The following is a question on Arrow's theorem with a pairwise majority decision. The bits I was unsure about was (bi) (is the 4th condition satisfied?) and also is (bii) correct? Thanks for your help
(a). State the definition of a preference profile, with properties, and a social welfare function.
The preference profile is the collection $(\geq_1, ..., \geq_m )$ of individual preference rankings $\geq _i$
A social welfare function is a function which operates on preference profiles, obeying comparability and transitivity, and yields a group preference $\geq_g$ which obeys comparability and transitivity
(bi). Citizens vote for one of of two political parties using pairwise majority voting. A party wins if it gets the largest share of the votes. Ties are broken by flipping coins. Show that this satisfies 4 axioms of group preferences.
- Unrestricted domain: Yes, because an ordering is given by ranking the proportion of the votes.
- No dictatorship: Yes, because each voter has only one vote of equal weighting, so cannot be a dictator.
- Pareto's condition: Yes because if everyone prefers party $A$ to party $B$ then $A$ will win. Indifference is not relevant since everyone votes for their Favorited or does not vote at all ( I am assuming...)
- Independence of irrelevant alternatives: Not sure how to prove this. The usual way is to consider what would happen if you removed options and if this would affect preferences, but there are only two preferences?
(bii). Does this result contradict Arrow's impossibility theorem?
Arrow's impossibility theorem asserts that if there are at least three rewards and at least two individuals then there is no SWF which meets all $4$ group conditions.
Is it true to say that since there are only two parties, these can be considered as 'rewards'? Hence, Arrow's impossibility theorem does not apply and there is no contradiction.
Yes, majority voting with two alternatives satisfies independence of irrelevant alternatives: the only irrelevant alternative is the loser, and eliminating it doesn’t change the winner.
And yes, your answer to (bii) is correct: the parties in the election correspond to the rewards in your version of Arrow’s theorem, and since there are only two of them, the theorem doesn’t apply. (Other statements of the theorem replace rewards by candidates, for example.)