Tutorials
Tutorial 1 -- Textbook 49.2(1);
Answers to 49.1 and 49.2(2)+(3): File
Tutorial 2 -- Lecture 2 Website Problem 3 (b) (Median Voter Theorem);
Answers for 391.3: File
Tutorial 3 -- Textbook 42.1 and 42.2 (b);
Answers for 27.1, 27.2, 31.2, 42.2 (a): File
Tutorial 4 -- 2013S-Q3 and 2014S-Q3;
Answers for 59.2, 392.1: File.
Tutorial 5 -- Past Midterm 2015F-Q1, 2014F-Q2, 2013F-Q3, 2015S-Q3;
Answers for 114.2, 114.3, 114.4: File.
Tutorial 6 -- Textbook 191.1;
Answers for 156.2, 161.1, 163.2, 173.2, 173.3, 173.4: File.
Tutorial 7 -- Textbook 174.2;
Answers for 177.1, 177.2, 183.2, 183.3, 183.4, 192.1: File.
Tutorial 8 -- Textbook 211.1, 431.1, and Past Final 2014S-Q1,
Answers for 211.2, 212.1, 214.1, 234.1: File.
Tutorial 9 -- Lecture 8 Website Problem 2;
Answers for 430.1, 431.2: File.
Tutorial 10 -- Textbook 282.3 and Past Final 2014S-Q1, 2011F-Q3, 2013F-Q3 and 2014S-Q3
Answers for 227.2, 227.3, 282.1: File.
Answer for 296.1: File.
Other
* Course Website: Link
* Facebook Group: Link
* Piazza: Link
* Email: yiyang.wu@mail.utoronto.ca
* Past Pages: Fall 2013; Winter 2014; Fall 2014; Winter 2015; Fall 2015.
* Lecture: Mondays 6:10 - 8:10, MP 137
* Tutorial: Mondays 8:15 - 9:00, MP 137
* Office Hour: Wednesdays 3:00 - 5:00, IN 313
* Midterm Extra Office Hour: 2:00 - 4:00, SS 2128
* Final Extra Tutorial / Office Hour: TBA
Questions from ChatBox
Questions from Emails and Office Hours
* Question 41 (W2015): Can the two methods to find NE other than SPE be used to find all NE of a game?
* Answer: 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).)
* Question 40 (W2015): For 191.1, why q2 = 14?
* Answer: 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.
* Question 39 (W2015): 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.
* Answer: C needs to dominate both A and B in order to be dominant. It means C dominates A but not B.
* Question 38 (W2015): 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).
* Answer: 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.
* Question 37 (W2015): 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.
* Answer: 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.
* Question 36 (W2015): [For Voting Games], I am confused as to how you decide on an action to test.
* Answer: 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.
* Question 35 (W2015): When finding NE, can we eliminate weakly dominated strategies?
* Answer: NO. Only strictly dominated actions.
* Question 34 (W2015): What are examples of NE for the voting game 391.3?
* Answer: (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.
* Question 33 (W2015): What's relationship between the set of NE and the rationalizable set?
* Answer: All NE are rationalizable, not all rationalizable actions are NE. (Only for rationalizable sets obtained by elimination of strictly dominated actions)
* Question 32 (W2015): Will you post videos for all the questions you do in the tutorials?
* Answer: 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
* Question 31 (W2015): Does A not weakly dominated by B imply A not strictly dominated by B?
* Answer: YES. This is just the contrapositive of A s.d. by B implies A w.d. by B.
* Question 30 (W2015): For a two-player game, A is strictly dominated iff A is a never-best-response?
* Answer: YES. Here, s.d. means possibly s.d. by a mixed strategy.
* Question 29 (F2014): "For 2013F, question 1e, wouldn't the answer be (1-delta^3k-1,delta^3k-1)? As player 2 has the last round for this infinite-finite game, the guaranteed payoff will be delta^3k-1 if he refuses all the offer, so as long as player 1 offers that in the first round he will accept. I wonder if my logic makes sense."
