Let's carefully analyze and solve the problem step-by-step. --- ## Problem Restatement: - Two players choose among three alternatives \( a, b, c \). - Player 1's preferences: \( a \succ_1 b \succ_1 c \) - Player 2's preferences: \( c \succ_2 b \succ_2 a \) - Move order: Player 1 moves first, then Player 2. - Player 1 can veto one alternative. - Player 2, after observing Player 1's veto, chooses one of the remaining two alternatives. --- ## Part a: Model as an extensive form game tree ### Step 1: Player 1 chooses an alternative to veto: - Player 1 chooses to veto either \( a \), \( b \), or \( c \). - After this veto, the remaining two alternatives are available to Player 2. ### Step 2: Player 2 chooses an alternative: - Player 2 observes Player 1’s veto and chooses from the remaining two alternatives. ### Game Tree Structure: 1. Player 1's move: veto \( a \), \( b \), or \( c \). 2. Player 2's move: choose one of the remaining two alternatives. --- ## Part b: Number of pure strategies for each player ### Player 1: - Player 1's strategy is to specify a veto choice. - **Strategies for Player 1**: Veto \( a \), or veto \( b \), or veto \( c \). - **Total strategies for Player 1**: **3**. ### Player 2: - Player 2 observes Player 1's veto and then chooses from the remaining 2 alternatives. - For each possible veto by Player 1, Player 2 has 2 choices. - **Strategies for Player 2**: For each of Player 1's 3 vetoes, pick one of the remaining 2 options. - Total strategies for Player 2: \( 2 \times 3 = 6 \). --- ## Part c: Find all pure-strategy Nash equilibria ### Step 1: List Player 1's possible veto strategies: - \( V_a \): Veto \( a \) - \( V_b \): Veto \( b \) - \( V_c \): Veto \( c \) ### Step 2: List Player 2's possible responses after each veto: | Player 1's veto | Remaining options for Player 2 | Player 2's choices | Player 2's choice | Player 2's payoff (Player 2 prefers \( c \succ b \succ a \)) | |-----------------|----------------------------------|----------------------|---------------------|--------------------------------------------------------------| | veto \( a \) | \( b, c \) | choose \( b \) or \( c \) | \( b \) or \( c \) | payoff based on preferences: \( c \succ_2 b \succ_2 a \) | | veto \( b \) | \( a, c \) | choose \( a \) or \( c \) | | veto \( c \) | \( a, b \) | choose \( a \) or \( b \) | ### Step 3: Determine Player 2's best response (highest payoff) in each case: - After veto \( a \): remaining \( b, c \). Player 2 prefers \( c \) over \( b \), so best response: **choose \( c \)**. - After veto \( b \): remaining \( a, c \). Player 2 prefers \( c \) over \( a \), so best response: **choose \( c \)**. - After veto \( c \): remaining \( a, b \). Player 2 prefers \( b \) over \( a \), so best response: **choose \( b \)**. ### Step 4: Player 1's decision: - Player 1 anticipates Player 2's responses and chooses the veto that maximizes Player 1's payoff. Recall Player 1's preferences: \[ a \succ_1 b \succ_1 c \] - **Veto \( a \)**: - Remaining: \( b, c \) - Player 2's best response: \( c \) - Player 1's payoff: \( c \) (least preferred, since \( a \succ_1 b \succ_1 c \)) - **Veto \( b \)**: - Remaining: \( a, c \) - Player 2's best response: \( c \) - Player 1's payoff: \( a \) (most preferred among remaining options) - **Veto \( c \)**: - Remaining: \( a, b \) - Player 2's best response: \( b \) - Player 1's payoff: \( a \) (most preferred among remaining options) ### Step 5: Final outcome: - Player 1 prefers to veto \( c \) to leave \( a, b \). - Player 2 then chooses \( b \) (their best response), resulting in the outcome \( b \). **Conclusion:** Player 1's **best veto** is \( c \), leading to the outcome \( b \), which gives Player 1 a payoff of \( b \) (second most preferred). Player 2's best response after veto \( c \) is \( b \). --- ## **Final Answer:** ### Extensive form game tree: - Player 1 chooses to veto \( a \), \( b \), or \( c \). - Player 2, observing the veto, chooses from remaining options, with the best responses as above. ### Number of strategies: - Player 1: **3**. - Player 2: **6**. ### Nash equilibria: - The **unique pure-strategy Nash equilibrium** is: \[ \boxed{ \text{Player 1 vetoes } c, \quad \text{Player 2 chooses } b } \] Outcome: **\( b \)** (Player 2's choice after Player 1 vetoes \( c \)).