Social welfare Let's consider social welfare functions first. What would be a good social welfare function F? 6
Social welfare ◼ Let’s consider social welfare functions first. ◼ What would be a good social welfare function 𝐹? 6
Desirable properties ■Unanimity:For every<∈L,F(<,.,<)=<. If everyone has the same preference list <then we should just use that
Desirable properties ◼ Unanimity: For every ≺∈ 𝐿, 𝐹 ≺, …, ≺ =≺. ❑ If everyone has the same preference list ≺, then we should just use that. 7
Desirable properties Independence of irrelevant alternatives:Va,b E A,H<1,…,<<i,,<n∈L,let<=F(<1,<n) and 'F(1,.,<).Then a<ib台a<b,i implies a<b台a<'b. The social preference between any a and b depends only on the voters'preferences between a and b. If each voter i changes his ranking from <to <as long as they each don't change the relative preference between a and b,then they won't change the final comparison between a and b. 8
Desirable properties ◼ Independence of irrelevant alternatives: ∀𝑎, 𝑏 ∈ 𝐴, ∀≺1 ,… , ≺𝑛, ≺1 ′ , … , ≺𝑛 ′ ∈ 𝐿, let ≺= 𝐹 ≺1 , … , ≺𝑛 and ≺ ′ = 𝐹 ≺1 ′ ,… , ≺𝑛 ′ . Then 𝑎 ≺𝑖 𝑏 ⇔ 𝑎 ≺𝑖 ′ 𝑏, ∀𝑖 implies 𝑎 ≺ 𝑏 ⇔ 𝑎 ≺ ′ 𝑏. ❑ The social preference between any 𝑎 and 𝑏 depends only on the voters’ preferences between 𝑎 and 𝑏. ❑ If each voter 𝑖 changes his ranking from ≺𝑖 to ≺𝑖 ′ , as long as they each don’t change the relative preference between 𝑎 and 𝑏, then they won’t change the final comparison between 𝑎 and 𝑏. 8
Impossibility 1 ■Arrow's theorem.IfA≥3，then only dictatorship satisfies both unanimity and independence of irrelevant alternatives. A dictatorship is a social welfare function F(<1,…,<n)=<i for some i∈[n]. It's not a voting any more. Arrow's theorem says that there is no good social welfare function. 9
Impossibility 1 ◼ Arrow’s theorem. If 𝐴 ≥ 3, then only dictatorship satisfies both unanimity and independence of irrelevant alternatives. ◼ A dictatorship is a social welfare function 𝐹 ≺1,… , ≺𝑛 =≺𝑖 for some 𝑖 ∈ [𝑛]. ❑ It’s not a voting any more. ◼ Arrow’s theorem says that there is no good social welfare function. 9
Social choice? Social choice needs to get only one winner. Easier task than social welfare. a Question:Is there a good social choice function? 10
Social choice? ◼ Social choice needs to get only one winner. ❑ Easier task than social welfare. ◼ Question: Is there a good social choice function? 10