(c) Which of the states are absorbing?

In a Markov chain, a state is absorbing if, once the system enters that state, it cannot leave. In the context of this problem, the absorbing states are those where all the individuals support the same political party, as the discussion ends in such cases.

• State $S_0$: This is the state where all three people support political party $V$ (W has 0 followers). Once this state is reached, no one can change their opinion, so this state is absorbing.
• State $S_3$: This is the state where all three people support political party $W$ (W has 3 followers). Similarly, once this state is reached, no one can change their opinion, so this state is also absorbing.

Therefore, the absorbing states are:

• $S_0$
• $S_3$

(d) What is the probability to eventually end in state $S_3$?

To solve this, we need to compute the probability of absorption into state $S_3$ starting from $S_1$ (since at $t = 0$, W has one follower).

This type of problem is typically solved using the fundamental matrix for absorbing Markov chains. However, we can also approach this problem using a recursive approach based on transition probabilities.

Transition Probabilities:

Let $p_i$ represent the probability of eventually being absorbed into $S_3$ given that the system is currently in state $S_i$.

• $p_0 = 0$, because if the system reaches $S_0$, it will never reach $S_3$.
• $p_3 = 1$, because if the system reaches $S_3$, it is already absorbed there.

The recursive relations for intermediate states $S_1$ and $S_2$ are based on the transition probabilities:

• From $S_1$, you move to:

  • $S_0$ with probability $\frac{2}{3}$
  • $S_2$ with probability $\frac{1}{3}$

  So the equation for $p_1$ is: $p_1 = \frac{2}{3} \cdot p_0 + \frac{1}{3} \cdot p_2 = \frac{1}{3} p_2$ (since $p_0 = 0$)

• From $S_2$, you move to:

  • $S_1$ with probability $\frac{2}{3}$
  • $S_3$ with probability $\frac{1}{3}$

  So the equation for $p_2$ is: $p_2 = \frac{2}{3} \cdot p_1 + \frac{1}{3} \cdot p_3 = \frac{2}{3} p_1 + \frac{1}{3}$

Solving the System of Equations:

Now, solve the system of equations:

1. $p_1 = \frac{1}{3} p_2$
2. $p_2 = \frac{2}{3} p_1 + \frac{1}{3}$

Substitute $p_1 = \frac{1}{3} p_2$ into the second equation: $p_2 = \frac{2}{3} \cdot \frac{1}{3} p_2 + \frac{1}{3}$ $p_2 = \frac{2}{9} p_2 + \frac{1}{3}$ $p_2 \left(1 - \frac{2}{9}\right) = \frac{1}{3}$ $p_2 \cdot \frac{7}{9} = \frac{1}{3}$ $p_2 = \frac{9}{7} \cdot \frac{1}{3} = \frac{3}{7}$

Now, substitute $p_2 = \frac{3}{7}$ into the equation for $p_1$: $p_1 = \frac{1}{3} \cdot \frac{3}{7} = \frac{1}{7}$

Thus, the probability of eventually ending in state $S_3$ starting from state $S_1$ is $\boxed{\frac{1}{7}}$.

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Let me know if you need further clarifications!

Related Questions:

1. What is the probability to end in state $S_0$?
2. Can this Markov chain have any other absorbing states if the number of people changes?
3. How would the transition matrix change if the persuasion probabilities were different?
4. What happens to the transition probabilities if there were more than two political parties?
5. Can this model be extended to more complex discussions involving more individuals?

Tip: In Markov chains, once the absorbing states are reached, no transitions occur, making them crucial for understanding long-term outcomes.