Gibbard's theorem

In the fields of mechanism design and social choice theory, Gibbard's theorem is a result proven by philosopher Allan Gibbard in 1973. It 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. A corollary of this theorem is Gibbard–Satterthwaite theorem about voting rules. The main difference between the two is that Gibbard–Satterthwaite theorem is limited to ranked (ordinal) voting rules: a voter's action consists in giving a preference ranking over the available options. Gibbard's theorem is more general and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to candidates (cardinal voting). Gibbard's theorem can be proven using Arrow's impossibility theorem. Gibbard's theorem is itself generalized by Gibbard's 1978 theorem and Hylland's theorem, which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve an element of chance. The Gibbard's theorem assumes the collective decision results in exactly one winner and does not apply to multi-winner voting. Consider some voters , and who wish to select an option among three alternatives: , and . Assume they use approval voting: each voter assigns to each candidate the grade 1 (approval) or 0 (withhold approval). For example, is an authorized ballot: it means that the voter approves of candidates and but does not approve of candidate . Once the ballots are collected, the candidate with highest total grade is declared the winner.