Download Game Theory Instructor: Michał Lewandowski, PhD Problem 1

Game Theory Problem 1 Instructor: Michał Lewandowski, PhD Agnieszka Radwanska plays tennis with Caroline Wozniacki. Agnieszka may choose to aim at Wozniacki’s backhand or Wozniacki’s forehand. If Wozniacki predicts correctly, which side will Radwanska choose, she will return the ball with higher probability. The payoff table in terms of probabilities of scoring a point is given below: a) Find all Nash equilibria of this game and calculate equilibrium payoffs of both tennis players. Nash equilibria: …………………..……………………………………………………………………………..………………….. Equilibrium payoffs:………………………………………………………………………………………………………………… b) Determine the Best response correspondences of both tennis players and draw them on the diagram below: Best response correspondences: ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… ……………………………………………………………… Problem 2 The following game is given: a) Find all Nash equilibria of this game ……………………………………………………………………………………. b) Find security levels of both Mr Raw and Mrs Column (payoffs that they can guarantee themselves) – first write the payoffs of Mr Raw and Mrs Column game:
Mr Raw’s game
A
A
B
B
Mrs Column’s game
A
B
A
B
Security levels: Mr Raw’s:………………………., Mrs Column:…………………………… c) Draw the payoff polygon for the original game and mark the Status Quo point (given by players’ security levels) and the negotiation set. d) Find the Nash arbitration solution …………………………………………………………………………………………… Problem 3 Consider the following duopoly game. We have two firms, producing an identical good. Each firm chooses its production quantity (x1 and x2). The price of the good is given by the inverse demand function p(x1,x2)=60‐3(x1+x2) (or 0 if the sum of production quantities exceeds 20). Marginal cost is constant and is 12 for both firms. Both players want to maximize profit, which is the difference between revenue and cost. a) (Cournot) Assume that the players make their choices of x1 and x2 simultaneously. Find the Nash equilibria and equilibrium profits …..…………………………………………………………………………………. ……………………………………………………………………………………… ……………………………………………………………………………………… b) (Stackelberg) Assume that player 1 makes his choice first, and then after that player 2 is making her choice after observing player 1’s choice. Timing of the game, payoffs and possible actions – all these elements are common knowledge in this game. Determine the equilibrium and equilibrium payoffs. ………..………………………………….……..…………………………………………..………………… ……………………………………..…………………………………………………………………………… ………………………………………………………………………………………………………………….. Problem 4 Consider the following game in extensive form: a) Transform this game into strategic form. b) Find all pure strategy Nash equilibria. …………………………………………………………………… c) Are there Nash equilibria, which are not Subgame Perfect Nash Equilibria in this game? …………………… d) How many subgames are there in the game tree above (the whole game is also a subgame)? ……………........
Problem 5 Consider the following game: Player 1
A
B
C
Player 2
X
4,2
1,1
2,3
Y
0,2
4,2
2,1
a) Solve the game by iterative elimination of dominated strategies. Each time, indicate what strategy dominates a given strategy. Determine the Nash equilibrium which is the solution of this process. ………………..…………………………………….………………………………………………………………… b) Did we loose some Nash equilibria by using procedure of iterative elimination of dominated strategies? If yes, write which one. …………………….…………………………………….………………………………………………………….. Problem 6 Consider the following ultimatum game. a) How many strategies does Player 1 have? How many strategies does Player 2 have? Player 1:………….., Player 2:……………. b) Find all Subgame Perfect Nash equilibria of this game ………………………………………………………………………………………………. c) Is the following strategy: “accept 3 and reject everything else” for Player 2 and “offer 3” for Player 1 a Nash equilibrium? If yes, why? ……………………………………………………………………………………… ……………………………………………………………………………………… Problem 7 Find all Evolutionarily Stable Strategies in the following game: Answer: ………………………………………………………………………………………………………………………………………………………….. Zadanie 8 Consider the following game from the lecture (Bank run game) a) Is there a separating equilibrium in this game in which a good type withdraws money and a bad type does not withdraw money from the bank? …………………………………………………………………………………………………………………………………………………….. b) Check the following strategies: Player 1: Does not withdraw; Player 2: good type: does not withdraw, bad type: Withdraw. Is it a Bayesian Nash equilibrium? If yes, for what beliefs of Player 1? …………………………………………………………………………………………………………………………………………………………………. Problem 9 (EXTRA) Alice, Beatrice, Cecil and Dylan, Ernie and Felix are classmates from the same school. After long discussions the boys agreed that the most important criterion to evaluate a given girl is that a blond girl has blue eyes and a brunette has dark eyes. The girls on the other hand agreed similarly that the most important criterion to evaluate a given boy is his height: the taller the better. The second criterion (less important) for both girls and boys turned out to be the hair color: any given girl wants the boy with a different hair color than her own and any given boy wants the girl with the same hair color as his own. The characteristics of both girls and boys is given in the following tables: Alice Beatrice Cecil Hair Blonde Brunette Brunette Eyes Blue Dark Blue Dylan Ernie Felix Hair Blonde Brunette Blonde Height Tall Tall Small a) Determine the rankings of both girls (concerning boys) and boys (concerning girls): I place II place III place I place II place III place Alice Dylan Beatrice Ernie Cecil Felix b) Is the following matching stable? If not, who can block this matching? Beatrice ‐ Dylan, Cecil – Ernie Alice ‐ Felix ………………………………………………………………………………………………………………………………………………………………………. c) Which matching will be found if we use Gale Shapley algorithm with boys proposing to girls? ……………………………………………. ……………………………………………. ……………………………………………. d) Can you gain by reporting false preferences? Hint: Use a Gale‐Shapley algorithm with boys proposing, when Alice lies about her true ranking and tells that she prefers Felix to Dylan. What will be the stable matching then? ……………………………………………. ……………………………………………. …………………………………………….