COS 445 Final Due online Monday, May 21st at 11:59 pm All problems on this final are no collaboration problems. You may not discuss any aspect of any problems with anyone except for the course staff. You may not consult any external resources, the Internet, etc. You may consult the course lecture notes on Piazza, any of the five course readings, past Piazza discussion, or any notes linked on the course webpage (e.g. the cheatsheet, or notes on linear programming). You may discuss the test with the course staff, but we will only answer questions on clarification and will not give any guidance or hints. You should feel free to ask any questions and let us judge whether or not to answer, but just know that we may choose to politely decline to answer. Please upload each problem as a separate file via MTA. You may not use late days on the final. You must upload your solution by May 21st at 11:59pm. If you are working down to the wire, upload your partial progress by in advance. There is no grace period for the final. In case of a true emergency where you cannot upload, email me (smweinberg@princeton.edu) your solutions asap and contact your dean. There are no exceptions, extensions, etc. to the final policy without dean involvement. Hint for the entire exam: Each of problems 2-4 will be easier if you can draw inspiration from a past PSet, exam, or lecture. But each of them will also clearly require substantial insight beyond what you can get just by understanding a related PSet/exam/lecture. Problem 1: COS 445 Speedrun (60 points) For each of the 12 problems below, you do not need to show any work and can just state the answer. However, if you simply state an incorrect answer with no justification, you will get no credit. You are encouraged to provide a very brief justification in order to receive partial credit in the event of a tiny mistake. Part a: Stable Matchings (5 points) Four students Alice, Bob, Claire and David are applying to summer internships at Apple, Bell Labs, Capital One and Dell (all of which need exactly one intern). Here are their preferences, sorted from favorite to least favorite: Alice: Apple, Dell, Bell, Capital One. 1
Bob: Apple, Bell Labs, Dell, Capital One. Claire: Capital One, Dell, Bell, Apple. David: Bell, Dell, Apple, Capital One. and the companies preferences: Apple: Alice, Claire, Bob, David. Bell Labs: Bob, Alice, Claire, David. Capital One: Alice, David, Claire, Bob. Dell: Alice, Bob, Claire, David. Find the stable matchings that arise from running the deferred acceptance algorithm when the students propose and when the companies propose. Your answer does not need to show work: presenting the correct matchings suffices. Part b: Voting Rules (5 points) The town of Princeton is holding an election between candidates Alice, Bob and Carol. 9/20 of the voters prefer A B C, 3/10 prefer C B A and 1/4 prefer B C A. State the candidate selected by each of the following voting rules: Borda, IRV, Plurality. Part c: Game Theory (5 points) Find a Nash equilibrium of the following game and state the expected payoff for both players. The first number in each box denotes the payoff to the row player, and the second number is the payoff to the column player. You do not need to show work, stating the answer suffices. A B C (4,3) (0,4) D (-2,1) (2,0) Part d: Extensive Form Games (5 points) Find a sub-game perfect Nash equilibrium of the following game. The number on the a non-leaf node represents the player whose turn it is, the label on each edge denotes the name of the action they may take at that node and the vector on any leaf node represents the pay off for player 1 and player 2, respectively. You do not need to show work, stating the answer suffices. Part e: Linear Programming (5 points) Write the dual of the following LP. You do not need to solve it. Maximize 2x + 3y, such that: x + y 10. 4x + y 2. x + 3y 5. x, y 0. 2
Part f: Scoring Rules (5 points) Suppose you are asked to predict tomorrow s weather. There s four possible outcomes: it will be sunny, rainy, cloudy or snowy. You will be paid according to the logarithmic scoring rule (S( x, i) = log 2 (x i )). Suppose you saw a cloud in the sky, and think that it will be cloudy with probability 1/2, rainy with probability 1/4, sunny with probability 1/8 and snowy with probability 1/8. Given your beliefs about the weather, what distribution should you report to maximize your expected payoff, and what is your resulting expected payoff? You do not need to show work, stating the answer suffices. Part g: Welfare-maximizing Auctions (5 points) Recall that in a general welfare-maximization setting, there is an arbitrary set A of possible outcomes, and each bidder i has some valuation function v i : A R, and the welfare of an outcome x is i v i(x). A welfare-maximization algorithm is subset-optimal if there exists some set A A such that on input v 1,..., v n, the algorithm selects the outcome arg max x A { i v i(x)}. Prove that, for all subset-optimal algorithms, there exists a payment scheme such that the resulting mechanism is dominant-strategy truthful. Formally, for all i, design a payment rule p i ( ) which takes as input n valuation functions and outputs a real number, such that the mechanism that selects outcome arg max x A { i v i(x)} and charges bidder i price p i (v 1,..., v n ) is dominant strategy truthful. You do not to prove that your mechanism is dominant strategy truthful, stating the correct payment scheme suffices. Hint: Recall the Vickrey-Clarke-Groves mechanism for inspiration. Part h: Revenue-maximizing Auctions (5 points) Suppose Alice is trying to buy a pen. Her value for the pen is drawn from the uniform distribution on [2, 4]. What is the optimal auction a seller should use to sell Alice the pen? You do not need to show work, stating the answer suffices. 3
