MANUAL
CHAPTER 2
SOLUTIONS 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
3. For the Team-project game, suppose a jock is matched up with a sorority girl as shown in FIGURE PR3.3.
FIGURE PR3.3
Low Low 3,0 2,2 1,6
Sorority girl
Moderate 4,1 3,4 2,5 High 5,2 4,3 3,4
Jock
Moderate High
a. Assume that both are rational and that the jock knows that the sorority girl is rational. What happens?
ANSWER: For the jock, both low and moderate strictly dominate high, and low
strictly dominates moderate. None of the sorority girls strategies is strictly dominated, however. After eliminating the strictly dominated strategies, the reduced game is as shown in FIGURE SOL3.3.1. As we dont know what the sorority girl believes about the jock, we cannot go any further. The answer is then that the jock chooses low and the sorority girl chooses low, moderate, or high.
FIGURE SOL3.3.1
Low
Sorority girl
Moderate 4,1 High 5,2
Jock
Low
3,0
b. Assume that both are rational and that the sorority girl knows that the jock is rational. What happens?
ANSWER: With the game shown in FIGURE SOL3.3.1, the sorority girl now knows the jock is rational and thus will play low. Hence, she should choose high as it strictly dominates both low and moderate. Hence, the jock chooses low effort and the sorority girl chooses high effort.
4. Consider the strategic form game shown in FIGURE PR3.4.
FIGURE PR3.4
x a
Player 2
y 1,1 2,2 1,2 z 0,2 1,0 3,0 1,3 3,1 0,2
Player 1
b c
a. Assume that both players are rational. What happens?
ANSWER: For player 1, a is strictly dominated by b. Neither b nor c is strictly
dominated. For player 2, z is strictly dominated by x. Player 1 plays either b or c and player 2 plays either x or y.
b. Assume that both players are rational and that each believes that the other is rational. What happens?
ANSWER: By the assumption, we can go two rounds of the iterative deletion of strictly dominated strategies (IDSDS). After eliminating the strictly dominated strategies, the game is as shown in FIGURE SOL3.4.1. Now b strictly dominates c for player 1. Neither of player 2s strategies is strictly dominated. Thus, player 1 chooses b and player 2 chooses either x or y.
FIGURE SOL3.4.2 Player 2
x y 2,2
Player 1 b
3,1
5. For the strategic form game shown in FIGURE PR3.5, derive the strategies that survive the iterative deletion of strictly dominated strategies.
FIGURE PR3.5
x a
Player 2
y 3,4 3,2 4,4 1,5 z 2,1 3,3 0,4 3,0 5,2 4,4 3,5 2,3
Player 1
b c d
ANSWER: For player 1, no strategy is strictly dominated. Between a and b, a is better when player 2 uses x, while b is better when player 2 uses z. Hence, a does not strictly dominate b and b does not strictly dominate a. With a and c, a is better when player 2 uses x, while c is better when player 2 uses y. With a and d, a is better when player 2 uses x, while d is better when player 2 uses z. With b and c, b is better when player 2 uses x, while c is better when player 2 uses y. With b and d, b is better when player 2 uses x, while d is just as good when player 2 uses z. (Note that b weakly dominates d but does not strictly dominate it.) Finally, with c and d, c is better when player 2 uses x, while d is better when player 2 uses z. Hence, none of player 1s strategies is strictly dominated. Turning to player 2, x strictly dominates z as it yields a strictly higher payoff for all strategies of player 1. We can then eliminate z. With x and y, x is better when player 1 uses b, but y is better when player 1 uses a. After the first round of the iterative deletion of strictly dominated strategies (IDSDS), we are left with {a,b,c,d} and {x,y}. The remaining game is then as shown in FIGURE SOL3.5.1. FIGURE SOL3.5.1 Player 2
x a
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y 3,4 3,2 4,4 1,5
5,2 4,4 3,5 2,3
Player 1
b c d
SOLUTIONS MANUAL
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
Now d is strictly dominated by a (as well as by b and c). One can show that no other strategies are strictly dominated. The surviving game is then as shown in FIGURE SOL3.5.2.
