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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14
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2answers
327 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
19 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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0answers
15 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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75 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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0answers
35 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 ...
3
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1answer
37 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 ...
2
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1answer
33 views

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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2answers
78 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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0answers
59 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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0answers
22 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
776 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 ...
1
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1answer
41 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
51 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
99 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-...
19
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2answers
421 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
84 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}{...
5
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1answer
33 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 ...
2
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0answers
40 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 ...
3
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0answers
205 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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0answers
64 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 ...
2
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0answers
73 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. ...
7
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1answer
108 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 ...
0
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0answers
42 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
45 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, ...
1
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1answer
54 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 ...
0
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1answer
46 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 $n$, ...
3
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1answer
213 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
161 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
124 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 ...
4
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1answer
91 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 ...
2
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1answer
29 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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0answers
46 views

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} \...
4
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2answers
123 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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0answers
57 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 ...
3
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1answer
571 views

In Arrow's Impossibility Theorem, what's the difference between a rank vote and a cardinal vote?

Arrow's impossibility theorem states that in any rank-based voting system involving three or more candidates, at least one of the following criteria will by necessity be violated: If every individual ...
5
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1answer
455 views

Probability of winning an election while losing the popular vote

Suppose a country with 'E' electorates and 'V' voters in each electorate, were to hold an election. Each vote is independent of all others, and has a 50% chance of being for party A and a 50% chance ...
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0answers
45 views

Are there corollaries (or generalised versions) of Arrow's theorem covering obvious fixes?

For a given set of alternatives $X=\{x_1,\dots,x_n \}$, let each individual have a total and transitive preference order among the alternatives $X$. The goal would be to have a system or rule to ...
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1answer
266 views

What is a good introductory book on Rational Choice Theory for a mathematician?

I'm interested in Rational Choice Theory as an approach to political science. Amongst other, related subjects, I'd like to know a thing or two about Arrow's impossibility theorem (and other aspects of ...
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1answer
433 views

Arrow's impossibility theorem implications [closed]

If we knew that in an unfair voting system (ie: it is known that IIA does not hold) then if (1) holds, how would we go about inferring that (2) holds and then is it possible to show that if (1) and (2)...
0
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1answer
225 views

Poisson vs Binomial for Voting

(this is my first post so I apologise if I am asking wrongly) I am trying to determine whether I should use poisson or binomial distribution to model american poll votes for the coming election. My ...
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2answers
38 views

Describe all irreducible voting systems with two outcomes

A situation where $n$ voters choose between two candidates can be modelized by a $n$-uple $(a_1,\ldots,a_n)\in\lbrace 0,1 \rbrace^n$, where $a_i$ denotes the wish of the $i$-th voter. So the voting ...
0
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1answer
281 views

Counterexample to each Arrow's axiom

I am trying to find ways that combine group preferences which hold for 3 out of the 4 Arrow's axioms. Here is what I have come up with so far: Unrestricted Domain Not sure what scheme would ...
4
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1answer
117 views

Analyse the US presidential election from the perspective of Arrow's theory

I want to consider the voting system against the 4 Arrow's axioms. So first the voting system simplifies to first-past-the post system where the highest polling candidate wins the election. How does ...
2
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1answer
76 views

name of a fairness criterion

I found the following fairness criterion in the exercises of Excursions in Modern Mathematics: If a majority of the voters have candidate X ranked last, then candidate X should not be a winner of the ...
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0answers
246 views

Show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms.

I want to show, by example, that we can find social welfare functions which satisfy any three of the four Arrow's axioms. Given at least three rewards, and at least two individuals, there is no ...
0
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1answer
744 views

Pairwise majority voting and Arrow's axioms

The following is a question on Arrow's theorem with a pairwise majority decision. The bits I was unsure about was (bi) (is the 4th condition satisfied?) and also is (bii) correct? Thanks for your help ...
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1answer
84 views

Boolean functions and characters

I'm trying to solve this, but I have no idea how to start! I'm not even sure what the question is really asking. Can anybody rephrase this and perhaps give me pointers/solutions for one of the parts?
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1answer
24 views

Condorcet winner in an examen subject with typo

I took a course in decisional models this semester. It is delightful because it allows to understand why are the weaknesses of every decision making process. Yet, the teacher gave us an old exam ...
11
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1answer
147 views

Can Continuous Time Markov Chains be used as a reasonable voting system?

I just compared a couple of example elections, as given on Wikipedia to show how Condorcet-methods differ from non-Condorcet ones, to what happens if you just interpret the underlying preference ...
3
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1answer
450 views

Algorithm for assigning users to “buckets” according to users' preferences and ranking

Suppose there is a set of $n$ users which must each be assigned to one, and only one, of $k$ mutually exclusive "buckets". However, the number of users allocated to the $i$-th bucket must be no lower ...