Suppose there are two criminals who are thinking about robbing either an insurance company or a liquor store. The take from the insurance company robbery would be Gh50,000 each, but the job requires two people (one to do the robbing and one to drive the getaway car). The take from robing a liquor store is only $1000 but can be done with one person acting alone or both. a. What are the strategies of these players (the two criminals)? b. Write this situation in a normal game form assuming they are acting simultaneously. c. What are the equilibria for this game? (Note: Both Pure strategy and Mixed strategy)
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- Let there be two players in a game, Player 1 and Player 2. Consider a jar containing 5 snakes. 3 of the snakes in the Kjar are venomous, while the remaining 2 are non-venomous. In the game, both the players have to put their hand in the jar one after the other and pick a snake out. Each snake, if picked out of the jar, will bite the player's hand. The event of picking a venomous snake, or equivalently, a venomous snake's bite will earn the player zero points. On the other hand, the event of picking a non-venomous snake, or equivalently, a non-venomous snake's bite will earn the player one point. Let X denote Player 1's pick and let Y denote Player 2's pick. Suppose Player 1 is the first to pick out a snake. The expected value of Player 1's pick is: E(X)= (Express your answer as a fraction or round your answer to two decimal places)You and I play the following game. Hidden from you, I put a coin in my hand: with probability p it is a 10 pence coin and otherwise it is a 20 pence coin. You now guess which coin is in my hand: you guess it is 20 pence with probability s and otherwise you guess it is a 10 pence coin. You get to win the coin if you guess correctly and otherwise win nothing. What (in terms of p and s) is your expected gain in pence from playing this game once with me? Challenge: suppose we are going to play repeatedly and you want to maximise your gain and I wish to minimise my loss. What value of p should I choose and what value of s should you choose? (This question is somewhat ill-defined, but it does have an interesting possible answer.) (Note: anything labelled "challenge" will not be part of the hand-in.)Consider the Stackelberg game depicted below in which you are the row player. R U 4,0 1,2 3,2 0,1 0,0 2,0 You may choose whether you want to be the leader (and commit to a possibly mixed strategy) or the follower in a game against the course staff (column player). You may trust that we maximize our expected payoff. The points awarded to you will equal one half of the expected payoff you obtain. If you want to be the leader, please submit your commitment strategy. For example, if you want to commit to [0.5: U, 0.2: M, 0.3: D], then submit: 0.5 0.2 0.3 If you want to be the follower instead, just submit: F
- Define a new game as a modified version of the game in Problem 1, which we will call the Matching Two Pennies game: each player choose heads (H) or tails (T) for two pennies. In this game, Player 1 wins if both coins show the same combination of heads and tails, while Player 2 wins if they do not. (a) Write the strategy sets S1 and S2 for this game. (b) Give the payoffs for each player in this Matching Two Pennies game, by defining T;(81, 82) for each player and for each strategy pair, either as a list or in matrix form.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?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?)
- 2. Paul and Stella play a game with three strategies each, T, M, and B for Paul, and L, C, and R for Stella. Both move simultaneously. The payoffs are given by the following form: Stella L с R Paul T (8,4) (10, 2) (2,3) M (4,2) (10, 1) (5,7) B (1,4) (10, 3) (9,-4) a. Which strategy is dominated? b. What is the pure-strategy Nash equilibrium? Identify all if there are more than one. c. If Paul moved first, so that Stella observed it, which strategy would Paul choose?Suppose that Jason and Chad each are thinking of opening up a diet coke stand on the fourth floor of this building. Suppose that potential customers are evenly spaced on a distance that is normalized to 1. Customers will buy a diet coke from whichever stand requires the least walking. If they are the same distance the customer will flip a coin. This is depicted below. 1/4 1/2 3/4 Suppose that Jason and Chad are simultaneously choosing the location of their stands, what is the Nash Equilibrium location? a. One of them puts a stand at 3/4 and the other puts a stand at 1/4 b. Chad and Jason put their stands right next to each other at 1/2 c. One of them puts a stand at 0 and the other puts a stand at 1 d. There is no Nash Equilibrium1.a) If the three executives of a fraudulent organization report nothing to the authorities, each gets a payoff of 100. If at least one of them blows the whistle, then those who reported the fraud get 28, while those who didn’t get -100. Suppose they play a symmetric mixed-strategy Nash equilibrium where each is silent (does not report fraud) with probability p. What is p?A, 0.1B, 0.28C, 0.5D, 0.8 b) In a two-player game, with strategies and (some known and some unknown) payoffs as shown below, suppose a mixed-strategy equilibrium exists where 1 plays C with probability 3/4, and Player 2 randomizes over X, Y, and Z with equal probabilities. What are the pure-strategy equilibria of this game? A, (A, Y) and (B, X)B, (A, Z) and (C, Y)C, (B, X) and (C, X)D, (C, X) and (C, Y)
- Please find herewith a payoff matrix. In each cell you find the payoffs of the players associated with a particular strategy combination: The first entry is the payoff of player 1, the second entry is the payoff of player2. Player 2 t1 t2 t3 Player 1 S1 3, 4 1, 0 5, 3 S2 0, 12 8, 12 4, 20 S3 2, 0 2, 11 1, 0 Suppose both players select their strategies (S1, S2 or S3 for player 1 and t1, t2 or t3 for player 2) simultaneously and that the game is played once. In your explanation to the questions below, please do refer to the figures in the matrix. Suppose player 2 could move before player 1 (i.e. has a first mover advantage). In your explanation to the questions below, please do refer to the figures in the matrix. What strategy would (s)he select? Is it really an ‘advantage’ for player 2 to move first? Or does player 2 benefit from being the second mover (and hence player 1 moving first)? I.e. for this question, do not make a comparison to the outcome of the…Write out the extensive form of a game between a farmer (playing in the first round) and nature (playing a mixed strategy in the second round). Assume that the farmer can either pay a cash rent of $1500 for land (English system) or 1/2 of crop production to the landlord (sharecropping). Assume the farmer is planting corn and will produce 2 tons of corn. Assume that nature has a 50% chance of playing a strategy in which the price of corn is $3500/ton and a 50% chance of playing a strategy in which the price of corn is $500/ton. The farmer keeps any money left after paying cash rent and sells any corn left after paying the landlord in sharecropping. What is the most that a risk neutral farmer would be willing to pay for an accurate prediction of the price of corn in problem 1 before choosing whether to pay cash rent or sharecrop?Consider the following game. You will roll a fair, 6-sided die either once or twice. You decide whether to do the second roll after you see how the first one lands. The payoff is $n, where n is the outcome of the last roll. For example, if the first roll lands 4 and the second lands 2, you win $2. If you only do one roll and it lands 4, you win $4. Suppose you make your decision about whether to go for a second roll based on expected monetary value. Then you will go for a second roll if (and only if) the first roll lands x or lower. What is x?