Anyone know about definition of weak dictator?

Question

I am trying to prove arrow's impossibility theorem in case which ties are allowed in individual preference lists and so is social preference list.

It says that if $p$ is a weak dictator, then we can not say anything about how society ranks $a$ versus $b$ if $p$ has $a$ and $b$ tied.

Can anyone explain why this is true?

I am so confused about what a weak dictator is.

Answer 1

If $p$ is a dictator, society’s ranking of any two alternatives is always exactly the same as $p$’s ranking of those two alternatives. This is no longer quite true if $p$ is a weak dictator. A weak dictator determines society’s choice only when the weak dictator actually has a preference. Thus, when we say that $p$ is a weak dictator, we mean that

• if $p$ ranks $a$ ahead of $b$, then so does society — so far this is just like dictatorship by $p$ — but
• if $p$ is indifferent between $a$ and $b$ (i.e., ranks them equally), society might rank them equally, might prefer $a$ to $b$, or might prefer $b$ to $a$: we can’t deduce anything about society’s ranking of $a$ and $b$ in this case.

Here’s a small example. There are four voters, $p,q,r$, and $s$. For any set of alternatives, we determined society’s ranking as follows. We start with $p$’s ranking. If it contains no ties, we use it for this little society as a whole. If $p$ has no preference between $a$ and $b$, we rank them according to $q$’s preference, if any. If $q$ also has no preference between $a$ and $b$, we rank them according to $r$’s preference. If $r$ also has no preference between $a$ and $b$, we rank them according to $s$’s preference. And if none of the four voters has a preference between $a$ and $b$, we leave them tied in society’s preference schedule. In this scheme $p$ is a weak dictator: if $p$ prefers an alternative $a$ to an alternative $b$, so will this little society, no matter how $q,r$, and $s$ rank $a$ and $b$. If $p$ is indifferent between $a$ and $b$, however, we can’t tell how society will rank $a$ and $b$ without knowing the other voters’ preferences.