1.	Create a two-by-two matrix that represents a two-person zero-sum game in which each player has two options and the game has a saddle point.	

2.	Create a two-by-two matrix that represents a two-person zero-sum game in which each player has two options and the game has no saddle point.	

3.	Create a two-by-two matrix that represents a two-person zero-sum game in which each player has two options and at least one player has a dominated strategy.	

4.	Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and the game has a saddle point.	

5.	Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and the game has no saddle point.	

6.	Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and at least one player has a dominated strategy.	

7.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddle point?  What is each player's minimax or maximin strategy?  Justify your response. 	

8.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddle point?  What is each player's minimax or maximin strategy?  Justify your response. 	

9.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddlepoint?  What is each player's minimax or maximin strategy?  Justify your response. 	

10.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddlepoint?  What is each player's minimax or maximin strategy?  Justify your response. 	

11.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddlepoint?  What is each player's minimax or maximin strategy?  Justify your response. 	

12.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to column Player II. 

 
Does this game have a saddlepoint?  What is each player's minimax or maximin strategy?  Justify your response. 	

13.	In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix:

 
Solve the game determining the best mix of selections for both batter and pitcher. 	

14.	In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: 

 
Solve the game determining the best mix of selections for both batter and pitcher. 	

15.	In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: 

 
Solve the game, determining the best mix of selections for both batter and pitcher. 	

16.	In the game of matching pennies, Player I wins a penny if the coins match and Player II wins if the coins do not match.  Present this game as a two-by-two matrix, where each player has two outcomes from which to select.	

17.	In the game of matching pennies, Player I wins a penny if the coins match and Player II wins if the coins do not match.  Is this a zero-sum game?  Is this a fair game? 	

18.	In a game, each player chooses one of three coins: penny, nickel, or dime.  If both players choose the same coin, both players loose their coin.  Otherwise, the player with the more valuable coin wins the less valuable coin from the other player.  Represent this game as a three-by-three matrix of ordered pairs.	

19.	In a game, each player chooses one of three coins: penny, nickel, or dime.  If both players choose the same coin, both players loose their coin.  Otherwise, the player with the more valuable coin wins the less valuable coin from the other player.  Is this a symmetric game?  Is there a dominated strategy? 	

20.	You want to carry insurance for your small business.  The annual policy  costs $1000 this year.  If you are sued and you have no insurance, you will pay $5000.  If you have insurance and are sued you pay nothing, and your partner gives you $500 for your wise foresight, so that the policy costs you only $500.  Represent this game as a two-by-two matrix.	

21.	If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000.  If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500.  If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties.  Statistically, how often should you cheat? 	

22.	If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000.  If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500.  If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties.  Statistically, how often should the government audit you? 	

23.	Consider the following partial-conflict game played in a non-cooperative manner.  The first payoff is to Player I; the second to Player II.

 
Discuss the players' possible strategies when this game is played. 	

24.	Consider the following partial-conflict game played in a non-cooperative manner. The first payoff is to Player I; the second is to Player II.

Player II	

Player I	
Choice A	
Choice B	

Choice A	
(6, 1)	
(3, 3)	

Choice B	
(5, 5)	
(1, 6)	

Discuss the players' possible strategies when this game is played. 	

25.	Consider the following partial-conflict game played in a non-cooperative manner. The first payoff is to Player I; the second is to Player II.

Player II	

Player I	
Choice A	
Choice B	

Choice A	
(1, 9)	
(3, 3)	

Choice B	
(5, 5)	
(9, 1)	

Discuss the players' possible strategies when this game is played. 	

26.	Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs.  What are the optimal strategies for this game? 

Sole survivor	
	
2 point	

One of two survivors	
	
4 points	

One of three survivors	
	
3 points	

Non-survivor	
	
1 point	

27.	Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs.  What are the optimal strategies for this game?

Sole survivor	
	
4 point	

One of two survivors	
	
3 points	

One of three survivors	
	
2 points	

Non-survivor	
	
1 point	

28.	Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs.  What are the optimal strategies for this game?

Sole survivor	
	
1 point	

One of two survivors	
	
4 points	

One of three survivors	
	
3 points	

Non-survivor	
	
2 point	

29.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to the column Player II. Does this game have a saddle point? What is each player's minimax or maximin strategy?

30.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to the column Player II. Does this game have a saddle point? What is each player's minimax or maximin strategy?

31.	In the following two-person zero-sum game, the payoffs represent gains to the row Player I and losses to the column Player II. Does this game have a saddle point? What is each player's minimax or maximin strategy?

Answer Key

1.	For example, 

2.	For example, 

3.	For example, 

4.	For example,

5.	For example,

6.	For example,

7.	Has a saddle point at 4.  Play first row and first column.	
8.	Has a saddle point at 5.  Play second row and first column.	
9.	No saddle point.  Play second row and first column.	
10.	Has a saddle point at 4.  Play first row and first column.	
11.	Has a saddle point at 3.  Play third row and first column.	
12.	Has no saddle point.  Play first row and first column.	
13.	Batter expects 1/2 fastball and 1/2 knuckleball.  Pitcher plays 1/4 fastball and 3/4 knuckleball.	
14.	Batter always expects fastball. Pitcher always plays knuckleball.	
15.	Batter expects 3/4 knuckleball and 1/4 fastball.  Pitcher plays 3/4 knuckleball and 1/4 fastball.	
16.	
	
17.	Yes.  Yes.	
18.	
	
19.	Yes.  No.	
20.	
	
21.	1/4	
22.	1/2	
23.	Dominating strategies: Player 1 chooses A, Player 2 chooses B.	
24.	Player I chooses A; Player II chooses B.	
25.	Player I chooses B; Player II chooses A.	
26.	First player shoots one of the other players.	
27.	No one shoots.	
28.	First player shoots one of the other players.	
29.	This game has a saddle point at 5. The maximin strategy of Player I is to play the 2nd row; the minimax strategy of Player II is to play the 2nd column.	
30.	There is no saddle point. The maximin strategy of Player I is to play the 3rd row; the minimax strategy of Player II is to play the 1st column.	
31.	This game has a saddle point at 6. The maximin strategy of Player I is to play the fourth row; the minimax strategy of Player II is to play the fourth column.