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beard strategy, Game Theory, Game Theory

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#questi1 A, Explain how a person can be free to choose but his or her choices are casually determined by past event

2 B , Draw the casual tree for newcomb''s problem when Eve can''t perfectly detect Adam''s casual history. The probabilities of Eve rightly or wrongly detecting whether adam will later open only the black box instead of opening both boxes are respectively denoted r and w. recal that L denotes the smaller amount of money always in the clear box and M denotes the larger amount of money that eve might might put in side the opaque box E A

C, Derive the two expected payoffs formulas E A (1B / r, w) and E A ( (2B /r,w) and use them to solve for another formula that equals the smallest value of M (denoted M*) required in order for Adam''s expected payoff from opening only the opaque box to exceed that from opening both boxes by a multiple of as least ( a sign that looks like derivative) L what is the resulting formula for M*. finally suppose (L, sign that looks like derivative I don''t know ) = (300, 95), (r,w)=(.58, .43) and use the formula for M* to calculate the numerical value of M* for this case

2. A, Suppose a CD player player tries to detect whether its partner is C player instead of a DD player by looking for external signals that are at least as typical for DD players than DD players than for cd players draw a diagram tp explain how two boundariesb.L and bu are optimally determined by the minimum likehood ration Lmin. Show on the diagram where it is optimal to respond C versus D. Also explain what happens to the boundries when detection becomes more cautious by raising the minimum likehood ration

b. What is meaning of the LDD detection strategy

c. What is the main problem with the green -beard strategy? Explain how the LDD strategy overcomes this problem

3. A. If CD players are able to use the LDD strategy better than pure chance then explain what happens to the signal reliability ration as a CD player detects more cautiously

b. Assume a population contains either CD ot DD players where each player is randomly matched with partner taken from the whole population. Also assume the fear and greed payoff differences are equal. What are the expected payoff formulas for CD players [ denoted E(DD/x CD ) ] depending on the fraction of CD players in the population, denoted x CD \

c. Use expected payoff formulas of part C to algebraically derive an inequality for the signal reliability ration r/w that determines when the CD players will outperform the DD players. Thenuse this inequality with Part A, to explain how CD players can always outperform DD players starting from any positive initial fraction of CD players x CD > 0.

4, A. Use the inequality derived for part C question 3; to obtain an inequality required x *CD = 1 to remain stable against DD invaders. Also draw the ROC diagram discussed in class for visually representing this stability inequality

B. Explain how a diagram similar to that shown in part A can be used to derive a prediction of what will happen to the CD players equilibrium probability of cooperating if the fear and greed pay off difference decrease relative to the cooperation payoff difference

C. Again explain how a diagram similar to that shown in Part A can be used to derive a prediction of what will happen to the CD player equilibrium probability of cooperating if they exchange email messages instead of talking talk face to faceon..

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