# 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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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 ...
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1 vote
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### 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 ...
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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 ...
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### What is the maximum possible variance between proportional representation and FPTP?

I wanted to calculate how big of a difference there can be between two specific electoral systems. To minimize complexity, the specific politicians that would be elected under both systems is not in ...
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### 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 ...
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### Are there characterizing theorems for veto and nomination?

We know that for some voting methods, there are characterizing theorems, for example, for majority vote we have May's theorem which states that: simple majority voting is the only anonymous, neutral, ...
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### 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 ...
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### 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{ \...
1 vote
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### 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 $R_1, \dots R_k$ ...
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### $n$ 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 ...
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1 vote
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### 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 ...
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### Shapley value of a parliament with infinitely many parties

Question: A parliament has $n$ parties. The two main parties (we call them $A$ and $B$) have $\frac{1}{3}$ of the total seats each. The remaining $\frac{1}{3}$ of the seats are equally divided among ...
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### 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 $A$ ...
1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 > ...
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### 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 ...
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### 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 $W$...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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1 vote
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### 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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1 vote
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### 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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### 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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### 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 ... 1 vote
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### 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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### 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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1 vote
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### 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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### 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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1 vote
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### 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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### 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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### 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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1 vote
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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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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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### 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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### 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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### 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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### 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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1 vote
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### 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 ...
1 vote
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### 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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### 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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