**1. Maximize profit:** Firm \(i\) earns \(\pi_i=(10-P_i+P_j)P_i\). Holding the rival’s price fixed, differentiate:
\[\frac{\partial\pi_i}{\partial P_i}=10-2P_i+P_j=0.\]
**2. Check the maximum:** The second derivative is \(-2<0\), and the best response is nonnegative for \(P_j\ge0\). Thus \(BR_i(P_j)=5+P_j/2\).
**Answer: C — \(5+P_j/2\).**
A: **Incorrect:** The rival-price coefficient is 1/2, not 1/3.
B: **Incorrect:** The rival-price coefficient is 1/2, not 2/3.
C: **Correct:** This is the profit-maximizing best response.
D: **Incorrect:** This doubles the correct rival-price coefficient.
Under the Nash equilibrium each player chooses P=15
6406535121094.
This game has unique Nash equilibrium
Published solution
**1. Solve mutual best responses:** \(P_1=5+P_2/2\) and \(P_2=5+P_1/2\) imply \(P_1=P_2=10\).
**2. Identify the false statement:** The equilibrium price is 10, so the claim that each firm sets 15 is false.
**Answer: C — The equilibrium price is not 15.**
A: **Incorrect:** True: exchanging the firms leaves actions and payoff rules unchanged, so do not select it.
B: **Incorrect:** True: the equilibrium (10,10) is symmetric, so do not select it.
C: **Correct:** False: equilibrium prices are 10 rather than 15; select this statement.
D: **Incorrect:** True: the two linear best responses intersect uniquely, so do not select it.
**(Choosing a route)** Four people must drive from A to B at the same time. Two routes are
available, one via X and one via Y (Refer to the left panel of following Figure). The roads from A to
X, and from Y to B are both short and narrow; in each case, one car takes 6 minutes, and each
additional car increases the travel time per car by 3 minutes. (If two cars drive from A to X, for
example, each car takes 9 minutes.) The roads from A to Y, and from X to B are long and wide; on
A to Y one car takes 20 minutes, and each additional car increases the travel time per car by 1
minute; on X to B one car takes 20 minutes, and each additional car increases the travel time per
car by 0.9 minutes. Formulate this situation as a strategic game and answer the given
subquestions.
Choose the correct alternative
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In Nash equilibrium two people take each route
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For the NE action profile, each person’s travel time is either 29.9 or 30 minutes
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Both In Nash equilibrium two people take each route & For the NE action
profile, each person’s travel time is either 29.9 or 30 minutes
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None
Published solution
**1. Express congestion:** If \(k\) drivers use A–X–B, their time is \(26+3.9(k-1)\); the \(4-k\) drivers on A–Y–B take \(26+4(3-k)\).
**2. Test deviations:** With two drivers on each route, times are 29.9 and 30 minutes. Switching would put three drivers on the other route and raise the switcher’s time to 34 or 33.8 minutes. Any other split gives a driver on the crowded route a profitable switch.
**Answer: C — Both statements.**
A: **Incorrect:** The equal split is true, but this option omits the other true claim.
B: **Incorrect:** The travel times are true, but this option omits the equal split.
C: **Correct:** Both claims hold at the equilibrium.
D: **Incorrect:** Both listed claims are true.
**(Choosing a route)** Four people must drive from A to B at the same time. Two routes are
available, one via X and one via Y (Refer to the left panel of following Figure). The roads from A to
X, and from Y to B are both short and narrow; in each case, one car takes 6 minutes, and each
additional car increases the travel time per car by 3 minutes. (If two cars drive from A to X, for
example, each car takes 9 minutes.) The roads from A to Y, and from X to B are long and wide; on
A to Y one car takes 20 minutes, and each additional car increases the travel time per car by 1
minute; on X to B one car takes 20 minutes, and each additional car increases the travel time per
car by 0.9 minutes. Formulate this situation as a strategic game and answer the given
subquestions.
Now suppose that a relatively short, wide road is built from X to Y, giving each person four options
for travel from A to B: A–X–B, A–Y–B, A–X–Y–B, and A–Y– X–B (As shown in the right panel of the
given figure). Assume that a person who takes A–X–Y–B travels the A–X portion at the same time as
someone who takes A–X–B, and the Y–B portion at the same time as someone who takes A–Y–B.
(Think of there being constant flows of traffic.) On the road between X and Y, one car takes 7
minutes and each additional car increases the travel time per car by 1 minute. Consider the Nash
equilibrium in this new situation.
Choose the correct alternative
6406535121099.
In any Nash equilibrium, two persons take A–X–B, one person takes A–X–Y–B,
and one person takes A–Y–B
6406535121100.
In equilibrium profile, every person’s travel time increases compared to the
case when new road was not there
6406535121101.
Both In any Nash equilibrium, two persons take A–X–B, one person takes
A–X–Y–B, and one person takes A–Y–B & In equilibrium profile, every person’s travel time increases
compared to the case when new road was not there
6406535121102.
