Questions tagged [voting-theory]

For questions regarding the mathematical analysis of voting systems and behavior. Examples include the median voter theorem or the Condorcet jury theorems.

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20 views

Does Borda count satisfies the participation criterion?

The Borda count, or average rank method, is said to satisfies the participation criterion. This means that this ranking method is free from the "no show paradox". Do you know any proof of ...
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36 views

The number of distinct piles of ranked ballots in Ranked-Choice Voting.

This is about Ranked-Choice Voting (RCV) where the ballot has $C$ candidates, and there are $C$ levels of ranking of preference. Equal ranking of candidates is not allowed and no voter is required to ...
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48 views

Shapley Value for Voters in Electoral College

Consider (a simplified version of) the USA electoral college. Each state has millions of voters and tens of electors. The candidate with the most electors wins and each state allocates electors ...
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1answer
40 views

How would you find the least number of popular votes to win an election given it is split in electoral districts?

I was given a problem to find the least number of popular votes to win an election through an electoral college. They gave me $N$ number of counties with $K$ required electoral votes to win. Each ...
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1answer
69 views

What is the probability of a sampled election to be wrong?

Suppose that only a subset of people who wants to vote is allowed to. The sampling of voters is fair, and anyone has the same chance of being selected to vote. Suppose $n$ people are allowed to vote, ...
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100 views

Probability there is no president

I know you may decide my question is off-topic but I will give it a try: I read this in a Facebook group, posted by a person who claims to be a mathematician. In the US presidential elections, for a ...
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1answer
52 views

Does this proof of a voting related lemma work? if so, how?

I'm attempting to read this PDF on Arrow's impossibility theorem and ultrafilters. I find myself unconvinced that the proof of their Lemma 13 demonstrates what they say it does. I'm hoping someone ...
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32 views

Are these two definitions of a filter on a set equivalent?

I'm attempting to read this PDF on Arrow's impossibility theorem and ultrafilters. The PDF says that there are non-principal ultrafilters on any infinite sets, but does not demonstrate this. So I ...
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77 views

Can we make a voting system where it is cryptographically hard to find a dictator

As Wikipedia says, Arrow's impossibility theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria: If every voter prefers ...
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2answers
62 views

Is there a known example of a voting system that does not satify the dictator fairness criterion but does satisfy the others?

As Wikipedia says, Arrow's impossibility theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria: If every voter prefers ...
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210 views

Probability of different winners in a two candidate election (range voting vs majority)

I was clicking through the xkcd comics, and I came upon xkcd 2225. I didn't know about "Range Voting", so decided to read about this voting system. I came up with the following problem based ...
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37 views

How to incentivize the members of a committee voting honesty?

Suppose you have a majority voting mechanism and a committee composed by a number of players $n$, $C = \{P_1,...,P_n\}$. Each player is required to compute the result of a certain function $f$ and ...
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Arrow's impossibility theorem simple proof and explanation

I am trying to prove the Arrow's Impossibility Theorem. I was searching on the internet but there is lots of different versions. I want to prove it for this statement: Arrow's Theorem: Consider a set ...
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Probability of different winner in two voting systems

There are two methods of election that I often hear about: first past the post (whichever candidate gets the most votes wins) and the two-round system. In the two-round system, the top two candidates ...
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62 views

Hamilton's paradox

We have the rule: We compare parties A and B in two different elections. If it happens that A wins votes and B loses votes, it cannot happen that A loses a seat and B wins a seat. Concretely, I ...
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1answer
22 views

Shapley-Shubik Power with no quota (weighted voting)

I'm learning weighted voting methods and MathLab does a terrible job explaining this question so I'm trying to get clarification. I don't understand how you can find a critical player without a quota....
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378 views

Honest vs. Dishonest Voting: does it matter?

