)] = q * So using this equation together with consumer 1 s best response function, we can solve for Consumer 1 s best response becomes:

Transcription

1 Econ8: Introdction to Game Theory ASWERS T PRBLEM SET a If yo decide to bring q i coins to the market, yor payoff is determined as: i P qi 60 qi [8 0.5q q q 5 ]q i 60 q i If firm i sbscript confses yo, then jst write the payoff to any firm, say firm and proceed. [8 0.5q q q 5 ]q 60 q 8q 0.5q 0.5q q q q 0.5q q [ 0.5q q 5 ] 60 We have 5 of these payoff fnctions. i.e. firm s, s, 5 s Firm s best response fnction is fond by taking the partial derivative of firm s payoff with respect to q and setting it eqal to zero q 0.5q q 5 0 q q 4 0.5q q 5 So we have 5 of these best response fnctions and since the problem is symmetric, in eqilibrim we know that each individal s first period qantity will be the same: q q q 5 q So sing this eqation together with consmer s best response fnction, we can solve for Consmer s best response becomes: q q q 8 The ash eqilibrim otpt levels are each firm ptting 8 coins in the market. q. b The collsion cartel otpt level of this game is fond by writing the payoff to the cartel and finding the that maimizes it. [8 0.5] Cartel s optimal otpt is fond by taking the derivative of the payoff fnction and setting it eqal to zero i.e. q q q 5 4 c In or eperiment in class, we observed that the individal otpt levels flctated arond 5 or which is close to the answer fond in part a.

2 In order to find the best response fnctions, we first need to write the payoff fnctions of PEC and non-pec prodcers. Payoff fnction of PEC: Payoff fnction of non-pec: ow we can find the best response fnctions of each prodcer by taking the derivative of the payoff fnction and setting it eqal to zero. PEC s best response fnction: on-pec s best response fnction: o b To find the ash eqilibrim qantities in the oil market, we wold solve the two best response fnctions simltaneosly and In this gessing an integer game player s strategy sets are given by: S {,,, 4, 5} and S {,,, 4, 5}. Therefore we have the following game matri: Player 4 5, -, - -, - -, -4 -, -5 -, -, -, - -, -4 -, -5 Player -, - -, -, -, -4 -, , - -4, - -4, - 4, 4-4, , - -5, - -5, - -5, -4 5, 5 Marking the best responses of each player we see that there are 5 ash eqilibria in this game located along the diagonal. We see that the strategy profile 5, 5 strictly Pareto dominates the other eqilibria. Therefore we can conclde that this is a pre coordination game.

3 6 a Player R P S R 0, 0 -,, - Player P, - 0, 0 -, S -,, - 0, 0 b There is no ash eqilibrim in pre strategies in this game. 8 a Since a player can send at most one nit to a given continent, and each player has soldiers, both players have strategies: AB, AC and BC where for eample AB means a player sent one soldier to A and one soldier to B. Player AB AC BC Player AB 0, 0, -, - AC -, 0, 0, - BC -, -, 0, 0 b The ash eqilibrim strategies for each player are AB, AB. The ash eqilibrim strategy AB for player is not a strictly dominant strategy. Becase when player chooses the strategy BC, playing strategy AB reslts in the same payoff as playing strategy AC. For all the other strategy choices of player, playing strategy AB reslts in a higher payoff. Therefore strategy AB is a weakly dominant strategy for player. Similarly strategy AB is a weakly dominant strategy for player. c If each player has one nit of soldiers, then strategies available to each player are: A, B, and C. Player A B C Player A 0, 0, -, - B -, 0, 0, - C -, -, 0, 0 The ash eqilibrim strategies for each player are A, A.

4 The ash eqilibrim strategy A for player is not a strictly dominant strategy. Becase when player chooses the strategy C, playing strategy A reslts in the same payoff as playing strategy B. For all the other strategy choices of player, playing strategy A reslts in a higher payoff. Therefore strategy A is a weakly dominant strategy for player. Similarly strategy A is a weakly dominant strategy for player. d If there are n continents with vales a > a > > an and ach player is endowed with k nits k < n of soldiers, then the weakly dominant strategy is to place yor k soldiers to the first highest vale continents a, a, a k. 9 a The payoff profit fnctions for firm and firm are given by: 0, ln ln ln ln , ln ln b Compte the best response fnction for Firm and Firm. Compte the ash Eqilibrim strategies., Solving the two best response fnctions together we get the.e. strategies: 48 c The sm of the firm payoffs is given by: U ln ln ln U U Solving these two eqations together we get 40.

5 In order to maimize their joint profit the two firms shold redce their catches from 48 to 40 each. d ow assme there are fishing firms. Calclate the ash eqilibrim strategies for each firm. 0 ln ln,,, There are best response fnctions. In ash eqilibrim we know that each firm s otpt choice will be the same: Solving one of the best response fnctions together with the above eqation we get:

6 q q a The best response crves are drawn. i If Firm chooses a qantity level of 4, he best response of Firm is 4. ii If Firm chooses a qantity level of, the best response of Firm is 4 b The ash eqilibrim otpt levels are 4, 4. c The two firms can be better off if they collde by redcing their otpt. d The strategy q dominate the strategy q.becase no matter what otpt firm chooses, by choosing q, firm attains a higher vale inner isoprofit crve than that of q. For eample sppose q, then q, q pts firm on a higher vale isoprofit crve than q, q. The strategy q dominate the strategy q. Becase no matter what otpt firm chooses, by choosing q, firm attains a higher vale inner isoprofit crve than that of q. For eample sppose q, then q, q pts firm on a higher vale isoprofit crve than q, q. e Sppose the government introdces a qantity ceiling to Firm of. i.e. Firm cannot prodce more than nits, whereas Firm can prodce any qantity in [0,6]. Firm s best response fnction nder this minimm prodction reqirement is drawn. The ash Eqilibrim qantity levels for this new game is.5,. Firm wold benefit from the qantity ceiling imposed on Firm. The new.e. q.5, q pts firm on a higher valed isoprofit crve compared to the old.e. q 4, q 4.