Problem #1. Develop a game tree for the game of matching pennies, the rules being that each player chooses to show head or tails. A wins $1  if both players make the same choice. B wins $1 if the choices are different. List pure strategies for both players and give the normal form.

I could have drawn a tree here. But that takes me a little time. So I used the following artifice
The root naode is labelled A. A tosses a coin. He gets a Head or Tail.
The two offspring nodes are labelled B Each of the two offspring has a left branch and a right branch. The left branch is Head and the right branch is Tail.

The event sequence
HH gets a payoff of +1 to A (match)
HT gets a payoff of  -1 to A (no match)
TH gets a payoff of -1 to A (no match)
TT gets a payoff of +1 to A (match)

Pure Strategies for A are:

A1: Choose Head
A2: Choose Tail

Pure Strategies for B are:

B1: Choose Head
B2: Choose Tail

Matrix form
 
 

	B1	B2
A1	1	-1
A2	-1	1

A's optimal strategy is (1/2, 1/2).
B's optimal strategy is (1/2, 1/2).

Problem #2. Develop a game tree for the game of matching pennies, the rules being the same as above except that each player tosses the coin. If possible,  list the players' pure strategies and give the normal form.

Neither player has any decision to make as the outcome is determined by a sequence of chance events. Therefore, what we have  is called a stochastic model. there are no pure strategies and no normal form.

The tree looks like this.

The root node  has two branches.The left branch is labelled Heads (1/2), the 1/2 being the probability of getting  ahed. The right branch is labelled Tails (1/2).
 

Problem 3. The British have decided to attack the American arsenal at Concord, and the Americans knew this. They do not know which way the British have chosen to come- whether by land or by sea. The American force is too small to defend both routes; they must choose to defend one or the other and take the consequences.

In fact the British are low on Ammunition, and if the two forces meet, the British will retreat. This scores one for the Americans. If the forces do not meet, the British reach Concord arsenal and then they have plenty of ammunition. Now both sides must plan what to do about the British return from Concord. The Americans can either lay an ambush on the known path of return or move up and attackthe British at the arsenal. The British can leave immediately by day or or wait for night.

If the British meet the ambush by day they wilol be destroyed (score one for the Americans). If the British meet the ambush by night they can filter through with some small loss (score one for the British). If the Americans attack the arsenal and the British have already left, the British score one for getting way free. But if the Americans attack and find the British waiting for night, both sides suffer heavy losses and score zero. Model the above situation as a 2-person zero sum game. Since it is arbitrary who is considered to make the first decision, assume that Americans = A. State the pure strategies for both sides and find the payoff matrix.

A= Americans
B = British

Root node is A (land option, sea option)

At each of the two child nodes, B has a sea option and a land option.

Both (land, land) and (sea, sea) have a payoff of +1 (Americans win)

When A gets the next turn, A has Ambush and Attack options.

At each of the four nodes, the British have an option to Leave or Wait.

Payoffs to Americans

(land, sea, ambush, leave) = 1
(land, sea, ambush, wait) = -1
(land, sea, attack, leave) = -1
(land, sea, ambush, wait) = 0
(land, land) = 1
(sea, sea) = 1
(sea, land, ambush, leave) = 1
(sea, land, ambush, wait) = -1
(sea, land, attack, leave) = -1
(sea, land, ambush, wait) = 0

Pure strategies for A

A1: Land; then Ambush
A2: Land, then attack
A3: Sea; then ambush
A4: Sea, then attack

Pure strategies for B

B1: Sea; then leave
B2: Sea, then wait
B3: Land; then leave
B4: Lamd, then wait

Matrix Form
 
 

	B1	B2	B3	B4
A1	1	-1	1	1
A2	-1	0	1	1
A3	1	1	1	-1
A4	1	1	-1	0

A's optimal strategy is: (1/6, 1/3, 1/6, 1/3)
B's optimal strategy is: (1/6, 1/3, 1/6, 1/3)
Value= 1/3