Exercises and Projects for Chapter 7: Normal Form and Strategies

Exercises

1. Consider the following game:
   Both Ann and Beth put one dollar in the pot. Ann gets a card from a stack of 4 queens and 4 kings and looks at it privately. Then Ann either folds, in which case Beth gets the money in the pot, or raises. Raising means that Ann has to put another dollar into the pot. When Ann has raised, Beth either folds, in which case Ann gets the pot, or Beth calls by putting also one additional dollar into the pot. If Beth calls, Ann gets the pot if she has a king, otherwise Beth gets the pot.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
2. Consider the following game:
   Both Ann and Beth put one dollar into the pot. Ann gets a card and looks at it privately. Then Ann either checks, in which case Ann gets the money in the pot if Ann's card is red, or Beth gets the pot if Ann's card is black. Ann can also raise by putting another dollar into the pot. Now Beth either folds, in which case Ann gets the pot, or Beth calls by putting one additional dollar into the pot. If Beth calls, Ann gets the pot if she has a red card, otherwise Beth gets the pot.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
3. Consider the following game:
   Ann starts the game by selecting (Ann doesn't draw, she chooses) two cards from a deck of cards containing four queens and four kings. Ann puts these cards face down in front of Beth. Beth is allowed to see one of them. Then Beth must guess whether the two cards are two kings, two queens, or one king and one queen. If she is right, she wins $1 from Ann, otherwise she has to pay $1 to Ann.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
   3. Eliminate weakly dominated strategies. Find the IEWD matrix, obtained by iterated elimination of weakly dominated strategies.
4. Consider the following game:
   Ann starts the game by selecting (Ann doesn't draw, she chooses) one card from a deck of cards containing four queens and four kings. Ann puts this card face down in front of Beth. Beth is not allowed to see it, but is allowed to see one card chosen randomly from the remaining deck of seven cards. Then Beth must guess whether the card face down in front of her is a king or a queen. If she is right, she wins $1 from Ann, otherwise she has to pay Ann $1.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
   3. Are there any weakly dominated strategies?
5. Consider the following game:
   Ann starts the game by selecting (Ann doesn't draw, she chooses) two cards from a deck of card containing four queens and four kings. Ann puts these cards face down in front of Beth. Beth is not allowed to see them, but is allowed to see one card chosen randomly from the remaining deck of six cards. Then Beth must guess whether the two cards are two kings, two queens, or king and queen. If she is right, she wins $1 from Ann, otherwise she has to pay Ann $1.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
   3. Eliminate weakly dominated strategies. Find the IEWD matrix, obtained by iterated elimination of weakly dominated strategies.

6. Consider the following game:

   KUHNPOKER: We play with a 8 cards deck, four "1"s and four "2"s. Every player gets one card and looks at it secretly. The start bet is $2. Ann moves first by either checking or raising.

   • If Ann checks, then Beth can check or raise.
     • If Beth checks, both cards are revealed and the player with the higher card wins the pot of $4. splitting it again equally in case of a draw.
     • If Beth raises, she increases the bet to $3. Then Ann has two options, she can either fold or call.
       • If Ann folds, Beth gets the pot money of $5, i.e. wins $2. Ann's card is not revealed in that case.
       • If Ann calls, she also increases her bet to $3. Then both cards are revealed again, and the player with the higher card gets the money of $6, i.e. wins $3. Again, in case of a draw the money is split equally.
   • If Ann raises, she increases the bet to $3. Then Beth has two options, she can either fold or call.
     • If Beth folds, Ann gets the pot money of $5, i.e. wins $2. Beth's card is not revealed in that case.
     • If Beth calls, she also increases her bet to $6. Then both cards are revealed again, and the player with the higher card gets the money of $6, i.e. wins $3. Again, in case of a draw the money is split evenly.
   1. Draw the extensive form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
   2. Find the normal form of the game.
7. How many pure strategies do both players have in NIM(6,2), with extensive form shown again? How many reduced pure strategies do the players have?
   
8. Find the normal form of the following game, which is rather similar to Myerson Poker:
   STRIPPED-DOWN POKER: Both Ann and Beth put one dollar in the pot. Ann gets a card from a stack of 4 Queens and 4 Kings and looks at it privately. Then Ann either folds, in which case Beth gets the money in the pot, or raises. Raising means that Ann has to put another dollar in the pot. When Ann has raised, Beth either folds, in which case Ann gets the pot, or Beth calls by putting also one more dollar in the pot. If Beth calls, Ann gets the pot if she has a King, otherwise Beth gets the pot.
9. Find the normal form of the variant of MYERSON POKER that is played with 4 queens and only 3 kings.
10. Find the normal form of the variant of STRIPPED-DOWN POKER that is played with 4 queens and only 3 kings.
11. a) How many pure strategies does Ann have in this game? How many pure strategies does Beth have? Describe one pure strategy for Ann and one for Beth.
    b) How many reduced pure strategies do Ann and Beth have?
    
12. a) How many pure strategies does Ann have in this game? How many pure strategies does Beth have? Describe one pure strategy for Ann and one for Beth.
    b) How many reduced pure strategies do Ann and Beth have?
    
13. Find the normal form of the variant of MYERSON POKER that is played with q queens and k kings.
14. Find the normal form of the variant of STRIPPED-DOWN POKER that is played with q queens and k kings.
15. a) How many pure strategies does Ann have in this game? How many pure strategies does Beth have? Describe one pure strategy for Ann and one for Beth.
    b) How many reduced pure strategies do Ann and Beth have?
    
16. a) How many pure strategies does Ann have in this game? How many pure strategies does Beth have? Describe one pure strategy for Ann and one for Beth.
    b) How many reduced pure strategies do Ann and Beth have?
    
17. a) How many pure strategies does Ann have in this game? How many pure strategies does Beth have? Describe one pure strategy for Ann and one for Beth.
    b) How many reduced pure strategies do Ann and Beth have?
    c) Find the normal form of the game.
    
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