* Answer: NO. The one you described is not SPE. It may be a NE, but the question was asking for SPE. You need to guess a pattern and use induction to prove it. Video here: Link.
* Question 28 (F2014): "Professor Peski said in the last lecture that auction will not be on the final, I wonder if that is true. It seems true from the past finals. I just want to confirm."
* Answer: Yes. There will not be direct auction questions on the exam, but auction is just an example of a game with incomplete information, and games with incomplete information will appear on the exam.
* Question 27 (F2014): "Can you explain how to do Q2 2013 by video?"
* Answer: Write the best response functions as usual, and guess the NE for Beth because she has discrete actions; there is no way of guessing Allan's actions, continuous. All possible ones are (0, 0), (0, 1), (1, 0), (1, 1) first number being the action of high type Beth and second number being the action of low type Beth. Then use Allan's best response function to compute his action, and check whether Beth is best responding to that: if both types of Beth are best responding, then that pair is a NE. Video here: Link; Video of 2013Q3 is here: Link.
* Question 26 (F2014): For F2014S-Q1, how to show any w* in [pi/4, 3pi/4] can be supported as an SPE payoff when N = 2?
* Answer: Note that any w in [pi/2, pi] can be supported as SPE payoff when N = 1 from part (a), therefore, start with the Firm proposal w = w*, then Union Accept if w >= w* and Reject when w < w*, then they play the (Lockdown, Strike) NE after all histories, then in the next period it is a N = 1 game again, use the SPE w = 2w* for w* in [pi/4, pi/2] and w = 2w* - pi/2 for w* in [pi/2, 3pi/4] after all histories. Need to check that this is indeed SPE, it is done here in the video: Here.
* Question 25 (F2014): In L9Q2, (OO, SOO) is NOT a BNE when 3/8 < p < 5/8
* Answer: YES. Since He(didn't hear dh type) is NOT best responding to She playing OO. He(dh) gets 0 from S and 3 from O, thus will not play S (even though we assumed it): Video.
* Question 24 (F2014): For 282.3, the objective function for player A is wrong.
* Answer: YES. The first line was correct, but the result is computed incorrectly as I suspected during the tutorial. The correct expression is Here, but in that file, the maximization is done incorrectly through FOC which gives the local min NOt the local max. The method used in the tutorial today was correct, yielding the same answer y* = 0.
* Question 23 (F2014): SPE = equilibrium in any periods?
* Answer: NO. SPE means equilibrium after ALL histories. Use backward induction to prove SPE for finite horizon games and one deviation principle to prove SPE for infinite horizon games. NE means equilibrium in all periods. Only need to show best responses "on equilibrium path" to prove NE in both finite and infinite horizon games.
* Question 22 (F2014, Obsolete): The starred questions that were not done in tutorials:
* Answer: 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
* Question 21 (F2014): For 177.1, is (30, A, 50) really a feasible NE outcome?
* Answer: NO! That was a mistake. (30, A, 50) is a Pareto Optimal outcome that Pareto Dominates the SPE. It is NOT a NE because 50 is not the best response after history (30, A): 35 is the best response, but (30, A, 35) does not Pareto Dominate the SPE. It is possible to prove the that no NE exist that Pareto Dominates the SPE, but there exists Pareto Optimal Nash Equilibrum, one example would be (0, (A after history {w = 0} and R after anything else, 50 after history {w = 0, A}), see video: here.
* Question 20 (F2014): "You discussed Q1 in Fall 2013 and showed that U^AM^1-A, L^BC^B is a nash equilbrium. You did not check the support condition for U^AM^1-A being a best response against L^BC^B. You have to see if U^AM^1-A gives higher payoff than D."
* Answer: D is strictly dominated so it will never be played in any NE (both pure and mixed) anyways.
