Or, a method for the fair and equitable distribution of political power.
Much thought has been given to methods of ensuring fair and accurate representation in a representative democracy. Ideally, if, for example, 40% of the population supports one party and 60% of the population, then 40% of the representatives in any given governing body should be from the first party and 60% from the second (or at least, 40% and 60% of the votes should come from each party, if we relax the assumption that one rep gets one vote), within the limits of rounding errors introduced by having a small finite number of representatives.
Let us assume, however, that you have instituted a perfect system of ranked-choice voting (or something similar) and a perfect and unbiased system for drawing districts from which representatives will be selected, so that you always have such perfect representation. Or, presume that you have a perfect direct democracy, so issues of proportional representation never come up in the first place. At that point, there is still another problem to be solved: the tyranny of the majority.
The trouble is that what we really want is not proportionate representation at all; it is proportionate distribution of power. Representation is merely a poor, but easier to measure, approximation for power. And in a perfect representative government where 60% of the constituency supports one party, and 60% of the governing votes are controlled by that party, they do not have 60% of the power- they have all of it, presuming you go with a simple majority voting scheme to pass legislation. If you require some larger plurality, however, the problem still does not go away; supermajority voting requirements simply mean that it takes a larger majority to become tyrannical, and in the meantime you have a roadblock: the minority party may not be able to pass anything, but it can keep the other side from passing anything either! Nobody getting anything done may be an equitable distribution of power, but only because any percentage of 0 is still 0.
It would be better if we could somehow guarantee that a 60% majority party would get what they want 60% of the time when in conflict with the minority, and a 40% minority would get what they want 40% of the time when in conflict with the majority. It turns out that there is a remarkably simple way to guarantee this result!
Rather than allowing the passing of legislation to be decided by a purely deterministic process, we introduce an element of randomness. When any issue is voted upon, the action to be taken should not be determined simply by whichever side gets the most votes; rather, the result of a poll is a random selection from a distribution of options, weighted by the number of votes cast for each one. If all representatives always vote strictly along party lines, then over a large number of votes on issues over which the two parties disagree, 60% will be decided in favor of the majority party, and 40% in favor of the minority party- but with no way to predict ahead of time which 60 or 40% they will be, such that it is impossible to game the system by strategically timing the introduction of certain bills.
Even if we imagine a more extreme split, like 90% vs. 10%, it is still impossible under this system for the majority to run away with tyrannical measures that harm the minority for very long. Should they try, it incentivizes the minority party to introduce ever more extreme countermeasure proposals more and more frequently, one of which is guaranteed to eventually pass! The majority party therefore has its own incentive to attempt to build consensus even with small minority factions, creating legislation that will benefit everyone.
Of course, it seems highly unlikely that any country could be convinced to run their government this way in real life! But, there are I think two good reasons for considering the idea. First, this could make for an interesting political backdrop in a sci-fi or fantasy story; perhaps your fantasy society finds the inclusion of randomness in their political process to be a religious imperative, as it is through the interpretation of random events that the will of the gods is revealed. Second, it highlights to mostly-overlooked but very important distinction between equitable representation, and equitable distribution of power. Hopefully, having looked a this simple solution to the problem will help someone to discover another, perhaps more practically tenable, one.