Consider the game matrix below:

Bob

Left

Right

Top

3.8, 5.8

8.8, 4.3

Amy

Bottom

1.8, 9.8

7.8, 6.8

What is the difference in total surplus between the Nash Equilibrium and the socially optimal outcome?

Answer:

To solve this problem, we need to determine the Nash Equilibrium and the socially optimal outcome in the game matrix. Then, we'll calculate the total surplus for both and find the difference.

### Step 1: Analyze the Game Matrix
The game matrix is as follows:

\[
\begin{array}{c|c|c}
 & \textbf{Bob - Left} & \textbf{Bob - Right} \\
\hline
\textbf{Amy - Top} & (3.8, 5.8) & (8.8, 4.3) \\
\hline
\textbf{Amy - Bottom} & (1.8, 9.8) & (7.8, 6.8) \\
\end{array}
\]

Here, the first number in each pair is Amy's payoff, and the second number is Bob's payoff.

### Step 2: Find the Nash Equilibrium
A Nash Equilibrium occurs when neither player can improve their payoff by unilaterally changing their strategy.

- **If Amy plays Top**:
  - Bob's payoffs are 5.8 (Left) and 4.3 (Right). Bob will choose Left because 5.8 > 4.3.
  - Given Bob chooses Left, Amy's payoff is 3.8.
  
- **If Amy plays Bottom**:
  - Bob's payoffs are 9.8 (Left) and 6.8 (Right). Bob will choose Left because 9.8 > 6.8.
  - Given Bob chooses Left, Amy's payoff is 1.8.

Comparing Amy's payoffs:
- If Amy plays Top, she gets 3.8.
- If Amy plays Bottom, she gets 1.8.

Since 3.8 > 1.8, Amy will choose Top. Therefore, the Nash Equilibrium is **(Top, Left)** with the payoffs (3.8, 5.8).

### Step 3: Find the Socially Optimal Outcome
The socially optimal outcome is the one that maximizes the total surplus (sum of payoffs for both players).

- **(Top, Left)**: Total surplus = 3.8 + 5.8 = 9.6
- **(Top, Right)**: Total surplus = 8.8 + 4.3 = 13.1
- **(Bottom, Left)**: Total surplus = 1.8 + 9.8 = 11.6
- **(Bottom, Right)**: Total surplus = 7.8 + 6.8 = 14.6

The socially optimal outcome is **(Bottom, Right)** with the total surplus of 14.6.

### Step 4: Calculate the Difference in Total Surplus
- Total surplus at Nash Equilibrium: 9.6
- Socially optimal total surplus: 14.6

**Difference in total surplus** = 14.6 - 9.6 = **5.0**

### Final Answer:
The difference in total surplus between the Nash Equilibrium and the socially optimal outcome is **5.0**.

Would you like further details or have any questions? 

Here are some related questions you might explore:
1. What is the significance of Nash Equilibrium in game theory?
2. How do you determine if a game is zero-sum or non-zero-sum?
3. Can there be multiple Nash Equilibria in a game?
4. How does the concept of Pareto efficiency relate to the socially optimal outcome?
5. What role does mixed strategy play in Nash Equilibrium?
6. How does the total surplus concept apply in economics?
7. How does the game change if one player’s strategy is dominant?
8. What are real-world examples of games similar to this matrix?

**Tip:** In games with multiple players, considering the incentives of each player can help identify Nash Equilibria and socially optimal outcomes effectively.

Explore the difference in total surplus between Nash Equilibrium and the socially optimal outcome in a game theory matrix. Understand how game theory concepts like Nash Equilibrium and social optimality affect strategic decision-making.

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