ms4023_2026T2_Q1_NA.pdf

**1. Write the profit:** \(\pi_i=(10+P_j)P_i-P_i^2\). **2. Differentiate:** \(\partial\pi_i/\partial P_i=10+P_j-2P_i=0\). **3. Check the maximum:** the second derivative is \(-2<0\). **Answer: C.** \[BR_i(P_j)=5+\frac{P_j}{2}.\] A: **Incorrect**: \(5+P_j/3\) does not satisfy the first-order condition in general. B: **Incorrect**: \(5+2P_j/3\) has the wrong coefficient on the other firm’s price. C: **Correct**: \(5+P_j/2\) satisfies the first-order condition and maximizes profit. D: **Incorrect**: \(5+P_j\) responds too strongly to the other firm’s price.

**1. Set both best responses:** \(P_1=5+P_2/2\), \(P_2=5+P_1/2\). **2. Solve:** these equations give the unique equilibrium \((10,10)\). **3. Test the statement:** against price \(15\), the best response is \(12.5\), not \(15\). **Answer: C.** “Each player chooses 15” is the incorrect statement. A: True, so not the requested option: both firms have the same strategy space and identical payoff formulas after exchanging player labels. B: **True**: the equilibrium \((10,10)\) is symmetric. C: False, hence the correct selection: equilibrium prices are \(10\), not \(15\). D: **True**: the two linear best-response equations have exactly one solution.

**Payoff table:** rows = player 1’s bid; columns = player 2’s bid. Each cell is \((u_1,u_2)\). \[\begin{array}{c|ccc} b_1\backslash b_2 & 0 & 1 & 2 \\ \hline 0 & (5/2,2) & (0,3) & (0,2) \\ 1 & (4,0) & (2,3/2) & (0,2) \\ 2 & (3,0) & (3,0) & (3/2,1) \end{array}\] **1. Compare player 1’s bids:** against opposing bids \(0,1,2\), bid 0 earns \((5/2,0,0)\), while bid 2 earns \((3,3,3/2)\). Bid 2 is strictly better in every case. **2. Compare player 2’s pure bids:** their payoff vectors are \(b_0:(2,0,0)\), \(b_1:(3,3/2,0)\), \(b_2:(2,2,1)\). No single pure bid strictly dominates another. **Answer: B — Player 2**, when domination is restricted to pure strategies. **Review note:** allowing mixed-strategy domination changes the answer. A 50–50 mixture of player 2’s bids 1 and 2 earns \((5/2,7/4,1/2)\), strictly dominating bid 0. Under that convention, C (“None”) is correct. A: Incorrect under the pure-strategy convention: player 1’s bid 0 is strictly dominated by bid 2. B: Correct under the pure-strategy convention: no single pure bid strictly dominates another for player 2. It is not correct if mixed-strategy domination is included. C: Incorrect under the pure-strategy convention because player 2 has no strictly dominated pure bid. Under the broader mixed-strategy convention, this option would be correct.

**Payoff table:** rows = player 1’s bid; columns = player 2’s bid. Each cell is \((u_1,u_2)\). \[\begin{array}{c|ccc} b_1\backslash b_2 & 0 & 1 & 2 \\ \hline 0 & (5/2,2) & (0,3) & (0,2) \\ 1 & (4,0) & (2,3/2) & (0,2) \\ 2 & (3,0) & (3,0) & (3/2,1) \end{array}\] **1. Eliminate player 1’s bid 0:** bid 2 gives a strictly larger payoff against every opposing bid. **2. Eliminate player 2’s bids 0 and 1:** against player 1’s remaining bids \(1,2\), player 2’s bid 2 gives \((2,1)\), beating bid 0’s \((0,0)\) and bid 1’s \((3/2,0)\). **3. Eliminate player 1’s bid 1:** against bid 2, bid 1 earns \(0\) but bid 2 earns \(3/2\). **Answer: C.** Only \((2,2)\) survives. A: **Incorrect**: player 1’s bid 0 is eliminated in the first round. B: **Incorrect**: both players’ bid 1 is eliminated in later rounds. C: **Correct**: successive elimination leaves only \((2,2)\). D: **Incorrect**: a surviving profile exists.

