Assignment

Exercise 1

In the following matrices, payo s are in alphabetical order.

For each game, identify

a) all strategy pro les which survive iterated elimination of strictly domi-nated strategies,

b) all strategy pro les which survive iterated elimination of weakly domi-nated strategies, and

c) all Nash equilibria (pure and mixed strategies).

Exercise 2

In the following trees, payo s are in alphabetical order.

For each tree, identify

a) all all pure strategy Nash equilibria, and

b) all pure strategy subgame perfect Nash equilibria.

Exercise 3

There are two chickens|Ann and Bob. As it is well known, their main activity is to cross the road. At least, this is what they do in this game: they cross the road and try to remain alive.

Crossing the road is risky because our chickens can be killed by drivers. Let pA denote the probability of Ann crossing the road alive. Let pB denote the probability of Bob crossing the road alive. Let a denote Ann's e ort she exerts while crossing the road. Let b denote Bob's e ort he exerts while crossing the road. E orts represent how much attention the chickens pay when crossing the road. We assume that a; b 2 [0; 1].

E orts determine players' probabilities of survival. In particular, we assume that pA = a and pB = b. For example, if Ann pays no attention at all (i.e., a = 0), then she dies for sure (i.e., pA = 0). We also assume that e orts are costly and each player has quadratic cost of e ort: 12 a2 for Ann and 12 b2 for Bob. If a chicken dies, then his/her utility is zero. If a chicken is the only one that survives, then his/her utility is 0.75. Finally, if two chickens survive, then each gets utility 0.5. Given all the information we have, we derive utility function of Ann, uA(a; b) = 0:5ab + 0:75a(1 b) 0:5a2, and utility function of Bob, uB(a; b) = 0:5ab + 0:75b(1 a) 0:5b2.

a) Assume that the chickens make choices simultaneously (i.e., static game). What is the Nash equilibrium (NE)?

b) Assume that the chickens make choices in stages (i.e., dynamic game). In par-ticular, in stage 1, Ann chooses her e ort a. Then, in stage 2, Bob observes Ann's choice a and chooses his e ort b. What is the sub-game perfect Nash equilibrium (SPNE)?