Let's analyze the **sequential game** step by step: --- ## **Game Structure Recap** - **Player I** chooses **L** or **R** at node 1. - If **L**, the game ends: - If **L** (at node 2): (3, 3) - If **R** (at node 3): (0, 5) - If **R** (at node 4), **Player II** chooses **L** or **R**. - If **L**, (–1, 0) - If **R** (at node 6), game proceeds to node 7 (Player I chooses **L** or **R**): - **L**: (4, 5) - **R**: (5, –1) --- # **a) Find all the sub-game Nash Equilibria and identify the Subgame Perfect Equilibrium (SPE):** ## **Step 1: Backwards Induction** ### **Node 7 (Player I chooses L or R)** - Choices: - **L:** (4, 5) - **R:** (5, –1) - **Player I** payoff: 4 (L), 5 (R). - **Player I** chooses **R** (since 5 > 4). ### **Node 6 (Player II chooses L or R)** - If **L**: (–1, 0) - If **R**: Player I will choose **R** at node 7 ⇒ (5, –1) - **Player II** payoff: 0 (L), –1 (R) - **Player II** chooses **L** (since 0 > –1). ### **Node 4 (Player I chooses L or R)** - If **L**: (3, 3) - If **R**: Go to node 5 (Player II chooses L or R) - If **L:** (–1, 0) - If **R:** Player I chooses **R** at node 7 ⇒ (5, –1) - From above, Player II chooses **L** at node 6, so payoffs are (–1, 0). - So, if Player I chooses **R** at node 4, payoff is (–1, 0). - Compare at node 4: - **L:** (3, 3) - **R:** (–1, 0) - **Player I** chooses **L** (since 3 > –1). --- ## **Step 2: Nash Equilibria (NE) and Subgame Perfect Equilibrium (SPE)** ### **NE:** - Strategy profiles: - **Player I:** (Node 1: L or R; Node 7: L or R) - **Player II:** (Node 6: L or R) ### **SPE:** - From backwards induction: - **Player I:** L at node 1, R at node 7 - **Player II:** L at node 6 So, **the subgame perfect Nash equilibrium (SPE) is:** - Player I: Choose **L** at node 1; if node 7 is reached, choose **R**. - Player II: Choose **L** at node 6. There may be other NE, but only this strategy profile is subgame perfect because it is consistent with backwards induction at every subgame. --- # **b) Most Likely Outcome (Payoff) by Backwards Induction** From above, the most likely outcome is: - **Player I chooses L at node 1.** - Final payoff: **(3, 3)** --- ## **Summary Table** | Player I Strategy | Player II Strategy | Outcome | Payoff | |-------------------------|-------------------|---------|--------| | L at 1, R at 7 | L at 6 | (L chosen at 1) | (3, 3) | | (other NE possible, but not SPE) | | | | --- ## **Final Answers** ### **(a)** - **All sub-game Nash Equilibria**: Multiple, but only one is **subgame perfect**: - **Player I:** L at node 1, R at node 7 - **Player II:** L at node 6 - **Subgame Perfect Nash Equilibrium (SPE):** - Player I: L at node 1, R at node 7 - Player II: L at node 6 ### **(b)** - **Most likely outcome (by backwards induction):** - **Player I chooses L at node 1.** - **Payoff:** (3, 3) --- Let me know if you need a full listing of all Nash equilibria or further explanation!