# Tagged Questions

For questions regarding the formal analysis of collective decision problems.

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### 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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If a planner obeys anonymity and strong Pareto principle (individual preferences carry over to the group), then the social utility function will be: $W(x)=\sum_{i=1, ..., n}U_i(x)$ i.e. summing ...
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### 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 ...
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### 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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### Arrow's theorem, strategic thinking and utilitarianism

I have this problem as part of a course on Decision Theory, and was not sure about question a (4th condition of Arrow's theorem) and question dii (utilitarianism). I have provided the whole question ...
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### Sen's theorem on minimal liberalism

I want to solve the following question regarding Sen's theorem and preferences orderings. The bit I am having trouble with is 2b Question 1: Describe Sen's theorem ('paradox') There is no social ...
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### does individually strategy proof implies coalitionally strategy proof?

Suppose $F$ is a social choice function \begin{equation*}F:N\rightarrow A\end{equation*} where $N=\{1,...,n\}$ is the set of agents and $A$ is a finite set of outcomes. suppose that $F$ is ...
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### Tail driven inequality [closed]

In this empirical model, Lj is a measure of left-tail driven inequality and Rj a measure of right-tail driven inequality. It represents what in this article? What exactly is the meaning of ''left-tail ...
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### Rational Fuctions, Choice Correspondence, Utility, Path Independence

I'm trying to prove that path independence implies that Sen's alpha holds. Can someone guide me on how i can approach this proof? Specifically Sen's alpha essentially states, that if for a choice ...
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### Impartiality axiom in Terry Tao's Arrow's Theorem proof

The short expository paper is here. On page 2, The notion of a quorum is well-defined; it is not possible for such a group to be able to force a vote some of the time and not at other times ...
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### Maximal clique problem

I understood what clique is all the nodes of the sub graph have to be connected to each other. In the following figure, it says that the maximal clique is {1,2,3,4,5}. But as per the definition of ...
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### Arrow’s Theorem

Suppose $k ≥ 3$ Recall that Arrow’s Theorem shows that any function $F:(S_k)^n\to S_k$ (the input is composed of n permutation of $[k]$ and the outcome is a single permutation of $[k]$ that satisfies ...
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I am in the early stages of planning my senior project and was wondering if anybody had some recommendations of literature about the applications of ultrafilters in social choice theory, along with ...
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### Theory of social choice - relation of preference

Let $X =\{ a,b,c,d\}$ be a set a of possible choices. Define a relation of preference $R$ which generates a rule of choice $C'$ such that it generates another relation $R'$ which is different from $R$ ...
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### Understanding the proof of Gibbard-Satterthwaite theorem

Let $n$ be the number of voters and $A$ be the set of alternatives. For voter $i$, we denote by $a \succ_i b$, if $i$ prefers $a$ to $b$, where $a,b \in A$. Let $L(A)$ denote the set of all strict ...
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### Arrow's Impossibility Theorem: example function

I am looking for a social welfare function which satisfies "unrestricted domain", "Pareto efficiency", and "independence of irrelevant alternatives". One of the known proofs for Arrow's theorem argues ...
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### Can a Condorcet winner be generally dispreferred on an individual basis?

Suppose that a Condorcet winner exists in an election. Certainly it is possible that an individual voter prefers some other candidates to the Condorcet winner. They might even prefer most, or all, ...
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### Anyone know about definition of weak dictator?

I am trying to prove arrow's impossibility theorem in case which ties are allowed in individual preference lists and so is social preference list. It says that if $p$ is a weak dictator, then we can ...
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### quasi rationality, interesting axiom of revealed preferences

So imagine there is a notion of rationality that captures the idea of "thresholds in preference." For example, let $\mathbb{Z}$ be the integers: $\mathbb{Z} = \{\dots, -10, -9, \dots, 0, 1, 2, \dots\}$...
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### Arrow's Impossibility Theorem Using Boolean Algebra

I am currently working on a research project which involves using Boolean matrices for the proof of Arrow's Impossibility Theorem and various other lemmas and results related to quasi ordered sets. In ...
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### Game Theory - Voting

In this setup there are 4 candidates running. For a candidate to be eliminated, the candidate needs to receive less than 1/3 of the votes when paired up with another candidate. This process ...
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### Are the following two definitions of Borda winner equivalent?

The Borda count is a method used to determine the winner object where people rank objects. For instance, imagine each person ranking 3 objects. The highest ranked object gets 2 points, the second gets ...
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### the proof of Arrow's Theorem

I read Philip J. Reny's paper (Arrow’s Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach) What I cannot understand is step 5 of the proof of arrow's theorem. I think figure 4 is a ...
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### Arrow's Impossibility Theorem and Ultrafilters. References

I need some references (far away from Wikipedia) about the proof using Ultrafilters of Arrow's Impossibility Theorem. Online resources are preferred.
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### Does risk aversion cause diminishing marginal utility, or vice versa?

