Math Problem Statement
ind the saddle points and values of the payoff matrices. (To enter the saddle points, order your answers from smallest to largest r, then from smallest to largest c.)
(a)
2
2
3
1
−2
−3
2
2
4
saddle points
(r, c) =
(r, c) =
(r, c) =
(r, c) =
value
Solution
To find the saddle points in a payoff matrix, we identify entries in the matrix that are simultaneously the largest in their row and the smallest in their column, or vice versa. Saddle points indicate stable strategies in a game, and their values represent the equilibrium payoffs.
For matrix (a):
2 & 2 \\ 3 & 1 \\ -2 & -3 \\ 2 & 2 \\ 4 & ? \end{matrix}$$ ### Step 1: Identify Maxima in Rows and Minima in Columns 1. **Row Maxima:** Determine the largest value in each row. 2. **Column Minima:** Determine the smallest value in each column. ### Step 2: Find Saddle Points Look for entries that match both a row maximum and a column minimum. ### Step 3: Arrange Saddle Points Order the saddle points as specified, from smallest to largest by row (*r*) and then by column (*c*). I’ll now calculate these values for this specific matrix layout.import numpy as np # Define the matrix matrix = np.array([[2, 2], [3, 1], [-2, -3], [2, 2], [4, np.nan]]) # Calculate row maxima (ignoring nan in the matrix) row_maxima = np.nanmax(matrix, axis=1) # Calculate column minima (ignoring nan in the matrix) column_minima = np.nanmin(matrix, axis=0) # Check for saddle points saddle_points = [] for i in range(matrix.shape[0]): # iterate over rows for j in range(matrix.shape[1]): # iterate over columns if matrix[i, j] == row_maxima[i] and matrix[i, j] == column_minima[j]: saddle_points.append((i, j, matrix[i, j])) saddle_points[(2, 0, -2.0)][(2, 0, -2.0)]successThe saddle point in matrix (a) is located at: $$(r, c) = (2, 0)$$ with a payoff value of $$-2$$. Would you like further details on how I derived the saddle point or have additional questions? Here are some related questions for further exploration: 1. What is the significance of a saddle point in game theory? 2. How does the presence of multiple saddle points affect the strategy in a game? 3. Can a matrix have no saddle points? If so, what does this imply? 4. What are pure and mixed strategies in the context of saddle points? 5. How does the minimax theorem relate to saddle points? **Tip:** In game theory, a saddle point often represents an equilibrium where neither player has an incentive to deviate from their chosen strategy.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Game Theory
Saddle Points
Payoff Matrix
Formulas
Saddle point condition: Entry must be maximum in its row and minimum in its column or vice versa
Theorems
Minimax Theorem
Suitable Grade Level
College-level Game Theory