Part i: Price of Anarchy (5 points) Consider the following network. There are two nodes, s and t, and one unit of flow traveling from s to t. There are two directed edges from s to t, one with cost c(x) = 2 and the other with cost c(x) = 1 + x 2. Compute the Price of Anarchy of this graph. You do not need to show work, stating the answer suffices. Part j: Cake cutting (5 points) Say there is a single cake, the unit interval [0, 1]. Come up with valuation functions for Alice, Bob, and Charlie such that the allocation that awards Alice the interval [0, 1/2], Bob the interval [1/2, 6/10], and Charlie the interval [6/10, 1] is proportional and equitable, but not envy-free (note that there are many possible answers). You do not need to show work, stating an answer suffices. Part k: Behavioral Economics (5 points) Recall the Von Neumann-Morgenstern axioms from Lecture 23. Let there be two possible deterministic outcomes, A and B. Define a preference ordering over the events {.5A+.5B,.75A+.25B, A} that is transitive, but violates independence of irrelevant alternatives. You do not need to show work, stating an answer suffices. Part l: Time-Inconsistent Planning (5 points) In the planning graph below, what path would be taken by a naive planner with present bias b = 2? What about a sophisticated planner with present bias b = 2? What is the shortest path from s to t? Your answer does not need to show work: presenting the correct paths for each case suffices. Problem 2: Stability, Unanimity, Non-Dicatorship Don t Mix (25 points) Recall from Midterm 1 that you were asked to provide a voting rule that was stable and unanimous and not a dictatorship (and then some) for m 3 candidates. Recall also that this is impossible. Here, we ll guide you through part of the proof. 4
Recall that two preferences, are a-stable if for all b, a b a b. A voting rule is stable if whenever f( 1,..., n ) = a, and i, i are a-stable for all i, then f( 1,..., n) = a as well. For all parts of this problem there are m 3 candidates, but only n = 2 voters. Part a (5 points) Let f be unanimous, and stable. Consider the votes where a 1 b 1... (a is the favorite, b is the second-favorite), and b 2 a 2... (b is the favorite, a is the second-favorite). Prove that f( 1, 2 ) {a, b}. Part b (5 points) Let f be unanimous, and stable. Let 1, 2 be defined as in part a. Prove that if f( 1, 2 ) = a, then f( 1, 2) = a, for any 2 satisfying the following condition: For all c a, d a, c 2 d c 2 d. Note that this condition is equivalent to saying that 2 can be obtained from 2 by only moving a, while keeping all other candidates in place. Part c (5 points) Let f be unanimous, and stable. Let 1, 2 be defined as in part a. Prove that if f( 1, 2 ) = a, then f( 1, 2) = a for all 2. Conclude further that for all 2, f( 1, 2) = a whenever a is ranked first by 1. Part d (5 points) Say that voter i is a dictator under f for candidate j if whenever voter i ranks j first in their ordering, f selects candidate j. Prove that no voting rule f has the property that voter 1 is a dictator for candidate a and voter 2 is a dictator for candidate b a. Part e (5 points) Let f be unanimous, and stable. Prove that f is a dictatorship (recall that n = 2 and m 3). 5
Problem 3: The all-pay auction (20 points) The all-pay auction has the following format: each bidder i submits a sealed bid b i. The highest bidder wins the item, and every bidder pays their bid. Hint: You may use the following facts without proof in your solution. These, and related facts, can be immediately derived from the CDF of the uniform distribution on [0, 1], but they are given as hints to save you time. The probability that a single draw from U[0, 1] is less than x [0, 1] is exactly x. The probability that n iid draws from U[0, 1] are all less than x [0, 1] is exactly x n. The expected value of a single draw from U[0, 1] is 1/2. The expected value of the maximum of n draws from U[0, 1] is Part a (10 points) n. n+1 Find a Bayes-Nash equilibrium of the all-pay auction when there are two bidders whose values are drawn independently from the uniform distribution on [0, 1]. Prove that it is a Bayes-Nash Equilibrium. Part b (10 points) Find a Bayes-Nash equilibrium of the all-pay auction when there are n bidders whose values are drawn independently from the uniform distribution on [0, 1]. Prove that it is a Bayes-Nash Equilibrium. Problem 4: Fair phone division (35 points) You and your n 1 best friends are deciding whether to buy a bundle deal on n new phones. In each of the following problems, there is a single bundle of n phones, and the total cost is B. Friend i has value v ij for phone j. Each valuation satisfies j v ij B for all i. Friend i gets payoff v ij p i if they receive phone j and pay p i. (This is true whether p i is positive or negative). Your job is to convince your friends to purchase the bundle by designing an envy-free allocation of phones as well as how much each friend will pay. That is: You must, taking as input v ij for all i, j, propose which friend will receive which phone. Your procedure need not be strategyproof. You must decide how much each friend will pay, and this sum must exactly cover the cost of the bundle: i p i = B. Note that you are allowed to have p i < 0 for some i if you want: (think of this as one friend paying another to be OK with a junk phone). Your procedure need not be strategyproof. Your final allocation/prices must be envy-free. That is, for all i, friend i must prefer their phone at the price they paid to any other phone at the price paid by its new owner. To be extra formal, if i(j) denotes the friend who received phone j, and j(i) denotes the phone received by friend i, we must have that for all i and all j, v ij(i) p i v ij p i(j). 6
Part a (5 points) Prove that if an allocation/payment is envy-free (but might fail to satisfy i p i = B), that for any c R, updating p i = p i + c for all i is still envy-free. Part b (5 points) Prove that if an allocation/payment is envy-free, and i p i = B, then v ij(i) p i 0 for all i. Part c (5 points) Design a protocol to find an envy-free allocation/payment with i p i = B when n = 2. Part d (10 points) Design a protocol to find an envy-free allocation/payment with i p i = B when n = 3. Part e (10 points) Design a protocol to find an envy-free allocation/payment with i p i = B when n = 4. 7