12. What factors do economists suppose influence the amount of education people choose to obtain? If the decision were based entirely on material considerations, how would people decide how much education they should get? Can you think of events that could influence this decision?

8. Suppose an industry has three identical firms competing on quantities with demand P = 100 - 20 and constant marginal costs of MC = 1. What are the firms' best response functions?

Question 1 [20%] Consider the following normal form game G: X Y A B 4,-47, -7 8, -86, -6 a. [6% ] Is there a pure Nash equilibrium in G? Justify your answer. b. [14% ] Compute a mixed equilibrium using the indifference conditions of the players. Present both the equilibrium and the analysis clearly. No coding is required.

Question 3 [25%] Consider the following normal form game G. Your task is to find the correlated equilibrium that max- imizes the sum of players' utilities, using Linear Programming in MATLAB. In your report, you need to present the equilibrium that you have computed, the linear program that you are solving (which should include the equilibrium conditions that are satisfied), and a screenshot of your MATLAB in- put AND output. Use the following ordering of variables when constructing your MATLAB input: PXA, PXB, PXC, PYA, PYB, PYC, PZA, PZB, PZC. A B с X 6,6 0,8 0,0 Y 8,0 2,2 0,0 Z 0,0 0,0 1,1

1 a) Calculate u (0.6X € 0.4Y, 0.7A Ⓒ 0.3B) [NOTE: Understanding notation, i.e., what you are asked to calculate, is a part of the problem] (b) List all best response strategies of the Row player to Column playing X.

2. Suppose that in a small town, the market for cement had five companies with market shares 0.3, 0.2, 0.2, 0.2, and 0.1. The following year, a new firm entered but the leading firm increased its share. Now the shares are 0.5, 0.1, 0.1, 0.1, 0.1, and 0.1. Did the market become more competitive or less competitive?

11. The labor supply curve is the relationship between the wage level and the quantity of labor that workers are willing to provide. Why is applying the usual ceteris paribus assumption more complicated in this case than when we use apply it to the product market?

Question 2 [25%] Ten commuters must decide simultaneously in the morning to use route A or route B to go from home (same place for all) to work (ditto). If a of them use route A, each of them will travel for 10a + 40 minutes; if b of them use route B, each of them will travel for 106 minutes. Everyone wishes to minimize his/her commuting time. Your tasks: 1 a. [12%] Describe the pure Nash equilibrium (or Nash equilibria) of this ten-person game. Compute the corresponding profile of commuting times. Explicitly list all equilibrium conditions that are satisfied. b. [6%] What is the traffic pattern (strategies) minimizing the total travel time of all commuters (the sum of their travel times)? Describe the corresponding profile of commuting times (individual payoffs/cost). c. [7% ] What does this mean about the Price of Anarchy of this game (assuming that the objective function is the total travel time)? No coding is required.

Question 4 [30%] Consider the following sponsored search auction instance I: • 2 slots. The top slot has a known click-through rate (CTR) ctr₁ = 1 and the bottom slot has a known click-through rate ctra=0.5. • 2 advertisers. Advertiser 1 has a private value-per-click v₁ = 2 and advertiser 2 has a private value-per-click U₂ = 1. • The payoff of advertiser i, (i is either 1 or 2), who is assigned to the top slot is (v₁ - Pi), where pi is the price charged per-click to i. The payoff of advertiser j (j is either 1 or 2 but different than i) who is assigned at the bottom slot is 0.5-(v₁-p;) where p, is the price charged per-click to j. P; and p; are defined by the auction rule, as follows. Consider the following auction rule (first-price auction): Advertisers are asked to declare their value per click (this doesn't mean that their declarations are truthful!). Advertisers are then ranked according to their declarations and the advertiser with the highest declaration is assigned to the slot with the highest CTR (top slot), the advertiser with the second highest declaration is assigned to the slot with the lowest CTR (bottom slot). In case of a tie, advertiser 1 is allocated to the top slot. The per-click payment of any advertiser is equal to their own bid. a. [6%] Compute the optimal/highest social welfare (sum of individual values) in I. b. [9% ] Let bi and b2 denote the bids placed by advertiser 1 and 2, respectively, and assume b2 > b₁. Formulate the conditions that need to be satisfied at equilibrium. The conditions should contain only variables V₁, V2, b₁ and b₂. c. [15% ] Write a function that takes as input the bid of advertiser 1 and calculates the best response, i.e. the strategy/bid of advertiser 2 that results in the highest possible utility for advertiser 2. You can (or not) follow a brute-force approach, i.e. consider all possible declarations/bids for adver- tiser 2, calculate the associated utility and keep the bid that maximizes that utility. Copy and paste your MATLAB code in your report.

1. In analyzing interactions among individuals as players in a game, what's the difference between a Nash equilibrium and a dominant strategy equilibrium? How are they similar and how are they different? Use specific examples. (3 points)