Two fishermen (Andy and Bob) are fishing in the same lake. Each can either fish for 5 or 6 hours. Their payoffs depend on how much time each of them fishes: - If they both fish for 6 hours each, their payoff is 2 fish for Andy and 2 fish for Bob; - if they both fish for 5 hours each, their payoff is 3 fish for Andy and 3 fish for Bob; - if one fishes for 5 hours and the other for 6, then the payoff of the former is 1 fish and of the latter is 4 fish. What are the payoffs at the Nash Equilibrium if they play sequentially, and Bob plays first? (Hint: Before you choose the correct answer, write the game in a tree form and find the Nash Equilibrium) Select one: a. Andy gets 2 and Bob gets 2 b. Andy gets 3 and Bob gets 3 c. Andy gets 1 and Bob gets 4 d. Andy gets 4 and Bob gets 1
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- We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does this game have a Nash Equilibrium? (If not, why not? If yes, what is the Nash Equilibrium?)We have a group of three friends: Kramer, Jerry and Elaine. Kramer has a $10 banknote that he will auction off, and Jerry and Elaine will be bidding for it. Jerry and Elaine have to submit their bids to Kramer privately, both at the same time. We assume that both Jerry and Elaine only have $2 that day, and the available strategies to each one of them are to bid either$0, $1 or $2. Whoever places the highest bid, wins the $10 banknote. In case of a tie (that is, if Jerry and Elaine submit the same bid), each one of them gets $5. Regardless of who wins the auction, each bidder has to pay to Kramer whatever he or she bid. Does Jerry have any strictly dominant strategy? Does Elaine?Suppose that two players are playing the following game. Player 1 can choose either Top or Bottom, and Player 2 can choose either Left or Right. The payoffs are given in the following table: Player 1 Player 2 Left Right Top 6 1 9 4 Bottom 2 4 5 3 where the number on the left is the payoff to Player 1, and the number on the right is the payoff to Player 2. D) What is Player 1’s maximin strategy?E) What is Player 2’s maximin strategy?F) If the game were played with Player 1 moving first and Player 2 moving second, using the backward induction method we went over in class, what strategy will each player choose?
- Cameron and Luke are playing a game called ”Race to 10”. Cameron goes first, and the players take turns choosing either 1 or 2. In each turn, they add the new number to a running total. The player who brings the total to exactly 10 wins the game. a) If both Cameron and Luke play optimally, who will win the game? Does the game have a first-mover advantage or a second-mover advantage? b) Suppose the game is modified to ”Race to 11” (i.e, the player who reaches 11 first wins). Who will win the game if both players play their optimal strategies? What if the game is ”Race to 12”? Does the result change? c) Consider the general version of the game called ”Race to n,” where n is a positive integer greater than 0. What are the conditions on n such that the game has a first mover advantage? What are the conditions on n such that the game has a second mover advantage?Theo and Addy are deciding what toys to pick out at the toy store. Depending on what toys they pick, they can play different games together, but they can’t coordinate their choices. They can’t talk to one another at all until after that make their choice. Below is their payout matrix which shows their utility for each choice. All the bold figures are for Theo and all the non bold figures are for Addy. Addy Strategies Theo Strategies Toy Gas Pump Jump Rope Toy food 20 10 10 3 Ball 7 3 9 4 a) If Theo chooses Toy Food, what would be the possible outcomes for Addy? What would be best for Addy? b) If Addy chose a Toy Gas Pump, what are the possible outcomes for Theo? What would be best for Theo? c) Does Addy have a dominant strategy? If yes, what is her strategy? If not how can you tell? d) Does Theo have a dominant strategy? If yes, what is her strategy? If not how…H2. One day, Sam and Ryan play odds/evens to see who gets the last doughnut. On command, they each extend one or two fingers. If the sum is odd Sam wins the doughnut, if the sum is even Ryan wins the doughnut. Suppose the payoff from winning the doughnut is 1 and the payoff from losing is 0. a) Illustrate this interaction as a game in matrix form. b) Suppose that Sam thinks that Ryan will play one finger for sure? What will Sam play? Does Sam have reason to think that Ryan will play one finger for sure? c) Do either of them have a strictly dominated strategy? d) Find the pure strategy Nash equilibria of the game, if any. e) Suppose that Sam thinks that Ryan will play one finger or two fingers with even odds. Will Ryan play one finger for sure, play two fingers for sure or play each strategy with even odds? Does Sam have good reason to believe that Ryan will play one finger or two fingers with even odds?
