Gibbard–Satterthwaite theorem

The Gibbard - Satterthwaite theorem is a statement in the social choice theory on group decisions, especially across the boundaries of preference elections. In preference elections, each group member joins a number of decision alternatives according to their individual advocacy.

The theorem states that each preferred choice for three or more decision alternatives can be manipulated by strategic voting behavior, if it satisfies the democratic values ​​that all people participate equally in the process and terms of the process, each alternative has the chance to be accepted.

Exact formulation

For a precise statement of the theorem two definitions are useful: A method is called dictatorial if there is an excellent person whose preference, the process decides. One method is called manipulable if there are situations in which a party - which is both the process and the voting behavior of all other stakeholders knows - can improve the chances of alternative by wrong for this, but for a different alternative, or the can worsen chances for an alternative by voting for them. With these definitions is the Gibbard - Satterthwaite theorem:

When three or more decision alternatives at least one of the following three conditions is met at every preference choice:

An example of

In the following example, the rules of instant - runoff voting shall apply:

It should be between four options A, B, C and D to decide. Among the voters, there are four groups, which rank the options as follows:

First Option B is deleted, then D. Thus A is prefers collectively by a majority of 61:39. Now one group but would like to avoid at all events, that candidate A wins. Because they think they know, for ideological reasons, the preferences of the other groups, they put A at the top of their order of preference:

Due to the new preferences B is first deleted and then C. Among the remaining candidates A and D D prefers collectively by a majority of 53:47.