avoid paying by credit card at all costs. In the event you do pay by credit card, never let it out of your sight and demand as many receipts as possible. There is typically a credit card carbon copy and an official shop receipt.
Never tell a vendor where you are staying and 'never tell a vendor how much you paid for any other purchases. Just say you got a good price and you want a good price from him or her too. And, above all, never be afraid to say 'No'.
It must also be said that, as for us buyers, not all sellers are actually very good at what they do. A vendor that is completely disinterested or even aggressive is unlikely to give a good price. Move on.