IUITY (Utils) While spending the weekend in New York City, Raphael, Susan, and their son, Alex, are lucky enough to hail the Cash Cab for their taxi ride. During their ride, they win $200 for correct answers and receive only one strike for a wrong answer, so at the end of the ride they are eligible for the video bonus question. Their vacation budget before entering the cab was $400, and based on their understanding of the type of bonus question they'll be asked, they believe they have a 80% chance of getting the question right. As explained in the article, if they answer the video bonus question correctly, they will double their winnings, but if they miss the video bonus question, they will lose all of what they had previously won. Alternatively, they can choose not to play for the bonus and walk away with their winnings from the cab ride. The following graph shows the cab riders' utility as a function of their total vacation budget. For simplicity, assume that all three passengers have the same preferences, and they only care about their joint budget since they are a family. Refer to the graph to answer the questions that follow. 7 8 9 10 u(x) -+-+-+- ? UTILITY (Utils) 2 8 10 9 0 100 200 -+-+-+ u(x) + 300 400 500 600 700 800 900 1000 VACATION BUDGET (Dollars) (?) If the family chooses not to do the video bonus question, they will receive a utility of To determine whether they should attempt the video bonus question, you first have to examine the payoffs the family will receive, depending on whether they get the video bonus question correct or incorrect. Complete the following table with the total vacation budget they will have in each situation and the utilities corresponding to each outcome. Answer correctly Answer incorrectly Vacation Budget (Dollars) Utility (Utils) Because the passengers believe they have a 80% chance of getting the question correct, their expected utility from taking the gamble is utils. (Note: Enter all decimal places, and do not round your answer.) Because the expected utility from taking the gamble is than the utility they receive if they do not do the video bonus question, Raphael, Susan, and Alex do the video bonus question. Suppose that the odds that passengers get the video bonus question right increase with the number of people they have in the cab and decrease with the number of strikes they received during their trip. If only Raphael and Susan had been in the cab, and they had received two strikes during the trip, they would have estimated their chance of getting the video bonus question right to be only 40%. Complete the first row of the following table with the guaranteed utility the passengers will receive if they walk away with their original winnings, the new expected utility from taking the gamble, and the answer to whether they should do the video bonus question in this case. Guaranteed Utility Expected Utility Do the Video Bonus? Scenario 40% chance of answering correctly $300 in winnings during the cab ride $200 initial vacation budget (Utils) (Utils) (Yes or No) Complete the first row of the following table with the guaranteed utility the passengers will receive if they walk away with their original winnings, the new expected utility from taking the gamble, and the answer to whether they should do the video bonus question in this case. Scenario 40% chance of answering correctly Guaranteed Utility Expected Utility Do the Video Bonus? (Utils) (Utils) (Yes or No) $300 in winnings during the cab ride $200 initial vacation budget Now suppose that the chance that Raphael, Susan, and Alex get the video bonus question right is again 80%, but that instead of winning $200 during their ride, the family won $300. Taking this new information into account, complete the second row of the previous table. Again, suppose that the chance the passengers get the video bonus question right is 80%, and the total winnings from the cab ride returns to $200. However, suppose the initial vacation budget Raphael, Susan, and Alex started with was only $200 instead of $400. Based on this information, complete the final row of the previous table