I was wondering which part I am misunderstanding about the individual-by-individual updating scheme from the book of Jackson M. (Social and Economic Networks, 2008) . The full transition matrix in the book is written as,
$$\text{Pr}(s_{t+1}=s+1|s_t=s)= \left( {n-s\over n} \right) p_s, \text{for} \ 0\le s \le n-1 \\ \text{Pr}(s_{t+1}=s-1|s_t=s)= \left( {s\over n} \right) (1-p_{s-1}), \text{for} \ 1\le s \le n \quad \text{(eq. 2)} \\ \text{Pr}(s_{t+1}=s|s_t=s)= \left( {n-s\over n} \right) (1-p_s)+\left( {s\over n} \right)p_{s-1}, \text{for} \ 0\le s \le n \\ \text{Pr}(s_{t+1}=s'|s_t=s)=0, \text{otherwise}. $$
I am trying to understand why it is $\left( {s\over n} \right) (1-p_{s-1})$ rather than $\left( {s\over n} \right) (1-p_s)$ in $\text{(eq. 2)}$ above since the state is originally from state $s$,
The book states that,
Consider a setting where at the beginning of a new period, one individual is picked uniformly at random and updates his or her action based on the current number of people in the society taking action $1$ or $0$. How an individual updates his or her action depends on the state of the system. In particular, let $\ p_s$ denote the probability that the individual chooses action $1$ conditional on $s$ out of the other $n-1$ agents choosing action $1$. This form of updating could be a form of best reply behavior, where an individual (myopically) chooses an action which gives a highest payoff given the current actions of the other individuals.
Would it be that each randomly chosen individual at each period is deducted from the set of players choosing action $1$? I mean looking from the individual's perspective, she decides as if she hasn't chosen an action yet and therefore the state is considered to be $s-1$?
Please let me know if I need to adjust my question to make it clearer. Thank you.
This transition matrix is consistent: gives positive probabilities, sums up to $1$ and behaves well on the boundaries $s = 0$ and $s = n$. It seems that $p_s$ and $p_{s-1}$ are just two parameters, so it would be hard to say without going through the paper why are they chosen to be distinct. In the end, it's the choice of a model, which is likely to be justified by the authors.