flexibeast.space

Maths resources

Arrow's impossibility theorem ... states that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting the specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives.

— Wikipedia: ‘Arrow's Impossibility Theorem’

Gibbard's theorem ... states that for any deterministic process of collective decision, at least one of the following three properties must hold:

1. The process is dictatorial, i.e. there exists a distinguished agent who can impose the outcome;

2. The process limits the possible outcomes to two options only;

3. The process is open to strategic voting: once an agent has identified their preferences, it is possible that they have no action at their disposal that best defends these preferences irrespective of the other agents' actions.

— Wikipedia: ‘Gibbard's theorem’

Sen's paradox ... shows that no means of aggregating individual preferences into a single, social choice, can simultaneously fulfill the following, seemingly mild conditions:

1. The unrestrictedness condition, or U: every possible ranking of each individual's preferences and all outcomes of every possible voting rule will be considered equally,

2. The Pareto condition, or P: if everybody individually likes some choice better at the same time, the society in its voting rule as a whole likes it better as well, and

3. Liberalism, or L (from which the theorem derives its gist): all individuals in a society must have at least one possibility of choosing differently, so that the social choice under a given voting rule changes as well. That is, as an individual liberal, anyone can exert their freedom of choice at least in some decision with tangible results.

— Wikipedia: ‘Sen's paradox’

[A]ggregating judgments with majority voting can result in self-contradictory judgments ... Philosopher Philip Pettit believes the discursive dilemma makes it impossible to make simple statements about the beliefs of a collective.

Wikipedia: ‘Discursive dilemma’