FIGURE SOL3.5.2 Player 2
x a 5,2 4,4 3,5 y 3,4 3,2 4,4
Player 1 b
c
No further strategies can be eliminated. All we can conclude is that player 1 will play either a, b, or c and player 2 will play either x or y.
6. Two Celtic clansthe Garbh Clan and the Conchubhair Clanare set to battle. (Pronounce them as youd like; I dont speak Gaelic.) According to tradition, the leader of each clan selects one warrior and the two warriors chosen engage in a fight to the death, the winner determining which will be the dominant clan. The three top warriors for Garbh are Bevan (which is Gaelic for youthful warrior), Cathal (strong in battle), and Duer (heroic). For Conchubhair, it is Fagan (fiery one), Guy (sensible), and Neal (champion). The leaders of the two clans know the following information about their warriors, and each knows that the other leader knows it, and furthermore, each leader knows that the other leader knows that the other leader knows it, and so forth (in other words, the game is common knowledge): Bevan is superior to Cathal against Guy and Neal, but Cathal is superior to Bevan against Fagan. Cathal is superior to Duer against
Conchubhair Clan
Fagan Bevan 2,1 3,0 1,2 Guy 1,2 0,1 1,0 Neal 2,0 1,2 0,0
Garbh Clan
Cathal Duer
7. Consider the two-player strategic form game depicted in FIGURE PR3.7.
SOLUTIONS MANUAL
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
3-5
FIGURE PR3.7
w a 1,2 1,3 2,3 3,4 b c d
Player 2
x 0,5 5,2 4,0 2,1 y 2,2 5,3 3,3 4,0 z 4,0 2,0 6,2 7,5
Player 1
a. Derive the strategies that survive the iterative deletion of strictly dominated strategies.
ANSWER: Examining player 1s strategies, first note that d is optimal for player 1 when player 2 is expected to use w. Thus, d cannot be strictly dominated since to be strictly dominated requires that there is another strategy that yields a higher payoff for all strategies of the other player. Since b is best for player 1 when 2 uses x, then b is not strictly dominated either. c is not strictly dominated since it yields a higher payoff than a and b when player 2 uses w and a higher payoff than d when player 2 uses x. a is strictly dominated by c (and also by d) in that c yields a higher payoff than a for any strategy of player 2. We then find that the set of strategies for player 1 that survive the first round of the iterative deletion of strictly dominated strategies (IDSDS) is {b,c,d}. Turning to player 2s strategies, x is best for player 2 when player 1 uses a, w and y are both optimal when player 1 uses b, and z is best when player 1 uses d. Thus, none of player 2s strategies is strictly dominated. After one round of IDSDS, the game is as shown in FIGURE SOL3.7.1. FIGURE SOL3.7.1
w b 1,3 2,3 3,4
Player 2
x 5,2 4,0 2,1 y 5,3 3,3 4,0 z 2,0 6,2 7,5
Player 1 c
d
Since we failed to eliminate any of player 2s strategies in the first round, we are unable to eliminate any of player 1s strategies in the second round (if you are unconvinced by this statement, check for yourself). Turning to player 2, w and y are best when player 1 uses b and z is best when player 1 uses d. Thus, w, y, and z are not strictly dominated. However, x is strictly dominated by w. After two rounds of IDSDS, the game is as shown in FIGURE SOL3.7.2.
FIGURE SOL3.7.2
w b
Player 2
y 5,3 3,3 4,0 z 2,0 6,2 7,5
1,3 2,3 3,4
Player 1
c d
For player 1, d is best when player 2 uses z and b is best when player 2 uses y. However, c is strictly dominated by d. Since none of player 1s strategies was eliminated in the second round, none of player 2s strategies can be eliminated in the third round. After three rounds of IDSDS, the game is as shown in FIGURE SOL3.7.3.
