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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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Maximising the difference between the ordering of pairs of elements in a collection of totally ordered sets

Suppose I have $M$ totally ordered sets, all of length $N$, who all share the same elements. For example, for $M=3$ and $N=4$, we can have $S_1 = \{A < B < C < D\}$, $S_2 = \{B < A < C &...
Samuel Fedida's user avatar
1 vote
1 answer
46 views

Finding weights for the Borda count to elect any given candidate

Suppose three candidates A, B, and C run for office and voters submit their preference orders. The typical Borda count awards 2 points for a first-place vote, 1 point for a second-place vote, and 0 ...
Jared's user avatar
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0 votes
0 answers
22 views

Pivotal's definition on arrow's impossible theorem

I'm trying to understand Arrow's impossible theorem, and I understand the idea of using extremal lemma. The pivotal is defined as the first individual who change his preference of from ranking $b$ to ...
Chang Henry's user avatar
2 votes
1 answer
62 views

Probability that a uniform random variable + a constant be greater than another uniform random variable

I am trying to understand a model of probabilistic voting from the following paper https://ideas.repec.org/a/eee/pubeco/v96y2012i1p10-19.html and the one component I struggle with goes as follows: ΔU ...
flafont11's user avatar
4 votes
1 answer
104 views

Which Property Did the Sport Climbing Point System Violate?

In the 2020 Summer Olympics Sport Climbing was added as a new Olympic Sport. Competitors had to compete in three categories: lead climbing, bouldering and speed climbing. Within each category the ...
Matthias Vegh's user avatar
2 votes
0 answers
42 views

Eigenvector of a matrix in which values normalise after multiplication

Definition: Suppose $x\in\mathbb R$. Then define $||x||$ the collapsing of $x$ by $||x||=0$ if $x<1$, and $||x||=1$ if $x\geq 1$. Definition: Suppose that $M$ is a matrix over the closed interval $...
Jim's user avatar
  • 538
0 votes
1 answer
52 views

Given a social system in which people are supporters of either a $G$- or a $H$-orientation, calculate the probability to have a $G$-person elected.

In this paper, Majority rule, hierarchical structures, and democratic totalitarianism: A statistical approach, I read: A social system is considered in which people are supporters of either a $G$-...
Mark's user avatar
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0 votes
0 answers
16 views

Best voting system where there are few voters and seferal candidates?

I'm looking at a scenario where there are few voters (2-3), and several to many candidates (20-30) where at most there is room for 3 winners. Approval voting seems like the way to go to avoid tactical ...
LavaHot's user avatar
  • 101
2 votes
0 answers
74 views

On the Condorcet Paradox

The OEIS sequence A277935 gives the number of ways 2n-1 people can vote on three candidates such that the Condorcet paradox arises. Is there a general formula for the number of ways $n$ people can ...
Poisson's user avatar
  • 372
1 vote
0 answers
42 views

Truthful budget partitioning

Suppose there is a budget $M$ that needs to be partitioned into two projects $A$ and $B$. There are two players, where the first player prefers to allocate $pM$ to A and $(1-p)M$ to $B$ (i.e., it ...
Doron's user avatar
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4 votes
0 answers
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Median Voter Models in Two Dimensions (computing area of a bounded sector)

I've been trying to work out a way of computing area for a two-dimensional median voter model I've been working on. A, B, and C are political parties that can choose where they want to be on the ...
loxm's user avatar
  • 51
2 votes
0 answers
142 views

What properties of a voting method is best for deciding what movie to watch among a small group of people? [closed]

Almost every night me and my friends get together to watch a movie. Current Method: Each person (<10) picks 5 movies they want to watch and we vote on them. Each person gets around 7 votes which ...
Jacob Edie's user avatar
0 votes
2 answers
609 views

Game theory - three voters for two candidates nash equilibrium

There are 3 voters (x, y, and z) and two candidates (Alice and Bob). For either Alice or Bob to win they need 2 votes. If Alice wins x gets 1, y and z get 0. If Bob wins, x gets 0 and y and z get 1 ...
bee761's user avatar
  • 11
4 votes
1 answer
60 views

Approximative formula for normal distribution being above threshold

Suppose that $X \sim \mathcal{N}(r + \frac{1}{N}, s) $ and $Y \sim \mathcal{N}(r, s)$ for some $r, s \approx 1$ and $N \approx 10^6$. What are good approximate formulas for the quantity $$\frac{ \...
Frederik Ravn Klausen's user avatar
2 votes
1 answer
50 views

Assessing the efficiency of a single vote in a multiparty presidential election

In a country there is a voting system where all parties get represented in parliament if they meet a bar of $n$ percent. Suppose that the parties are grouped into two groups of red $R_1, \dots R_k $ ...
Frederik Ravn Klausen's user avatar
3 votes
0 answers
31 views

$n$ voters ranks $m$ candidates, what is the probability of the Smith set having cardinality $k$?

Let’s say there are $n$ voters who vote on $m$ candidates. Each voter creates a list where they rank the candidates from most favorite to least favorite. There are $m!$ different possible lists each ...
Mettek's user avatar
  • 437
1 vote
1 answer
143 views

Ask a binomial distribution problem in the voting setup

Two candidates were running for a post. The voting machines recorded 520,000 votes for the first candidate and 480,000 votes for the second one. Afterwards it became apparent that the voting machines ...
Jonathen's user avatar
  • 1,088
2 votes
0 answers
27 views

Help me find a way of sorting pairs of candidates for ranked pairs voting when not all voters rank all candidates

(Background: For ranked pairs voting when all voters rank all candidates, pairs of candidates $(A, B)$ are sorted by the margin of victory between the two candidates (that is, if $x$ voters prefer $A$ ...
Luke Thornburgh's user avatar
1 vote
0 answers
19 views

How do I formally prove this single transferable vote rule is independent of covered alternatives when electing a single winner?

