We are in a season where some major elections are happening (e.g. U.S. elections) and I find it interesting to address.
Objective
When we decide "better", we need to define an objective. To be clear, my definition of better is as follows:
Definition: A voting method $m_1$ is better than another voting method $m_2$ only if $m_1$ leads to higher mean lifespan expectancy in the upcoming 20 years for those who vote.
Note 1: this definition is not subject to change, and is not a question. Kindly assume that this objective is true when giving any answer. Therefore, suggesting alternative objectives is off-topic.
Note 2: suppose that elections are actually meaningful. E.g. do not discuss facts such as lobbyiests.
A Common Voting Method: Simple Majority Voting
The population is defined to be the set of all living citizens that are eligible for voting. We then choose the president that the majority want.
The assumption here is that the president that is chosen by the majority is one that will lead to maximize our objective better than any other candidate president.
This means that over-represented kinds of people in the population will have their votes translate into actions. If the majority of the population are of kind "idiot", then we will choose a decision that idiots think are best. If kind "genius" is a minority, then its vote will not translate into actions (unless geniuses happen to agree with idiots; possibly for different reasons, but if wrong thinking leads to right decision then who cares).
Alternative Voting Method 1
The population is the space of possible decisions. In this case this set is $\mathcal{D} = \{choose president_1, choose president_2, \ldots, choose president_n\}$.
Additionally, we define another population of citizens $\mathcal{C}$. We partition this set into various strata based on the kind of citizens. Suppose that all citizens in $\mathcal{C}$ are of kinds that are definiable by the binary parameters $p_1, p_2, \ldots, p_m$ only. This means that we will have $2^m$ partitions/strata. Our strata is $\mathcal{S} = \{s_1, s_2, \ldots, s_{2^m}\}$.
Then, for any stratum $s_i \in \mathcal{S}$, we find the majority vote of citizens that fall within stratum $s_i$ (i.e. the most popular president candidate in stratum $s_i$. Let $v_i$ be the majority vote of stratum $s_i$.
Finally, we list all strata votes $v_1, v_2, \ldots, v_{2^m}$, and choose the most popular vote from this list.
This alternative voting method assumes that all kinds of voters have equal power in deciding the right action. Therefore, "geniuses", "almost-geniuses", "normal", "almost-idiots", and "idiots" are guaranteed to be equally powerful.
Alternative Voting Method 2
This is identical to the alternative voting method 1, except for the following distinction: instead of assuming that vote $v_i$ of stratum $s_i$ has an equal power in deciding the final chosen president as any other stratum, we follow a weighted approach such that $v_i$ has a higher power in choosing the president than another vote $v_l$ only the stratum $s_i$ contains citizens of a kind that is smarter than the kind in the stratum $s_l$.
Other Voting Methods
Feel free to define other voting methods if they help answering the questions.
Questions
- Q1) How to decide which voting method is a better method to maximize the optimization objective?
Q2)
- A1) Using the method in the answer of (Q1) and assuming that there is a deterministic answer: Which voting method is one that maximizes the optimization objective above?
- A2) Using the method in the answer of (Q1) but assuming that there is no deterministic answer: Which voting method is one that is more likely to maximize the optimization objective?
Q3) Justify your answer for questions (Q1) and (Q2).
Hints on How You Might Want to Answer This
This question is fundamentally a statistical inference problem where your goal is finding the decision that is shall maximize the optimization objective.
For example, by asking the votes of people from the population, you are doing an identical job to ensemble learning algorithms, such as Random Forests where votes are averaged from various randomly grown decision trees. Except that you replace humans by decision trees.
Such inference usually happens by:
- Analyzing some dataset (if you have one),
- OR (if you don't have such dataset, or don't find it convenient to build one) theoretically discuss the pros and cos of the various inference methods, and then try to justify why you think that the pros and cons of some inference method are better suited for the scenario at hand.
Similar to other machine learning problems, this is a question that requires some thinking and the addition of extra assumptions. There is no theorem or a deterministic equation that you can use solve this deterministically without assuming extra things. Thus you are free to suggest the addition of extra assumptions and constraints to justify your solution.
Then, if your assumptions and constraints seem plausible, I will consider your answer to be the correct answer.
