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ECON 0200 Summer 2012 University of Pittsburgh

Michael LeGower

Homework #2 Write your answers to the following questions on separate sheets of paper. Show all of your work. Your answers are due in class on Wednesday, May 30th, 2012. 1. Write the following games out in the form requested. Be sure to label strategies and players. a. Stag hunt. Two hunters, Pete and Joe, are out in the woods, hunting game. If they combine their efforts, they can bring down a deer providing each with plenty of meat (payoff = 9). However, if one chooses to hunt smaller game--- say a rabbit--- while the other is hunting the deer, the rabbit hunter will catch his rabbit (payoff=8), but the deer hunter will be left empty-handed (payoff=0). Finally, if both hunt rabbits independently, they can both catch rabbits, albeit providing less meat than if they split a deer (payoff=7). Write this simultaneous-move game in normal form. Joe Pete

Hare Stag

Hare 7,7 0,8

Stag 8,0 9,9

b. Chicken. Alice and Bob drive toward each other in the middle of a road. As they approach the impact point, each has the option of continuing to drive down the middle of the road or to swerve. Both believe that if only one driver swerves, that driver loses face (payoff = 0) and the other gains in self- esteem (payoff = 2). If neither swerves, they are maimed or killed (payoff = -10). If both swerve, no harm is done to either (payoff = 1). Write this simultaneous-move game in normal form. Bob Swerve Nerve Alice Swerve 1,1 0,2 Nerve 2,0 -10,-10 c. Prisoner’s dilemma two ways. Nature flips a coin. If the coin comes up heads, Walter and Jesse must play a simultaneous-move prisoner’s dilemma game. If the coin comes up tails, Walter and Jesse must play a sequential-move prisoner’s dilemma game. For both prisoner’s dilemma games, if both “Confess” both get plea bargains (payoff = 3). If both “Deny”, both get off with a slap on the wrist (payoff = 5). If one chooses “Confess” and the other “Deny”, the confessor gets out immediately (payoff = 10) and the denier goes to prison for life (payoff = 1). In the sequential version of the game, Walter moves first. Write this game entirely in extensive form. Remember to include the move of Nature and the associated probabilities of each action (Heads and Tails). Remember also the role of information sets in writing simultaneous-move games.

ECON 0200 Summer 2012 University of Pittsburgh

Michael LeGower

Nature Heads, w/ prob 0.5

Tails w/ prob 0.5

Walter Confess

Walter Deny

Jesse

Confess Jesse

Deny

Jesse

Confess

Deny

Confess

Deny

3,3

10,1

1,10

5,5

Confess

Jesse Deny 10,1

3,3

Confess 1,10

Deny 5,5

2. Hawk-Dove. Two birds arrive simultaneously at a nesting area. Each can choose one of two strategies: “Hawk”, which means fight; or “Dove”, which means don’t fight. Suppose the value of the nesting area is V > 0. If both birds play “Dove”, they share the nesting area and earn a payoff of V/2 each. If one plays “Hawk” and the other plays “Dove”, the bird playing “Hawk” wins exclusive use of the nesting area, earning a payoff of V, and the bird playing “Dove” must fly away, earning a payoff of 0. Finally, if both birds play “Hawk” they earn a payoff of W > 0. a. Write this game in normal form, indicating the players, strategies, and payoffs. Bird 2 Bird 2

Hawk Dove

Hawk W,W 0,V

Dove V,0 V/2 , V/2

b. Make and defend an assumption about the value of W, i.e. whether W > V/2 or W < V/2. Carefully explain your reasoning. If the two birds are evenly matched and both play Hawk, then each would have a 50% chance of winning the resulting fight. Say that if a bird wins it would get exclusive use of the nesting area, which we have established provides a payoff of V. Say if a bird loses, it suffers some injury which brings with it a payoff of –C < 0. So if both play Hawk, the expected payoff to each bird is W = .5(V) – .5(C). So the W payoff should be less than V/2, what they could both get by sharing the nesting area and avoiding possible injury. c. Given your assumption in part b, what is/are the pure strategy Nash equilibrium/equilibria of this game? Does either bird have a dominant strategy? Both birds have a dominant strategy: Hawk: if the other bird plays Hawk, the best response is Hawk (W>0) and if the other bird plays Dove, the best response is Hawk (V>V/2). The pure strategy equilibrium is (Hawk, Hawk).

