Let's solve the game step by step. ### **Payoff Matrix** \[ \begin{array}{c|cc} & \text{S1} & \text{S2} \\ \hline \text{S1} & (-6, -6) & (0, -10) \\ \text{S2} & (-10, 0) & (-1, -1) \\ \end{array} \] (First entry: Player X, second entry: Player Y) --- ### **(a) Find the Nash equilibrium of this game. Is it Pareto optimal?** #### **Step 1: Best Responses** - **Player X's choices** (rows): - If Y plays S1: X gets -6 (S1), -10 (S2) → **Best: S1** - If Y plays S2: X gets 0 (S1), -1 (S2) → **Best: S1** - **Player Y's choices** (columns): - If X plays S1: Y gets -6 (S1), -10 (S2) → **Best: S1** - If X plays S2: Y gets 0 (S1), -1 (S2) → **Best: S1** #### **Step 2: Nash Equilibrium** A Nash equilibrium is where both are playing their best response. - Both choose **S1** (first row, first column): **(-6, -6)** This is the **only Nash equilibrium** in pure strategies. #### **Pareto Optimality** - An outcome is Pareto optimal if there is no way to make one player better off without making the other worse off. - Compare (-6, -6) with (-1, -1): Both players are better off at (-1, -1). **So, (-6, -6) is NOT Pareto optimal.** --- ### **(b) What is the general name of this type of game?** This is a version of the **"Prisoner's Dilemma"**: - Both players have a dominant strategy that leads to a worse outcome for both than if they cooperated. --- ### **(c) Is the outcome (S2, S1) Pareto optimal?** - (S2, S1) = (-10, 0) - Is there another outcome that makes one player better off without making the other worse off? - (S1, S1): (-6, -6) → Y is worse off. - (S1, S2): (0, -10) → X is better off, Y is worse off. - (S2, S2): (-1, -1) → X is better off, Y is worse off. So, **(S2, S1) is NOT Pareto optimal** because moving to (S2, S2) makes X better off with only a small loss to Y, and moving to (S1, S2) makes X much better off, but Y worse off. However, strictly speaking, **(S2, S1) is not Pareto optimal** because (S2, S2) is strictly better for X without harming Y. --- ### **(d) Is it a zero-sum game? Is it a symmetric game?** - **Zero-sum?** - For each outcome, sum the payoffs: - (-6) + (-6) = -12 - (0) + (-10) = -10 - (-10) + 0 = -10 - (-1) + (-1) = -2 - Not zero in any case → **NOT a zero-sum game.** - **Symmetric?** - If swapping payoffs between players gives the same matrix, it's symmetric. - The matrix is symmetric across the diagonal: (-6, -6), (0, -10), (-10, 0), (-1, -1). - Yes, **it is a symmetric game**. --- ## **Summary Table** | Part | Answer | |------|--------| | a) | Nash equilibrium: (S1, S1) with payoffs (-6, -6); **Not Pareto optimal**. | | b) | **Prisoner's Dilemma**. | | c) | (S2, S1) is **not Pareto optimal**. | | d) | **Not a zero-sum game**; **It is a symmetric game**. |

Let's carefully analyze the problem step-by-step. --- ### **Payoff Matrix** \[ \begin{array}{c|cc} & \text{S1} & \text{S2} \\ \hline \text{S1} & (-6, -6) & (0, -10) \\ \text{S2} & (-10, 0) & (-1, -1) \\ \end{array} \] *(First entry: Player X, second: Player Y)* --- ### **(a) Find the Nash equilibrium. Is it Pareto optimal?** **Step 1: Determine best responses** - **Player X:** - If Player Y plays **S1**: - X's payoffs: S1 = -6, S2 = -10 → **Best: S1** - If Player Y plays **S2**: - X's payoffs: S1 = 0, S2 = -1 → **Best: S1** - **Player Y:** - If Player X plays **S1**: - Y's payoffs: S1 = -6, S2 = -10 → **Best: S1** - If Player X plays **S2**: - Y's payoffs: S1 = 0, S2 = -1 → **Best: S1** **Conclusion:** Both players always prefer **S1** regardless of what the other does. **Nash equilibrium:** **(S1, S1)** with payoffs **(-6, -6)**. --- ### **Pareto Optimality** - Compare **(S1, S1)** with other outcomes: - **(S2, S2): (-1, -1)** Both players are better off here than at **(S1, S1)** since -1 > -6. Therefore, **(S1, S1)** is **not** Pareto optimal because **(S2, S2)** Pareto dominates it. **Answer:** **The Nash equilibrium is (S1, S1). It is not Pareto optimal.** --- ### **(b) What is the general name of this type of game?** This is a classic example of the **Prisoner's Dilemma**: - Both players have a dominant strategy (S1 for both). - The outcome (S1, S1) is not Pareto optimal. - Mutual defection leads to worse payoffs than mutual cooperation (S2, S2). **Answer:** **Prisoner's Dilemma** --- ### **(c) Is the outcome (S2, S1) Pareto optimal?** - **(S2, S1): (-10, 0)** **Check if there's another outcome that makes at least one player better off without hurting the other:** - **(S2, S2): (-1, -1)** - X: from -10 to -1 (better) - Y: from 0 to -1 (worse) - So, Y is worse off; this doesn't Pareto dominate **(S2, S1)**. - **(S1, S2): (0, -10)** - X: from -10 to 0 (better) - Y: from 0 to -10 (worse) - Y worse off; no. - **(S1, S1): (-6, -6)** - X: from -10 to -6 (better) - Y: from 0 to -6 (worse) - Y worse off; no. - **(S2, S1): (-10, 0)** is **not** Pareto dominated by any other outcome because moving to any other outcome makes at least one player worse off. **Conclusion:** **(S2, S1) is Pareto optimal.