In a situation in which there are many sellers but a single monopoly supplier can adjust the supply and price of a good at will, the monopolist will adjust the price so that his profit is maximized given the amount that is demanded at that price. A similar analysis using supply and demand can be applied when a good has a single buyer, a monopsony, but many sellers.
Where there are both few buyers or few sellers, the theory of supply and demand cannot be applied because both decisions of the buyers and sellers are interdependent – changes in supply can affect demand and vice versa. Game theory can be used to analyze this kind of situation.
The supply curve does not have to be linear. However, if the supply is from a profit maximizing firm, it can be proven that supply curves are not downward sloping (i.e. if the price increases, the quantity supplied will not decrease). Supply curves from profit maximizing firms can be vertical or horizontal or upward sloping.
Standard microeconomic assumptions cannot be used to prove that the demand curve is downward sloping. However, despite years of searching, no generally agreed upon example of a good that has an upward sloping demand curve has been found (also known as a giffen good). Non-economists sometimes think that this would not be the case for certain goods. For example, some people will buy a luxury car because it is expensive. In this case the good demanded is actually prestige, and not a car, so when the price of the luxury car decreases, it is actually changing the amount of prestige so the demand is not decreasing since it is a different good.