Test 2

First Version

  1. (4 points) Analyze the following sequential game with randomness for two players Ann and Beth. The vertices without a label are either the random vertices or the terminal vertices. The possible random moves are labeled as R1, R2, ... and the probabilities are also given for them. Ann's possible moves are labeled as A1, A2, ..., and Beth's possible moves as B1, B2, ....


    • As can be seen in the Figure, Ann's expected payoff is 8/3 and Beth's 11/3.
    • Ann chooses A2, A3, A6, and A7 in the corresponding situations.
    • Beth chooses B1, B3, B5, and B8 in the corresponding situations.
  2. (3 points) Consider the following game given by the extensive form as shown below:

    • Ann has 3 information sets. The single vertex where Ann decides between A6 or A7 is its own information set.
    • Beth has two information sets.
    • Beth has 2 · 2= 4 pure strategies. One of them consists in choosing B1 in the first information set, and B4 at the second.
  3. (1 point) You have six $1-bills, three $5-bills, and two $10 bills in your pocket. You randomly choose one of them to give the taxi driver a tip. What is the expected value?
    The expected value is (6/11)·1 + (3/11)·5 + (2/11)·10 = 41/11 = 3.7.
  4. (1 point) When does a sequential game have perfect information?
    A sequential game has perfect information if every player, when about to move, knows all moves done by the other players (including the random player) so far.
  5. (3 points) How does Card Counting in Black Jack work? Which of the following five Casino rules or features determine whether a player counting cards can on the long run win some money, and how much? (more than one answer may apply) What else determines the expectations of the card counter?
    See text. The ratio and how many stacks are used are important. Furthermore it is important when the cards are reshuffled.
  6. (3 points) Mark that sentences that are true:
    1. Every game is either sequential or simultaneous.
    2. Only games with perfect information have an extensive form.
    3. In games of perfect information, all information sets contain exactly one vertex.
    4. Only sequential games have an extensive form.
    5. In games without randomness, all information sets contain just one vertex.
    6. No vertices belonging to different players may occur in the same information set.

    1. False
    2. False
    3. True
    4. False
    5. False, randomness or not is irrelevant for information.
    6. True

Second Version

  1. (4 points) Analyze the following sequential game with randomness for two players Ann and Beth. The vertices without a label are either the random vertices or the terminal vertices. The possible random moves are labeled as R1, R2, ... and the probabilities are also given for them. Ann's possible moves are labeled as A1, A2, ..., and Beth's possible moves as B1, B2, ....


    • As can be seen in the Figure, Ann's expected payoff is 11/3 and Beth's 3.
    • Ann chooses A1, A4, A6, and A7 in the corresponding situations.
    • Beth chooses B1, B3, B6, and B7 in the corresponding situations.
  2. (3 points) Consider the following game given by the extensive form as shown below:

    • Ann has 3 information sets. The single start vertex and the vertex where Ann decides between A5 or A6 are both own information set.
    • Beth has tthree information sets.
    • Ann has 2 · 2 · 2 = 8 pure strategies. One of them consists in choosing A1 in the first information set, A5 in the second information set, and A4 in the third one.
  3. (1 point) You have seven $1-bills, two $5-bills, and three $10 bills in your pocket. You randomly choose one of them to give the taxi driver a tip. What is the expected value?
    The expected value is (7/12)·1 + (2/12)·5 + (3/12)·10 = 47/12 = 3.9.
  4. (1 point) When does a sequential game have perfect information?
    A sequential game has perfect information if every player, when about to move, knows all moves done by the other players (including the random player) so far.
  5. (3 points) How does Card Counting in Black Jack work? Which of the following five Casino rules or features determine whether a player counting cards can on the long run win some money, and how much? (more than one answer may apply) What else determines the expectations of the card counter?
    See text. The ratio and how many stacks are used are important. Furthermore it is important when the cards are reshuffled.
  6. (3 points) Mark that sentences that are true:
    1. Every simultaneous game could also be formulated as a sequential game with imperfect information.
    2. No two different information sets can contain the same vertex.
    3. Only games with perfect information have an extensive form.
    4. In games of perfect information, all information sets contain exactly one vertex.
    5. The extensive form of a game is its bimatrix.
    6. Information sets contain always two or more vertices.

    1. True
    2. True
    3. False
    4. True
    5. False
    6. False