FIGURE SOL3.7.3
w
Player 2
y 5,3 4,0 z 2,0 7,5
Player 1
b d
1,3 3,4
SOLUTIONS MANUAL
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
Since none of player 2s strategies was eliminated in the third round, none of player 1s strategies can be eliminated in the fourth round. Since w and y are both optimal when player 1 uses b, and z is optimal when player 1 uses d, then we cannot eliminate any of player 2s strategies. Given that no strategies are eliminated in this round, no strategies can be eliminated in any further rounds. We conclude that { } b,d for player 1 and {w,y,z} for player 2 are the strategies that survive the IDSDS. b. Derive the strategies that survive the iterative deletion of weakly dominated strategies. (The procedure works the same as the iterative deletion of strictly dominated strategies, except that you eliminate all weakly dominated strategies at each stage.)
strategy for player 1 when player 2 is expected to use w. Thus, d cannot be weakly dominated since to be weakly dominated requires that there is another strategy that yields at least as high a payoff for all strategies of the other player and a strictly higher payoff for some strategies of the other player. Since b is the unique optimal strategy for player 1 when player 2 uses x, then b is not weakly dominated either. c is not weakly dominated since it yields a strictly higher payoff than a and b when player 2 uses w and a strictly higher payoff than d when player 2 uses x. a is weakly (and strictly) dominated by c (and also by d). We then find that the set of strategies for player 1 which survive the first round of the iterative deletion of weakly dominated strategies (IDSDS) is {b,c,d}. Turning to player 2s strategies, x is the unique optimal strategy for player 2 when player 1 uses a and z is the unique optimal strategy when player 1 uses d. w weakly dominates y since it yields an identical payoff when 1 player uses a, b, or c and a strictly higher payoff when player 1 uses d. w is not weakly dominated by x since it yields a strictly higher payoff when player 1 uses b, and it is not weakly dominated by z since it yields a strictly higher payoff when player 1 uses a. Therefore, the set of strategies for player 1 that survive the first round of the IDSDS is {w,x,z}. After one round of IDSDS, the game is as shown in FIGURE SOL3.7.4.
ANSWER: Examining player 1s strategies, first note that d is the unique optimal
FIGURE SOL3.7.4
w b
Player 2
x 5,2 4,0 2,1 z 2,0 6,2 7,5
1,3 2,3 3,4
Player 1
c d
d is not weakly dominated since it is the unique optimal strategy when player 2 uses w, and b is not weakly dominated since it is the unique optimal strategy when player 2 uses x. c is not weakly dominated by b since it yields a strictly higher payoff when player 2 uses w, and it is not weakly dominated by d since it yields a strictly higher payoff when player 2 uses x. None of player 1s strategies is eliminated in the second round of IDSDS. Turning to player 2, w is the unique optimal strategy when player 1 uses b and z is the unique optimal strategy when player 1 uses d. x is strictly and thus weakly dominated by w. After two rounds of IDSDS, the game is as shown in FIGURE SOL3.7.5.
FIGURE SOL3.7.5 Player 2
w b 1,3 2,3 3,4 z 2,0 6,2 7,5
Player 1
c d
For player 1, d strictly and therefore weakly dominates both b and c. Since none of player 1s strategies was eliminated in the second round, none of player 2s strategies can be eliminated in the third round. After three rounds of IDSDS, the game is as shown in FIGURE SOL3.7.6.
SOLUTIONS MANUAL
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
3-7
FIGURE SOL3.7.6 Player 2
w z 7,5
Player 1 d
3,4
There is nothing left for player 1 to do. Since z yields a strictly higher payoff than w when player 1 uses d, then z strictly and therefore weakly dominates w. 8. Consider the three-player game shown in FIGURE PR3.8. Player 1 selects a row, either a1, b1 or c1. Player 2 selects a column, either a2 or b2. Player 3 selects a matrix, either a3 or b3. The first number in a cell is player 1s payoff, the second number is player 2s payoff, and the last number is player 3s payoff. Derive the strategies that survive the iterative deletion of strictly dominated strategies.
