INSTRUCTIONS: all the calculations on the separate piece of paper which you do not hand in. GOOD LUCK!

Transcription

1 INSTRUCTIONS: 1) You should hand in ONLY THE ANSWERS ASKED FOR written clearly on this EXAM PAPER. You should do all the calculations on the separate piece of paper which you do not hand in. 2) Problems 1-3 are for everyone (There is 90 minutes time to solve these problems). Problems A and B are only for those who wish to improve their midterm results (There is 20 minutes time to solve these problems). GOOD LUCK!

2 (30 p) Problem 1 (Big John and Little John) Big John and Little John eat coconuts, which dangle from a lofty branch of a palm tree. A palm produces only one fruit per tree. To get the coconut, at least one of them must climb the tree and knock the coconut loose so that it falls to the ground. A coconut is worth 10 Kc (kilocalories) of energy, the cost of climbing up the tree is 2 Kc for Big John and is negligible for Little John, who is much smaller. If both individuals climb the tree, shake the palm, run down the tree and eat the coconut, Big John gets 7 Kc and Little John gets 3 Kc. If only Big John climbs the tree while Little John waits on the ground for the coconut to fall, Big John gets 6 Kc and Little John gets 4 Kc. If only Little John climbs the tree, Big John gets 9 Kc and Little John gets 1 Kc. What will Big John and Little John do if each wants to maximize net energy gain? Consider three cases: 1. Big John decides first what to do: climb (C) or wait (W) 2. Little John decides first 3. Both individuals decide simultaneously For each of the above cases: a) Draw an extensive-form game (write payoff pairs in the same order as the order of moves) b) Indicate how many strategies does each player have 1. BJ: LJ: 2. BJ: LJ: 3. BJ: LJ: c) Find Subgame Perfect Nash Equilibria of these game (specify its equilibrium payoffs) 1. SPNE: 2. SPNE: 3. SPNE:

3 d) Write this game in normal-form e) Determine all pure strategy Nash Equilibria (for each equilibrium be sure to specify equilibrium payoffs) which equilibrium contains an incredible threat (who threatens?) 1. NE: 2. NE: 3. NE: f) For the case of a simultaneous game (3.) determine additionally the mixed strategy equilibrium and payoffs (if it exists). In this game only, specify Best Response correspondences for each player and draw them on a diagram. Mixed strategy equilibrium:.. Equilibrium payoffs:. Best response correspondences: (15 p) Problem 2 (Nash Equilibrium vs ESS) Consider the following game. Pairs of members of a single population engage in the following game. Each player has three actions, corresponding to demands of 1, 2, or 3 units of payoff. If both players in a pair make the same demand, each player obtains her demand. Otherwise the player who demands less obtains the amount demanded by her opponent, while the player who demands more obtains aδ, where a is her demand and δ is a number less than 1/3. Find the set of pure strategy symmetric Nash equilibria of the game, and the set of pure evolutionarily stable strategies.

4 Nash Equilibria:. Evolutionarily Stable Strategies:. (25p) Problem 3 (Fight) Asterix and Artifis meet each other. Artifis does not know whether Asterix drunk the magic potion (is strong) or not (is weak). He assigns probability p to Asterix being strong. Asterix is fully informed. Each person can either fight or yield. The fight takes place only if they both decide to fight. If Asterix is strong, he defeats Artifis in a fight. If Asterix is weak, Artifis defeats him. They both have the following preferences: if I win in a fight I get +2, if I lose in a fight I get -2. If I yield, I get 0 no matter what my opponent does. If I fight and my opponent yields, I get +1. a) Formulate this situation as a Bayesian game. b) Is there any Bayesian equilibrium in which a strong type of Asterix chooses to yield? Why? c) For what beliefs p of Artifis is there a pooling Bayesian Nash equilibrium in this game? Find this equilibrium. d) For what beliefs p of Artifis is there a separating Bayesian Nash equilibrium in this game? Find this equilibrium.

5 Problem A (Iterative Elimination of Dominated Strategies) Consider the following game: Player 1 Player 2 L R U 6,2 0,2 M 1,3 6,2 D 3,1 2,3 a) Solve the game by iterative elimination of dominated strategies. Each time, indicate which strategy dominates a given strategy. Determine the Nash equilibrium which is the solution of this process. (Hint: Mixed strategies are also allowed in the procedure of elimination.)... b) Did we lose some Nash equilibria by using procedure of iterative elimination of dominated strategies? If yes, write which one..... Problem B (SPNE) Consider the following game in extensive form: a) How many subgames does this game have? No. of subgames:... b) Find all pure-strategy Subgame Perfect Nash equilibria, specify equilibrium strategies and payoffs: