MA 125 - In-Class Assignment 2 Key - Spring 2008
The name’s ;but my friends call me :
1. For a particular at bat in a baseball game, we de…ne success as the batter get a hit and failure
if the batter does not get a hit. Imagine the following scenario. The manager of the team
at bat can let Player A hit, or substitute Player B for Player A. The manager of the team
in the …eld can then choose to stay with Pitcher X, or substitute either Pitcher Y or Pitcher
Z for Pitcher X. Finally, if Player A was not originally replaced, the manager of the team
at bat can again let Player A hit, or subsitute Player B for Player A. The success rates (as
percentages) of the hitters against the three pitchers are given in the table below.
Pitcher X Pitcher Y Pitcher Z
Player A 20% 29% 27%
Player B 35% 23% 24%
(a) Construct a game tree for the managers’choices.
Start
A
B
X
Y
Z
Y
Z
X
A
B
A
B
A
B
Man. 1 Man. 2 Man. 1 Outcome
27%
20%
35%
29%
23%
24%
35%
23%
24%
(b) Determine the strategies each manager should follow, assuming best play by both man-
agers.
If Manager 1 opts to go with Player B at the beginning, Manager 2 will bring in Pitcher
Y, and the batter will be successful only 23% of the time.
If Manager 1 stays with Player A, then if Manager 2 stays with Pitcher X, Manager 1
will bring in Player B be successful 35% of the time. If Manager 2 brings in Pitcher Y,
then Manager 1 will stay with Player A and be successful 29% of the time. If Manager
2 brings in Pitcher Z, Manager 1 will stay with Player A and be successful 27% of the
time.
So, Manager 1 will not switch to Player B immediately because the outcomes for staying
with Player A at the start are better. Given that Manager 1 will stay with Player A,
Manager 2 will switch to Pitcher Z because this gives the lowest percentage of success
to the other team. Finally, Manager 1 will stick with Player A to face Pitcher Z.
