Part 1: Voting Systems Consider the following (true) preference orderings of 4 agents on 4 candidates. Agent #1: A > B > C > D Agent #2: B > C > D > A Agent #3: C > B > D > A Agent #4: D > C > A > B Assume that agents #2, #3, and #4 are truthful while agent #1 may or may not be truthful in reporting his preference ordering. For each of the following voting systems, find the best response of agent #1 among the given strategy profiles. Ties are broken alphabetically (for example, A beats B in a tie). Q1: Borda Rule. Among the three orderings below, choose the one that achieves the best outcome (among the four) for agent #1. a) A > B > C > D b) B > A > D > C c) C > D > A > B A1: b. Explanation: First, calculate Borda scores of four candidates given by the three invariant agents (2, 3, 4). A: 0 + 0 + 1 = 1 B: 3 + 2 + 0 = 5 C: 2 + 3 + 2 = 7 D: 1 + 1 + 3 = 5 Now, for each of the four choices, we can determine the winner by adding scores to candidates. a: (A > B > C > D) gives Borda scores (1+3=4, 5+2=7, 7+1=8, 5+0 = 5) and the winner is C. b: (B > A > D > C) gives Borda scores (1+2=3, 5+3=8, 7+0=7, 5+1 = 6) and the winner is B. This is the best outcome agent #1 can achieve. c: (C > D > A > B) gives Borda scores (1+1=2, 5+0=5, 7+3=10, 5+2 = 7) and the winner is C. Q2: Pairwise Elimination with ordering C, A, B, D. Among the three orderings below, choose the one that achieves the best outcome (among the four) for agent #1. a) A > B > C > D b) C > D > A > B
c) C > D > B > A A2: a. Explanation: a: if (A > B > C > D) is submitted, then C wins against A with 3 votes. C and B are matched in the next round, yet they are tied with two votes each -- B wins the tie-breaker. Finally, B and D are matched, and B wins with three votes. b: if (C > D > A > B) is submitted, then C wins against A with 4 votes. Then C advances and wins against B with three votes. Finally, C wins against D with three votes again, and C is the winner. c: if (C > D > B > A) is submitted, then C wins against A with 4 votes. Then C advances and wins against B with three votes. Finally, C wins against D with three votes again, and C is the winner. Part 2: Social Welfare Function Consider the following social welfare function W: given (strict) orderings of agents on the candidates, W returns the lexicographically lowest ordering among the n orderings submitted by n agents. For example, if two agents vote on three candidates with A > B > C and B > A > C, then W chooses A > B > C which comes lexicographically before B > A > C. Specify whether this social welfare function, W, satisfies each of the following properties (in general, so for every number of candidates, outcomes, etc..). Q1: Pareto Efficiency. a) Yes. b) No. Explanation: a. If everyone prefers one candidate over another (say, A > B), then the returned preference ordering of W must have that A > B (because it picks an ordering from the submitted orderings). Q2: Dictatorship. a) Yes. b) No.