**1. Write the payoff:** \(v_i=(a+e_j)e_i-2e_i^2\). **2. Differentiate and set to zero:** \(a+e_j-4e_i=0\). **3. Check the maximum:** the second derivative is \(-4<0\). **Answer: A.** For \(a\ge0\), \[BR_i(e_j)=\frac{a+e_j}{4}.\] **Boundary note:** in general, nonnegative effort requires \(BR_i(e_j)=\max\{0,(a+e_j)/4\}\). A: Correct when \(a+e_j\ge0\): \((a+e_j)/4\) maximizes the payoff. Otherwise the constrained optimum is zero. B: **Incorrect**: differentiating \(-2e_i^2\) produces \(-4e_i\), so the denominator is 4, not 2. C: **Incorrect**: a best response is expressed in terms of the opponent’s effort \(e_j\), not the player’s own unknown effort \(e_i\). D: **Incorrect**: the coefficient of \(e_j\) in the derivative is 1, not 2.

**1. Set both best responses:** \(4e_1=a+e_2\), \(4e_2=a+e_1\). **2. Subtract:** \(5(e_1-e_2)=0\), so \(e_1=e_2\). **3. Substitute:** \(3e_1=a\); thus \(e_1=e_2=a/3\), uniquely. **Answer: C — Both statements**, assuming \(a\ge0\). **Review note:** the passage does not specify the sign of \(a\). If \(a<0\), the unique feasible equilibrium is \((0,0)\), and B is correct instead. A: True for \(a\ge0\), but incomplete because the uniqueness statement is also true. For \(a<0\), the claimed negative effort is infeasible. B: **True**: the equilibrium is unique. For \(a\ge0\), choose the combined option; if negative \(a\) is allowed, this is the correct standalone option. C: Correct under the intended assumption \(a\ge0\): both \(e_1=e_2=a/3\) and uniqueness hold. D: **Incorrect**: uniqueness holds, so at least one of the statements is true.

**Payoff table:** rows = player 1; columns = player 2. Each cell is \((u_1,u_2)\). \[\begin{array}{c|cc} & \mathrm{Left} & \mathrm{Right} \\ \hline \mathrm{Top} & (5,4) & (0,3) \\ \mathrm{Bottom} & (3,2) & (1,7) \end{array}\] **1. Player 1’s best responses:** Left → Top \((5>3)\); Right → Bottom \((1>0)\). **2. Player 2’s best responses:** Top → Left \((4>3)\); Bottom → Right \((7>2)\). **Answer: A.** The two mutual best responses are \((\mathrm{Top},\mathrm{Left})\) and \((\mathrm{Bottom},\mathrm{Right})\). A: **Correct**: exactly two pure-strategy Nash equilibria exist. B: **Incorrect**: there are two, not one, pure-strategy equilibria. C: **Incorrect**: a game cannot simultaneously have exactly two and exactly one pure-strategy equilibrium. D: **Incorrect**: the first statement is true.

**Payoff table:** rows = player 1; columns = player 2. Each cell is \((u_1,u_2)\). \[\begin{array}{c|cc} & \mathrm{Left} & \mathrm{Right} \\ \hline \mathrm{Top} & (5,4) & (0,3) \\ \mathrm{Bottom} & (3,2) & (1,7) \end{array}\] **1. Make player 1 indifferent:** \(5q=3q+(1-q)\), so \(q=1/3\). **2. Make player 2 indifferent:** \(4p+2(1-p)=3p+7(1-p)\), so \(p=5/6\). **3. Calculate the requested value:** \[\frac{1}{p-q}=\frac{1}{5/6-1/3}=2.\] **Answer: 2.**

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Check what each player knows:** all previous actions are observed at every decision node. **Answer: A — Perfect information.** Every information set contains just one decision node. A: **Correct**: each player knows the preceding actions when making a choice. B: **Incorrect**: no unobserved earlier choice or nonsingleton information set is specified. C: **Incorrect**: the game satisfies the definition of perfect information. D: **Incorrect**: the sequential structure given is sufficient to determine the information structure.

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. List the possible endings:** \(A\), \(BC\), \(BDE\), \(BDF\). **Answer: C — 4 terminal nodes.** Each complete history ends at one distinct leaf of the game tree. A: **Incorrect**: this counts only three of the four possible endings. B: **Incorrect**: the game can end in more than two ways. C: **Correct**: the terminal histories are A, BC, BDE and BDF. D: **Incorrect**: the game specifies only four terminal histories.

