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Chapter 9: Problem 13

Find the steady-state vector for the transition matrix. $$ \left[\begin{array}{lll} 0 & \frac{1}{8} & 1 \\ 1 & \frac{5}{8} & 0 \\ 0 & \frac{1}{4} & 0 \end{array}\right] $$

Short Answer

Expert verified
The given matrix does not have a steady-state vector since the eigenvalue 位=1 is missing.

Step by step solution

01

Find the eigenvalues

First, we need to find the eigenvalues of the given matrix. To do this, we need to solve the characteristic equation: det(A - 位I) = 0, where A is the given matrix, 位 is the eigenvalue, and I is the identity matrix. For the given matrix A = \[ \begin{pmatrix} 0 & \frac{1}{8} & 1 \\ 1 & \frac{5}{8} & 0 \\ 0 & \frac{1}{4} & 0 \end{pmatrix} \] we have the characteristic equation: det(A - 位I) = \( \begin{vmatrix} -\lambda & \frac{1}{8} & 1 \\ 1 & \frac{5}{8}-\lambda & 0 \\ 0 & \frac{1}{4} & -\lambda \end{vmatrix} \) = 0 Calculating further: \(-\lambda((\frac{5}{8}-\lambda)(-\lambda)-0)-(\frac{1}{8})(0)+1(0)=-(\lambda^3-\frac{5}{8}\lambda^2) = 0\) Solve for 位: \(\lambda^3 - \frac{5}{8}\lambda^2 = \lambda^2(\lambda - \frac{5}{8}) = 0\) We have two eigenvalues: 位_1=0 and 位_2=\(\frac{5}{8}\). However, the steady-state vector is always corresponding to the eigenvalue 位=1. In this case, 位=1 is missing, which means the matrix does not have a steady-state vector.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transition Matrix
A Transition Matrix is a square matrix used to describe the transitions of a Markov chain. Each element of the matrix represents the probability of moving from one state to another in a system. Transition matrices are fundamental in various fields such as economics, game theory, and stochastic processes.

The given matrix in the exercise is a three-state transition matrix with elements indicating the transition probabilities between different states over a single time step. For example, the element in the first row and third column represents the probability of transitioning from state 1 to state 3.

It's pivotal for a transition matrix to meet certain criteria. Specifically, the sum of the probabilities in any given row must equal 1, accounting for all possibilities of transitioning from that state to all others, including itself. Analyzing a transition matrix can reveal long-term behavior such as steady states, which are the probabilities that are constant over time.
Eigenvalues
The term eigenvalues is central to the study of linear algebra and is deeply connected to various mathematical areas and practical applications. An eigenvalue of a matrix is a number, often denoted by \( \lambda \) (lambda), that, when associated with a particular linear transformation, yields a scaled version of an eigenvector of that transformation. In simple terms, it's the factor by which the eigenvector is scaled when multiplied by the matrix.

The concept of eigenvalues is crucial in determining the long-term behavior of a system, particularly when analyzing transition matrices in Markov chains. In the context of Markov chains, an eigenvalue of 1 corresponds to a potential steady-state condition where the system's state remains unchanged after transitions. For a matrix to depict a stable long-term behavior, there must be an eigenvalue equal to 1.

However, the absence of such an eigenvalue, as seen in the matrix from the exercise, signifies that there is no steady state, implying the system does not reach a point where the state probabilities are constant through the transitions.
Characteristic Equation
The Characteristic Equation springs from the realm of linear algebra and is instrumental in finding the eigenvalues of a matrix. It is formed by setting up the equation det(\(A - \lambda I\) = 0, where \( A \) is the matrix in question, \( \lambda \) the eigenvalue, and \( I \) the identity matrix of matching dimensions.

Solving this equation, often through expansion, yields the eigenvalues. In the provided exercise, the characteristic equation is derived from the transition matrix and restructured to determine eigenvalues that could lead to finding a steady-state vector.

The calculation process involves expanding the determinant of the matrix after subtracting \( \lambda \) times the identity matrix from it. It's important to get a polynomial equation where the powers of \( \lambda \) determine the potential eigenvalues. As demonstrated in the exercise, if the eigenvalues do not include 1, it concludes that a steady-state vector does not exist for the given matrix. This insight is pivotal for understanding the behavior of the system characterized by the transition matrix.

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Most popular questions from this chapter

Let $$ \left[\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right] $$ be the payoff matrix with a \(2 \times 2\) matrix game. Assume that either the row player uses the optimal mixed strategy \(P=\left[\begin{array}{ll}p_{1} & p_{2}\end{array}\right]\), where $$ p_{1}=\frac{d-c}{a+d-b-c} \text { and } p_{2}=1-p_{1} $$ or the column player uses the optimal mixed strategy $$ Q=\left[\begin{array}{l} q_{1} \\ q_{2} \end{array}\right] $$ where $$ q_{1}=\frac{d-b}{a+d-b-c} \text { and } q_{2}=1-q_{1} $$ Show by direct computation that the expected value of the game is given by \(E=P A Q\).

$$ \left[\begin{array}{rrr} 2 & 4 & 2 \\ 0 & 3 & 0 \\ -1 & -2 & 1 \end{array}\right] $$Determine whether the two-person, zero-sum matrix game is strictly determined. If a game is strictly determined, a. Find the saddle point(s) of the game. b. Find the optimal strategy for each player. c. Find the value of the game. d. Determine whether the game favors one player over the other. $$ \left[\begin{array}{rr} 1 & 2 \\ 0 & 3 \\ -1 & 2 \\ 2 & -2 \end{array}\right] $$

Robin and Cathy play a game of matching fingers. On a predetermined signal, both players simultaneously extend 1,2 , or 3 fingers from a closed fist. If the sum of the number of fingers extended is even, then Robin receives an amount in dollars equal to that sum from Cathy. If the sum of the number of fingers extended is odd, then Cathy receives an amount in dollars equal to that sum from Robin. a. Construct the payoff matrix for the game. b. Find the maximin and the minimax strategies for Robin and Cathy, respectively. c. Is the game strictly determined? d. If the answer to part (c) is yes, what is the value of the game?

Compute the steady-state matrix of each stochastic matrix. \(\left[\begin{array}{ll}\frac{3}{5} & 0 \\ \frac{2}{5} & 1\end{array}\right]\)

Find the expected payoff \(E\) of each game whose payoff matrix and strategies \(P\) and \(Q\) (for the row and column players, respectively) are given. \(\left[\begin{array}{rrr}2 & 0 & -2 \\ 1 & -1 & 3 \\ 2 & 1 & -4\end{array}\right], P=\left[\begin{array}{lll}.2 & .6 & .2\end{array}\right], Q=\left[\begin{array}{l}.2 \\ .6 \\\ .2\end{array}\right]\)
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