Example 11.4 The President of the United States tells person A his or her intention to run or not to run in the next election. Then A relays the news to B, who in turn relays the message to C, and so forth, always to some new person. We assume that there is a probability a that a person will change the answer from yes to no when transmitting it to the next person and a probability b that he or she will change it from no to yes. We choose as states the message, either yes or no. The transition matrix is then
\[ \begin{array}{l@{{}={}}c} \text{P = } & \left(\begin{array}{@{}ccccc@{}} & yes & no\\ yes & 1-a & a\\ no & b & 1- b \end{array}\right) \end{array} \]
The initial state represents the President’s choice.
Answer -
By definition 11.1 of an Absorbing Markov Chain -
Definitionn 11.1 A state \(s_i\) of a Markov chain is called absorbing if it is impossible to leave it (i.e., \(p_{ii}\) = 1). A Markov chain is absorbing if it has at least one absorbing state, andif from every state it is possible to go to an absorbing state (not necessarily in one step).
The only way for the transition matrix above to reach an absorbing state is for \(P_{yes,yes}\) = 1 or for \(P_{no,no}\) = 1. That is, for a or b to be equal to 0. This means that this will only happen if there is a 100% probability that the answer won’t be changed from yes to no (a) or no to yes (b) when it is relayed to the next person.