I invented a new STV rule that appears to be independent of covered alternatives for a single-winner case (IRV). I am not sure what basic approach to use to formally prove this is independent of ...
John Moser's user avatar
1 vote
0 answers
139 views

Formal application of Pigeonhole principle on voting and candidates

I am reading about the Pigeonhole principle and the following problem under that section: A state has $7$ counties. In one year, three candidates run in a statewide election. Is it possible that in ...
Jim's user avatar
  • 1,609
2 votes
1 answer
55 views

What to do when in Coombs voting method there two equal weights for candidates to be elimenated?

I've read about Coombs method on Wikipedia. I understand that we eliminate candidate with the most last-place votes. But what do we do when, for example, two candidates A and B have equal number of ...
Lily Sanders's user avatar
0 votes
0 answers
28 views

Multiple conditional probabilities in a simple voting model

I am struggling with a voting model problem that is set up as follows: Suppose there is a binary issue where policy $L$ and policy $R$ are equally likely to be optimal. There are three voters who vote ...
BDot35's user avatar
  • 15
5 votes
1 answer
368 views

Borda Count vs Average Ranking

Recently, I was explaining to my high school class what the Borda Count was. We had taken a class survey on something and everyone ranked their choices in order of preference. I calculated the Borda ...
ruferd's user avatar
  • 554
1 vote
1 answer
56 views

Can strategy be used to win a majority in an election which uses cumulative voting? This is a voting theory question. [closed]

Scenario: A town is having an election for its Board of Trustees. There are six seats on the Board. A bloc of voters wants to win a majority, or four seats. The election uses cumulative voting, an ...
naiose's user avatar
  • 23
1 vote
1 answer
283 views

How to use the hamilton method (largest remainder method) when the maximum amount of seats is limited

I hope this is the right place to ask even though the math behind it is quite easy, i have trouble with the application of the hamilton method on my problem. The hamilton method/hare niemeyer method ...
Inselino's user avatar
  • 113
3 votes
0 answers
234 views

Which voting algorithm to use to assign N number of people to G groups based on their ranked choice preference

I've been looking through social choice theory textbooks and videos trying to find the right sort of algorithm for this, but struggling. Basically I have N (say 21) people that I need to assign into G ...
Culi's user avatar
  • 31
3 votes
1 answer
57 views

If the real preferences of a population include a condorcet winner, does that mean that any condorcet method is invulnerable to manipulation?

This is probably true, and easy to prove, but I am not coming up with a proof... Say we have a population, with each individual casting a ballot of preferences Individual1 might say, eg, A > B > ...
josinalvo's user avatar
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0 votes
0 answers
55 views

Voting Behaviour and Random Samples in Excel

I am trying to figure out if there is a way to detect suspicious voting behavior in my country in South America. The way elections work is that each region has a several election centers with voting ...
dwk2021's user avatar
0 votes
1 answer
24 views

Is a fixed welfare function that outputs the same answer regardless of the inputs independence of irrelevant alternative?

I'm taking this course to learn game theory and I'm confused about a question in Unit 1.5. Background. In game theory, independence of irrelevant alternatives (IIA) says the social welfare function $W$...
Maybe's user avatar
  • 436
3 votes
1 answer
279 views

Yes/No voting alternatives

I have found many ressources on alternative voting system for multiple candidates (highest median, maximal lotteries, etc..), but very few on a "two candidates situation", or a yes/no ...
Albert Beton's user avatar
1 vote
1 answer
60 views

How are hypergraphs related to voting games?

The Wikipedia page on hypergraphs says In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory. I have not ...
dfrankow's user avatar
  • 115
2 votes
0 answers
63 views

What is the best voting system where all voters are omniscient and perfectly logical?

We can model each voter's preferences by assigning a real number, called a score, on the interval $[0, 1]$ to each candidate. The goal of each voter is to elect a candidate with the maximal score ...
Kainoa B's user avatar
  • 181
1 vote
1 answer
75 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 ...
Adrien Pavao's user avatar
1 vote
2 answers
114 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 ...
robert bristow-johnson's user avatar
4 votes
1 answer
178 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 ...
ericf's user avatar
  • 248
0 votes
1 answer
61 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 ...
user avatar
1 vote
1 answer
132 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, ...
lvella's user avatar
  • 715
3 votes
1 answer
116 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 ...
Pradeep Suny's user avatar
  • 1,603
1 vote
1 answer
68 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 ...
Ryan1729's user avatar
  • 335
4 votes
1 answer
151 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 ...
Ryan1729's user avatar
  • 335
1 vote
2 answers
222 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 ...
Ryan1729's user avatar
  • 335
8 votes
2 answers
277 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 ...
Varun Vejalla's user avatar
0 votes
0 answers
41 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 ...
Lorenzo's user avatar
  • 23
8 votes
2 answers
1k views

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 ...
EvaMGG's user avatar
  • 486
1 vote
0 answers
26 views

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 ...
Varun Vejalla's user avatar
3 votes
0 answers
84 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 ...
EvaMGG's user avatar
  • 486
0 votes
1 answer
109 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....
prime90's user avatar
  • 101
16 votes
2 answers
416 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 ...
jamaicanworm's user avatar
  • 4,554
2 votes
1 answer
367 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 ...
Billy.G's user avatar
  • 21
0 votes
0 answers
23 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),\...
porton's user avatar
  • 5,093