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)Therefore, draw the game tree of this game and round the payoffs to whole numbers. 4)Is the equilibrium separating, semi-separating or pooling? 5)Reconsider the case where Coke with low marginal cost be 10 instead of 5. Keeping everything else the same, 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. 6)Recalculate the profits of Coke and Pepsi respectively in the new duopoly game, in consideration of both scenarios where Coke is high cost and low cost. 7)Therefore, draw the new game tree of this game and round the payoffs to whole numbers. 8)What are the possible equilibriums and are there any conditions, if any? P=25−Q

D1) Consider the simplified poker game below: 1)What are the possible equilibriums and are there any conditions, if any? 2)Express this game in normal form and calculate the payoffs E) A company wish to distinguish between type A and type C students from a pool of students. It is willing to pay a salary of $160,000 and $60,000 to type A and type C students respectively. Type A and type C students can find an alternative job with salary $125,000 and $30,000 respectively. Suppose the cost of taking a tough course is $3,000 and $14,000 to type A and type C students respectively, 1)What is the minimum number of tough courses the company should set as a benchmark to separate type A and type C students? 2)Suppose the proportion of type A students is p. What will be the pooling equilibrium and what are the conditions for both types to be better off than separating equilibrium? 3)Is this equilibrium stable?

F) There are a total of 2 players in a first-priced, sealed bid auction. Each player has their own private value for the object. The object is of value to player 1. The unknown private value of a player is uniformly distributed between 0 and 1. 1)Assuming player 2 adopts the bidding function where is the unknown private value of player 2, what is the best response for player 1? 2)What is the Nash equilibrium of this game? 3)What is the expected payment for each player in terms of their private value? What will be the expected revenue for the seller? G) In an all-pay auction, the common value of object to bidders is 5. 1)Can there by any pure strategy Nash equilibrium? Why or why not? 2)Suppose there are 3 bidders, what will be the Nash equilibrium for mixed strategy? 3)Repeat the calculation for mixed strategy equilibrium for n bidders. What will be the mixed strategy equilibrium? What will be the expected revenue for the seller? v b(x)=x 2 x

Example A Compute all the Nash equilibria in this game

Answer: The Nash equilibria are , , , , , ,

Explanation: Nash equilibrium is a list of strategies, one for each player, such that all the strategies are mutual best responses to one another. Therefore, we intend to graph the best responses to find the mutual best response strategies which are the Nash equilibria, with steps as such: Assuming P2’s strategy is , we find the P1’s expected payoff for each pure strategy:

(A,A)(B,B)(C,C)(1 2A⊕1 2C,1 2A⊕1 2C) (2 3A⊕1 3B,2 3A⊕1 3B)(2 5B⊕3 5C,2 5B⊕3 5C) (1 2A⊕3 10B⊕1 5C,1 2A⊕3 10B⊕1 5C) (xA⊕yB⊕(1−x−y)C) E(A)=2x−y E(B)=3y−(1−x−y) =x+4y−1 E(C)=x+1−x−y =−y+1

Calculating the conditions where each strategy is the best response (BR):

Combining results in a graph:

The area shaded indicates the conditions where the strategy in particular is the BR given a strategy mix of the other player (given by coordinates). Since this game is symmetric, i.e the BR diagram for P1 and P2 are the same, we can therefore identify the mutual BR strategies from the diagram and they will be the Nash equilibria: 3 pure, 4 mixed (represented by the black dots).

Example A1 Compute all the Nash equilibria in this game: Answer: The Nash equilibria are , , , , , ,

(C,M)(D,R)(U,L)(3 4C⊕1 4D,2 3M⊕1 3R) (2 3U⊕1 3D,4 5L⊕1 5R)(3 7U⊕4 7C,4 7L⊕3 7M) (7 18U⊕5 9C⊕1 18D,6 11L⊕14 33M⊕1 33R)

Explanation: Nash equilibrium is a list of strategies, one for each player, such that all the strategies are mutual best responses to one another. Therefore, we intend to graph the best responses to find the mutual best response strategies which are the Nash equilibria, with steps as such: Assuming P2’s strategy is , we find P1’s expected payoff for each pure strategy:

Calculating the conditions where each pure strategy is the best response (BR):

(xA⊕yB⊕(1−x−y)C)