Let $A$ be the set of possible states of the world, or possible preferences a person could have. Let $G(A)$ be the set of "gambles" or "lotteries", i.e. the set of probability distributions over $A$. ...
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### Counting in Arrow's theorem

I seem to be really confused with the counting system in Arrow's theorem. Can I have a simple explanation how they determine the outcome? I can't determine the outcome using rules from my notes. It ...
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### Why not n=2 in Arrow's theorem

Why in the statement of Arrow's impossibility theorem we omit the case n=2? I will appreciate it if you can explain it in easy words. I'm by no means an expert in the area (I think it's very much ...
I am doing research on problems of location of a public facility on a network which lead me to the following question. Is there an interesting way to characterize the class of functions $f : \mathbb{... 2answers 386 views ### What does Arrow's theorem say about Kaldor-Hicks social welfare functions with von Neumann-Morgenstern utility? Let$A$be the set of all possible states of the world, let$G(A)$be the set of all "lotteries" or "gambles", i.e. the set of all probability distributions over$A$. Now consider an individual with ... 1answer 326 views ### How can a social welfare function be a linear combination of von Neumann-Morgenstern utility functions? The von Neumann-Morgenstern axioms were an attempt to characterize rational decision-making in the presence of risk. The von Neumann-Morgenstern utility theorem says that if someone is vNM-rational, ... 2answers 83 views ### Decide the most favorable candidate Consider an election voting process where people need to elect a representative among n number of candidates. Is there an approach to determine the most favorable option? Voting just a single person ... 1answer 94 views ### Stability under supremum of sets of social choice function with single peaked preferences Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ... 1answer 130 views ### Using limit argument with non-continuous social-choice functions This question is related to another question of mine Invariance of strategy-proof social choice function when peaks are made close from solution, and it revolves around the use of limit arguments with ... 1answer 163 views ### Invariance of strategy-proof social choice function when peaks are made close from solution A question emerging from reading Schummer, J., & Vohra, R. V. (2002). Strategy-proof Location on a Network. Journal of Economic Theory, 104(2), 405–428. The setting is as follows: A finite set ... 2answers 145 views ### Does Arrow's Theorem apply when choosing a single best candidate? According to Wiki, Arrow's Impossibility Theorem proves that we cannot create a social welfare function that obeys unanimity, non-dictatorship, and IIA. However, in real elections, we want to choose ... 1answer 146 views ### What is the general formula for electoral districts tying. I apologize if this question is a bit of a read. (You might want to get a frosty beverage.) Professor Alan Natapoff of MIT demonstrated, if 9 Voters are districted into 3 electoral districts of 3 ... 1answer 1k views ### Gibbard–Satterthwaite Theorem versus Arrow Theorem Arrow Theorem is a very classical result in social choice theory, stating very roughly that any reasonable voting procedure is either dictatorial or subject to tactical voting. More precisely, there ... 0answers 131 views ### Can the Nash bargaining solution be applied in repeated game? I am trying to develop a model involving two agents who interact strategically to set an optimal time for a joint work. These agents will have to meet repeatedly. I want to derive the optimal time for ... 2answers 214 views ### Is there a voting method with a sane strategy? Is there a voting method where the best strategy for strategic voters can be explained in a sane way? According to Gibbard–Satterthwaite, there is no "strategy-free" (and reasonable) voting method. ... 1answer 116 views ### Does it make sense to run Meek STV with more choices than seats? During the current batch of moderator elections at Gaming, it has been argued that since only 2 seats are up for grabs in this round of elections, it only makes sense to cast ballots by only ranking ... 1answer 176 views ### Bayesian analysis of the Venice Doge elections Does anyone know of a Bayesian (or a classical) analysis of the Venetian Doge election system? I am looking mainly for chances of subversion, chances for a candidate to be elected at each stage, or ... 1answer 84 views ###$N\setminus S$is not$\beta$-effective for$A\setminus B$, and$S$is not$\beta$-effective for$B$Given a social choice function$F$, a subset$B\subset A$of the candidates and a coalition$S\subset N$of the voters,$\beta$-effectiveness of$S$for$B$is equivalent to$N\setminus S$not being$\...
We showed in class that every strongly, exactly consistent s.c.f is strongly firm (I don't know if this is the right translation - we defined strong firmness as the equivalence of $*,\alpha,\beta$ ...