Let's say I have a website where people can rate movies on a scale of 0-10. We say people vote honestly when the rating they give to a movie is what they actually think. People vote dishonestly when ...
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1answer
99 views

Creating a Majority Graph from multiple preference orders

I can't find much on voting theory on this exceptional site. I am trying to find a way of constructing a majority graph based on a few preference. When I try to construct one, I end up breaking the ...
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18 views

A problem about voting on a numeric value

Let $w = f(v)$ where $f$ is a strictly increasing continuous function $[0;+\infty)\to [0;1)$. Let $A(w_1,\dots,w_n)$ be some average. Then we want $f^{-1}(A(f(cv_1),\dots,f(cv_n))) = cf^{-1}(A(f(v_1),\...
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84 views

Problem in downvote system

Problem For my game, I'm building a system where players have power/weight, and they can downvote each other, players with 66% of downvote weight are banned. The weight of the votes is calculated ...
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47 views

Simple Algorithm to Find the Inner Smith Set

I've been writing a library to tabulate results of ranked choice ballots by multiple methods. For Condorcet Methods I would like to quickly reach the smallest possible Dominant (Smith) Set. The steps ...
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2answers
58 views

Organise “all possible voting schemes”

I’m currently reading 1, 2, and 3. Wikipedia lists some of the better-known voting schemes (Borda count, approval voting, run-offs, …), a few of which have actually been tried in reality (e.g., the ...
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1answer
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Simple Ranked Voting

So I'm looking at trying to solve this problem: I have N-individuals who're trying to determine the relative priority of M-projects. We decided to start off with a voting scheme which was something ...
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84 views

Voting system with probability

Three independent algorithms are executed in parallel. The role of each algorithm is to give an answer (Yes or no) with a certain probability to a certain number of questions (say 100). Example: ...
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73 views

Prove Nanson's method will select Condorcet Winner

I somehow cannot prove this, so requesting assistance: In the voting method known as the Nanson's method, the Borda counts of all candidates are computed. Next, those candidates with below average ...
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24 views

Why Is the Jefferson Method Not Susceptible to the Population Paradox?

When apportioning the seats of a legislative body, why does using the Jefferson method, where $P_i$ is population in state $i$, $N$ is total seats available, and we vary $Q>0$ until $$\sum_i{\bigg\...
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1answer
867 views

Mathematics solution for Gerrymandering problem?

Gerrymandering is a practice intended to establish a political advantage for a particular party or group by manipulating district boundaries, and can create large disproportions in voting results as ...
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1answer
42 views

Voting with weights: Proof that that the person with weight one actually had no longer the right to vote.

I found this interesting problem on the internet. Can someone help? When a proposal has to be approved, there must be at least 5 more yes votes than no votes. If not, the proposal is rejected. ...
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1answer
53 views

In terms accessible to someone new to electoral systems, what does the Schulze system do in case of no condorcet winner?

The title pretty much summarizes it. I am new to the idea of different electoral systems. Schulze is one that I find very interesting, but I find it difficult to understand what it does in case of no ...
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2answers
147 views

Difference between Arrow and Gibbard-Satterthawite theorem

I have a question related to two very important theorems from Social Choice Theory. What is the difference between Arrow Theorem and Gibbard-Saterthwaite theorem? I mean, the obvious one is that in G-...
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438 views

Minimizing the probability of a draw in a democratic poll

A group of $k$ people wants to choose democratically between $n$ possible options. They arrange a poll in which every person votes for $r$ out of the $n$ options without repetition, meaning there are $...
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1answer
90 views

Enforcing quota in any proportional voting system: frunction box

I've researched voting systems lately. I haven't encountered this idea, so I'd be grateful if somebody shows books/articles that include it or something similar. Take a voting result: $$\begin{array}{...
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1answer
34 views

Fair Rank Voting When Some Members Do Not Vote

I am currently trying to work out a problem with a voting system for a local club. Competitions are held frequently and the current system works as such: Everbody who enters the competition gets to ...
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43 views

Voting theory when NO is an option

Has anyone mathematically studied voting systems in which NO is one of the options, or more precisely in which one of the possible outcomes is that none of the candidates win? If so, where can I read ...
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259 views