None
Published solution
**1. Find the equilibrium split:** One driver uses A–X–B, one uses A–Y–B, and two use A–X–Y–B; nobody uses A–Y–X–B. Loads on A–X and Y–B are 3 each, and the new road has load 2.
**2. Calculate and check:** The route times are \(12+20=32\), \(20+12=32\), and \(12+8+12=32\) minutes. A direct-route driver switching to the shortcut takes 35 minutes; a shortcut driver switching to a direct route takes 32.9 or 33 minutes. The reverse shortcut is still longer. Checking the other integer route splits leaves only this equilibrium load pattern.
**3. Compare:** All drivers now take 32 minutes, exceeding the original 29.9 or 30 minutes: a Braess-type congestion effect.
**Answer: B — Every driver’s travel time increases.**
A: **Incorrect:** The correct split is one direct driver on each route and two on the shortcut, not two on A–X–B.
B: **Correct:** Every driver’s time rises to 32 minutes.
C: **Incorrect:** The travel-time statement is true, but the proposed split is false.
D: **Incorrect:** The travel-time statement is true.
An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each
player can bid either 0, 1 or 2. If player **i** bids more than player **j** then **i** wins the good and pays his
bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to
determine who the winner is, who gets the good and pays his bid while the loser pays nothing.
Formulate the game in matrix form and answer the given subquestions.
Which player does not have strictly dominated strategy?
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Player 1
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Player 2
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None
A published solution is not available for this question yet.
An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each
player can bid either 0, 1 or 2. If player **i** bids more than player **j** then **i** wins the good and pays his
bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to
determine who the winner is, who gets the good and pays his bid while the loser pays nothing.
Formulate the game in matrix form and answer the given subquestions.
Which strategies survive Iterated Elimination of Strictly Dominated Strategy?
6406535121143.
(0, 0)
6406535121144.
(1, 1)
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(2, 2)
6406535121146.
None
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|ccc} & 0 & 1 & 2 \\ 0 & (1.5,2.5) & (0,4) & (0,3) \\ 1 & (2,0) & (1,2) & (0,3) \\ 2 & (1,0) & (1,0) & (0.5,1.5)\end{array}\]
**1. Eliminate player 2’s bid 0:** Bid 2 strictly dominates it. With columns 1 and 2 left, player 1’s bid 2 strictly dominates bid 0.
**2. Continue:** With rows 1 and 2 left, player 2’s bid 2 strictly dominates bid 1. Against the remaining column 2, player 1’s bid 2 yields 0.5 whereas bid 1 yields 0.
**Answer: C — (2, 2).**
A: **Incorrect:** Both zero bids are eliminated.
B: **Incorrect:** Both bids of 1 are eliminated in later rounds.
C: **Correct:** Only bid 2 for each player survives.
D: **Incorrect:** A surviving pair does exist.
Consider the following normal form game and answer the given
This game has
6406535121103.
A unique Nash equilibrium in pure strategy
6406535121104.
A unique Nash equilibrium in mixed strategy
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Both are true
6406535121106.
None is true
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & Left & Right \\ Top & (8,2) & (1,6) \\ Bottom & (3,5) & (4,1)\end{array}\]
**1. Check pure profiles:** Player 1 prefers Top against Left and Bottom against Right; player 2 prefers Right against Top and Left against Bottom. No cell contains mutual best responses.
**2. Make both players indifferent:** If \(p=\Pr(Top)\) and \(q=\Pr(Left)\), then \(1+7q=4-q\) and \(5-3p=1+5p\), giving \(q=3/8\) and \(p=1/2\). Both probabilities are interior and uniquely determined.
**Answer: B — A unique Nash equilibrium in mixed strategy.**
A: **Incorrect:** There is no pure-strategy equilibrium.
B: **Correct:** There is exactly one mixed equilibrium, with p=1/2 and q=3/8.
C: **Incorrect:** The pure-equilibrium assertion is false.
D: **Incorrect:** The unique mixed-equilibrium assertion is true.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
This is a game of
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Perfect information
6406535121109.
Imperfect information
6406535121110.
None
6406535121111.
Can’t be determined
Published solution
\[\begin{array}{ccccccc}&&\text{Player 1}&&&&\\ &\swarrow A&&\searrow B&&&\\ (2,0)&&&&\text{Player 2}&&\\ &&&\swarrow C&&\searrow D&\\ &&(3,1)&&&&\text{Player 1}\\ &&&&&\swarrow E&\searrow F\\ &&&&&(0,0)&(1,2)\end{array}\]
Each terminal pair is (player 1 payoff, player 2 payoff).
**1. Identify observations:** Every player observes the preceding actions; each decision node is its own information set.
**Answer: A — Perfect information.**
A: **Correct:** Every decision maker knows the current history.
B: **Incorrect:** No distinct decision nodes are joined by an information set.
C: **Incorrect:** Perfect information applies.
D: **Incorrect:** The observations are specified.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
How many terminal nodes does this game have?
