When several people have to decide about a single yesno

When several people have to decide about a single yes/no issue*, the natural decision rule to use is the majority rule.

But when there are many issues to decide upon, the majority rule is "unfair" in the following sense: it is possible that the majority's opinion will be accepted on all topics and the minority's opinion will not be accepted on any topic. As an extreme example, it is possible that 51% of the population will decide about 100% of the issues.

I am looking for a decision rule which prevents this unfairness.

Formally, define a "uniform group" as a group of people who always vote in the same way. Define the "acceptance rate" of a uniform group as the percentage of issues on which the opinion of the uniform group got accepted.

Define a "fair decision rule" as a rule for which, for every uniform group containing X percent of the population, the acceptance rate tends to X when the number of issues tends to infinity.

MY QUESTION IS: Does there exist a fair division rule as defined above?

(* I restrict the question to yes/no issues, since when the issues are not binary the problems are much more complicated).

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