Student Q&A for Homework 2
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| Question No. 1 |
| Question: I must have set up the extensive form for question 1 incorrectly (but I can't find my error), because it appears that for the little monkey, subsequent to the big Monkey's failure to get the fruit by Banging the Tree, Climbing strictly dominates Banging the Tree. But then, part c) and d) don't make much sense, particularly part d) since there is no answer. |
| Abswer: An updated version has been posted. I changed the cost of C for little monkey (m) from 35 to 50. That should make things work. |
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| Question No. 2 |
| Question: In question 3, I think both Alice and Bob should not trust either prediction agency, since these agencies predict upward-sloping market demand curves! |
| Abswer: You are absolutely right. Change that to 1 - (q_a + q_b) and 1 - 2(q_a + q_b), respectively. |
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| Question No. 3 |
| Question: For Problem 4(a). There are three routes from X1 to X5. So by the hint, I can assume some of the three routes have non-zero traffic and solve for the amount of traffic by equating the costs on these routes. However, by doing this, I get a couple different solutions. How can I find NE among them? (I guess I'm confused about how to reason whether some agents would deviate or not.) |
| Abswer: You are asked to find one Nash equilibrium. There are three routes and there is a NE where all three get non-zero traffic. You can solve for this if you want. |
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| Question No. 4 |
| Question: In question 3, the problem of computing the Bayes-Nash equilibrium is effectively a constrained, non-linear optimization problem if we were to consider the requirements that the price predictions by both agencies as well as the production quantities of both producers are non-negative. The above problem yields a trivial unconstrained optimum for only a small range of α. Are we required to compute the bayes-nash equilibrium across the entire range of α Є [0,1]? |
| Abswer: I'm not sure it's really that hard. Can you send me an e-mail or come by my office hours to explain? |
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| Question No. 5 |
| Question: Is there a systematic way to find the optimal network flow in a routing game? |
| Abswer: I don't understand your use of the word "systematic." You have to find the "split" that minimizes cost. In node X_1 send "a" to X_3 and "1-a" to X_2, and in X_3 send "b" to X_5 and "1-b" to X_4. Then compute the total cost as a function of a and b, and find the values that minimize total cost. |
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| Question No. 6 |
| Question: Is there a mistake in question 8? I think I'm analyzing it correctly, but I am getting that the equivalent discount rate is (1-\sigma)\rho, not (\rho+\sigma). Would you mind taking another look at the problem? Thanks. |
| Abswer: I think the question is ok. Maybe you're confusing the "discount rate" with the "discount factor," which is (1/(1+discount_rate)). |
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| Question No. 7 |
| Question: TIP FOR QUESTION 8 (posted by Dimitrios) |
| Abswer: Suppose the agent is at round i. Then his net present value V, starting from the next round is (\pi + (1-\sigma)V)/(1 + \rho), in the case of dying. This is because V is time-invariant, since all rounds give the same payoff \pi, so we can thus define it recursively. Do this similarly for the non-dying case.
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| Question No. 8 |
| Question: Is problem 2 really an infinitely repeated game? My interpretation of the problem is that the moment someone accepts an offer, the game ends, and there are no more repetitions. |
| Abswer: True, the game ends once an offer is accepted, but the game can go on indefinitely if the offers keep getting rejected. |
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| Question No. 9 |
| Question: For problem 1, when a monkey climbs the tree and gets 90% of the fruit, does the other monkey get the remaining 10%, or does it go to waste? It's not clear to me from the problem description. |
| Abswer: It gets the remaining 10%. |
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| Question No. 10 |
| Question: In question 3, can the price ever be negative? I'm getting that if Alice trusts agency Y, then her best response to Bob is to produce at a level (based on the amount q_b produced by Bob) such that the total production q_a+q_b is greater than 1/2. Then agency X would predict that the price is negative in this situation. Am I doing something wrong, should negative prices be allowed, or should we assume that if the predicted price is negative then the agency predicts a price of 0 instead? |
| Abswer: Assume that prices are non-negative. (My solutions, however, do not suffer from this problem...) |
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| Question No. 11 |
| Question: For question 10, I thought in chapter 7, the theories of learning in multiagent systems are divided into two categories: descriptive theories and prescriptive theories. Are these the "learning agendas" this question is referring to? But there are only two? |
| Abswer: Well the term "agenda" stands for something broader. For example there used to be (and still are) computational goals in the learning literature. |
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| Question No. 12 |
| Question: Just to check, on page 214, the equation for the change in the fraction of agents playing a at time t is incorrect right? |
| Abswer: I don't have the textbook with me, but the change in the fraction of type-A species (x_A') should be equal to x_A * (payoff(A) - avg_payoff), where payoff(A) is the average payoff of A-types in the current population state. |
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| Question No. 13 |
| Question: In part (c) of question 6, are "stable points" what the textbook calls steady states (page 214)? |
| Abswer: I don't have the textbook with me, but I think yes. We mean the fractions that will not change under the replicator dynamics. |
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| Question No. 14 |
| Question: In the hint for problem 8, I think the confusion is over the meaning of "die". If an agent can still receive the payoff \pi during a round that he dies in, then the recursion is ok. If we say that the after the agent dies then he can't receive any payment, then the formula in the problem is wrong. |
| Abswer: Exactly. There is also another assumption that the payoff of each round is received after the round and thus discounted. |
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| Question No. 15 |
| Question: "The change in the fraction of type-A species (x_A') should be equal to x_A * (payoff(A) - avg_payoff), where payoff(A) is the average payoff of A-types in the current population state."
In the given problem, doesn't this imply the change in fraction for species A is (80%)(6.6-5.8) = 64%? So then the new fraction for species A is 80%+64% > 1? Am I confusing something here? |
| Abswer: The "change" as I put it is the derivative, not necessarily the true value. Assume continuous time. Sorry if that was not clear. |
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| Question No. 16 |
| Question: In the definition of relevance graph in problem 5, should it be "there is an edge from D_2 to D_1 if and only if D_2 is s-reachable from D_1" (instead of D_1 is s-reachable from D_2)? |
| Abswer: You are correct. I apologize for the typo. The qualitative results do not change though (that is, the relevance graph under the wrong definition will be exactly the "true" relevance graph with the directionality of all edges inverted---cycles that exist in one graph will exist in the other as well). Again, I'm sorry if this caused confusion. |
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| Question No. 17 |
| Question: CORRECTION FOR QUESTION 3 of Q&A |
| Abswer: There is no Nash equilibrium with all three routes receiving non-zero traffic. There is one with 2 routes receiving non-zero traffic. Thanks to Sinbae for pointing this out. |
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| Question No. 18 |
| Question: CORRECTION FOR QUESTION 10 of Q&A |
| Abswer: In some cases prices are negative. You may take the assumption that this is OK, i.e., maximize the profit function, not max{0,profit}. This makes the problem simpler. |
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| Question No. 19 |
| Question: For 1e, are we simply supposed to draw the sequence form game table and fill in the entries, or are we supposed to do something with sequence form (like find a realization plan)? |
| Abswer: No just fill in the table. |
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| Question No. 20 |
| Question: I'm not sure how to do apply sequence form to a game with nature nodes for 1e. I don't see any notes on this in the book. Can you give us any tips in this regard? |
| Abswer: I have sent an e-mail with a suggestion: Just plug in numbers for all pairs of sequences that have well-defined expected utilities for the two agents. Leave the rest zeros. |
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| Question No. 21 |
| Question: whats the biggest state in the us |
| Provide answer |
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