- Two street racers are playing a simultaneous game of chicken. They have to race towards each other and whoever swerves first is chicken and faces shame, a loss of 8. while the winner enjoys a gain of 3. If neither stop, they would crash into each other, a loss of 10. If both of them swerve at the same time, they are both chicken and face a loss of 5 each If player B destroys his own brakes before the race, and player A sees that, what would the new Nash equilibrium be in this case? a. Player A stops, Player B does not b. Player B stops, Player A does not c. Both players stop d. Neither players stopExercise 7: The facility location game. Our example is a game in which two firms compete through their choice of locations. Suppose that two firms A and B are each planning to open a store in one of six towns located along six consecutive exits on a highway. We can represent the arrangement of these towns using a six-node graph as in Figure 1 Now, based on leasing agreements, Firm 1 has the option of opening its store in any of towns 1, 3, or 5, while Firm 2 has the option of opening its store in any of towns 2, 4, or 6. These decisions will be executed simultaneously. Once the two stores are opened, customers from the towns will go to the store that is closer to them. So for example, if Firm A open its store in town 3 and Firm B opens its store in town 2, then the store in town 2 will attract customers from 1 and 2, while the store in town 3 will attract customers from 3, 4, 5, and 6. If we assume that the towns contain an equal number of customers, and that payoffs are directly…Q14. Do players have perfect information in the above game? Yes, all of them have perfect information No, player 2 has imperfect information No, player 3 has imperfect information No, no player has perfect information Q15. If we want to describe the above game with a strategic form representation, what would the strategy sets for the three players be? Player 1={a, b, c} ; Player 2={x, y}; Player 3={r, s} Player 1={a, b, c} ; Player 2={xx, xy, yx, yy}; Player 3={r, s}
- Two oil companies are deciding how much oil to extract from their properties, which lie above the same underground reservoir. The faster that oil is extracted, the less total oil is extracted. Letting x denote the extraction rate for company X and y denote the extraction rate for company Y, we assume that the total amount of oil extracted is 1/(x + y) million gallons of oil. Of the total amount that is extracted, the share going to company X is x/(x + y), and the share to company Y is y/(x + y); that is, a company’s share depends on how fast it extracts compared with the other company. The price of oil is $100 per gallon. Each company chooses its extraction rate from the interval [1,10] in order to maximize the monetary value of the oil that it extracts. Find the Nash equilibrium extraction rates. (Note: You can assume that the payoff function is hill shaped.)Suppose Rashard and Alyssa are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Rashard chooses Up and Alyssa chooses Down, Rashard will receive a payoff of 4 and Alyssa will receive a payoff of 5. Alyssa Up Down Rashard Up 8, 4 4, 5 Down 5, 4 6, 5 In this game, the only dominant strategy is for to choose . The outcome reflecting the unique Nash equilibrium in this game is as follows: Rashard chooses and Alyssa chooses.Assume the following game situation: If Player A plays UP and Player B plays LEFT then Player A gets $1 and Player B gets $3. If Player A plays UP and Player B plays RIGHT then Player A gets $2 and Player B gets $5. If Player A plays DOWN and Player B plays LEFET then Player A gets $4 and Player B gets $2. If Player A plays DOWN and Player B plays RIGHT then Player A gets $1 and Player B gets $1 What is the Mixed Strategy Equilibrium for Player B? O. (LEFT, RIGHT) = (1/8, 3/8) O. (LEFT, RIGHT) = (1/4, 3/4) O. (LEFT, RIGHT) = (1/2, 1/2) O. (LEFT, RIGHT) = (3/8, 1/8)