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There are two Philly Cheese Steak restaurants in South Philadelphia, Pat's and Sal's. They produce premium grade sandwiches for wholesale distribution. There are three choices for the wholesale price of a sandwich: $4, $4.50, $5.00. The cost of making a sandwich is constant and is the same for both vendors. By popular acclaim, Pat's Steaks are the best. The payoff table, or strategic form of the game is shown below.  In any cell, the first entry correpsonds to sandwiches sold by Pat, the second entry corresponds to sandwiches sold by Sal.

Units sold		Sal
		$4	$4.50	$5
Pat	$4	750, 250	800, 200	1000, 0
	$4.50	700, 300	750, 250	800, 200
	$5.00	600,   400	700, 300	750, 250

1. Suppose that Pat has announced that in order to preserve brand identity he will always charge $5.00. What price should Sal charge if he wants to miximize unit sales?

   
   Sal will choose that price that maximizes unit sales, given Pat's announced intention to charge $5.  So Sal should charge $4.

2. Suppose that Sal announces that he will always take the low road and charge $4, no matter what. How much should Pat charge if he wants to maximize unit sales?

   
   Pat will take Sal at his word.  Pat's best strategy then is to choose to price his steak sandwiches at $4.

3. Now suppose that Pat and Sal must announce their pricing decisions simultaneously. They no longer adhere to the pricing rules that they may have used in the previous two questions. What price strategy should Sal use in order to maximize unit sales?
   
   
   Sal looks to see if there is a dominant strategy.  He finds that a price of $4.50 dominates a price of $5 since his sales at the lower price are always greater.  Similarly, for any strategy chosen by Pat, Sal sees that his sales at a price of $4 are always greater than the sales when he charges $4.50.  His dominant strategy is $4.
   
4. Pat and Sal must announce their pricing decisions simultaneously. They no longer adhere to the pricing rules that they may have used in the first two questions. What price strategy should Pat use in order to maximize unit sales?

   
   Using the same reasoning, Pat choose $4 as his dominant strategy. 

5. (Extra Creit) Reconstruct the table with the payoffs stated in terms of dollar sales rather than unit sales.  If Pat and Sale are revenue maximizers rather than unit sales maximizers, what prices will they charge.  (Hint: First find Sal's best strategy, regardles of what Pat chooses.  Because Pat has the table available to him he knows what strategy Sal will choose, and so Pat makes his choice on that basis.)
   Sal's choice   Pat's Choice
   
   Restating the table in dollar sales:

   $ sales		Sal
   		$4	$4.50	$5
   Pat	$4	3000, 1000	3200, 900	4000, 0
   	$4.50	3150, 1200	3375, 1125	3600, 1000
   	$5.00	3000, 1600	3500, 1350	3750, 1250

Sal's dollar sales are best when he chooses to sell at $4, no matter what Pat does.  So Sal will choose $4.  Pat has this table and he knows that Pat's dominant strategy is $4, therefore Pat will pick that price that maximizes his unit sales given Sal's price of $4.  Pat will choose to sell at $4.50.