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5

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 "benevolent social planner" who makes decisions for the society). Let $X$ be some space of "social outcomes". (Social outcomes could be anything. But ...

5

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, $m=2$ and $n=3$. Then $$(\mathbb{R}^m\big)^n=(\mathbb{R}^m\big)^n$$ =\big(\mathbb{R}^2\big)^3=\Big\{\big((x_1,x_2),(x_3,x_4),(x_5,x_6)\big)\mid (x_1,x_2)\in\...

4

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 information is not even specified in a profile of preferences.

3

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 society's preferences by $\succ^*$. If the individual rankings are: 1: A $\succ$ B $\succ$ C $\succ$ D 2: B $\succ$ D $\succ$ A $\succ$ C, then B $\succ^*$ A. ...

3

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 at all --- instead you talk in more general terms about "rights that cannot be taken ... in any situation". You mention "Harsanyi's axioms"...

2

A direct revelation mechanism is one in which a player's type space is also their action space ($X_i=T_i$ for all $i$) and the outcome function is the same as the social choice function ($a(t)=f(t)$ for all $t\in T_1\times\cdots\times T_n$).

1

I solved it; here's my solution: Let $f:L^n\to A$ be an incentive compatible, non-dictatorial social choice function and let $F:L^n\to L$ be its extension. To show that $F$ is non-dictatorial, then for any voter $i\in\{1,\dots,n\}$ we need to find $\prec_1,\dots,\prec_n\in L$ such that $\prec_i\,\neq\,\prec$, where $\prec\,=F(\prec_1,\dots,\prec_n)$. So we ...

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