3. Two runners are training for a race (the 100 meter dash!). Lets refer to them as Runner 1 and Runner 2. They both need to decide how much to Train for the race; that is, how much Effort to put forward while training. The winner of the race will receive a prize equal to $100! They simultaneously choose effort, xi , between 0 and 100. Assume that the cost of effort for each runner are given by the following functions:

c1(x1) = x1

c2(x2) = 2 x2

The probability that running i wins the race is given by the following function:

pi = ( xi/X if X > 0

(1/2 if X = 0

where X = x1 + x2.

Note: this is not a symmetric problem since the costs of effort differ between the two runners.

A. Set up the maximization problem for each runner.

B. What is the equilibrium effort decision for each runner? That is, find e1 and e2

C. Does your answer in part B make sense? Explain.

4. In Homework 1, you found the Normal Form game of BM and LM to be:

	C	NC
C	3,5	1,9
NC	4,4	00

where C was Climb and NC was Not Climb and LM was the Row Player and BM was the Column Player. Find the Mixed Strategy Nash Equilibrium (MSNE) of this game assuming that BM plays strategy C with probability and LM plays strategy C with probability .

6. Consider the following situation between two friends, Susan and Adam. Susan and Adam are sitting at a magical table, and in front of Susan is $0.50. The money is split into a large pile, $0.40, and a small pile, $0.10. Susan has only two options to choose from: Pass (P) or Defect (D). If she decides to Defect, the game is over and the table tells her (since the table can somehow talk) that she gets the larger pile and Adam gets the smaller pile. However, if Susan decides to Pass each pile doubles in size and are magically moved in a position in front of Adam. Now, it is Adams turn to decide whether or not to Pass or Defect. Again, if Adam chooses to Defect he gets the larger of two piles, and if he Passes the game continues until the sum of both piles equals $8.00. Remember, if at any point a player decides to Defect, the game ends. (Please ask me if you are having trouble with this question...I can see how it would be hard to set up).

A. Set up the extensive form of this game.

B. Find the SPNE for this game.

7. (Biology Application) Many species of fish are attacked by parasites that attach themselves to gills and inner mouth parts. Often such fish will form a symbiotic relationship with a smaller species of fish for whom the parasite is a major food source. Mutual trust is involved, however, since the larger fish must avoid the tempation of eating the smaller fish, and the smaller fish must avoid the temptation of taking a chunk out of the larger fish, thereby obtaining a meal with much less work than picking around for the tiny parasites. This scenario, which is doubtless even more common and more important for humans than for fish, is explored in the following problem.

	c	d
c	5,5	-3,8
d	8,-3	0,0

where C is choosing to Cooperate and D is choosing to Defect (little fish taking the chunk or the big fish eating the little fish). Like we talked about in class, assume that each fish discount future utility with a discount factor (remember, 0 < < 1). Suppose the fish play this game inifinitely many times.

Definition: A strategy that dictates following one course of action until a certain condition is met and then following a different strategy for the rest of the game is called a Trigger Strategy.

A. Find the PSNE and MSNE of this game (I just want you to practice some more!).

B. Define the Trigger Strategy that exists in this game.

C. Prove that the Trigger Strategy in part A is a Subgame Perfect Nash Equilibrium (SPNE).