課程名稱︰賽局論 課程性質︰選修 課程教師︰古慧雯 開課學院：社會科學院 開課系所︰經濟系 考試日期（年月日）︰103/6/20 考試時限（分鐘）：110分鐘 試題 : 賽局論期末考 (6/20/2014) 古慧雯 總分35分。答題皆須附說明，未做解釋的答案概不計分。 1. Reconsider the fitness game exemplified by the story of dodos. There are two types of genes, s_1 and s_2. The fertility rate of a mother depends on how she interacts with the opponent mother, and is given as follows: │ s_1 s_2 ───────── s_1│ 0,0 1,0 s_2│ 0,1 2,2 (a) (2 points) Please find all Nash equilibrium of this fitness game. (b) (2 points) Please find all evolutionarily stable strategies. (c) (3 points) Let p_i denote the fraction of the population hosting gene s_i. It is proved that the replicator equations are: _ (p_i)' = p_i(f_i(p) – f(p)), i = 1, 2, _ where f_i(p) is called "fitness" of type i, and f(p) is the average fitness. Please give the replicator equation of p1 using data in the above matrix.(No points are given for the following two questions, if you do not answer this question correctly.) (d) (2 points) What are rest points (p1, p2) of the replicator equations? (e) (2 points) Find all asymptotic attractor(s) of the replicator dynamics. 2. The following game will be repeated twice: │ slow speed ──────────── slow │ 4,4 1,5 speed│ 5,1 -2,-2 The repeated game payoffs are just the sum of the stage-game payoffs. Consider a strategy s that tells you to choose slow at the 1st stage and to use slow with probability p at the 2nd stage, unless the two players have failed to use the same actions in the 1st stage. If such a coordination failure has occurred in the past, s tells a player to use whatever action that person didn't play at the 1st stage. (a) (3 points) Which value of p will make (s, s) a Nash equilibrium? (Credit is granted only when clear arguments are made.) (b) (2 points) What does s ask a player to do if (speed, speed) is played in the 1st stage? Is (s, s) a subgame perfect equilibrium (with p you calculated above)? (c) (2 points) Give a different subgame perfect equilibrium other than (s, s). 3. The following stage game will be repeated infinitely many times, and two players care about the long-run average payoffs. │ t_1 t_2 ────────── s_1│ 0,2 0,-3 s_2│ -1,-1 3,0 Instead of using strategies prescribed by finite automata, both players consider to use mixed strategies. (a) (2 points) The row player considers to coordinate their moves, and she warns the column player that if he does not follow the plan, she will punish him in the future. What is the row player's most sever punishing strategy? (b) (4 points) How should the Folk Theorem presented in the textbook be modified in this situation? Draw a graph with two players' payoffs on two axes and illustrate the result. 4. (5 points) Reconsider the finite version of the duel problem in the textbook. Two players are armed with pistols loaded with just one bullet, and they walk towards each other. The initial distance between them is D. Each player is allowed to fire at either distance D, or at distance 0. (Note they could fire at the same time.) The following table shows the probability that player i will hit his target when he fires at distance d, p_i(d), i = 1, 2. d = │ 0 D ────────── p_1(d)│ 1/2 1/3 p_2(d)│ 1/3 1/4 For each player, the primary goal is to maximize his own survival probability, and the secondary goal is to minimize his opponent's survival probability. In a subgame perfect equilibrium, what will player 1 do at distance D? 5. Alice confiscates Bob's and Carol's wallets and uses an English auction to sell their combined contents back to whoever bids higher. Bob's wallet contains b dollars, and Carol's wallet contains c dollars, but they know only how much was in their own wallet. We shall solve step by step a symmetric equilibrium in which a person whose wallet contained x dollars plans to bid up to B(x), where B(x) is a strictly increasing function of x. (a) (2 points) If Carol quits first when the price is raised to $10, what condition about b and B(.) can you infer (besides (B(b) > 10)? (b) (4 points) Let p denote the price that Bob plans to quit, i.e. p = B(b). From your answer above, write a condition that helps solve B(.) and solve it. --

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