Quantitative Methods for Business - Utility and Game Theory

Problem 7 A casino offers people the chance to play the following game: flip two fair coins. If both come up heads, the gambler wins $1. If both come up tails, the gambler wins $3. If one is heads and one is tails, the gambler gets nothing. The game costs $1.25 to play. Your friend, Richard, who has not taken a probability course and thus doesn't know any better, goes to this casino and plays the game 600 times. Estimate the probability that your friend loses between $132 and $195 over the course of the 600 games. (You need to provide a number instead of an expression involving NA(a,b)).

You are evaluating the possibility that your company bids $150,000 for a particular construction job. (a) If a bid of $150,000 corresponds to a relative bid of 1.20, what is the dollar profit that your company would make from winning the job with this bid? Show your work. (b) Calculate an estimate of the expected profit of the bid of $150,000 for this job. Assume that, historically, 55 percent of the bids of an average bidder for this type of job would exceed the bid ratio of 1.20. Assume also that you are bidding against three other construction companies. Show your work.

Suppose that the point spread for a particular sporting event is 10 points and that with this spread you are convinced you would have a 0.60 probability of winning a bet on your team. However, the local bookie will accept only a $1000 bet. Assuming that such bets are legal, would you bet on your team? (Disregard any commission charged by the bookie.) Remember that you must pay losses out of your own pocket. Your payoff table is as follows:     STATE OF NATURE DECISION ALTERNATIVES YOU WIN YOU LOSE BET $1000 -$1000 DON’T BET $0 $0   What decision does the expected value approach recommend? What is your indifference probability for the $0 payoff? (Although this choice isn't easy, be as realistic as possible. It is required for an analysis that reflects your attitude toward risk.) What decision would you make based on the expected utility approach? In this case are you a risk taker or a risk avoider? Would other individuals assess the same…

A friend offers you a chance to play a game in which there are only two outcomes, each with equal probability.  If you get the "good" outcome, you win $80.  But if you get the bad outcome, you only win $20. What price to play would make this a fair game (fair bet)?  Carefully follow all mathematical instructions in your answer.

Suppose that you and a friend are playing cards and you decide to make a friendly wager. The bet is that you will draw two cards without replacement from a standard deck. If both cards are spades, your friend will pay you $7. Otherwise, you have to pay your friend $2. Step 1 of 2: What is the expected value of your bet? Round your answer to two decimal places. Losses must be expressed as negative values.

Question 2   A potential criminal enjoys a gain of g > 0 from committing an illegal act. If the potential criminal does not commit the act, he gets a gain equal to g = 0. The expected punishment upon conviction is p (e.g. number of years in prison), but there is the possibility of legal error. In particular, if the criminal commits the act, there is a probability r that he will not be punished, and if the criminal did not commit the act, there is a probability q that he will be punished regardless. a. Derive the net gain (i.e. gain minus expected punishment) for the criminal from not committing the illegal act. b.   Use your answer in a. to derive the condition for the criminal to commit the illegal act. Show and explain every step of your calculations. c.   Lawmakers decided to further increase the standard of proof for a criminal conviction. How will this change a ect the criminal’s decision to commit the illegal act in your answer in b.?

In a large casino, the house wins on its blackjack tables with a probability of 50.9%. All bets at blackjack are 1 to 1, which means that if you win, you gain the amount you bet, and if you lose, you lose the amount you bet. a. If you bet $1 on each hand, what is the expected value to you of a single game? What is the house edge? b. If you played 150 games of blackjack in an evening, betting $1 on each hand, how much should you expect to win or lose? c. If you played 150 games of blackjack in an evening, betting $3 on each hand, how much should you expect to win or lose? d. If patrons bet $7,000,000 on blackjack in one evening, how much should the casino expect to earn? a. The expected value to you of a single game is $ (Type an integer or a decimal.)

