# Questions tagged [social-choice-theory]

For questions regarding the formal analysis of collective decision problems.

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### How to say these two distinct functions have the same structure?

Yesterday, I posted this question, which remains unanswered. In this related question, I ask a different yet more precise question that may help me solve the other question. Let $N=\{1,2\}$ be a two-...
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I am working on a social choice problem that must allow for alternative sets of different sizes. In the paper I'm struggling to write, the environment is a $5$-tuple $(N,A_\tau,\mathcal{P}(A_\tau),\... • 1,226 -1 votes 0 answers 60 views ### Can someone inductively prove this recursion formula? [duplicate] This is admittedly directly related to this other question but it's a little different. Let: $$a_n = \Big\lfloor (e-1)n! \Big\rfloor - 1 \qquad n \in \mathbb{Z} > 1$$ where$e \approx$2.... 1 vote 0 answers 44 views ### maximal/best element of a set of natural numbers where$x\mathbin{\mathrm{R}}y$defined as “$x$is a multiple of$y$” I am doing a mock exam in social choice theory our teacher gave us and one question is: Let$X = \{1, 2, 3, \dots\,\}$be the set of natural numbers and define$x\mathbin{\mathrm{R}}y$as “$x$is a ... • 11 1 vote 1 answer 50 views ### Finding weights for the Borda count to elect any given candidate Suppose three candidates A, B, and C run for office and voters submit their preference orders. The typical Borda count awards 2 points for a first-place vote, 1 point for a second-place vote, and 0 ... • 31.7k 1 vote 1 answer 25 views ### Properties of rules of preference I am working on aggregation of preferences and something is unclear concerning one property of rules of preference namely the monotonicity. A$\mathcal{W}$rule of preference is a function$F$from$\...
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Can someone please explain to me the steps for the calculation of the Condorcet paradox probability $q$ in the impartial culture model? I don't understand how does the Cauchy distribution arises from ...
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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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### 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 ...
255 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 ...
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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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### 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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### Improving description of vector systems and aggregation rules for social sciences

I am working on a simple individual based model that aggregates information. I am not a mathematician, but I would like to be as precise as possible with the terminology used to describe the system ...
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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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### Gibbard–Satterthwaite Theorem clarification

recently I read about this theorem on math websites which talked about voting systems. since I'm interested in politics and math I'm interested to understand this theorem but it requires higher level ...
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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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### 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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### 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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### Reference request for a generalization of the Johnston power index

Background: Recall that the Johnston power index is a variation of the more well-known Banzhaf power index. While the Banzhaf power allocates 1 point to each time a player is in a winning coalition, ...
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### How do I interpret the social choice function noted by Felix Brandt?

How do I read this: Let $\mathscr{U}$ be a universe of alternatives Let $\mathcal{F}(\mathscr{U}):\forall\mathscr{A}\in\mathcal{F}(\mathscr{U}),\mathscr{A}\subseteq\mathscr{P}_{\geq1}(\mathscr{U})$ A ...
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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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### Invariant for Gale–Shapley algorithm (Mating Ritual Algorithm)?

I found the following invariant for Mating Ritual algorithm (Lehman, Leighton and Meyer, Mathematics for Computer Science, §6.4) while going through MIT reading material: Definition. Let $P$ be the ...
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### Can the Borda count be used to select a distribution and not just a single choice?

Suppose I have n individuals and n unique, indivisible objects of potential value. I want to allocate those objects so as to make total welfare as great as possible, subject to the constraint that no ...
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### Rational choices payment plans

I am trying to research what a rational choice constitutes; a financially sound decision based on critically examining a set of data and concluding that the expected value for a given choice is higher ...
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### Finding the Dictator in Arrow's Impossibility Theorem

Arrow's Impossibility Theorem states that if we have at least three different social states and a finite number of individuals (voters), any social welfare function that satisfies the conditions of ...
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### 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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### Choice rule and path independence

Let a choice function be defined as a function $C:2^X \setminus \{\emptyset\} \rightarrow 2^X$ such that $C(A) \subseteq A$ for all $A \subseteq X$. Here, $2^X$ denotes the power set of $X$. We say a ...
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### Efficiency in Mechanism Design - Myerson-Satterthwaite

I was reading some online notes on the 4 conditions that cannot all hold given the Myerson-Satterthwaite theorem and this definition of efficiency confused me: An efficient mechanism selects the ...
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### 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 ...
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### 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 ...
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### Arrow's impossibility theorem and the independence of irrelevant alternatives

I have a question about the axiom of independence of irrelevant alternatives (IIA). According to the Wikipedia page on IIA, Arrow's formulation of IIA is as in here. I do not quite get this. In ...
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### 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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### Checking choice correspondence satisfy Weak Axiom of Revealed Preferences

I have a question whether the below choice correspondence satisfies WARP or not. I know how to check a choice satisfies WARP when set X is in he form of X={x,y,z} however i cannot figure out when it ...
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