(3, 4), which yields payoff (7, 0).
Including the cost of the current step for A, the payoff for the
remainder of the game is then (6, 0).
(c)
Now suppose that the position is (5, 4) and it is
A
’s turn to move.
What is the backward induction
solution to the game?
If A takes one step, then B will take one step to get to (4, 3), which is a winning position for B and a
losing position for A.
But if A takes two or more steps, A can achieve a winning position.
So A should
take two steps to get to (3, 4), which has payoff (7-4, 0) = (3, 0).
For parts (d) to (f), d
efine the “trigger zone” as all the points where the firm moving first can ensure
victory. (That is, it will get the patent and achieve a positive payoff.) and
the “safety zone for
A
” as all
the points where, regardless of which firm moves first, A can ensure victory.
Define the safety zone for
B
similarly.
(d)
Consider all the points where both firms are at a distance 5 or less from the finish line.
Which points
lie in the trigger zone, which lie in the safety zone for
A
, and which lie in the safety zone for
B
.
If one firm is 3 steps or less from the finish and the other is more than 3 steps from the finish, then the
first firm will win the game.
So (1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5) are in the safety zone for A and
similarly (4, 1), (5, 1), (4, 2), (5, 2), (4, 3), (5, 3) are in the safety zone for B.
If both firms are 4 or 5 steps from the finish, then the next firm to move can make a move that leaves
a position in its safety zone (at cost at most 4).
So (4, 4), (4, 5), (5, 4), (5, 5) are in the trigger zone.