1.Two members of a criminal-gang are arrested and imprisoned. The prosecutors lack sufficient evidence to convict the pair on the principal charge. However, the prosecutors offer each prisoner a bargain simultaneously. Each prisoner is given the opportunity either to: betray the other by testifying that the other committed the crime, or to remain silent. Let us call them Prisoner A and Prisoner B, and let us analyze how Prisoner A should make a decision. Prisoner A has two alternatives as mentioned above. What he does not know is Prisoner B’s action. But he knows that 1) if they both remain silent, then they are both sentenced to a year in prison on a lesser charge, 2) if he betrays and B remains silent, then he goes free while B will be sentenced to three years in prison, 3) if he remains silent and B betrays, then he will be sentenced to three years in prison while B goes free, and 4) if they both betray the other, then they are both sentenced to two years in prison. Now, remember that they are in solitary confinement with no means of communicating with the other. Furthermore, they are both self-interested and have no loyalty to each other whatsoever. Determine the appropriate decision under uncertainty for Prisoner A using the following decision rules, respectively: a. Maximax b. Maximin c. Equally likely Hint: The events (states of nature) to A are B’s possible actions.
2. Toy Story: You are the owner of a toy stand in a mall and you need to decide how many units of a particular toy you will order to sell at your stand. You know that the demand will either be 20, 40, 60, 80 or 100 with equal likelihood. The unit cost of the toy is $4 and every unit you sell earns revenue of $10 per unit. Leftover units cannot be salvaged or re-sold and are worthless. You have only one opportunity to buy in anticipation of demand. Because the toys are shipped in batches, you must order in units of 20. a. For each combination of your choice (order quantity) and demand realization, find your profit. Construct a decision table with the profits you find. b. How many toys should you order to achieve highest expected profit? Hint: you only need to order one of the demand realizations. Hence, both the alternatives and the events are the same, namely the five possible demands. The outcome of any combination of your order quantity and the demand is your resulting profit.
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