Explanation: b. W is non-dictatorial. Suppose some agent x is a dictator. If x submits C > B > A but someone else submits A > B > C, then W does not return agent x's preference ordering. Q3: Independence of Irrelevant Alternatives (IIA). a) Yes. b) No. Explanation: b. Because W is Pareto-efficient and non-dictatorial, we know by Arrow's theorem that W must not be IIA. Selling a single item: 1. Suppose you are selling a single item to n bidders with valuations drawn uniformly from the range [1,2]. What is the optimal reserve price? a. 0.5 b. 1 c. 1.5 d. (n-1)/n+1 e. 1.5^(n-1) 2. Consider a second-price auction (with random tie-breaking) where bidder 1 has valuation 4 and bidder 2 has valuation 10. Let (a,b) denote a strategy profile where bidder 1 bids a and bidder 2 bids b. Which of the following are Nash equilibria of the game? (Select the largest set that doesn t contain any non-equilibrium strategy profiles.) a. (0,4) b. (0,5),(4,10),(10,10) c. (4,10) d. (4,10),(9,10),(10,4),(10,9) e. (9,10),(10,4) 3. Consider an all-pay auction with three bidders, each of which has a valuation drawn uniformly from the range $[0,1]$. Write how each bidder should select his bid b (as a function of his valuation v) in a symmetric Bayes-Nash equilibrium. a. b=0 b. b=2v/3 c. b=2v^3/3 d. b=(2/3)^3v^2 e. b=3v^2 Selling more than one item: 1. A condominium developer wants to sell 4 identical units to a group of bidders. Each bidder wants only a single unit, and their valuations are 2,32,46,50,52,76 and 80. If the developer uses VCG to allocate the units, how much money will she receive? a. 76 b. 130 c. 178 d. 184 e. 200 Solutions Optimal reserve: 1 Equilibria of second price-auctions: (9,10),(10,4) All-pay: b=2v^3/3 Multi-item VCG: 184
Final Questions for Week 2 Q1. Mechanism Implementations Consider first-price and second-price auctions with 2 risk-neutral bidders whose private values are IID and drawn from a uniform distribution on [0,1]. (i) Select the auction mechanisms that are implementations in dominant strategies: a) Only first-price auctions. b) Only second-price auctions. c) Both first-price and second-price auctions. d) Neither first-price nor second-price auctions. Answer: b. (ii) Select the auction mechanisms that are implementations in Bayes-Nash equilibrium: a) Only first-price auctions. b) Only second-price auctions. c) Both first-price and second-price auctions. d) Neither first-price nor second-price auctions. Answer: c. Q2. Mechanism Properties Consider a voting with 3 players (1, 2, 3) and 2 alternatives (a, b). Each player has a strict preference, and prefers a or b with equal probabilities. Each player gets 5 if his/her top choice (based on the true preference) is selected and 0 otherwise. 2votingrules: inbothcases,eachplayersubmitonevotev i 2 {a, b} (i 2 {1, 2, 3}). Rule (1): the winning alternative w is uniformly randomly chosen from 3 votes submitted by 3 players. Players with v i = w pay 1, and players with v i 6= w pay 0. Rule (2): w is selected as in Rule (1). Players with v i = w pay 0, and players with v i 6= w get 1 (pay -1). 1
(i) Which voting rule(s) is truthful? a) Both (1) and (2) b) Only (1) c) Only (2) d) Neither Answer: a. (ii), Which voting rule(s) is budget balanced? a) Both (1) and (2) b) Only (1) c) Only (2) d) Neither Answer: d. (iii) Which voting rule earns a higher expected revenue? a) (1) b) (2) c) Their expected revenues are the same. Answer: a. (iv) Which voting rule gives a higher maxmin fairness? a) (1) b) (2) c) Their maxmin fairnesses are the same. Answer: b. 2
Alice currently has one serving of chocolate, which she values at $5, and Bob has one serving of ice cream, which he values at $7. Carol values the chocolate at $7, the ice cream at $10, and both together at $12. Suppose that we run VCG to determine whether some of Alice and Bob s desserts should be transferred to Carol. The following table illustrates the payo s to each agent. ; {chocolate} {ice cream} {chocolate, ice cream} Alice 0 5 0 5 Bob 0 0 7 7 Carol 0 7 10 12 1. What outcome will the mechanism choose? a) No transfers. b) Chocolate transferred to Carol. c) Ice cream transferred to Carol. d) Chocolate and ice cream transferred to Carol. Answer: (c) 2. What payments will the mechanism impose? a) (0, 0, 0) b) ( 5, 0, 5) c) (0, 7, 7) d) ( 5, 7, 12) e) ( 6, 8, 11) f) (0, 8, 7) g) ( 6, 0, 5) Answer: (f) 3. Which conditions does this environment violate? a) Choice set monotonicity b) No single-agent e ect c) No negative externalities d) Ex-post individual rationality e) All of the above f) None of the above Answer: (b) 1
4. How much could Alice, Bob, and Carol extract from the mechanism in payments if they colluded? a) At most $0 b) At most $1 c) At most $2 d) At most $3 e) At most $5 f) At most $7 g) At most $10 h) At most $12 i) None of the above Answer: (i) 2