9 votes
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Good books to learn social choice theory

Start with: A primer in Social Choice Theory, by Wulf Gaertner. If you want more, have a look at: Welfare Economics and Social Choice Theory by Allan Feldman and Roberto Serrano To dig deeper: ...
Martin Van der Linden's user avatar
6 votes

Arrow's impossibility theorem

Let the set of alternatives be $A = \left\{a_1,a_2,...,a_k\right\}$. Let the number of players be $n$. Let the set of preference orderings over $A$ be $\mathcal{P}$. Then the set of preference ...
Giskard's user avatar
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6 votes
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Utility representation of single peaked preferences

No. Basically, you can encode a form of lexicographic preferences, probably the most familiar example of non-representable preferences, as single-peaked preferences on $\mathbb{R}$. Define $\succeq$ ...
Michael Greinecker's user avatar
5 votes

Arrow's impossibility theorem

The Pareto criterion has two effects: It guarantees that every ranking can occur as a social ranking and it connects social rankings to individual rankings. If one drops the Pareto criterion but keeps ...
Michael Greinecker's user avatar
5 votes
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gibbard-satterthwaite theorem and median voting

Gibbard and Satterthwaite insist that the social choice function must be defined over all rational preferences over outcomes. That is, if voters' preferences could be anything (subject to the ...
Theoretical Economist's user avatar
5 votes
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Do any social welfare functionals, other than maximin, meet all of Arrow's conditions plus invariance regarding ordinal level comparability?

There are at least two other examples of SWFs that satisfy these conditions. The first is a positional dictatorship. Let N be the number of individuals (assume it is fixed). For any k between 1 and ...
Marcus Pivato's user avatar
5 votes
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Is $(\mathbb{R}^m)^n$ the real coordinate space of dimension $m\cdot n$?

No. And yes. For any set $X$ we have (by definition) $$X^k=\underbrace{X\times\cdots\times X}_{k\text{-times}}=\{(x_1,x_2,\ldots,x_k)\mid x_i\in X\text{ for }i=1,\ldots,k\}.$$ Now let, for example, $m=...
Michael Greinecker's user avatar
5 votes

Question about Social Welfare Function and Social Profile

In its most general formulation, a social welfare function is just a utility function representing the preferences of "society as a whole" (or the preferences of a hypothetical "...
Marcus Pivato's user avatar
5 votes

Choice Theory Book Reccomendations

The title "Choice Theory" is a bit ambiguous, because it could refer to Social Choice Theory (the theory of collective decisions), but it could also refer to the broader concept of Rational ...
Marcus Pivato's user avatar
4 votes
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Direct revelation mechanism's sets of strategies and types

In mechanism design you are free to choose the rules of the game. The designer can determine $(S, g)$, i.e., what players can do and what happens when players played some strategy profile $s \in S := \...
Bayesian's user avatar
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4 votes
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Difference between social choice function and mechanism outcome function

A social choice function presumes the individuals' preference parameters $\theta_i$'s are observable, whereas in a mechanism, such knowledge is not presupposed. Therefore, in a mechanism, the ...
Herr K.'s user avatar
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4 votes
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Difference between social choice functions and social decision functions?

Let $X$ be the set of alternatives. A social decision function maps profiles of preference orderings to relations on $X$ such that every nonempty subset of $X$ has at least one maximum under this ...
Michael Greinecker's user avatar
4 votes
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Which of Arrow's four desirable properties' is violated in this scenario?

Note that an alternative being highest-ranked by an agent involves comparisons with all other alternatives, which raises doubts that your method satisfies independence of irrelevant alternatives. ...
Michael Greinecker's user avatar
4 votes

Rawlsian SWF and Arrow Impossibilty Theorem

Independence of irrelevant alternatives prevents you from using the information needed to implement a Rawlsian SWF; the information who is society's worst-off cannot be used. Indeed, the relevant ...
Michael Greinecker's user avatar
4 votes
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Arrow's Dictator

