MAT109 · Erich Prisner · Franklin College · 2007-2009

Final Exam

Good Luck!

Question 1

(5 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 probabilities of the random moves are given. Ann's possible moves are labeled as A1 ... A7, and Beth's possible moves as B1, ... B7.

  1. What is Ann's expected payoff at the beginning of the game? What is Beth's expected payoff?
  2. Which one of A1 or A2 would Ann choose? Which one of A3, A4 A5? How would she decide between A6 and A7?
  3. Which one of B1 or B2 would Beth choose? Which one of B3 or B4? How would she decide between B5, B6, and B7?
  1. Ann and Beth both expect 2.
  2. Ann selects A2, A4, A7.
  3. Beth selects B2, B4, B7.

Question 2

(2 points)) Consider the following game given by the extensive form as shown below:

  1. How many information sets do Ann and Beth have in this game?
  2. How many pure strategies does Ann have? How many pure strategies does Beth have?
  1. Ann has 2 information sets, Beth has 3.
  2. Ann has 2·3 = 6 pure strategies, Beth has 2·2·3 = 12 pure strategies.

Question 3

(6 points) Consider the following zero-sum game given by the extensive form as shown below:

  1. Is this a game of perfect or imperfect information? Why?
  2. Finish the Normal Form by filling out the four empty entries. Remember that we have a zero-sum game and therefore only Ann's payoffs are shown.
     B1B3  B1B4  B2B3  B2B4 
    A1A3 2  0  .....  -2 
    A1A4 -1  -1  -1  -1 
    A2A3 .....  0  1  0 
    A2A4 -2  .....  0  ...... 
  3. For the following three questions: If you were unable to fill in the four entries, just fill in any of the numbers -2, -1, 0, 1, 2 and then answer the next three questions with this completed matrix.
  4. Is there any weak domination? Which strategies weakly dominate which?
  5. Draw the Best Response Digraph
  6. Are there any Nash equilibria in pure strategies?
  1. It is a game of imperfect information since at least one information set contains more than one vertex.
  2. See here for Excel help.
     B1B3  B1B4  B2B3  B2B4 
    A1A3 2  0  0  -2 
    A1A4 -1  -1  -1  -1 
    A2A3 1  0  1  0 
    A2A4 -2  -1  0  1 
  3. ---
  4. Ann's second pure strategy A1A4 is weakly---even strictly---dominated by A2A3.
    Remember that for checking weak domination for Beth's pure strategies, we have to look at Beth's payoffs, which are the opposites of the numbers given. Beth's third pure strategy B2B3 is weakly dominated by B1B4.
  5. See the digraph to the right.
  6. Yes, A2A3 versus B1B4.

Question 4

(2 and 1/2 points) Backward Induction can always be used for ... (more than one may apply!)

  1. ... Zero-sum 2-player sequential games with randomness and perfect information.
  2. ... Zero-sum 2-player simultaneous games with randomness.
  3. ... Non-zero-sum 2-player sequential games with randomness and perfect information.
  4. ... Zero-sum 3-player sequential games with randomness and perfect information.
  5. ... Zero-sum 2-player sequential games with randomness and imperfect information.
  1. Yes.
  2. No, simultaneous games are like sequential games with imperfect information.
  3. Yes. Non-zero-sumness doesn't matter here.
  4. Yes. Number of players doesn't matter here.
  5. No. It cannot be used when we have imperfect information.

Question 5

(2 points) True or false: (All games considered are finite, with complete but not necessarily perfect information.)

  1. Every game has a Normal Form.
  2. Simultaneous games do not have an Extensive Form
  3. Every game has a pure Nash equilibrium
  4. Every sequential game with randomness and perfect information has a pure Nash equilibrium
  1. True.
  2. False, they do.
  3. False.
  4. True. It is the solution found by Backward Induction.

Question 6

(4 points))

  1. What did Nash prove in his 1950 Ph. D. Thesis?
  2. State the "Law of Large Numbers"
  3. Explain what a Nash equilibrium is.
  4. Explain what a mixed strategy is.
  1. Every game has at least one Nash equilibrium in mixed strategies.
  2. If an experiment is repeated often, the empirical probability (relative frequency) of an event converges to the theoretical probability of that event.
  3. Strategies, pure or mixed, one for each player, such that every player gets less or the same payoff when he or she changes the strategy whereas all other stick to theirs.
  4. Assigning probabilities to all pure strategies such that all these probabilities add up to 1.

Question 7

You are in a card game, and a card is about to be dealt randomly from the stack of cards. You know that the stack of cards contains 22 cards, and that 6 of them are face cards. If the cards dealt is a face card, you win the game and all the money in the pot. Otherwise somebody else will win. The pot contains $1300, but 250 of these dollwars you have put in. That means, if you win you win $1050, but if you lose, you lose $250.

  1. How high is your probability of winning?
  2. What is the expected payoff for you?
  1. 6/22 = 3/11 = 27.27%
  2. (3/11)·1050 + (8/11)·(-250) = 1150/11 = $104.54 .