ECON 0200 Summer 2012 University of Pittsburgh

Michael LeGower

3. Find all of the pure strategy Nash equilibria in the following two-player, simultaneous move games (there may be more than one such equilibrium in each game). Also state whether either player 1 or player 2 has a dominant strategy and whether the Nash equilibrium is an equilibrium in dominant strategies. Remember: a Nash equilibrium is a list of strategies for all players. a. Player 2 Player 1

X Y

A 0,0 1*,2*

B 2,100* 3*,1

Player 1 has a dominant strategy: Y. Player 2 does not have a dominant strategy. The NE is (Y,A). b. Player 2 Player 1

X Y

A 3*,5* -4,-11

B 4,1 10*,-9*

Neither Player 1 nor Player 2 has a dominant strategy. The NE are (X,A) and (Y,B). c. Player 2 Player 1

X Y Z

A 10*,3 0,1 8,8

B 6*,4* 4,5* 2,9*

Player 1 has a dominant strategy: X. Player 2 does not have a dominant strategy. The NE is (X,B). d.

Player 1

X Y Z

A 15,15 30,40 35*,25

Player 2 B 40*,35 25,25 50,30

C 20,45* 40*,50* 35,35*

Player 1 does not have a dominant strategy. Player 2 has a dominant strategy: C. The NE is (Y,C).

ECON 0200 Summer 2012 University of Pittsburgh

Michael LeGower

e.

X Y Z

Player 1

Player 2 B 6*,2 4,4 4,6

A 4*,4* 2,6* 4*,0

C 0,4* 6*,4 4,4*

Neither Player 1 nor Player 2 has a dominant strategy. The NE is (X,A). 4. Odd or even. Two players simultaneously call out one of two numbers: 1 or 2. One player’s name is Odd; he wins if the sum of the two numbers chosen is odd. The other player’s name is Even; she wins if the sum of the two numbers chosen is even. The amount paid to the winner by the loser is always the sum of the two chosen numbers in dollars. a. Write this game in normal form. Be careful in calculating the payoffs to each player and in labeling the players and their strategies. Odd Even

1 2,-2 -3,3

1 2

2 -3,3 4,-4

b. A zero-sum game is one in which, for every outcome, the sum of all players payoffs adds to zero. Is this a zero-sum game? Explain why or why not. This is a zero sum game. In every outcome, the payoffs for the two players sum to zero because the loser must pay the winner his or her winnings. c. Is there a pure strategy Nash equilibrium for this game? If so, what is it? If not, how do you think the game should be played? There is no pure strategy Nash equilibrium for this game. It is a discoordination game: Even wants the players to coordinate and Odd wants the players not to coordinate. As such, if you play a pure strategy (1 or 2), the other play will be able to ensure a win. So you should randomize over your strategies when playing this game. 5. Jack and Jill. Two players, Jack and Jill, are put in separate rooms and told the rules of the game. Each is to pick one of six letters; G, K, L, Q, R, or W. If the two happen to choose the same letter, both get prizes as follows. Letter Jack’s Prize

G 3

K 2

L 6

Q 3

R 4

W 5

ECON 0200 Summer 2012 University of Pittsburgh Jill’s Prize

6

Michael LeGower 5

4

3

2

1

If they both choose different letters, they each get a payoff of 0. They have common knowledge of the schedule of prizes, i.e. this schedule of prizes is revealed to both players and both know that the schedule is revealed to both players. a. Write this game in normal form. Jill

