Question

Two firms set prices in a market with demand curve Q = 6 − p, where p is the lower of the two prices. If firm 1 is the lower priced firm, then it is firm 1 that meets all of the demand; conversely, the same applies to firm 2 if it is the lower priced firm. For example, if firms 1 and 2 post prices equal to 2 and 4 dollars, respectively, then firm 1–as the lower priced firm–meets all of the market demand and, hence, sells 4 units. If the two firms set the same price p, then they each get half of the market, that is, they each get (6−p )/2 . Suppose that prices can only be quoted in dollar units, such as 0, 1, 2, 3, 4, 5, or 6 dollars. Suppose, furthermore, that costs of production are zero for both firms. Finally, suppose that firms want to maximize their own profits.

Show that when we restrict attention to the prices 1,2, and 3 dollars, the (monopoly) price of 3 dollars is a dominated strategy.

Answer 1

The following game matrix shows the pricing game between two firms - Firm 1 and Firm 2 .Both the firms have option to choose between the prices 1 to 6 where the whole market demand will shift to that firm whose pricing strategy is lower than the other form and if both the firm chooses the same pricing strategy , then both the firms will get equal market demand . The game matrix of the above is as follows:-

In the above game ,there are two firm 1 and firm 2 . If the firm one chooses to play first and choose the pricing strategy of let's say $1 then the most dominant strategy for Firm 2 will be to choose $1 as all other strategies will lead Firm 2 a payoff of 0 .

Now if Firm 1 chooses a pricing strategy of $3 , Firm 2 also choose a pricing strategy of Dollar 3 as all other pricing strategy would lead it to a payoff of zero.

Similarly if Firm 2 has a chance to choose first , then again. the most dominant strategy for Firm 1 will be to choose the same pricing strategy as Firm 2 .

Thus we can consider that pure strategies for both the firm is to choose what the other firm chooses leading to equal division of market demand between the two firms .