Select one question to analyze. Referring to the three fundamental principles and three precepts of finance below, integrate a principle, a precept, or both in your response. In addition, integrate 1-2 external credible references to support your thoughts. FP1: The value of any asset is equal to the present value of the cash flows the asset is expected to produce over its economic life. FP2: There is a direct relationship between risk and return; as perceived risk increases, required return will also increase (and vice versa), holding other things constant. FP3: There is an inverse relationship between price and yield; if an asset's price increases, its return will decrease (and vice versa), holding other things constant. PR1: The present value of a cash flow (or an asset) is inversely related to its discount rate; increasing the discount rate decreases the present value (and vice versa), holding other things constant. PR2: The timing of the cash flows of an asset is important; sooner is better (later cash flows are more heavily discounted, reducing their present value). PR3: The present value of a cash flow (or an asset) is inversely related to its perceived risk; the higher the risk, the higher the discount rate, and therefore the lower the present value. 1. You win a lottery that pays $5,000 per month for 20 years, starting today. You have the option of taking a lump sum today instead of the monthly payments. The lottery people tell you that, if you take the monthly payments, the money you won is expected to earn 7.25% APR, compounded monthly, for the entire 20 years it will be invested. Given these data, what is the lump sum you would expect to receive today? Discuss the pros and cons of receiving a lump sum or monthly payments

4. Stage Game: Alice and Bob simultaneously choose contributions a E [0, 1] and b E [0, 1], respectively, and get payofs UA = 2b - a and UE = 2a - b, respectively. (a) (5 points) Find the set of rationalizable strategies in the Stage Game above. (b) (10 points) Consider the infnitely repeated game with the Stage Game above and with discount factor 5 E (0, 1). For each 5, fnd the maximum (a*, b*) such that there exists a subgameperfect equilibrium of the repeated game in which Alice and Bob contribute a* and b*, respectively, on the path of equilibrium. (c) (10 points) In part (b), now assume that at the beginning of each period t one of the players (Alice at periods t = 0, 2, 4,... and Bob at periods t = 1, 3, 5,...) ofers a stream of contributions ba = (at, at+1,...) and bb = (bt, bt+1,...) for Alice and Bob, respectively, and the other player accepts or rejects. If the ofer is accepted then the game ends leading the automatic contributions ba = (at, at+1,...) and bb = (bt, bt+1,...) from period t on. If the ofer is rejected, they play the Stage ( ) Game and proceed to the next period. Find (aA, bA), (aE, bE), and a, b b b such that the following is a subgameperfect equilibrium: s* : When it is Alice's turn, Alice ofers (aA, aA,...) and (bA, bA,...) and Bob accepts an ofer (ba,bb) if and only if (1 - 5) [2at - bt + 5 (2at+1 - bt+1) + ... ] 2 2aA - bA. When it is Bob's turn, Bob ofers (aE, aE,...) and (bE, bE,...) and Alice accepts an ofer (ba,bb) if and only if (1 - 5) [2bt - at + 5 (2bt+1 (- a ) t+1) + ... ] 2 2bA - aA. If there is no agreement, in the stage game they play a, b . bb Verify that s* is a subgame perfect equilibrium for the values that you found. (If you fnd it easier, you can consider only the constant streams of contributions ba = (a, a, . . .) and bb = (b, b, . . .).)

1. (15 Minutes - 20 Points) Answer each of the following subquestions BRIEFLY. (a) (5 points) In my second lecture I defined a solution concept called "pure strategy iterated strict dominance." In my fifth lecture I defined a more powerful version of iterated strict dominance. What was the difference betweeen them and why was the game below a useful example? L C R U M D 10, 9 10, 6 10, 10 -5, 9 15, 10 11, 12 -35, 10 10, 7 15, 5 (b) (5 points) Find all pure strategy Nash equilibria of the game below. X Y Z A B C 4, 3 5, 2 5, 1 1, 7 6, 6 4, 3 2, 3 7, 3 5, 3 (c) (5 points) In class I discussed the general discrete choice model of price competition between N firms: the firms choose prices p1, . . . , pn and each consumer i decides to purchase from the firm j for which vpj +Eij is largest. Describe briefly what happens to equilibrium prices as the number of firms N goes to infinity both for uniformly distributed Eij and under general distributions. (d) (5 points) Describe precisely an example of a game that has no pure or mixed strategy Nash equilibrium. Describe as well as you can a theorem that provides conditions under which a game with an infinite number of pure stategies must have a Nash equilibrium. What conditions of your theorem are violated in your example? 1 2. (25 Minutes - 30 Points) When my daughter Anna was 3 years old, she liked to play Rock-Paper-Scissors. However, she faced a difficulty - three year olds find it hard to make "scissors" with their fingers. Suppose that we capture this problem by treating her playing Rock-Paper-Scissors against her older sister using the asymmetric 3 3 game shown below (with Anna as player 1). R P S R P S 0, 0 -1, 1 1, -1 1, -1 0, 0 -1, 1 1 c, 1 1 c, -1 c, 0 (a) (13 Points) Consider first the version of this game where c > 1. (You can think of this as a model for the extreme situation where Anna is physically incapable of playing scissors.) Find a mixed-strategy Nash Equilibrium of this game. (b) (3 points) What is Anna's expected payoff in the equilibrium you found in part (a)? (c) (14 points) Consider now the version of this game with 0 < c < 1. Find a mixed strategy Nash equilibrium of this game in which both players play every strategy with positive probability. 2 3. (15 Minutes - 22 Points) Two students are deciding how long to spend studying for 14.12 on the night before the exam. Let ei be the fraction of the available time student i devotes to studying with 0 ei 1. Assume that the students' utilities are u1(e1, e2) = log(1 + 3e1 e2) e1 u2(e1, e2) = log(1 + 3e2 e1) e2 (A story for this would be that the first term reflects the benefits they get from learning and getting a good grade, whereas the second reflects the opportunity cost of time. The negative effect of e2 on student 1's utility could reflect that student 1 will get a lower grade if student 2 studies more and does better on the exam.) (a) (5 points) What is player 1's best response to a choice of e2 by player 2. (b) (13 points) Find a pure strategy Nash equilibrium of the game where players 1 and 2 choose e1 and e2 simultaneously. (c) (4 points) Is this game solvable by iterated strict dominance? How do you know this? 3 4. (25 Minutes - 28 Points) Suppose Prof. Ellison decides to run a classroom experiment to teach about mixed strategy equilibrium (and make some money). He chooses two students from the class. Each student is required to write down an integer from 1 to 100 inclusive. The rules of the game are that the student who writes down the smaller number must pay Prof. Ellison that number of dollars. The student who writes down the larger number pays nothing. If both students write down the same number assume that both pay. Assume that both students are risk-neutral and self-interested so that this game can be represented as S1 = S2 = {1, 2, . . . , 100} with s if s s u1(s1, s2) = 1 1 2 0 if s1 > s2 s2 if s2 s1 u2(s1, s2) = 0 if s2 > s1 (a) (3 points) Are any strategies in this game strictly dominated? (b) (7 points) This game has two pure strategy Nash equilibria. What are they? (c) (3 points) Discuss briefly why you should expect given what I told you in part (b) that this game would also have a mixed strategy Nash equilibrium. (d) (15 points) Find a symmetric mixed strategy Nash equilibrium of this game