The object of Arrow's theorem are social welfare functions, which maps profiles of strict preference relations to to preference relations, $f:\mathcal{P}^n\to\mathcal{R}$. There are some variations in ...
Michael Greinecker's user avatar
4 votes
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White Flight from Asian Immigration: Evidence from California Public Schools - is the methodology robust?

is the methodology robust? They used panel fixed effects to control for unobservable and IV for potential reverse causality, which is decent methodology for their research question. They follow ...
1muflon1's user avatar
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3 votes
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Majority Rule and Single Peakedness

Suppose that A={a,b,c,....,z} is a finite set of social alternatives, and let P={>1,>2,....,>N} be a profile of strict preference orders on $A$ (where the set {1,2,...,N} indexes the voters). ...
Marcus Pivato's user avatar
3 votes
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Anonymity in social welfare function

Anonymous/impartial SWFs focus only on the pattern of well-being, and not the identities of the people who end up at particular well-being levels. Identities here simply means names. When applying ...
Herr K.'s user avatar
  • 15.4k
3 votes

Minimalist example of voter profiles yielding different outcomes

First, let me state that this is a beautiful problem! Here is a proof that $n=4$. To prove that $n \leq 4$, consider the following example: $m+4$ voters have preference $a \succ d \succ b \succ c$ $...
Oliv's user avatar
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3 votes

Social choice theory

You are correct that it is not possible to violate Weak Pareto and Nondictatorship at the same time. But your explanation (second paragraph) is a bit muddled. Here is how I would put it. To prove, "...
Marcus Pivato's user avatar
3 votes
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Rawlsian SWF and Arrow Impossibilty Theorem

Let's say there are individuals 1 and 2, and alternatives A, B, C, and D. Society uses the Rawlsian SWF and thus ranks alternatives according to their maximal rank within individuals' rankings. Denote ...
VARulle's user avatar
  • 6,805
3 votes

Axiom of Minimal Liberalism & Sen's Theorem of Paretial Liberal

Your question is a bit confused, because it mixes together several different things. For example, in the title, you mention Sen's Minimal Liberalism, but in the actual question, you don't mention Sen ...
Marcus Pivato's user avatar
3 votes
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Possibilities in social choice: number of possible collective decision function?

I can't think of any theorems which specifically focus on the number of social aggregation rules (of some type) satisfying axioms X, Y, and Z. However, aside from impossibility theorems, most of the ...
Marcus Pivato's user avatar
3 votes

White Flight from Asian Immigration: Evidence from California Public Schools - is the methodology robust?

Here's an analysis of that paper... Link to Marginal Revolution blog. A quote... However, I don’t think the headline result in this paper is particularly credible. First, there isn’t a well-...
H2ONaCl's user avatar
  • 943
2 votes

What are good mathematics books to learn decision theory?

For decision theory: Gilboa, I. (2009). Theory of decision under uncertainty. Cambridge University Press. Parmigiani, G., & Inoue, L. (2009). Decision theory: Principles and approaches. John ...
Martin Van der Linden's user avatar
2 votes

What are good mathematics books to learn decision theory?

Decisions with Multiple Objectives: Preferences and Value Tradeoffs by Ralph L Keeney
decision maker's user avatar
2 votes

How does GNU software development sustain economically?

Open-source software development is done for diverse reasons, but it's a common misunderstanding that it's done primarily by hobbyists or professionally but as a side-project. I'm answering this ...
user149485's user avatar
2 votes
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Minimalist example of voter profiles yielding different outcomes

This is not mathematically rigorous but I believe the reasoning is correct. We assume that the allocation of votes between voter types is such that ties are not allowed, where it matters. Also, ...
Alecos Papadopoulos's user avatar
2 votes
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Weighted voting: Vote of representative weighted by number of election votes

You ask an excellent question, as it has been partly but not fully discussed in the scientific literature. Your exact proposal (weighting representatives by the amount of votes they got in the ...
AM.'s user avatar
  • 36
2 votes
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Is a social choice aggregation rule defined for a set of weightings over the set of voters (N)?

First it is important to define our terms. The concept of "weighted voters" does not necessarily make sense for an arbitrary social choice rule. So presumably we are interested in social choice ...
Marcus Pivato's user avatar

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