21) Consider the patent race in which two firms (R and S) are involved. Either firm can move 1, 2, or 3 steps closer to the end in any one period. It costs $2 (million) to move one step forward, $7 (million) to move two steps forward, and $15 (million) to move three steps forward. Whichever firm completes all the steps first gets the patent; the patent is worth $20 (million). a) Solve the game by backward induction, and draw a location space figure that shows Trigger Zone I and Trigger Zone II, as well as Safety Zone I and II for R and S. Also assign the locations at the southwest corners of the Trigger Zones I and II. b) Suppose that the patent is now worth $25, instead of $20. The costs of R&D are $2, $7, and $15 million for moving 1,2, and 3 steps, respectively. Solve the game by backward induction, and draw a location space figure that shows Trigger Zone I and Trigger Zone II, as well as Safety Zone I and II for R and S. Also assign the locations at the southwest corners of the Trigger Zones I and II.