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3
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2
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4
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5
Published solution
\[\begin{array}{ccccccc}&&\text{Player 1}&&&&\\ &\swarrow A&&\searrow B&&&\\ (2,0)&&&&\text{Player 2}&&\\ &&&\swarrow C&&\searrow D&\\ &&(3,1)&&&&\text{Player 1}\\ &&&&&\swarrow E&\searrow F\\ &&&&&(0,0)&(1,2)\end{array}\]
Each terminal pair is (player 1 payoff, player 2 payoff).
**1. Count endings:** The four terminal histories are A, BC, BDE, and BDF.
**Answer: C — 4 terminal nodes.**
A: **Incorrect:** This omits one terminal history.
B: **Incorrect:** There are more than two endings.
C: **Correct:** There are four terminal histories.
D: **Incorrect:** No fifth ending exists.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
How many information sets does this game have?
6406535121116.
2
6406535121117.
3
6406535121118.
4
6406535121119.
1
Published solution
\[\begin{array}{ccccccc}&&\text{Player 1}&&&&\\ &\swarrow A&&\searrow B&&&\\ (2,0)&&&&\text{Player 2}&&\\ &&&\swarrow C&&\searrow D&\\ &&(3,1)&&&&\text{Player 1}\\ &&&&&\swarrow E&\searrow F\\ &&&&&(0,0)&(1,2)\end{array}\]
Each terminal pair is (player 1 payoff, player 2 payoff).
**1. Count decision nodes:** Player 1 acts at the root and after BD; player 2 acts after B. Perfect information makes these three singleton information sets.
**Answer: B — 3 information sets.**
A: **Incorrect:** This counts players rather than decision information sets.
B: **Correct:** There are three singleton information sets.
C: **Incorrect:** Terminal nodes are not information sets.
D: **Incorrect:** The whole game is not one information set.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
How many pure strategies does player 1 have?
6406535121120.
1
6406535121121.
2
6406535121122.
3
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4
Published solution
\[\begin{array}{ccccccc}&&\text{Player 1}&&&&\\ &\swarrow A&&\searrow B&&&\\ (2,0)&&&&\text{Player 2}&&\\ &&&\swarrow C&&\searrow D&\\ &&(3,1)&&&&\text{Player 1}\\ &&&&&\swarrow E&\searrow F\\ &&&&&(0,0)&(1,2)\end{array}\]
Each terminal pair is (player 1 payoff, player 2 payoff).
**1. Specify complete plans:** Player 1 chooses A or B initially and E or F if BD occurs. A pure strategy specifies both choices, even off the realized path.
**2. Count:** \(2\times2=4\): AE, AF, BE, BF.
**Answer: D — 4 pure strategies.**
A: **Incorrect:** A pure strategy is not just the realized path.
B: **Incorrect:** This counts only one of player 1’s decisions.
C: **Incorrect:** All four combinations are feasible.
D: **Correct:** Two binary decisions give four strategies.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
How many pure strategies does player 2 have?
6406535121124.
1
6406535121125.
2
6406535121126.
3
6406535121127.
4
Published solution
\[\begin{array}{ccccccc}&&\text{Player 1}&&&&\\ &\swarrow A&&\searrow B&&&\\ (2,0)&&&&\text{Player 2}&&\\ &&&\swarrow C&&\searrow D&\\ &&(3,1)&&&&\text{Player 1}\\ &&&&&\swarrow E&\searrow F\\ &&&&&(0,0)&(1,2)\end{array}\]
Each terminal pair is (player 1 payoff, player 2 payoff).
**1. Count complete plans:** Player 2 has one information set with actions C and D, hence two pure strategies.
**Answer: B — 2 pure strategies.**
A: **Incorrect:** Both C and D are available.
B: **Correct:** One binary decision gives two strategies.
C: **Incorrect:** There is no third action.
D: **Incorrect:** Player 2 has only one binary decision.
Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A
while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game
ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and
then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1),
following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:
How many pure strategy Nash equilibria does this game have?
6406535121128.
1
6406535121129.
2
6406535121130.
3
6406535121131.
4
Published solution
Player 1 owns rows; player 2 owns columns. Each pair is (player 1 payoff, player 2 payoff).
\[\begin{array}{c|cc} & C & D \\ AE & (2,0) & (2,0) \\ AF & (2,0) & (2,0) \\ BE & (3,1) & (0,0) \\ BF & (3,1) & (1,2)\end{array}\]
**1. Check mutual best responses:** Against C, player 1 chooses BE or BF; against D, player 1 chooses AE or AF. Player 2 prefers C after BE and D after BF, and is indifferent when A is chosen.
**2. List intersections:** The pure Nash equilibria are (AE,D), (AF,D), and (BE,C).
**Answer: C — 3 pure Nash equilibria.**
A: **Incorrect:** There are three intersections of best responses.
B: **Incorrect:** There are three, not two, equilibria.
C: **Correct:** The three listed profiles are mutual best responses.
D: **Incorrect:** (BF,C) fails because player 2 prefers D.