* Question 19 (F2014): "For Q1 Fall 2013, you wanted to see what values of A guarantee that R is a best response against U^AM^1-A. You checked the support condition by seeing R> L^BC^1-B. That is too complicated. You can compare U2(R,U^AM^1-A)>=U2(L,U^AM^1-A) and U2(R,U^AM^1-A)>=U2(C, U^AM^1-A). You will find that easily that A = 1/2 without doing proof by contradiction."
* Answer: This is TRUE! See Question 16 in this list.
* Question 18 (F2014): "For the food truck problem, for the interior case, can you take the max(the solution, pi-1,pi+1). Or in the fall 2014 last question, can you take the max(p-i+5, .5*(ci+p-i+5) like the professor did."
* Answer: NO. That technique only works for this type of profit function where the interior is concave and boundary is flat of monotonic "towards" the interior. It does NOT work for all profit function structures. (Please don't ask me for the details, use the method I had in tutorial 4 if you are not really clear about how this trick works.)
* Question 17 (F2014): When proving an action is strictly dominated by a mixed strategy, how to find the mixing probabilities?
* Answer: You can figure out a range of probabilities and use any probability in this range as an example: on the exam, as long as you provide the correct range (i.e. prove that such mixing exists), you do NOT need to provide an example. Similarly, if you can guess an example correctly, you do NOT need to compute the range of all feasible mixing. Video explanation of how to find the range: here.
* Question 16 (F2014): For 2013FQ1, how to prove (UM, LC) and (UM, R) are mixed strategy NE?
* Answer (updated): (UM, LCR) is the easy mixed NE, you only need the indifference condition (condition 1). For (UM, LC), you need the support condition (condition 2), i.e. to show that playing a mixed strategy of LC is preferred to playing R. Video explanation here. For (UM, R), you need to show that R is preferred to any mix between L and C (which is a bit harder). Video explanation here.
--> For (UM, R), you can show R is preferred to L and C individually, because if player 2 mix between L and C, the payoffs are always linear in player 2's probabilities, so the optimal mix for player 2 is always either play L with probability 1 or play C with probability 1, i.e. R only needs to be preferred to L and C individually.
* Question 15 (F2014): "For Fall 2011, how do you get pure strategy nash equilibrium in d algebraically?" (Last question)
* Answer: You cannot solve a question with discrete actions algebraically. We just propose a NE and prove that it is. In the traffic game, we propose any NE with 79 or 80 people driving on the direct route, then (1) for the people who are driving on the direct route, switching to the long way will increase the time from less than 1 hour to 1 hour; (2) for the people who are driving on the long way, switching to the direct route will increase the the time for 1 hour to more than 1 hour. Therefore, no one wants to switch and it is a NE.
* Question 14 (F2014): "I am confused about the answer for Q1 a) in 2013F. What I get is a range for Alfa (2/3, 3/4), and I do not know how can we narrow it down to Alfa=0.7? I also tried other numbers in that range, and I found out D is still dominated, instead of only when Alfa=0.7"
* Answer: You are right, 0.7 is just an example. On the exam, as long as you have an example, OR prove that an example exists, then it's ok. You never have to find all the mixing that work to dominate another action.
* Question 13 (F2014): How to model the swimming with sharks problem (Q114.4)?
* Answer: The actions on the second day is fixed so you do not need them as a part of the game. Then each player will only choose to Swim or Not swim on the first day. The payoffs are given in the question. Video explanation here.
* Question 12 (F2014): Will there be voting problems on the midterm?
* Answer: I can't answer that: but there will not be questions like 49.2 where the voters can vote for multiple candidates, which requires you to use set operations to find the winning candidates (as done in tutorial 1).
* Question 11 (F2014): "The player has three actions U, M, and D while the other player's actions are L and R. He wanted to show D is strictly dominated. He considered the mixed strategy U^alpha M^1-alpha instead of U^alpha1 M^alpha2 D^alpha3 when getting inequalities for alpha."
