Social Choice

Motivation. Social choice is about converting many people’s preferences into one big social preference. This is obviously how many collective decision making works: elections, markets, auctions, govn’t policty, etc. The problem is, as we will prove, social choice is really hard, and it’s really hard to avoid dictatorships. But there are mechanisms to avoid this too.

Formalizing Social Choice §

def. Social Choice Situation. We have to choose between candidates (=alternatives) in set $A$:

• An ordering $\succ =(\text{total order on A})$.
• $L=\{\text{all possible ordering of A}\}$. It is simply a permutation set on $A$, thus $|L|=|A|!$
• There are $n$ voters. Each voter $i$ has one ordering ${\succ }_i$.
• def. Social Welfare Function $F:L^n\to L$ i.e. given everybody’s prefs, come up with a collective pref
• def. Social Choice Function $f:L^n\to A$ i.e. given everybody’s prefs, come up with a collective single candidate Two functions have a simple distinction:
• Welfare function: full ordering, collectively.
• Choice function: just choose one collectively. We first discuss the welfare function in detail.

Impossibility (Welfare) §

Properties of a social welfare function are:

1. Unanimity: if all voters have the same preference $\succ$, then that is the collective order too. $F(\prec ,\dots ,\prec )=\prec$
2. Dictatorship: if voter $i$’s preferences are just taken as the collective order, disregarding everybody else’s preferences, it’s a dictatorship. Dictator $i$: $\forall {\prec }_1,\dots ,{\prec }_n\in L$ just take $F(\dots )={\prec }_i$
3. Independence of Irrelevant Alternatives (IIA). Let candidates $a,b\in A$ and preferences of voters ”$\dots$”, and observe $F$ which ${\prec }_{\ast }=F(\dots )$.
   1. Suppose we create a new rankings ${\prec }_{\ast }^{\prime }=F({\dots }^{\prime })$, where each voter $i$ voted $a{\prec }_{i}b$ and maintains their preference ranking of $a$ and $b$ in the new ranking: $a{\prec }_{i}b⟺a{\prec }_i^{\prime }b$
   2. Then collective ranking between $a$ and $b$ should also be maintained: $a{\prec }_{\ast }b⟺a{\prec }_{\ast }^{\prime }b$ thm. Arrow’s Impossibility Theorem. Every social welfare function $F$ over choices $|A|\ge 3$, that satisfies unanimity and IIA, must be a dictatorship. Proof.