FIGURE PR3.8
a3 a2 3,1,0 0,3,1 1,0,2 b2 2,3,1 1,1,0 1,2,1 a2 3,1,1 2,0,2 1,1,1
b3 a1 b1 c1 b2 1,3,2 2,2,1 0,2,0
a1 b1 c1
ANSWER: To begin, consider player 1. Neither a1 nor b1 is strictly dominated, as a1 yields the highest payoff for player 1 when players 2 and 3 choose (a2 ,a3), while b1 is best when players 2 and 3 choose (b2 ,b3). However, a1 strictly dominates c1. Thus, the surviving strategies for player 1 are a1 and b1. Turning to player 2, neither of her strategies is strictly dominated since a2 is best when players 1 and 3 choose (b1,a3) and b2 is best when players 1 and 3 choose (a1, a3). Finally, neither of player 3s strategies is strictly dominated as a3 is best when players 1 and 2 choose (c1,a2) and b3 is best when players 1 and 2 choose (a1,a2). After the first round, the reduced game is as shown in FIGURE SOL3.8.1. FIGURE SOL3.8.1 a3
a2 a1 b1 3,1,0 0,3,1 b2 2,3,1 1,1,0 a1 b1 a2 3,1,1 2,0,2
b3
b2 1,3,2 2,2,1
Since no strategies of players 2 and 3 were eliminated in the first round, no strategies of player 1 can be eliminated in the second round. Neither of player 2s strategies is strictly dominated, as a2 is best when players 1 and 3 choose (b1, a3) and b2 is best when players 1 and 3 choose (a1, a3). For player 3, b3 strictly dominates a3. After the second round, the reduced game is as shown in FIGURE SOL3.8.2.
FIGURE SOL3.8.2 b3
a2 a1 b1 3,1,1 2,0,2 b2 1,3,2 2,2,1
In round 3, neither of player 1s strategies is strictly dominated, but for player 2, b2 strictly dominates a2. After the third round, the reduced game is as shown in FIGURE SOL3.8.3.
w
z 7,5
Player 1 d
3,4
There is nothing left for player 1 to do. Since z yields a strictly higher payoff than w when player 1 uses d, then z strictly and therefore weakly dominates w. 8. Consider the three-player game shown in FIGURE PR3.8. Player 1 selects a row, either a1, b1 or c1. Player 2 selects a column, either a2 or b2. Player 3 selects a matrix, either a3 or b3. The first number in a cell is player 1s payoff, the second number is player 2s payoff, and the last number is player 3s payoff. Derive the strategies that survive the iterative deletion of strictly dominated strategies.
FIGURE PR3.8
a3 a2 3,1,0 0,3,1 1,0,2 b2 2,3,1 1,1,0 1,2,1 a2 3,1,1 2,0,2 1,1,1
b3 a1 b1 c1 b2 1,3,2 2,2,1 0,2,0
a1 b1 c1
ANSWER: To begin, consider player 1. Neither a1 nor b1 is strictly dominated, as a1 yields the highest payoff for player 1 when players 2 and 3 choose (a2 ,a3), while b1 is best when players 2 and 3 choose (b2 ,b3). However, a1 strictly dominates c1. Thus, the surviving strategies for player 1 are a1 and b1. Turning to player 2, neither of her strategies is strictly dominated since a2 is best when players 1 and 3 choose (b1,a3) and b2 is best when players 1 and 3 choose (a1, a3). Finally, neither of player 3s strategies is strictly dominated as a3 is best when players 1 and 2 choose (c1,a2) and b3 is best when players 1 and 2 choose (a1,a2). After the first round, the reduced game is as shown in FIGURE SOL3.8.1. FIGURE SOL3.8.1 a3
a2 a1 b1 3,1,0 0,3,1 b2 2,3,1 1,1,0 a1 b1 a2 3,1,1 2,0,2
b3
b2 1,3,2 2,2,1
Since no strategies of players 2 and 3 were eliminated in the first round, no strategies of player 1 can be eliminated in the second round. Neither of player 2s strategies is strictly dominated, as a2 is best when players 1 and 3 choose (b1, a3) and b2 is best when players 1 and 3 choose (a1, a3). For player 3, b3 strictly dominates a3. After the second round, the reduced game is as shown in FIGURE SOL3.8.2.
