MAT109 · Erich Prisner · Franklin College · 2007-2009

Exercises and Projects for Chapter 9: Behavior 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 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. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?
    4. What are the behavior strategies for Ann and Beth?

    a) The extensive form of the game is shown to the right. Ann has two information sets, holding a King or holding a Queen. In both cases she has the choice to raise or fold, thus she has the pure strategies "RR", "RF", "FR", and "FF". Beth has only one information set with the options to call ("C") or to fold ("F"), therefore only two pure strategies "C" and "F".

    b) The normal form of the game looks like this:
     C  F 
    RR01
    RF1/20
    FR-3/20
    FF-1-1

    c) A Nash equilibrium in mixed strategies is Ann using "RR" with probability 1/3 and "RF" with probability 2/3, wheras Beth plays "C" with probability 2/3 and "F" with probability 1/3. The expected payoff when both players play this is 1/3 for Ann.

    d) Ann raises always when receiving a king, and in 1/3 of the cases when holding a queen. Beth plays "C" with probability 2/3 and "F" with probability 1/3.

  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?
    2. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?
    4. What are the behavior strategies for Ann and Beth?

    a) Let the first information set for Ann mean she has a red card resulting with a probability of 1/2, and the second one that she has a black card, having the same probability of 1/2. Beth has just one information set---the situation where Ann raises and Beth doesn't know whether Ann's card is red or not.

    b) Let us just desribe how to compute one entry, "RC" versus "C", the others are done in the same way. If Ann gets a red card, she raises, and Beth calls, therefore Ann gets a payoff of 2 in this case. If Ann gets a black card, she checks, in which case her payoff is -1. Therefore her expected payoff for this pair of pure strategies is 1/2·2 + 1/2·(-1) = 1/2.
     C  F 
    RR01
    RC1/20
    CR-1/21
    CC00

    c) A Nash equilibrium in mixed strategies is Ann using "RR" with probability 1/3 and "RC" with probability 2/3, wheras Beth plays "C" with probability 2/3 and "F" with probability 1/3. The expected payoff when both players play this is 1/3 for Ann.

    d) Ann raises always when receiving a king, and in 1/3 of the cases when holding a queen. Beth plays "C" with probability 2/3 and "F" with probability 1/3.

  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?
    2. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?
    4. What are the behavior strategies for Ann and Beth?


    a) Ann has one vertex, which forms her information set. She has three moves, serving "KK" or "KQ" or "QQ". Beth's decision which card to see is really not a decision (since it is based on no information at all) but rather a random move. Beth has two information sets, depending on whether she sees a King or a Queen. She has three moves in each information set, guessing KK, KQ, or QQ, therefore Beth has 32=9 pure strategies.

    b) We only show Ann's payoffs, since we have a zero-sum game.
     KK,KK  KK,KQ  KK,QQ   KQ,KK  KQ,KQ  KQ,QQ   QQ,KK  QQ,KQ  QQ,QQ 
    KK-1-1-1111111
    KQ1010-10101
    QQ11-111-111-1

    c) One Nash equilöibrium consists of Ann playing "KK" in 1/4 of the cases, "KQ" in 1/2 of the cases, and "QQ" in 1/4 of the cases, and Beth playing "KK,KQ", "KK,QQ", "KQ,KQ", and "KQ,QQ" each one in 1/4 of the cases. The expected payoff for both Ann and Beth is 0.

    d) Since Ann has only one position, her behavior strategy is just the mixed strategy mentioned above. Beth's behavior strategy would consist in playing "KK" or "KQ" each in 1/2 of the cases when seeing a king, and "KQ" and "QQ" each in 1/2 of the cases when seeing a queen. This behavior strategy also seems to be rather natural.

  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?
    2. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?
    4. What are the behavior strategies for Ann and Beth?


     KK  KQ  QK  QQ 
    K-11/7-1/71
    Q11/7-1/7-1

    c) A Nash equilibrium in mixed strategies is Ann using both strategies with equal probability and Beth using just "QK". The expected outcome when both players play this is -1/7 for Ann.

    d) Ann's behavior strategy is the above-mentioned mix of 50% "K" and 50% "Q". Beth selects "Q" when seeing a king, and "K" when seeing a queen.

