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Game Theory and Optimization: Baseball Manager Decisions and Rock Hound Profits - Prof. Jo, Assignments of Mathematics

Solutions to two game theory problems. The first problem involves a baseball manager making decisions about which player to use and which pitcher to face based on success rates. The second problem deals with two rock hounds splitting and selling their meteor chunks. Game trees, payoff matrices, and equilibrium points.

Typology: Assignments

Pre 2010

Uploaded on 08/04/2009

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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.
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MA 125 - In-Class Assignment 2 Key - Spring 2008

The nameís ; but my friends call me :

  1. For a particular at bat in a baseball game, we deÖne 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 Öeld 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 managersíchoices.

Start

A

B

X

Y

Z

Y

Z

X

A B A B A B

Man. 1 Man. 2 Man. 1 Outcome

(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.

  1. Two rock hounds each have one chunk of meteor. If the Örst rock hound splits her chunk into 4 pieces and the second splits her chunk into 4 pieces as well, each piece can be sold for $25. Each rock hound can split her chunk into as many as three additional pieces without the other knowing (until the go to sell the pieces). However, for each additional piece above the 8 total pieces, the price per piece at which each can be sold for decreases by $3.

(a) Construct the payo§ matrix that gives the proÖts for each rock hound for all the possi- bilities.

Rock Hound 2 4 5 6 7 4 (100; 100) (88; 110) (76; 114) (64; 112) Rock Hound 1 5 (110; 88) (95; 95) (80; 96) (65; 91) 6 (114; 76) (96; 80) (78; 78) (60; 70) 7 (112; 64) (91; 65) (70; 60) (49; 49)

(b) Find all the equilibrium points of this matrix. The equilibrium points are (80; 96) and (96; 80) : (c) Find the reduced payo§ matrix.

Rock Hound 2 5 6 Rock Hound 1 5 (95; 95) (80; 96) 6 (96; 80) (78; 78)

(d) Describe the best strategies for each rock hound. The best strategy for each rock hound is to select to make either 5 or 6 pieces.

  1. Aaron and Alex engage in a pistol duel to the death. Each has one shot. Aaron may Öre when he is 60 paces, 40 paces, or 20 paces away, while Alex may Öre when he is 50 paces, 30 paces, or 10 paces away. Assume that, if one of the duelists Öres and misses, the other one knows this and will walk up and kill the person who missed. Below are the percentages of success for each duelist when Öring at the various number of paces.

Aaron Alex Strategy Percentage of Strategy Percentage of Success Success 60 paces 20% 50 paces 25% 40 paces 40% 30 paces 35% 20 paces 70% 10 paces 75%

We can view the duel as a zero-sum game, with the payo§s computed as follows. Decide who will Öre Örst and compute that personís payo§. For example, if Aaron choose to Öre at 60 paces, then Aaron shoots Örst, no matter what. He wins 20% of the time and loses 80% of the time, meaning his payo§ will be 20% 80% = 60%:

(a) Find the payo§ matrix, with Aaron as the row player.

Alex 50 30 10 60 60% 60% 60% Aaron 40 50% 20% 20% 20 50% 30% 40%