Game theory: Approval Voting not weakly dominated strategies

"In the system of “approval voting”, a citizen may vote for as many candidates as she wishes. If there are two candidates, say A and B, for example, a citizen may vote for neither candidate, for A, ...
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69 views

Embedding preference orders in 2D Euclidean space

PROBLEM I have a set of candidates $C=\{c_1,c_2,...,c_m\}$ and a set of voters $V=\{v_1,v_2,...,v_n\}$. Each voter ranks all of the candidates (e.g. $v_1: c_1 \succ c_2 \succ c_3 \succ \dots \succ ...
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74 views

Probability of a different electoral system returning a different result

Consider an election with three candidates: A, B and C. Each voter orders the candidates randomly (ABC, ACB, BAC, BCA, CAB or CBA), such that there is a 1/6th chance of voting for each of the options. ...
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1answer
115 views

Probability of winning an election while losing the popular vote: electorates of size 3

Suppose we live in a country with an interesting electoral system: each electorate has exactly 3 voters. 2 parties run for office, and each voter has a 50/50 chance of voting for each party. Whoever ...
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47 views

eVoting with Damgard-Jurik-Cryptosystem

I am trying to implement a secure elecetronic voting system. Therefore I found the Damgard and Jurik Cryptosystem. In their paper the authors describe a secure protocol for "A Length-Flexible ...
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1answer
54 views

Math of voting as single block

So the only time your vote actually affects anything is when your vote decides an election. If your vote does not decide an election, it wouldn't have turned out differently if you haven't voted, ...
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1answer
58 views

Why is it continuous?

I came across this proof that I don't understand while reading the book "Majority Judgement" by Balinski and Laraki. To simplify, we have a function $f$ which is continuous and monotonic. $f$ takes n ...
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1answer
49 views

A fair transparent voting system

Let's say we have a group of $n$ people voting about which one of them should be their leader. They choose to vote by ranking, that is, by giving each member a number of points between $1$ and $n$, ...
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1answer
297 views

When is there always a Condorcet winner?

I found the following in a book about voting systems, and it is claiming that in this particular situation there always is a Condorcet winner: "Suppose there is an odd number of voters and all voter's ...
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1answer
211 views

Biased Ballot Theorem

Classical Ballot theorem states if candidate $A$ receives $a$ votes and $B$ receives $b$ votes $(b < a)$, and assume that each candidate is equally likely to get a vote, then the probability that ...
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1answer
209 views

The way to divide a group into two equal halves according to their preferences

I know that does not sound like a math question, but I believe there should be a really simple solution to this somewhere in combinatorics (I am sorry in advance if I am wrong). There is a group of 6 ...
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1answer
123 views

Weighted test - make sure all questions can impact the result

I'm making a weighted test, i.e. a test with N yes/no questions where the score differs from question to question. For simplicity assume that the sum of all individual scores is 100 and assume the ...
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1answer
32 views

Minimal number of supporters need to win a multi-level election

On July 27th, Max Alekseyev posted a sequence to the OEIS: A290323: Minimal number of supporters among total of n voters that may make (but not guarantee) their candidate win in a multi-level ...
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What is this multi-winner voting problem related to? Load balancing? Quadratic Constraint? Integer optimization? Gurobi solves it? Can it go faster?

Problem Statement Let's run an election. $i \in \text{voters}$ $j \in \text{candidates}$ $x_j \in \{ 0, 1 \}$ The candidate is chosen by setting this to 1. This is the election result. $b_{i,j} \...
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2answers
134 views

The splitting team earnings by vote problem

For a situation with $n$ team members having to share a prize of $x$ dollars, is there a mathematical formulation of how a voting system can be used to split the money in a way that would satisfy the ...
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58 views

Numerical voting algorithm

This question relates closely to other posts. See note at the bottom. Problem: Suppose that a committee with $n$ members needs to vote on whether to accept a proposition. Each member in the ...