  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?
    2. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?
    4. What are the behavior strategies for Ann and Beth?


    a) Ann has one vertex, which forms her information set. She has three moves, serving "KK" or "KQ" or "QQ". Beth's decision which card to see is really not a decision (since it is based on no information at all) but rather a random move. Beth has two information sets, depending on whether she sees a King or a Queen. She has three moves in each information set, guessing KK, KQ, or QQ, therefore Beth has 32=9 pure strategies.

    b) We only show Ann's payoffs, since we have a zero-sum game.
     KK,KK  KK,KQ  KK,QQ   KQ,KK  KQ,KQ  KQ,QQ   QQ,KK  QQ,KQ  QQ,QQ 
    KK-11/31/3-1/311-1/311
    KQ1010-10101
    QQ111/3111/3-1/3-1/3-1

    c) One Nash equilöibrium consists of Ann playing "KK" in 30% of the cases, "KQ" in 40% of the cases, and "QQ" in 30% of the cases, and Beth playing "KQ,KK" in 25% of the cases, "KQ,KQ" in 15% of the cases, "QQ,KK" in 35% of the cases, and "QQ,KQ" in the remaining 25 of the cases. The expected payoff for Ann is 1/5.

    d) Since Ann has only one position, her behavior strategy is just the mixed strategy mentioned above. Beth's behavior strategy would consist of playing 40% of "KQ", 60% of "QQ" when facing a king, and 60% of "KK" and 40% of "KQ" when facing a queen.

  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. Draw the Normal Form of the game.
    3. Analyze the game. How should Ann and Beth play?


    a) Since Ann has 4 information sets with two moves possible in each, she has 16 pure strategy. The same holds for Beth.
  7. Assume there are the following two behavior strategies in MYERSON POKER (shown to the right) for Ann and Beth. Ann would always raise when having a red card, and with a blue card she would raise in 1/3 of the cases. Beth would meet in 2/3 of the cases. What are the expected payoffs for An and Beth then? Translate this into a mixed strategies for Ann and Beth.
    ...
  8. Also for MYERSON POKER, Ann has the four pure strategies (raise, raise), (raise,check), (check,raise), and (check,check), where the first entry refers to Ann's red card vertex, and the second one to the blue card vertex. Beth has the two pure strategies (meet) and (pass). Translate Ann's mixed strategy using all four pure strategies with probability 1/4 into a behavior strategy.
    ...

  9. The next six questions refer to the MINI LE HER(3,2) game already considered in previous exercises. This game is played with a 6 cards deck, two Aces, two Kings, two Queens. Ann and Beth get both a card at which they look without showing it to their opponent. A third card is put, back up, on the desk. Now Ann can decide whether she wants to exchange cards with Beth. Next Beth has the opportunity to exchange her card with that lying on the desk. Now both players reveal their cards, the higher one wins.

    Let Ann play the behavior strategy of exchanging the card always if it is a Queen, exhanging with probability 1/3 if it is a King, and never changing an Ace. What is the corresponding mixed strategy for Ann?


    ...
  10. Beth plays the following behavior strategy: If Ann didn't exhange her card, then Beth exchanges always a Queen, exchanges with probability 2/3 a King, and never exchanges an Ace. If Ann did exchange then there are 9 different situations: Beth will then never exhange if she has an Ace, always if she has a Queen. If she has a King, she will not exchange if Ann has a Queen of King, but always if Ann has an Ace. Compute the corresponding mixed behavior for Beth.
    ...
  11. Let XQXKXA be a pure strategy for Ann, where XQ refers to the case that she has a Queen, and XK and XA to the cases where she has a King respectively Ace. The entries are "C" for changing cards with Beth, and "N" for not changing cards. What is an equivalent behavior strategy for choosing "CCN" in 30% of the cases, "NNN" in 30% of the cases, and "CNN" in the remaining 40% of the cases?
    ...
  12. Pure strategies for Beth have 12 entries, since she has 12 different information sets. We combine Beth's decision into a 12-letter word XQXKXAXQQXQKXQAXKQXKKXKAXAQXAKXAA, with letters "C" or "N" depending on whether she exchanges with the unknown card on the desk. The letters Xj refer to cases where Ann didn't exchange cards, and Beth has card j, and the letters Xij refer to cases where Ann did exchange and Ann has card i and Beth card j. Assume Beth chooses "CCNCNNCNNCCN" with probability 1/2, and "CCNCNNCCNCCN" and "CNNCNNCNNCCN" both with probability 1/4. Find an equivalent behavior strategy.
    ...
  13. Show that Ann's strategy described in question 9 is never part of a Nash equilibrium.
    ...
  14. Show that Beth's strategy described in question 10 is never part of a Nash equilibrium.
    ...
  15. .....
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