# 2 economics questions.

*label*Economics

*timer*Asked: Nov 18th, 2016

**Question description**

Q1. In the penny-in-a-box game, two players P and Q alternate over three rounds as follows:

i. P places a penny in a box, either head up or tail up, at his discretion. Then, P seals the box in such a way that Q cannot detect the penny orientation, and hands it over to Q.

ii. At this point, Q can either ﬂip the box or leave it as is, and hands it back to P, who had no way of observing Q's action.

iii. Finally, P decides whether to ﬂip the box or leave it as is. The two players open the box. If the penny comes up head P wins, otherwise P loses. For this game,

a. Write the game in extensive form, and then convert the extensive form into strategic form.

b. Model the strategic form game as a linear program and ﬁnd the optimal mixed strategies for the two players.

c. Assume that P uses his optimal mixed strategy. Describe a deterministic strategy that Q can use to maximize her winnings.

d. Demonstrate that the last move is superﬂuous by showing the equivalence with a game for P and Q consisting only of the ﬁrst two moves. 5

Q2. Assume that there are four chips of value 6, 5, 2, 7. There are two players- player 1 and player 2. The players can pick any chip one at a time. The one who will have maximum value chips at the end will win. Thus the game ends in two rounds.

a. Write the game in extensive form, and then convert the extensive form into strategic form.

b. Model the strategic form game as a linear program and ﬁnd the optimal mixed strategies for the two players.

c. Find an arrangement of at least three rounds for which the optimal strategy is deterministic for both players