Maths resources

General

Mathematical constraints on society/politics

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:

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

The process limits the possible outcomes to two options only;

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:

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,

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

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’