4. At the same carnival, another game is played by flipping the same weighted coin in #3 and rolling a fair dice (that is equally likely to land on each of the six sides). The player wins if the coin lands on heads and the dice lands. on an odd number. What is the probability that a player wins this game? Show the modėl and/or calculation(s) that lead to your answer. A. 3/20 B. 1/4 C. 7/20 D. 7/10
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- You and a coworker are assigned a team project on which your likelihood or a promotion will be decidedon. It is now the night before the project is due and neither has yet to start it. You both want toreceive a promotion next year, but you both also want to go to your company’s holiday party that night.Each of you wants to maximize his or her own happiness (likelihood of a promotion and mingling withyour colleagues “on the company’s dime”). If you both work, you deliver an outstanding presentation.If you both go to the party, your presentation is mediocre. If one parties and the other works, yourpresentation is above average. Partying increases happiness by 25 units. Working on the project addszero units to happiness. Happiness is also affected by your chance of a promotion, which is depends on howgood your project is. An outstanding presentation gives 40 units of happiness to each of you; an aboveaverage presentation gives 30 units of happiness; a mediocre presentation gives 10 units…7. Suppose the only game in town involves flipping a fair coin (so Heads and Tails are equally likely), with a $x bet. If Heads comes up, the payoff is $0.9x; if Tails comes up, you lose the $x. You have $10,000, and must win at least $5,000 by tomorrow morning to pay off a debt to a mean dude. a. Compute the likelihood of winning at least $5000 by making a single bet of $10,000. b. Compute the likelihood of winning at least $1000 by playing the game 10,0000 times and betting a dollar each time. What is the likelihood of not losing money? Message learned?2. Consider the following Bayesian game with two players. Both players move simultaneously and player 1 can choose either H or L, while player 2's options are G, M, and D. With probability 1/2 the payoffs are given by "Game 1" : GMD H 1,2 1,0 1,3 L 2,4 0,0 0,5 and with probability 1/2 the payoffs are according to "Game 2" : G |M|D H 1,2 1,3 1,0 L 2,4 0,5 0,0 (a) Find the Nash Equilibria when neither player knows which game is actually played. (b) Assume now that player 2 knows which one among the two games is actually being played. Check that the game has a unique Bayesian Nash Equilibrium.
- 9. You have a chance to play a coin flip game where if the coin comes up heads you get $5,000; but lose $1,000 if it comes up tails. If you are risk averse (A>0), how much would you need to be paid to walk away and i not play the game? Explain.2) Veronica is going to sell two vases to a dealer. She can sell each vase for $10. In order to bring her vases to the dealer, Veronica needs to take a bus. There is a 50% chance that all the vases carried on one bus trip will be broken during the trip and a 50% chance that none of the vases carried on one bus trip will be broken during the trip. A roundtrip bus ticket costs $1. Veronica's net income is given by the revenues from the sale of her vases minus the cost of her bus trip(s). 2a) How many trips would Veronica make if she were risk-neutral? 2b) Suppose that Veronica's utility from income is u(w)=100ln(x+w), and that she has no wealth other than the vases (also allow her to have negative wealth as in when she takes two vases on the same trip and both break while she paid the fare). What is the expected utility of net income when Veronica takes both vases on one bus trip? And when she makes two trips? What strategy is preferred? 2c) Would Veronica's preferred strategy change if…1. Use the following game tree to answer parts a & b. At a chance node (Nature's turn) or when a player is indifferent between strategies, each branch has equal likelihood (50% chance they take either path). Nature receives no payoff. Assume players are mistake proof. Use a complete strategy when discussing the results of this game and provide a payoff, even if that payoff is an expected one. Recall that an expected payoff is the "average" payoff: for example if with 50% chance we receive a payoff of 5 and with 50% chance we receive a payoff of 2, the expected payoff is .5 x 5+ .5 x 2 = 3.5 a. What is the outcome of the game? Assume players are risk neutral (i.e., they see no difference in getting a payoff of 5 vs an expected payoff of 5). Red 0, 1, 3, 3 3, 2, 3, 2 Dorothy Blue Top 1, 4, 3, 2 Scarecrow 50% Green Nature Right Bottom 0, 2, 3, 1 Lion 50% 1, 3, 4, 2 3, 4, 3, 2 50% Left Nature 3, 2, 1, 2 Up 50% Red. 1, 3, 4, 2 Down Tinman Blue 0, 3, 3, 3 Yellow Dorothy 4, 3, 4, 2…
