For questions regarding the formal analysis of collective decision problems.

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If $X$ is finite and $R$ is a complete and reflexive binary relation on $X$, then $M(R, S) \neq \emptyset$ on any $S \subset X$ iff $R$ is acyclic.

Could you help me to verify my proof and my writing? Definition 1: A binary relation $R$ on $X$ is complete if, for all $x, y \in X$ such that $x \neq y$,$xRy$ or $yRx$ or both and reflexive if, for ...
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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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1answer
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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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1answer
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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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1answer
23 views

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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23 views

Questions about arrow's impossibility theorem.

If we had chosen to work in a context where ties are allowed in the individual preference lists, then Arrow’s Theorem no longer holds with the conclusion that there is a person whose list is always ...
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1answer
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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, ...
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Comparison of non-order based voting methods (reference request)

There is plenty written on the relative merits of various voting systems where the voters submit ordered lists of preferences. However, there are several reasonable voting systems not using such a ...
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1answer
86 views

Game Theory in relation to economics and sociology [closed]

I know some algebra and calculus, and have been reading about Linear Programming/Game Theory. How are the models in this field, even the infinite calculus models, usable in macro economics. Even ...
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1answer
53 views

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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1answer
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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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1answer
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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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1answer
147 views

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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1answer
76 views

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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2answers
104 views

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 ...
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2answers
201 views

Functions minimized at the median of their arguments

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 : ...
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2answers
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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 ...
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1answer
235 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, ...
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2answers
78 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 ...
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1answer
79 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 ...
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1answer
104 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 ...
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1answer
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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 ...
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2answers
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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 ...
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1answer
135 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 ...
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1answer
645 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 ...
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97 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 ...
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2answers
175 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 ...
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1answer
107 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 ...
3
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1answer
153 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 ...
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1answer
83 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 ...
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Question on social choice functions

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 $*,\alpha,\beta$ ...