6. Imagine a lottery with a jackpot consisting of sixteen $1 coins. There are just 2 people who can buy tickets in this lottery, Adam and Bea. The price of a lottery ticket is $1, and both Adam and Bea have five $1 coins. Let T_A and T_B denote the number of the lottery tickets that Adam and Bea buy, respectively. Since each person has five $1 coins, both strategy sets are {0, 1, 2, 3, 4, 5}. If either T_A > 0 or T_B > 0, the probability that Adam wins the jackpot, P_A, is T_A=(T_A + T_B), and the probability that Bea wins the jackpot, P_B, is T_B=(T_A + T_B). If T_A = T_B = 0 then P_A = P_B = 0. The payoff functions are ?_A = 16P_A - T_A for Adam and ?_B = 16P_B - T_B for Bea.

(a) If you were either Adam or Bea, how many tickets would you buy?

(b) What is the Nash equilibrium of this game?

(c) What are the Pareto-optimal strategy combinations?

(d) Now suppose that Adam has only three $1 coins to spend on lottery tickets, and that Bea still has five. What is the Nash equilibrium of this revised game? What are the Pareto-optimal strategy combinations?

7. There are two friend, Quick and Slow, taking tough Econ 301 course. Both need to decide whether to do the assignment themselves or wait until just it is due and the copy what the other has done. Quick finds the material easy and will get 8 out of 10 questions correct if he does the work while Slow will get only 5 out of 10. If both wait, hoping to copy the other's work, they will

both get zero.

(a) Write out this game in its strategic form.

(b) Do either have the dominant strategy, and if so, what they are?

(c) What outcome would you expect. Explain.

8. Let's follow a story of an impending naval engagement in the Persian Gulf. The grid below shows the positions and the choices of the combats. An Iraqi ship at the point I is about to fire a missile, intending to hit an American ship at A. The missile's path is programmed at the launch; it can travel in a straight line, or make sharp right-angled turns every 20 second. If the Iraqi

missile flew in a straight line from I to A, American missile defences could counter such a trajectory very easily. Therefore the Iraqis will try a path with some zigzags. All such paths that can reach A from I lie along the grid shown. Each length like IF equals the distance the missile can travel in 20 seconds.

The American ship's radar will detect the launch of the incoming Iraqi missile, and the computer will instantly launch an antimissile. The antimissile travels at the same speed as the Iraqi missile, and can make similar 90 degree turns. So the antimissile's path can also be set along the same grid starting at A. However to allow for enough explosives to ensure a damaging

open-air blast, the antimissile has only enough fuel to last one minute, so it can travel just three segments (e.g., A to B, B to C, and C to F, which we can be written as ABCF). If, before or at the end of the minute, American antimissile meets the incoming missile, it will explode and neutralize the threat. Otherwise Iraqi missile will go on to hit American ship. The question is, how should the

trajectories of the two missiles be chosen?