미시경제학 게임이론 문제 질문
-
게시물 수정 , 삭제는 로그인 필요
① Consider an industry in which two firms compete by simultaneously and independently selecting prices for their goods. Firm 1's product and firm 2's product are imperfect substitutes, so that an increase in one firm's price will lead to an increase in the quantity demanded of the other firm's product. In particular, each firm i faces the following demand curve:
qi = max{0, 24 − 2pi + pj},
where qi is the quantity that firm i sells, pi is firm i's price, and pj is the other firm's price. (The maximum operator is needed to ensure that quantity demanded is never negative.)
Suppose that the maximum possible price is 20 and the minimum price is zero, so the strategy space for each firm i is Si = [0, 20]. Both firms produce with no costs.
(a) What is rm i's prot as a function of the strategy profile (p1, p2)? Let each firm's pay off be defined as its profit.
(b) Show that for each firm i, prices less than 6 and prices greater than 11 are dominated strategies, whereas prices from 6 to 11 are not dominated.
(c) Consider a reduced game in which firms select prices between x and y, where x < y.
What strategies are undominated in such a game? Use your answer to compute the set of strategies profiles in their survives iterated dominance in the original game.
② Two players are involved in a dispute over an object. The value of the object to player i is vi > 0. Time is modeled as a continuous variable that starts at 0 and runs indefinitely. Each player chooses when to concede the object to the other player; if the first player to concede does so at time t, the other player obtains the object at that time. If both players concede simultaneously, the object is split equally between them, player i receiving a pay off of vi/2. Time is valuable: until the first concession each player loses one unit of pay off per unit of time.
(a) Formulate this situation as a normal-form game.
(b) Show that in all Nash equilibria one of the players concedes immediately
한국어로 설명좀 해주세요 ㅜㅜ 어떻게 푸는지 모르겠어요..
qi = max{0, 24 − 2pi + pj},
where qi is the quantity that firm i sells, pi is firm i's price, and pj is the other firm's price. (The maximum operator is needed to ensure that quantity demanded is never negative.)
Suppose that the maximum possible price is 20 and the minimum price is zero, so the strategy space for each firm i is Si = [0, 20]. Both firms produce with no costs.
(a) What is rm i's prot as a function of the strategy profile (p1, p2)? Let each firm's pay off be defined as its profit.
(b) Show that for each firm i, prices less than 6 and prices greater than 11 are dominated strategies, whereas prices from 6 to 11 are not dominated.
(c) Consider a reduced game in which firms select prices between x and y, where x < y.
What strategies are undominated in such a game? Use your answer to compute the set of strategies profiles in their survives iterated dominance in the original game.
② Two players are involved in a dispute over an object. The value of the object to player i is vi > 0. Time is modeled as a continuous variable that starts at 0 and runs indefinitely. Each player chooses when to concede the object to the other player; if the first player to concede does so at time t, the other player obtains the object at that time. If both players concede simultaneously, the object is split equally between them, player i receiving a pay off of vi/2. Time is valuable: until the first concession each player loses one unit of pay off per unit of time.
(a) Formulate this situation as a normal-form game.
(b) Show that in all Nash equilibria one of the players concedes immediately
한국어로 설명좀 해주세요 ㅜㅜ 어떻게 푸는지 모르겠어요..
