3. Decision Theory Example. A decision maker (named D) attends a fairground. There is a game there where you select one of two coins one is double headed and the other has one head and one tail. The stall owner (named S)charges D 5 to play the game, and then lets D pick one of the coins (by pointing to them from a distance so D cant see which coin is which.) S then tosses the chosen coin, and if it comes up Tails, D wins 10. If the coin comes up Heads, D gets a consolation prize of 2. In either case, S then gives D the option of double or quits with the coin just tossed. Thus D can get 4 if it comes up Heads again for a second time in a row, or 20 if it comes up a second Tails in a row. If it doesnt come up the same then D gets nothing. D, who studied a course in Management Science, thinks this is an excellent deal. The reasoning is thus: either you get Tails (with a good payoff) or else you get Heads in which case you will go double or quits (since its likely a double Headed coin). Either way you win a minimum of 4, and thus the true stake is only 1 (5-4). And with the chance to win 10 on the toss of a coin, D highlights this is definitely a positive expected value game and thus suggests its worth a play. Ds friend, C, has always been more considered, and studied Mathematics instead, taking Decision Theory as an option. C does a quick calculation, and thinks it isnt so clear, and maybe, in fact, the game is just about fair and so the stall owner is not giving away anywhere near as much as D thinks. At the end of the evening, C notes that S has indeed made somewhat of a profit.

a. For this situation, describe the errors, if any, that D has made.

b. From the perspective of Cs logic, provide a decision tree encapsulating stochastic and deterministic nodes, and outline the optimal strategies that should be adopted. In particular, calculate the expected value of the game.