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Many of you will be familiar with the children's game Rock-Scissors-Paper. In Rock-Scissors-Paper, two people simultaneously choose either "Rock," "Scis sors," or "Paper," usually by puttjng their hands into the shape of one of the three choices. The game is scored as follows. A person choosing Scissors beats a person choosing Paper (because scissors cut paper). A person choosing Paper beats a person choosing Rock (because paper covers rock) . A person choosing Rock beats a person choosing Scissors (because rock breaks scis sors). If two players choose the same object, they tie. Suppose that each indi vidual play of the game is worth 10 points. The following matrix shows the possible outcomes in the game:
PLAYER 2
Rock
Scissors
Paper
PLAYER 1
0
10
-10
(a) Suppose that Player 2 announced that she would use a mixture in which her probability of choosing Rock would be 40%, her probability of choos ing Scissors would be 30%, and her probability of Paper, 30%. What is Player l's best response to this strategy choice of Player 2? Explain why your answer makes sense, given your knowledge of mixed strategies.
(b) Find the mixed-strategy equilibrium of this Rock-Scissors-Paper game.
Two players, Amy and Beth, take turns choosing numbers; Amy goes first. On her turn, a player may choose any number between 1 and 10 Who will win the game? What are the optimal strategies (complete plans of action) for each player?
A market research firm conducts studies regarding the success of new products. The company is not always perfect in predicting the success.
In a two player, one shot simultaneous move game each player can select strategy A or B. If both players select strategy A, each receives a payoff of $500.
One airport had 28% late arrivals. After a new directing system was installed, a sample of 1200 flights had 322 late arrivals. At the .01 level, did the new system lower the rate of late arrivals? List the null and alternate hypotheses.
Find a real-world example where the uncertain quantity you are interested in can be modeled as a random variable with exponential distribution. Explain why the memoryless property indeed holds.
A jar contains four white marbles, five red marbles, and six black marbles. If a marble were selected at random, what is the probability that it is white or black?
Draw a game tree for a game in which First has two possible actions (Up or Down) at each node and Second has three possible actions (Top, Middle, or Bottom) at each node.
Assuming that the systolic blood pressures of deep sea divers are normally distributed, the doctor would perform a chi-square test to test her research hypothesis. In that case, what is the test value that she would compute.
How much effort do the students exert in the Nash equilibrium of the game introduced by Amalia and what value v for the cake induces the students to exert the efficient effort level?
There is 0.02% probability that a credit card holder of a company reports the loss or theft of the credit card each month. The company has 15,000 credit cards in Memphis. Use the Poisson distribution to answer the following questions.
Construct the normal form of game assuming consumers move simultaneously and choose between two strategies: "Adopt" or "Don't Adopt." Solve for any (pure strategy) Nash equilibria.
Construct a 3 x 3 game in which there is only one Nash equilibrium in pure strategies and the vector of payoffs in this equilibrium is "worse" than some other vector of payoffs in the game.
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