Exercises
Page 3
8. Diego has just completed a summer internship with the Bureau for Bureaucratic Affairs in Washington, DC, and has been offered regular employment after he graduates next year with a starting salary of $36,000. The employment offer is valid for 90 days. If Diego does not accept it by then, the offer will be withdrawn. But Diego has other options. His aunt, for one, has told him that he can always work as a salesperson in her insurance agency if he so desires. This position typically pays $24,000 in salary and commissions in the first year, but the earning potential increases considerably with time. Diego, however, would rather work with the prestigious firm of Goldman & Cash, whose starting salary is $50,000 and dress in three-piece suits. Diego believes there is an 80% chance that G&C will hire him if he achieves an A average in his senior year. If G&C does not hire him, he could apply again to the BBA in DC, but Diego has no idea if the position will be available then. On the other hand, if he fails to achieve an A average in his senior year, Diego believes that the likelihood that G&C would hire him drops to a mere 10%. He assesses the probability of achieving an A average as 0.7. Assuming that Diego’s sole criterion is to maximize his first year’s earnings, determine his optimal employment-decision strategy. (Remember Diego’s aunt.)
9. Nautical Chicks is a club of women college students who learn sailing and open-sea navigation in the Caribbean. One good day while anchored near Gran Tortuga they discovered the remains of what seemed to be a very old shipwreck. At first they thought it was a sunken World War II cargo steamer, which was of no major interest. However, one of the girls, a history major, suggested it might be none other than the legendary Spanish galleon, El Conde del Chorizo, which sank in these waters in the midst of the terrible hurricane Santa Rabieta in 1622 with $100 million in gold doubloons.
After several reconnaissance dives, the Chicks concluded that the probability that the wreck was that of El Conde was only one in a hundred. A quick marine-radio call to the Caribbean Salvage Company revealed that the cost of retrieving the contents of a wreck such as they had described was about $1 million. The Chicks then contacted Shipwrecks @ Key West, who are known for their expertise in identifying ancient vessels throughout the seven seas. The manager of Shipwrecks told them that the likelihood of their correctly identifying El Conde was 80% whereas that of misidentifying any other vessel as El Conde was 10%. The cost of conducting the identification in the high seas was $300,000.
A. Determine Nautical Chicks’ decision strategy. (Four-digit precision recommended.)
B. There are 20 ladies in Nautical Chicks. To raise the money needed to conduct the identification and recover the contents of the wreck, if any, each Chick would have to contribute $65,000. Compare this number to the net expected value given imperfect information accruing to each Chick. Would you invest your money in this venture? If your answer conflicts with the model’s recommendation, explain the discrepancy.
Nuestra Señora de Atocha
History of the Atocha
Treasures of the Atocha
The Story of the Atocha
The Shipwreck of the Atocha
The Atocha and Santa Margarita Story
The Atocha and the Tierra Firme Fleet
10. Chess Strategy
Julia is playing chess with Diego. At this point in the game, they have the same pieces and are basically even in the strength of their deployments. Julia is to move and among her alternatives are two worthy of note. One involves an ordinary exchange of knights with no appreciable advantage ensuing. The other, however, presents a possible strategic opportunity. If she were to move her bishop to square h5, Diego might then position his knight such that on his following move he could check Julia’s king and subsequently capture her rook. However, if Diego pursued this “rook option,” Julia would maneuver her knight so that it would confine (though not check) Diego’s king. Diego could capture Julia’s knight with his, but would thereby forfeit the opportunity of taking her rook. Julia would thus wind up losing a knight (although she’d still retain the offensive). Now, if Diego went ahead and checked Julia’s king, she could move her king to safety, sacrifice her rook, and then checkmate Diego’s king with a bishop-knight pincer attack. Of course, Diego could forgo the “rook option” altogether and attack Julia’s bishop at the outset, in which case Julia would be forced to retreat and hand over the offensive to Diego. Hence, Julia’s strategy will be successful if and only if Diego (a) moves to position his knight for a subsequent check of Julia’s king and (b) goes on to capture her rook, ignoring the threat from her knight. (Clearly, Diego’s prosecution of this “rook option” is tantamount to his unawareness of Julia’s checkmate strategy.)
A. Knowing Diego’s playing habits, Julia believes that the probabilities that he will opt for moves (a) and (b) are 0.7 and 0.6, respectively. Determine Julia’s optimal strategy using relative piece values as the sole criterion. (Tactical considerations aside, relative piece values are 3 points for knights and bishops, and 5 for rooks. Assume that a checkmate is valued at 10.)
B. Determine Julia’s optimal strategy also assuming that handing over the offensive to Diego because of a forced retreat at this point in the game is equivalent to losing a rook.
C. Determine Julia’s optimal strategy assuming now that this is her first game with Diego and that she knows nothing about his playing habits or skills. Assume B above as well.
D. Determine Julia’s optimal strategy assuming she knows Diego is a capable player and therefore the probabilities that he will opt for moves (a) and (b) are at best both 0.2.
E. Based on these analyses, would you say that knowledge of an adversary’s ability and behavioral dispositions is necessary in a player’s decision-making process? Explain why.
F. Having modeled this problem, do you think decision analysis provides a practical way of devising strategies for entire chess games? Explain your reasoning.