* Answer: NO. If you want to show that D is strictly dominated, you should NOT include D in the mix, you want a mix between only L and C to strictly dominate D. Including D will not change the result whether D is dominated or not, but it is not necessary to do that at all.
* Question 10 (F2014): In Bertrand duopoly, how to write the best response function?
* Answer: Think about what is the optimal pricing given the other firm is pricing at (1) below cost ==> anything strictly larger than the opponent's price, (2) equal to cost ==> anything larger or equal to cost, (3) between cost and monopoly price ==> a tiny bit smaller than the opponent's price, and (4) above monopoly price ==> set price at the monopoly price.
* Question 9 (F2014): What's the difference between the best response function notation b_i(a_-i) and B_i(a_-i)?
* Answer: b_i is a function: it maps an element in A_-i to an element in A_i ; B_i is a correspondence: it maps an element in A_-i to a set in A_i.
* Question 8 (F2014): How to prove the quantity produced by each firm should be equal in Cournot duopoly?
* Use the best response: q_i = (α - c - Q_-i) / 2, and q_j = α - c - Q_-j. Subtracting these two equations will give q_i - q_j = (Q_-i - Q_j) / 2 = - (q_i - q_j) / 2, which means q_i - q_j = 0, for any i, j, i.e. all the q's are equal.
* Question 7 (F2014): For the 2/3 of the average problem, are {0, 0, ..., 0} and {1, 1, ..., 1} the only Nash Equilibria?
* Answer: If the number of players is divisible by 4, then NO! There is a NE where 1/4 of the players choose 0 and 3/4 of the players choose 1. In this case, 2/3 of the average is 1/2 which is equidistant from 0 and 1, so no one will have incentive to deviate.
* Question 6 (F2014): For 391.3, should the rationalizable actions be "voting for the favorite candidate"?
* Answer (updated): NO! The correct rationalizable actions are A(2)_ABC = A(2)_ACB = A(2)_CAB = {A} and A(2)_BAC = A(2)_BCA = A(2)_CBA = {B}, as stated in the tutorial. Try to construct the table for a player with preference C > B > A, if she knows that C will never win (because more than 2/3 of the voters will never vote for C), then voting for B is actually better than voting for C. So "voting for the favorite candidate" is not always optimal.
* Question 5 (F2014): For 49.2, how to show S is weakly dominated by S\{c_k}?
* Answer: Here is a video (also for people who are not familiar with the set notation used) explaining this: link.
* Question 4 (F2014): For 2/3 of the average game (discrete), why is the action {1} rationalizable?
* Answer: After everything larger than 1 is eliminated, {0, 1} is the rationalizable set of actions for every player. If everyone else chooses {1}, 2/3 of the average will be 0.66666 if I choose 1; and "0.66666 - a bit" if I choose 0. The distance between 0.66666 and 1 is smaller than the distance between "0.66666 - a bit" and 0, therefore I am better off choosing 1, i.e. {1} is not dominated and cannot be eliminated.
* Question 3 (F2014): "Can we say prisoner's dilemma is a situation where if both agree to maximize their collective payoffs leads to a better outcome for both players than if they decided to follow their own dominant strategies?"
* Answer: YES. PD games is any two-player two-action game with a (strict) dominant strategy equilibrium which is not Pareto optimal (i.e. players can jointly deviate to both get higher payoffs).
* Question 2 (F2014): Can I take photos of the blackboard during the tutorials instead of copying the notes down?
* Answer: YES, as long as I do NOT appear in your photos.
* Question 1 (F2014): "I was wondering if you could move the T.A tutorial on another day. The reason is have I class from 2 to 4 on Monday. I get really tired after 2 hours. I think this applies to other students." ...
* Answer: NO. (But I do agree with the part that people get really tired (and hungry) after 2 hours of game theory on a Monday morning.)
Last Updated: February 28, 2019 at 3:43 PM
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