A wheel of fortune in a gambling casino has 54 different slots in which the wheel pointer can stop. Four of the 54 slots contain the number 9. For $3 bet on hitting a 9, if he or she succeeds, the gambler wins $16 plus return of the $3 bet. What is the expected value of this gambling game? (Present your answer in dollars with 2 decimal places but without $ sign)

In a final round of a MegaMillion TV show, a contestant has won $1 millionand has a chance of doubling the reward. If he loses his winnings drop to$500,000. The contestant thinks his chances of winning are 50%. Should heplay? What is the lowest probability of a correct guess that will make his betprofitable? Show work

You ask someone about their preferences over the following pairs of lotteries: Lottery A Lottery B Payoff Probability Payoff Probability $200 25% $200 5% $100 45% $100 90% 30% $0 5% Lottery C Lottery D Payoff Probability Payoff Probability $200 20% $200 0% $100 0% $100 45% $0 80% $0 55% The individual says they prefer lotter A over lottery B, and lottery D over lottery C. Is this person using expected utility theory to evaluate these lotteries? How can you know for certain?

Cost-Benefit Analysis Suppose you can take one of two summer jobs. In the first job as a flight attendant, with a salary of $5,000, you estimate the probability you will die is 1 in 40,000. Alternatively, you could drive a truck transporting hazardous materials, which pays $12,000 and for which the probability of death is 1 in 10,000. Suppose that you're indifferent between the two jobs except for the pay and the chance of death. If you choose the job as a flight attendant, what does this say about the value you place on your life?

Well explained answer and well labeled

Topic B Problem: Imagine you have two competing athletes who have the option to use an illegal and/or dangerous drug to enhance their performance (i.e., dope). If neither athlete dopes, then neither gains an advantage. If only one dopes, then that athlete gains a massive advantage over their competitor, reduced by the medical and legal risks of doping. However, if both athletes dope, the advantages cancel out, and only the risks remain, putting them both in a worse position than if neither had been doping. What class concept best describes this situation? Using this class concept, what outcome do we expect from these two athletes? Are there any factors that could change the outcome predicted by this course concept?

sub question : a, b, c

Here is my question!

What type of risk behavior does the person exhibit who is willing to bet $60 on a game where 20% of the time the bet returns $100, and 80% of the time returns $50?  Is this a fair bet? Explain.

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)

Need alternate and original answer with no AI used

1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…

2. In this question, your goal will be to understand whether learning in games is always valuable for players. Consider the following incomplete-information game. First, nature chooses between one of the following two A and B tables, each with probability 0.5: A L R B L R U 0,0 6,-3 U-20,-20 -7,-16 D -3,6 5,5 D -16,-7 -5,-5 Then, players 1 and 2 simultaneously choose U or D and L or R, respectively, and obtain payoffs according to the table chosen by nature. Parts I-III present variations of this game under different assumptions about what players know about nature's move.

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Suppose your wealth is 8 and you are considering a bet of a 50% chance of winning $4 and a 50% chance of losing $4. What is your risk premium given you have In utility ? Draw a diagram and label the relevant values.

Problem 1 You have been tasked with planning the annual gala for the United Way of Salt Lake. The event is intended to be a formal, but fun-filled evening with great food and entertainment. Prominent members (and donors) of the community regularly attend. The Gala is your largest fundraiser of the year. Consider the following “game against nature” in which mother nature has no payoffs. You must decide whether to hold the gala outdoors or indoors. The amount you can raise is determined by whether you choose indoors or outdoors and whether or not mother nature decides to rain. Historically, outdoor galas have proven to raise more money than indoor galas, but rain certainly tends to dampen donor moods. The payoffs and expected probabilities are provided in the following matrix:       Mother Nature       Rain (1/3) No rain (2/3) United Way Indoors 200,000 200,000   Outdoors 100,000 300,000 Should you hold the gala outdoors or indoors? Why? Show your work.

Note: this is an economics question. Game theory and Nash equilibria section.  *Discuss the attached case in terms of eBay's strategic move.

Betting on Events Suppose there is a 75% chance of rain tomorrow. You are offered a "contract" that will pay you $1 if it rains tomorrow and $0 if it doesn't. Suppose there is a 75% chance of rain tomorrow. You are offered a "contract" that will pay you $1 if it rains tomorrow and $0 if it doesn't. Question 4 What is the expected (average) value of the contract's payout in dollars?