FIGURE SOL3.8.2 b3
a2 a1 b1 3,1,1 2,0,2 b2 1,3,2 2,2,1
In round 3, neither of player 1s strategies is strictly dominated, but for player 2, b2 strictly dominates a2. After the third round, the reduced game is as shown in FIGURE SOL3.8.3.
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SOLUTIONS MANUAL
CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
FIGURE SOL3.8.3 b3
b2 a1 b1 1,3,2 2,2,1
In the fourth round, b1 strictly dominates a1 and this solves the game since each player has only one strategy remaining. Thus, by the iterative deletion of strictly dominated strategies, we conclude that the strategy profile that will be played is (b1,b2 ,b3).
9. A gang controls the drug trade along North Avenue between Maryland Avenue and Barclay Street. The city grid is shown in FIGURE PR3.9a. The gang leader sets the price of the drug being sold and assigns two gang members to place themselves along North Avenue. He tells each of them that theyll be paid 20% of the money they collect. The only decision that each of the drug dealers has is whether to locate at the corner of North Avenue and either Maryland Avenue, Charles Street, St. Paul Street, Calvert Street, or Barclay Street. The strategy set of each drug dealer is then composed of the latter five streets. Since the price is fixed by the leader and the gang members care only about money, each member wants to locate so as to maximize the number of units he sells.
FIGURE PR3.9a
Maryland Avenue
Charles Street
St. Paul Street
Calvert Street
Barclay Street North Avenue
For simplicity, assume that the five streets are equidistant from each other. Drug customers live only along North Avenue and are evenly distributed between Maryland Avenue and Barclay Street (so there are no customers who live to the left of Maryland Avenue or to the right of Barclay Street). Customers know that the two dealers set the same price, so they buy from the dealer that is closest to them. The total number of units sold on North Avenue is fixed. The only issue is whether a customer buys from drug dealer 1 or drug dealer 2. This means that a drug dealer will want to locate so as to maximize his share of customers. We can then think about a drug dealers payoff as being his customer share. FIGURE PR3.9b shows the customer shares or payoffs. Let us go through a few so that you understand how they were derived. For example, suppose dealer 1 locates at the corner of Maryland and North and dealer 2 parks his wares at the corner of Charles and North. All customers who live between Maryland and Charles buy from dealer 1, as he is the closest to them, while the customers who live to the right of St. Paul buy from dealer 2. Hence, dealer 1 gets 25% of 3 the market and dealer 2 gets 75%. Thus, we see that (1 , 4 ) are the payoffs for strategy pair 4 (Maryland, St. Paul). Now, suppose instead that dealer 2 locates at Charles and dealer 1 at Maryland. The customer who lies exactly between Maryland and Charles will be indifferent as to whom to buy from. All those customers to his left will prefer the dealer at Maryland, 7 and they make up one-eighth of the street. Thus, the payoffs are (1 , 8 ) for the strategy pair 8 (Maryland, Charles). If two dealers locate at the same street corner, well suppose that customers divide themselves equally between the two dealers, so the payoffs are (1 , 1 ). Using 22 the iterative deletion of strictly dominated strategies, find where the drug dealers locate.
FIGURE PR3.9b Drug Dealers Payoffs Based on Location Dealer 2's location
Maryland Maryland Charles St. Paul Calvert Barclay
11 , 22 71 , 88 31 , 44 53 , 88 11 , 22
Charles
17 , 88 11 , 22 53 , 88 11 , 22 35 , 88
St. Paul
13 , 44 35 , 88 11 , 22 35 , 88 13 , 44
Calvert
35 , 88 11 , 22 53 , 88 11 , 22 17 , 88
Barclay
11 , 22 53 , 88 31 , 44 71 , 88 11 , 22
Dealer 1's location