H3 Game Theory Personal Notes

Sample Paper: A) Compute all the Nash equilibria in this game

A1) Compute all the Nash equilibria in this game:

B) Coke demand function ; Pepsi demand function , where denotes the unit price of Coke and denotes the unite price of Pepsi. Both have same marginal cost 8 of producing per can of soda. 1)From the demand function, are Coke and Pepsi complements or substitutes? 2)Compute the Nash equilibrium price and profits. 3)What will be the maximum profit for Coke and Pepsi respectively if they were to collude? 4)Consider a game where each of the the two firms can either price cooperatively, i.e collude or price aggressively, i.e. maximising own profit. Construct the 2x2 game matrix. 5)Does this game suit the features of a prisoners’ dilemma? Explain. 6)This game is being played repeatedly for 100 games. What is the sub-game perfect equilibrium? 7)This game is being played repeatedly infinitely. Write down the strategy for Grim Trigger for each player. 8)What is the condition for Grim Trigger to be a sub-game perfect equilibrium strategy? 9)Consider a stick-and-carrot strategy as follow: Cooperative state: Play the collusive price and remain in the cooperative state if no deviation occurs, otherwise switch to punishment state Punishment state: Price 8 and switch to cooperative state if no deviation occurs, otherwise remain in the punishment state If both players adopt such strategy, what condition is required for it to be a sub-game perfect equilibrium? 10) What if in the competition state, firms price 0? Rewrite the stick-and-carrot strategy, and repeat the calculations to find the condition. Q=44−2p+q Q=44−2q+p p q

C) Consider the game below:

1)By best response analysis, find the Nash equilibrium of this game. 2)By iterated elimination of strictly dominated strategies, find the Nash equilibrium of this game. 3)This game is being played repeatedly infinitely. Write a stick-and-carrot strategy where the cooperative strategy leads to a Pareto efficient outcome, the punishment leads to a Minmax payoff. 4)Assuming the probability that the game will end in a single round is p, what is the condition for such stick-and-carrot strategy to be a sub-game perfect equilibrium?

D) Coke and Pepsi try to maximise profits in the soda industry. The marginal cost of Coke can either be 5 or 15, with probability p of a low cost. In the first stage, Coke can choose to price in a way that maximises low cost profit or high cost profit. In the second stage, the Pepsi choose to whether enter the market not. The marginal cost of Pepsi is 10 and Pepsi also need to incur a fixed cost of 40 to enter. 1)The demand function of the soda industry is given by . Calculate the profits of Coke with high cost and low cost in a monopoly game without bluffing, and the profits of high cost Coke with bluffing. 2)Calculate the profits of Coke and Pepsi respectively in the duopoly game, in consideration of both scenarios where Coke is high cost and low cost. 3)Ther