1. Consider the Rock-Paper-Scissors game: Two players simultaneously throw their right arms up and down to the count of “one, two, three.”(Nothing strategic happens as they do this.) On the count of three, each player quickly forms his or her hand into the shape of either a rock, a piece of paper, or a pair of scissors. Abbreviate these shapes as R, P, and S, respectively. he players make this choice at the same time. If the players pick the same shape, then the game ends in a tie. Otherwise, one of the players wins and the other loses. he winner is determined by the following rule: rock beats scissors, scissors beats

paper, and paper beats rock. Each player obtains a payof of 1 if he or she wins, −1 if he or she loses, and 0 if he or she ties. (a)  [1pt] Represent this game in extensive form. Does the order Answer:  See fgure below.  he order does not mater.

of players ma#er here?

0: 0

2 

1: −1

1: −1

−1: 1

−1: 1 

2

0: 0

(b)  [1pt] Represent this game in normal form.

Answer: See table below.

R        P        s

R    0: 0   −1: 1   1: −1

P   1: −1     0: 0   −1: 1

s  −1: 1   1: −1     0: 0

2. Consider a modifcation of the Rock-Paper-Scissors game in which P2 is an expert in body language tells.  In particular, assume that P2 can tell whether P1 is about to play R or not. At the same time, P2 cannot tell whether P1 is about to play P or S. In other words, P2 can partially observe P1’s choice.

(a)  [1pt] Represent this game in extensive form.                            (b)  [1pt] Represent this game in normal form.

Answer: See Figure 1 below.                                                    Answer: See table below.

3. Consider the following normal form game.

L      R

R        

P       -J: J

s          J: -J

J: -J

-J: J

-J: J

2

s\             0: 0

Figure 1: Answer to 2a.

In parts (a-d): Is player 1’s strategy sJ dominated? If your answer is yes, describe a strategy eJ that dominates it. If your answer is no, describe a belief 9S to which sJ is a best response.

(a)  [0.5pt] sJ = A. Answer:  No. 9S = R.

(b)  [0.5pt] sJ = B. Answer:  Yes. eJ = (0,Q: 0: 0: 0,寸)

(c)  [0.5pt] sJ = c. Answer:  Yes. eJ = (0,?: 0: 0: 0,\)

(d)  [0.5pt] sJ = D. Answer:  No. 9S = L.

(e)  [1pt] Find all inefcient outcomes. For each such outcome s, provide another outcome s\ which is more efcient than s. Answer:  he set of inefcient outcomes is I = s/{(A: L): (D: L)}. (D: L) is more efcient than any outcome except (A: L) and (D: L).

4. A driver (D) and a policeman (P) play the following simultaneous-move game. he driver chooses her speed (the maximal car speed is 150 MPH) and the policeman chooses whether to monitor the road with his radar or not. If the car’s speed is s 90 and if the policeman monitors, then the driver gets a ticket. When the driver’s speed is x, her payof is Sx if she does

not get a ticket, and -x if she does. he best outcome for the policeman is that he monitors the road and catches a speeding driver. All the other outcomes are his second best (that is, the policeman has only two utility levels).

(a)  [1pt] Represent this game in normal form.

I = {D: P }

sD = [0: J?0]

sP = {m: n}

-x:   sP = m and x s o0 uD (x: sP ) = Sx:    sP = n or x s o0

J:   sP = m and x s o0

uP (x: sP ) = 0:   sP = n or x s o0

(b)  [1pt] Find all efcient outcomes.

Hint: Use the defnition: Take an arbitrary outcome (x: sP ) and see if you can fnd a more efcient outcome. You might fnd it useful to examine some arbitrary outcomes, say, (?0: m): (J00: m): (?0: n): (J00: m) and then try to generalize your observations.

Answer:  he only efcient outcome is (J?0: n). For any other s e s, there is another outcome s\ that is more efcient:

● (J?0: n) is more efcient than (x: n) when x < J?0

● (J?0: n) is more efcient than (x: m) when x s o0

● (x - .: m) is more efcient than (x: m) when x s o0 and . = (x - o0)\S

(c)  [1pt] For each player, fnd the best response mappings restricted to simple beliefs. hat is, for the driver, calculate BRD(sP ) for each sP e sP and, for the policeman, calculate BRP(x) for all x e sD .