# Questions from Emails
(+ some questions that I imagine you want to ask)
(Top) Last term: The starred questions that were not done in tutorials:
A: The following are very brief answers to some of them:
➭ - 156.2 + 173.2
(a) (C, F after C, G after D)
(c) (Karl:E, Rosa:B after R, B after (E, B), H after (E, H), Ernesto:H after E, B after (R, B), H after (R, H))
(173.2) NE are (C, EG) (D, EG) (C, FH) (D, FH) (C, EH) (D, FG)
➭ - 161.1 (Not an interesting question)
➭ - 173.4 With bridge (A, R) Without bridge (N, F)
➭ - 177.2 max c(a) + (p(a) - c(a)) / 2 --> FOC: p'(a) + c'(a) = 0, which is not equal to max c(a) where FOC is c'(a) = 0
➭ - 183.2 either ($0, (always A)) or ($0.01, (R after history 0, A otherwise))
➭ - 183.3 (0) for dictator game; (0, always A) or (0, (R after history 0, A otherwise))
➭ - 183.4 ((1/2 , 1/2), (A after histories (1 - x, x) such that x - beta_2 | 1 - 2x | >= 0, R otherwise) [action of player 1 is given in the form (x1, x2)]
➭ - 192.1 (a) (b) ...
(c) [Everything expressed in cents] Firm 1: p_1 = c or p_1 = c + 1
Fimr 2: p_2 = any integer in [c, infinity) after p_1 < c; any integer after p_1 = c; c + 1 after c + 1; p_1 - 1 after p_1 > c + 1; p_m after p_1 > p_m (where p_m is the monopoly price).
➭ - 211.2 (at least two buyers bid v [everyone else bid < v], seller always accept price = max of the bids)
➭ - 212.1 (???)
➭ - 214.1 (???)
➭ - 234.1 SPE = (always S, always S); NE = ((S in first period, S or C afterwards), (S in first period, S or C afterwards))
➭ - 430.1 K-period punishment is NE if x / (1 - delta) >= (y - 1) / (1 - delta ^ (k + 1)) + 1 / (1 - delta)
➭ - 431.2
(a) Need x / (1 - delta) > (1 + delta)(y - 1) + 1 / (1 - delta)
(b) Same as (a)
(c) Need x / (1 - delta) > (y + delta) / (1 - delta)^2
➭ - 227.2 If L > H (1 - p), then (demand L, always Accept); if L <= H (1 - p), then (demand H, (Accept after (H, H), Reject after (H, L))). History is specified as (demand of union, move of chance).
➭ - 227.3 Use one deviation principle
(12) Office Hour: Can the two methods to find NE other than SPE be used to find all NE of a game?
A: No. You could apply two methods to the NE you found to find more NE but that still will not cover all possible NE. (The two methods are: (1) Deviation off equilibrium path that will not change the equilibrium path (i.e. second player do not need to behave optimally in situations that will never be reached in equilibrium anyways); (2) Deviation on equilibrium path that will change the equilibrium path (i.e. non-credible threat: second player can lie about what they will do in the original equilibrium action so that the first player will switch to another action).)
(11) Office Hour: For 191.1, why q2 = 14?
A: There is nothing special about the number 14: anything larger than or equal to 12 (alpha) works safely. There are numbers less 12 that could work there too, but it does not work for all histories, so anything larger than 12 is better.
(10) Email: Lecture 1 Problem Set - Q4 Voting for the best Alternative
Parts 2 and 3 confused me because in Part 2 we showed that voting for C is not weakly dominant, but then in part 3 it says that voting for A is weakly dominated by voting for C. I'm not sure how this is logically possible.
A: C needs to dominate both A and B in order to be dominant. It means C dominates A but not B.
(9) Email: For 49.2, I understand what you wrote for the first part, but then you began formally working in terms of k, with k1,k2,k3 be a partition of k. I was just wondering if this is necessary, or if during the exam I can just show the first part (where you worked in terms of ci).
A: To get full mark, it is necessary to give formal proofs. The way I solved it first is not very precise: it is clear how the comparison works in my cases.
(8) Email: For Q391.3 from the textbook, I am confused about how you proved that B is w.d. by A. It seems equally likely that A and B win based on what you wrote on the board in class.
A: In that question, we know that C is least preferred by more than 2/3 of the voters, so C will never win. Then the number of cases that are left is smaller than usual, allowing B to be dominated by A.
(7) Email: [For Voting Games], I am confused as to how you decide on an action to test.
A: You need to guess which actions are dominated by which. For voting games, usually the votes that help the most preferred candidate to win dominates the votes that help the least preferred.
(6) Office Hour: When finding NE, can we eliminate weakly dominated strategies?
A: NO. Only strictly dominated actions.
(5) Office Hour: What are examples of NE for the voting game 391.3?
A: (i) Everyone voting for his favorite candidate. (ii) Everyone voting for the same candidate (any candidate is ok). There are other ones too, guess the equilibrium and check that no voter has incentive to deviate and vote for another candidate than who he is supposed to vote for in the equilibrium.
(4) Office Hour: What's relationship between the set of NE and the rationalizable set?
A: All NE are rationalizable, not all rationalizable actions are NE. (Only for rationalizable sets obtained by elimination of strictly dominated actions)
(3) Tutorial: Will you post videos for all the questions you do in the tutorials?
A: NO! I will not post videos for anything that I did in the tutorial. I use the youtube channel to answer questions in emails that are hard to answer with writing.
List of the relavent ones for the midterm:
Part of 49.2 I didn't do in tutorial:
Link.
Midterm 2013 Fall Q1 on solving for mixed strategy NE:
Link and
Link.
Midterm 2014 Winter Q1 on showing an actions is strictly dominated by a mixed strategy:
Link
(2) Office Hour: Does A not weakly dominated by B imply A not strictly dominated by B?
A: YES. This is just the contrapositive of A s.d. by B implies A w.d. by B.
(1) Office Hour: For a two-player game, A is strictly dominated iff A is a never-best-response?
A: YES. Here, s.d. means possibly s.d. by a mixed strategy.
Last Year:
Link