a) Rock-Paper-Scissors-Dynamite: This game is like Rock-Paper-Scissors but with the addition of Dynamite which beats Rock and Paper but loses to Scissors (which cut the wick of the dynamite). Express this as a game (find A matrix) and derive equilibrium strategies. b) Weighted Rock-Paper-Scissors: This game is like Rock-Paper-Scissors but with weights, i.e., Rock beats Scissors by 5 points, Paper beats Rock by 3 points and Scissors beats Paper by 1 point. Each player wants to maximize their total points. Express this as a game (find A matrix) and derive equilibrium strategies.
The Correct Answer and Explanation is :
Solution for Rock-Paper-Scissors-Dynamite (RPSD)
Game Matrix Representation
We have four strategies: Rock (R), Paper (P), Scissors (S), and Dynamite (D). The payoff matrix for Player 1 (where Player 2 receives the negative of each value) is:
[
A =
\begin{bmatrix}
0 & -1 & 1 & -1 \
1 & 0 & -1 & -1 \
-1 & 1 & 0 & 1 \
1 & 1 & -1 & 0
\end{bmatrix}
]
- Rock (R) beats Scissors (+1), loses to Paper and Dynamite (-1).
- Paper (P) beats Rock (+1), loses to Scissors and Dynamite (-1).
- Scissors (S) beats Paper and Dynamite (+1), loses to Rock (-1).
- Dynamite (D) beats Rock and Paper (+1) but loses to Scissors (-1).
Equilibrium Strategy
To find a mixed-strategy Nash equilibrium, let Player 1 play R, P, S, and D with probabilities (x_1, x_2, x_3, x_4). Similarly, Player 2 plays with probabilities (y_1, y_2, y_3, y_4). The optimal strategy is to make the opponent indifferent among their choices, leading to:
[
x_1 = x_2, \quad x_3 = x_4, \quad x_1 + x_2 + x_3 + x_4 = 1
]
Numerically solving, the equilibrium probabilities satisfy:
[
x_1 = x_2 = \frac{1}{3}, \quad x_3 = x_4 = \frac{1}{6}
]
Solution for Weighted Rock-Paper-Scissors (WRPS)
Game Matrix Representation
The given weights translate into the payoff matrix:
[
A =
\begin{bmatrix}
0 & -3 & 5 \
3 & 0 & -1 \
-5 & 1 & 0
\end{bmatrix}
]
- Rock (R) beats Scissors by 5 points, loses to Paper by 3 points.
- Paper (P) beats Rock by 3 points, loses to Scissors by 1 point.
- Scissors (S) beats Paper by 1 point, loses to Rock by 5 points.
Equilibrium Strategy
Let Player 1 play R, P, S with probabilities (p_1, p_2, p_3), and Player 2 plays with (q_1, q_2, q_3). The expected payoff must be equalized across choices:
Solving:
[
p_1 = \frac{3}{9}, \quad p_2 = \frac{5}{9}, \quad p_3 = \frac{1}{9}
]
This means players should favor Paper the most, then Rock, and play Scissors the least due to the scoring differences.
Conclusion
- RPSD has a balanced 1/3, 1/3, 1/6, 1/6 probability mix.
- WRPS favors Paper with 5/9 probability, Rock with 3/9, and Scissors with 1/9.
Now, I will generate an image illustrating the game matrices and strategies.
Here is the generated image illustrating the game matrices and equilibrium strategies for Rock-Paper-Scissors-Dynamite (RPSD) and Weighted Rock-Paper-Scissors (WRPS). Let me know if you need any modifications or further clarifications!