Jack

G K L Q R W

G 3,6 0,0 0,0 0,0 0,0 0,0

K 0,0 2,5 0,0 0,0 0,0 0,0

L 0,0 0,0 6,4 0,0 0,0 0,0

Q 0,0 0,0 0,0 3,3 0,0 0,0

R 0,0 0,0 0,0 0,0 4,2 0,0

W 0,0 0,0 0,0 0,0 0,0 5,1

b. What is/are the pure strategy Nash equilibrium/equilibria of this game? The pure strategy equilibria are any strategy combinations where Jack and Jill choose the same letter: (G,G); (K,K); (L,L); (Q,Q); (R,R); (W,W). c. Can one of the combinations of strategies be regarded as a focal point? Which one? Explain. I would argue that the focal strategy combination is the one that gives both Jack and Jill positive and equal payoffs: (Q,Q). 6. Cournot duopoly game. Suppose the market for widgets is dominated by two firms: firm A and firm B. The market price of Widgets depends on the combined quantity or widgets produced by firms A and B, such that,

, where

quantity produced by firm A, and cost of making each widget ( given by

is the price of widgets,

is the

is the quantity produced by firm B. Assume further that the ) is the same for both firms. Note that firm A’s profits are

(and firm B’s profits are given analogously). Suppose that each firm can

choose to produce just four quantities of widgets: 100, 120, 140, or 180. a. First, construct a 4x4 table showing the market price of widgets for each combination of firm A and firm B quantities. Since each firm has 4 possible quantities, there should be 16 prices. Firm B 100

100 100-200/4= $50

120 100-220/4= $45

140 100-240/4= $40

180 100-280/4= $30

ECON 0200 Summer 2012 University of Pittsburgh Firm A

120 140 180

100-220/4= $45 100-240/4= $40 100-280/4= $30

Michael LeGower 100-240/4= $40 100-260/4= $35 100-300/4= $25

b. Using this price information and the fact that

100-260/4= $35 100-280/4= $30 100-320/4= $20

100-300/4= $25 100-320/4= $20 100-360/4= $10

, write this game in normal form by

calculating the profits earned by each firm for each combination of quantities and filling in the appropriate 4x4 matrix.

Firm A

Firm B 100 120 140 100 $4,200; $4,200 $3,700; $4,440 $3,200; $4,480 120 $4,440; $3,700 $3,840; $3,840 $3,240; $3,780 140 $4,480; $3,200 $3,780; $3,240 $3,080; $3,080 180 $3,960; $2,200 $3,060; $2,040 $2,160; $1,680

180 $2,200; $3,960 $2,040; $3,060 $1,680; $2,160 $360; $360

c. Using the table constructed in part b, find the Nash equilibrium of this quantity choice game. What quantity of widgets does each firm produce in equilibrium? The NE of the game is for each firm to produce 120 units. That is the only combination of quantities where neither firm can unilaterally increase or decrease production and earn higher profits. 7. Pitt versus Penn State. Pitt and Penn State must decide on next year’s in-state tuition for fulltime undergraduate students. Surprisingly, neither university uses cost accounting methods to determine what the percentage increase in tuition should be. Instead, Pitt plans to increase Pitt tuition by 4% plus one-half of Penn State’s percentage increase in tuition. Similarly, Penn State plans to increase Penn State tuition by 3% plus two-thirds of Pitt’s percentage increase in tuition. a. Write the reaction functions (best response functions) for both universities. Let ∆TPitt be Pitt’s tuition increase in percentage terms and ∆T PSU be Penn State’s tuition increase in percentage terms: ∆TPitt=4+(1/2)* ∆TPSU ∆TPSU=3+(2/3)*∆TPitt b. Find the percentage increase in tuition at both universities and explain why these comprise a Nash equilibrium. Whose tuition increase is the highest? The Nash equilibrium is the pair of tuition increases where each university is best responding to the tuition increase of the other university. As a result, it will correspond to the values of ∆TPitt & ∆TPSU where the two reaction functions intersect (or equivalently the values that solve the 2 equation system above). ∆TPitt=4+(1/2)* (3+(2/3)*∆TPitt)

ECON 0200 Summer 2012 University of Pittsburgh ∆TPitt=5.5+1/3*∆TPitt (2/3)*∆TPitt=5.5 ∆TPitt=8.25 ∆TPSU=3+(2/3)* 8.25=8.5 Penn State’s tuition increase is higher.

Michael LeGower...