I am new to Markov chains since I am doing my own studying on it recently. I was doing some questions and came across this one that got me stuck.
Suppose there are four employees and they need to vote which of the 3 designs for their upcoming project to select. Once they reach a consensus the design is then chosen. Define a shift toward the consensus as one of the following 211 -> 310, 220 -> 310, 310 -> 400. Assume that an employee who can make a shift toward the consensus is twice as likely to make a vote change as any other voter and that if such a voter changes her or his vote, all changes are equally likely. Also, assume that all other voters are equally likely to change their votes and that all choices are equally likely. Draw the transition matrix.
I know that the diagram would be: 
but I am not sure how to derive the probabilities from the following assumptions.
Let's say you're in the 211 state. Based on your assumptions the two singular opinion holders are twice as likely to switch. So two thirds of the time one of those guys switches. If they switch then you're equally likely to go to 310 or 220. So $$p_{211 \to 310} = p_{211 \to 220} = (2/3)(1/2)$$
If one of the other guys switches you have to go back to 211 so $$ p_{211 \to 211} =1/3$$
As a sanity check it's good that summing over all destination states gives us 1.
Just continue in that manner for all start states. The 220 state is really easy, since everyone is equally likely to switch and the all pick destinations with equal likelihood you're equally likely to go to 311 or 211. The start from 400 case is of course trivial. So you just have one case left. I'll leave that for you.