MAT109 · Erich Prisner · Franklin College · 2007-2009

## Exercises and Projects for Chapter 6: Extensive Form of General Games

### 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 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.
1. Draw the Extensive Form of the game. How many pure strategies does Ann have, and how many pure strategies does Beth have?
2. Consider the following game:
Both Ann and Beth put one dollar in 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 in the pot. Now Beth either folds, in which case Ann gets the pot, or Beth calls by putting one more dollar in 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?
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 a 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?
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 cards face down in front of Beth. Beth is not allowed to see it, but is allowed to see one card of the remaining deck of seven cards. Then Beth must guess whether the card face down in fron 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?
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 of the remaining deck. Then Beth must guess whether the two cards are two Kings, two Queens, or one King and 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?
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?
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