### Solution to the Game by Dominance Method
The given game is a 4x4 matrix game, where Player A has four strategies (A.I, A.II, A.III, A.IV) and Player B has four strategies (B.I, B.II, B.III, B.IV). The payoffs for each strategy are given in the problem statement.
**Step 1: Write the payoff matrix**
The payoff matrix for the game is:
| | B.I | B.II | B.III | B.IV |
| --- | --- | --- | --- | --- |
| **A.I** | 6 | 8 | 3 | 13 |
| **A.II** | 4 | 1 | 5 | 3 |
| **A.III** | 8 | 10 | 4 | 12 |
| **A.IV** | 3 | 6 | 7 | 12 |
**Step 2: Apply the dominance method**
We will apply the dominance method to find the optimal strategy for Player A.
**Row dominance:**
A strategy of Player A is said to be dominated if there exists another strategy that gives a better payoff or equal payoff for all strategies of Player B. We will check for row dominance.
* A.I is dominated by A.III (since 6 ≤ 8, 8 ≤ 10, 3 ≤ 4, and 13 ≤ 12)
* A.II is not dominated
* A.IV is dominated by A.III (since 3 ≤ 8, 6 ≤ 10, 7 ≤ 4, and 12 ≤ 12)
**Column dominance:**
A strategy of Player B is said to be dominated if there exists another strategy that gives a better payoff or equal payoff for all strategies of Player A. We will check for column dominance.
* B.I is not dominated
* B.II is dominated by B.IV (since 1 ≤ 3, 10 ≤ 12, 6 ≤ 12, and 6 ≤ 12)
* B.III is dominated by B.IV (since 3 ≤ 13, 5 ≤ 3, 4 ≤ 12, and 7 ≤ 12)
**Step 3: Reduce the game matrix**
After applying row and column dominance, we can reduce the game matrix to:
| | B.I | B.IV |
| --- | --- | --- |
| **A.II** | 4 | 3 |
| **A.III** | 8 | 12 |
**Step 4: Find the optimal strategy**
The reduced game matrix has a saddle point at (A.III, B.I), which means that the optimal strategy for Player A is A.III and for Player B is B.I. The optimal payoffs are 8 and 8, respectively.
**Conclusion:**
The optimal strategy for Player A is A.III, and for Player B is B.I, with optimal payoffs of 8 and 8, respectively.
**Tools and concepts for further analysis:**
* To analyze the game further, we can use graphical methods to identify the optimal strategy.
* We can also use linear programming to solve the game, which would provide an alternative approach to finding the optimal strategy.
* In addition, we can apply other game theory concepts, such as Nash equilibrium, to analyze the game and find the optimal strategy.
I hope this solution helps you understand the dominance method and its application to game theory!
Uppercopy
Mini tools