- 4.6. A person purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on one trip will be broken during the trip. This person con- siders two strategies: Strategy 1: Take the dozen eggs in one trip. Strategy 2: Make two trips, taking six eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on aver- age, six eggs make it home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. c. Could utility be improved further by taking more than two trips? How would the desirability of this possibility be affected if additional trips were costly?3. Suppose we play the following game. I give you $100 for your initial bankroll. At each time n, you decide how much of your current wealth to bet. You cannot borrow money. You can only play with the money I gave you in the beginning or any money that you have won so far. The game is simple. At each time n ≥ 1, you decide the amount to bet. I will roll a fair die. If the die comes up 1,2,3,..., or 5, you win; if the die comes up 6, then you lose. IOW, if you bet $10 on the first roll, you will either have $90 or $110 after the first roll. (a) Suppose you wish to maximize your profit on the first roll. How much should you bet? (Most of you will get this wrong.) (b) What is the expected profit on the first roll if your bet is b with 0 ≤ b ≤ 100? (c) Suppose you wish to maximize your expected profit on the first roll. How much should you bet? (d) Suppose you wish to maximize your expected profit betting on the nth roll. How much of your current wealth do you bet? (e) Let X₂, be your…2. Kier, in The scenario, wants to determine how each of the 3 companies will decide on possible new investments. He was able to determine the new investment pay off for each of the three choices as well as the probability of the two types of market. If a company will launch product 1, it will gain 50,000 if the market is successful and lose 50,000 if the market is a failure. If a company will launch product 2, it will gain 25,000 if the market is successful and lose 25,000 if the market will fail. If a company decides not to launch any of the product, it will not be affected whether the market will succeed or fail. There is a 56% probability that the market will succeed and 44% probability that the market will fail. What will be the companies decision based on EMV? What is the decision of each company based on expected utility value?
- 1. Mr. Smith can cause an accident, which entails a monetary loss of $1000 to Ms. Adams. The likelihood of the accident depends on the precaution decisions by both individuals. Specifically, each individual can choose either "low" or "high" precaution, with the low precaution requiring no cost and the high precaution requiring the effort cost of $200 to the individual who chooses the high precaution. The following table describes the probability of an accident for each combination of the precaution choices by the two individuals. Adams chooses low precaution Adams chooses high precaution Smith chooses low precaution Smith chooses high precaution 0.8 0.5 0.7 0.1 1) What is the socially efficient outcome? For each of the following tort rules, (i) construct a table describing the individuals' payoffs under different precaution pairs and (ii) find the equilibrium precaution choices by the individuals. 2) a) No liability b) Strict liability (with full compensation) c) Negligence rule (with…2. Christiaan can go hiking, or he can stay at home. Hiking would be fun if nothing bad happens, but there is a risk if he goes hiking that he will meet a bear (not fun) or get bitten by a snake (very not fun). Christiaan decides that if there is a 5% chance of meeting a bear and a 1% chance of getting bitten by a snake, he would prefer to go hiking rather than stay at home. However, if the chance of meeting a bear is 10% and the chance of a snake bite is 5%, he definitely would rather stay at home. then (a) Consider the utility function: U (stay home) = 25, U (hike no event) = 100, U (hike & snake) -1000, U (hike & bear) = -200. Does this utility function represent Christiaan's pref- erences? Explain. (b) Suppose that the utility function in (a) does represent Christiaan's preferences. Would Christiaan prefer to hike or stay home if the probability of meeting a bear is 6% and the probability of being bitten by a snake is 4%? Show your work.$11. Anne and Bruce would like to rent a movie, but they can't decide what kind of movie to get: Anne wants to rent a comedy, and Bruce wants to watch a drama. They decide to choose randomly by playing "Evens or Odds." On the count of three, each of them shows one or two fingers. If the sum is even, Anne wins and they rent the comedy; if the sum is odd, Bruce wins and they rent the drama. Each of them earns a payoff of 1 for winning and 0 for losing "Evens or Odds." (a) Draw the game table for "Evens or Odds." (b) Demonstrate that this game has no Nash equilibrium in pure strategies.