There are N sites that need protection (number them 1 to N). Someone is going to pick one of them to attack, and you must pick one to protect. Suppose that the attacker is going to attack site i with probability q;. You plan on selecting a site to protect, with probability p¡ of selecting site i. If you select the same site to protect that the attacker chooses to attack, you successfully defend that site. The choice of {4i} and {p;} represent the attacker's and defender's strategy, respectively. 1) What is the probability that you successfully prevent the attack, given strategies {4;}, {p:}?? 2) If you knew {qı,...,qN} in advance, how should you choose {p;} to maximize the probability you successfully prevent an attack?
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- Matthew is playing snooker (more difficult variant of pool) with his friend. He is not sure which strategy to choose for his next shot. He can try and pot a relatively difficult red ball (strategy R1), which he will pot with probability 0.4. If he pots it, he will have to play the black ball, which he will pot with probability 0.3. His second option (strategy R2) is to try and pot a relatively easy red, which he will pot with probability 0.7. If he pots it, he will have to play the blue ball, which he will pot with probability 0.6. His third option, (strategy R3) is to play safe, meaning not trying to pot any ball and give a difficult shot for his opponent to then make a foul, which will give Matthew 4 points with probability 0.5. If potted, the red balls are worth 1 point each, while the blue ball is worth 5 points, and the black ball 7 points. If he does not pot any ball, he gets 0 point. By using the EMV rule, which strategy should Matthew choose? And what is his expected…Each of the two players independently (and simultaneously with the other) decides whether to go to a play or a concert. Each would rather go with the other to a concert than with them to a play, but prefers this to not being together, in which case they don't care where they go alone. Additionally, each is indifferent between attending the play together and participating in a lottery where both go to the concert with a probability of ¾ and to different events with a probability of ¼. Describe the game in matrix form and find all its equilibria under the assumption that the players have von Neumann-Morgenstern preferences.A Bank has foreclosed on a home mortgage and is selling the house at auction. There are two bidders for the house, Zeke and Heidi. The bank does not know the willingness to pay of these three bidders for the house, but on the basis of its previous experience, the bank believes that each of these bidders has a probability of 1/3 of valuing it at $800,000, a probability of 1/3 of valuing at $600,000, and a probability of 1/3 of valuing it at $300,000. The bank believes that these probabilities are independent among buyers. If the bank sells the house by means of a second- bidder, sealed-bid auction, what will be the bank’s expected revenue from the sale? The answer is 455, 556. Please show the steps in details thank you!
- You are considering a $500,000 investment in the fast-food industry and have narrowed your choice to either a McDonald’s or a Penn Station East Coast Subs franchise. McDonald’s indicates that, based on the location where you are proposing to open a new restaurant, there is a 25 percent probability that aggregate 10-year profits (net of the initial investment) will be $16 million, a 50 percent probability that profits will be $8 million, and a 25 percent probability that profits will be −$1.6 million. The aggregate 10-year profit projections (net of the initial investment) for a Penn Station East Coast Subs franchise is $48 million with a 2.5 percent probability, $8 million with a 95 percent probability, and −$48 million with a 2.5 percent probability. Considering both the risk and expected profitability of these two investment opportunities, which is the better investment? Explain carefully.Obi-Wan is considering whether to buy a lightsaber. With probability 0.50 he will value the lightsaber at $4,000, and with probability 0.50 he will value it at $1,000. If new lightsabers sell for $2,500, then buying a new lightsaber is a: Multiple Choice fair gamble. better-than-fair gamble. less-than-fair gamble. less-than-fair gamble if Obi-Wan risk neutral.The mixed stratergy nash equalibrium consists of : the probability of firm A selecting October is 0.692 and probability of firm A selecting December is 0.309. The probability of firm B selecting October is 0.5 and probability of firm selecting December is 0.5. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?
- How much is his risk on any random day due to late arrival?Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary’s boat wins, Donna would give him $31. If Gary’s boat does not win, Gary would give her $31. Gary’s utility function is p1x^21+p2x^22, where p1 and p2 are the probabilities of events 1 and 2 and where x1 and x2 are his wealth if events 1 and 2 occur respectively. Gary’s total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). Taking the bet would reduce his expected utility. Taking the bet would leave his expected utility unchanged. Taking the bet would increase his expected utility. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. The information given in the problem is self-contradictory.Clancy has difficulty finding parking in his neighborhood and, thus, is considering the gamble of illegally parking on the sidewalk because of the opportunity cost of the time he spends searching for parking. On any given day, Clancy knows he may or may not get a ticket, but he also expects that if he were to do it every day, the average amount he would pay for parking tickets should converge to the expected value. If the expected value is positive, then in the long run, it will be optimal for him to park on the sidewalk and occasionally pay the tickets in exchange for the benefits of not searching for parking. Suppose that Clancy knows that the fine for parking this way is $100, and his opportunity cost (OC) of searching for parking is $20 per day. That is, if he parks on the sidewalk and does not get a ticket, he gets a positive payoff worth $20; if he does get a ticket, he ends up with a payoff of
- Arielle is a risk-averse traveler who is planning a trip to Canada. She is planning on carrying $400 in her backpack. Walking the streets of Canada, however, can be dangerous and there is some chance that she will have her backpack stolen. If she is only carrying cash and her backpack is stolen, she will have no money ($0). The probability that her backpack is stolen is 1/5. Finally assume that her preferences over money can be represented by the utility function U(x)=(x)^0.5 Suppose that she has the option to buy traveler’s checks. If her backpack is stolen and she is carrying traveler’s checks then she can have those checks replaced at no cost. National Express charges a fee of $p per $1 traveler’s check. In other words, the price of a $1 traveler’s check is $(1+p). If the purchase of traveler’s checks is a fair bet, then we know that the purchase of traveler checks will not change her expected income. Show that if the purchase is a fair bet, then the price (1+p) = $1.25.For the following questions consider this setting. The deciding shot in a soccer game comes down to a penalty shot. If the goal-keeper jumps in one corner and the striker shots the ball in the other, then it is a goal. If the goalie jumps left and the striker shoots left, then it is a goal with probability 1/3. If the goalie jumps right and the striker shots right, it is goal with probability 2/3. Say the goalie's strategy is to jump left with probability 1 and the striker shoots left with probability 0.5, then the probability of a goal is (round to two digits) If the striker shoots in either corner with probability 0.5 and the goalie likewise shoots in either corner with probability 0.5, then the probability of a goal is (round to 2 digits)For the following questions consider this setting. The deciding shot in a soccer game comes down to a penalty shot. If the goal-keeper jumps in one corner and the striker shots the ball in the other, then it is a goal. If the goalie jumps left and the striker shoots left, then it is a goal with probability 1/3. If the goalie jumps right and the striker shots right, it is goal with probability 2/3. QUESTION Say the goalie's strategy is to jump left with probability 1 and the striker shoots left with probability 0.5, then the probability of a goal is (round to two digits) QUESTION If the striker shoots in either corner with probability 0.5 and the goalie likewise shoots in either corner with probability 0.5, then the probability of a goal is (round to 2 digits)