** --- ### **(d) Is it a zero-sum game? Is it symmetric?** - **Zero-sum?** Sum payoffs at each outcome: - (S1, S1): -6 + (-6) = -12 - (S1, S2): 0 + (-10) = -10 - (S2, S1): -10 + 0 = -10 - (S2, S2): -1 + (-1) = -2 Since payoffs do **not** sum to zero in general, **it is not a zero-sum game**. - **Symmetric?** The matrix is symmetric across the diagonal: \[ \begin{array}{c|cc} & \text{S1} & \text{S2} \\ \hline \text{S1} & (-6, -6) & (0, -10) \\ \text{S2} & (-10, 0) & (-1, -1) \\ \end{array} \] Swapping players' roles (interchanging payoffs) yields the same matrix structure, indicating **symmetry**. **Answer:** - **Not** a zero-sum game. - **Yes**, it is a symmetric game. --- ## **Final summarized answers:** | Part | Answer | |---------|--------------------------------------------------------------| | (a) | Nash equilibrium: **(S1, S1)** with payoffs **(-6, -6)**; **not Pareto optimal** | | (b) | **Prisoner's Dilemma** | | (c) | **(S2, S1)** **is Pareto optimal** | | (d) | **Not zero-sum**; **symmetric game** | --- ### **Note on the correction:** In the **below** solution, it was claimed that **(S2, S1)** *is not* Pareto optimal because **(S2, S2)** supposedly benefits X "without harming Y." **This is incorrect**: at **(S2, S2)**, Y's payoff is **-1**, worse than 0 at (S2, S1). Thus, **(S2, S1)** **is Pareto optimal** because no other outcome makes at least one player better off without hurting the other. --- **I hope this detailed step-by-step explanation clarifies the problem!**

Let's carefully analyze and solve the game step-by-step. --- ### **Payoff Matrix** \[ \begin{array}{c|cc} & \text{S1} & \text{S2} \\ \hline \text{S1} & (-6, -6) & (0, -10) \\ \text{S2} & (-10, 0) & (-1, -1) \\ \end{array} \] *(First entry: Player X, second: Player Y)* --- ### **(a) Find the Nash equilibrium. Is it Pareto optimal?** **Step 1: Determine best responses** - **Player X's choices:** - If Y plays **S1**: - X's payoffs: S1 = -6, S2 = -10 → **Best: S1** - If Y plays **S2**: - X's payoffs: S1 = 0, S2 = -1 → **Best: S1** - **Player Y's choices:** - If X plays **S1**: - Y's payoffs: S1 = -6, S2 = -10 → **Best: S1** - If X plays **S2**: - Y's payoffs: S1 = 0, S2 = -1 → **Best: S1** **Conclusion:** Both players always prefer **S1** regardless of what the other does. **Nash equilibrium:** **(S1, S1)** with payoffs **(-6, -6)**. --- ### **(b) Is the outcome (S2, S1) Pareto optimal?** - **(S2, S1): (-10, 0)** **Check if there's another outcome that makes at least one player better off without making the other worse off:** - **(S2, S2): (-1, -1)** - X: from -10 to -1 (better) - Y: from 0 to -1 (worse) - Y is worse off, so **(S2, S2)** does **not** Pareto dominate **(S2, S1)**. - **(S1, S2): (0, -10)** - X: from -10 to 0 (better) - Y: from 0 to -10 (worse) - Y worse off, no. - **(S1, S1): (-6, -6)** - X: from -10 to -6 (better) - Y: from 0 to -6 (worse) - Y worse off, no. **Conclusion:** No other outcome makes at least one player better off without harming the other. **Therefore, (S2, S1) is Pareto optimal.** --- ### **(c) Is the outcome (S2, S1) Pareto optimal?** **Answer:** Yes, as shown above, **(S2, S1)** is Pareto optimal because no other outcome improves one player's payoff without decreasing the other's. --- ### **(d) Is it a zero-sum game? Is it symmetric?** - **Zero-sum?** Sum payoffs at each outcome: - (S1, S1): -6 + (-6) = -12 - (S1, S2): 0 + (-10) = -10 - (S2, S1): -10 + 0 = -10 - (S2, S2): -1 + (-1) = -2 Since these sums are not all zero, **it is not a zero-sum game**. - **Symmetric?** To check symmetry, see if swapping players' roles yields the same payoff structure: \[ \begin{array}{c|cc} & \text{S1} & \text{S2} \\ \hline \text{S1} & (-6, -6) & (0, -10) \\ \text{S2} & (-10, 0) & (-1, -1) \\ \end{array} \] The matrix is symmetric across the diagonal: swapping Player X and Player Y's payoffs yields the same structure. **Answer:** - **Not** a zero-sum game. - **Yes**, it is a symmetric game. --- ## **Final Summary** | Part | Answer | |---------|--------------------------------------------------------------| | (a) | **Nash equilibrium:** (S1, S1) with payoffs (-6, -6). **Not Pareto optimal.** | | (b) | **Prisoner's Dilemma** | | (c) | **(S2, S1)** **is Pareto optimal.** | | (d) | **Not zero-sum;** **it is a symmetric game.** | --- ### **Important clarification:** In the initial solution, there was a mistake regarding whether **(S2, S1)** is Pareto optimal. - **(S2, S1): (-10, 0)** is **Pareto optimal** because no other outcome improves one player's payoff without hurting the other. - For example, moving to **(S2, S2)** improves X's payoff from -10 to -1 but harms Y (from 0 to -1). - Moving to **(S1, S1)** would hurt Y (from 0 to -6). - Moving to **(S1, S2)** would hurt Y (from 0 to -10). Thus, **(S2, S1)** is Pareto optimal. --- This completes the detailed step-by-step solution.