1. Draw the causal network for Newcomb's Problem when player 1 cannot perfectly detect player 2's causal history. The probabilities of player 1 rightly or wrongly detecting whether history H(1B) instead of history H(2B) will influence player 2's later decision are respectively denoted r and w. Recall that L denotes the smaller amount of money always in the clear box, and M denotes the larger amount of money that player 1 might put inside the opaque box. What are the expected payoff formulas for player 2 conditional on actually opening only the opaque box, or conditional on actually opening both boxes? - where these formulas depend on player 1' 2 2 s (r, w) signal probabilities; denoted E (1B*r, w) and E (2B*r, w). So in other words, use the causal network for Newcomb's Problem with imperfect forecasting, to 2 2 solve for the formulas for E (1B*r, w) and E (2B*r, w). Notice that a 2B decision guarantees at least the clear money, L. But a 1B decision may result in less or more money than L: - either zero money or the larger amount of money M. So the 1B decision is inherently "riskier" than the 2B decision. Also remember that most people are "risk averse" , meaning they prefer a guaranteed return to a variable ("riskier") return that on average equals the guaranteed return. So they won't voluntarily prefer a riskier return unless its average value exceeds some positive multiple of a guaranteed return. For example, a risk averse person might say, "I won't choose a riskier return over a guaranteed return unless its average value is more than double the guaranteed return. So define to be the "risk aversion multiple". 2. Notice 2 2 that E (1B*r, w) - E (2B*r, w) represents the average benefit from a 1B versus 2B decision. Then let represent the "risk aversion multiple" defined above, meaning the number of times this average benefit must exceed the guaranteed money in order to motivate a risk averse person to prefer the riskier 1B decision. This corresponds to the following inequality: E2 2 (1B*r, w) - E (2B*r, w) $ L (1) Then 2 2 use inequality (1), and the formulas for E (1B*r, w) and E (2B*r, w) derived in Question 1, to solve for the formula that equals the smallest value of M (denoted M ) required in order for min the expected payoff from opening only the opaque box to exceed that from opening both boxes by a multiple of as least times L. What is the resulting formula for M . min 3. Finally, suppose (L, ) = (100, 5), (r, w) = (.65, .45) and use the formula for M to calculate the min numerical value of M for this case.