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Count decision nodes:** player 1 initially, player 2 after \(B\), and player 1 after \(BD\): three nodes. **2. Use perfect information:** each decision node is its own information set. **Answer: B — 3 information sets.** Terminal nodes are not counted. A: **Incorrect**: two players does not imply only two information sets; player 1 moves at two distinct nodes. B: **Correct**: each of the three decision nodes forms one information set. C: **Incorrect**: the four terminal nodes are not decision-node information sets. D: **Incorrect**: the three decisions are not grouped into one information set.

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Count choices at both of player 1’s nodes:** two initial choices and two later choices. **2. Form complete plans:** \(AE,AF,BE,BF\); even unreached decisions must be specified. **Answer: D.** \(2\times2=4\) pure strategies. A: **Incorrect**: player 1 has several available complete action plans. B: **Incorrect**: this counts choices at only one of player 1’s two decision nodes. C: **Incorrect**: one of the four complete plans is missing. D: **Correct**: AE, AF, BE and BF are the four pure strategies.

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Count player 2’s choices:** there is one decision node, with actions \(C\) and \(D\). **Answer: B — 2 pure strategies.** A: **Incorrect**: player 2 has two alternative actions. B: **Correct**: the pure strategies are C and D. C: **Incorrect**: no third action or additional decision node for player 2 is specified. D: **Incorrect**: four strategies belong to player 1, who has two decision nodes; player 2 has only one.

**Normal-form payoff table:** player 1 chooses a complete plan (row); player 2 chooses C or D (column). Payoffs are \((u_1,u_2)\). \[\begin{array}{c|cc} & C & D \\ \hline AE & (2,0) & (2,0) \\ AF & (2,0) & (2,0) \\ BE & (3,2) & (2,1) \\ BF & (3,2) & (1,2) \end{array}\] **1. Find player 1’s best responses:** to \(C\): \(BE,BF\), both earning 3. To \(D\): \(AE,AF,BE\), all earning 2. **2. Find player 2’s best responses:** to \(AE,AF\), both \(C,D\) earn 0; to \(BE\), only \(C\) is best \((2>1)\); to \(BF\), both earn 2. **3. Match mutual best responses:** \((AE,D),(AF,D),(BE,C),(BF,C)\). **Answer: D — 4 pure-strategy Nash equilibria.** A: **Incorrect**: backward induction identifies one subgame-perfect equilibrium, but the question counts all pure-strategy Nash equilibria. B: **Incorrect**: indifference and actions at unreached nodes produce four equilibria, not two. C: **Incorrect**: there are four mutual best-response profiles. D: **Correct**: the four profiles are (AE,D), (AF,D), (BE,C) and (BF,C).

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Work backward from the final node:** player 1 chooses \(E\), since \(2>1\). **2. At player 2’s node:** \(C\) earns 2, while \(D\) followed by \(E\) earns 1; choose \(C\). **3. At the initial node:** player 1 chooses \(B\), earning 3 instead of 2 from \(A\). **Answer: C.** The unique subgame-perfect equilibrium is \((BE,C)\). A: **Incorrect**: in the subgame after B, player 2 would prefer C to D when player 1 plans E. B: **Incorrect**: choosing F is not optimal at player 1’s final node, where E gives a larger payoff. C: **Correct**: B, C and the contingent choice E are optimal in every subgame. D: **Incorrect**: F is not optimal at the final node, even though that node is unreached under C.

**Game tree:** P1 and P2 identify the player who moves. A–F label actions; terminal pairs are \((u_1,u_2)\). \[\begin{array}{ccccccc} & \mathrm{P1} & & & & & \\ \swarrow & & \searrow & & & & \\ A:(2,0) & & B:\mathrm{P2} & & & & \\ & \swarrow & & \searrow & & & \\ C:(3,2) & & & & D:\mathrm{P1} & & \\ & & & \swarrow & & \searrow & \\ & & E:(2,1) & & & & F:(1,2) \end{array}\] **1. Locate valid starting nodes:** the initial node, the node after \(B\), and the node after \(BD\). Each begins a subgame. **Answer: C — 3 subgames**, including the entire game. **Convention note:** excluding the entire game leaves two proper subgames. A: **Incorrect**: this counts only the entire game and ignores its two continuation subgames. B: This counts the two proper subgames, excluding the entire game; it is not the inclusive convention used by the supplied key. C: Correct under the inclusive convention: the whole game and the two continuation subgames. D: **Incorrect**: terminal nodes do not create additional decision subgames.