Game Theory: Analyzing a Number Guessing Game with Symmetries and Domination

1. List of all 14 strategies for P2:

• Guess 1,2,3,4 in order
• Guess 4,3,2,1 in order
• Guess 2,1,4,3 in order
• Guess 3,4,1,2 in order
• Guess 1,3,2,4 in order
• Guess 3,1,4,2 in order
• Guess 2,4,1,3 in order
• Guess 4,2,3,1 in order
• Guess randomly
• Guess 2,3,1,4 in order
• Guess 4,1,3,2 in order
• Guess 1,4,2,3 in order
• Guess 2,4,3,1 in order
• Guess 3,1,2,4 in order

2. Payoff matrix:

| | Guess 1 | Guess 2 | Guess 3 | Guess 4 | |------|---------|---------|---------|---------| | 1 | 0 | -1 | -2 | -3 | | 2 | -1 | 0 | -1 | -2 | | 3 | -2 | -1 | 0 | -1 | | 4 | -3 | -2 | -1 | 0 |

3. Reduced payoff matrix:

| | Guess 1,2,3,4 | Guess 4,3,2,1 | Guess 2,1,4,3 | Guess 3,4,1,2 | Guess 1,3,2,4 | Guess 3,1,4,2 | Guess 2,4,1,3 | |------|---------------|---------------|---------------|---------------|---------------|---------------|---------------| | 1 | 0 | -3 | -1 | -2 | -2 | -1 | -3 | | 2 | -3 | 0 | -2 | -1 | -1 | -2 | -3 |

4. Reduced 2x3 payoff matrix:

| | Guess 1,2,3,4 | Guess 4,3,2,1 | Guess 2,1,4,3 | |------|---------------|---------------|---------------| | 1 | 0 | -3 | -1 | | 2 | -3 | 0 | -2 |

The reduced pure strategies of P2 are Guess 1,2,3,4, Guess 4,3,2,1, and Guess 2,1,4,3. The corresponding linear functions are y = 3x, y = -3x, and y = 2x - 5. The value of the game is the average of the intersection points of these three functions, which is -1.

5. All the winning strategies for P1 are to choose numbers randomly with equal probability. All the winning strategies for P2 are to guess randomly.

6. To best play the game, P1 should choose numbers randomly with equal probability, and P2 should guess randomly. This ensures that both players have an equal chance of winning in the long run.