Counter-threats and Strategic Manipulation under Voting Schemes

Gibbard [5] has shown that if there are at least three possible outcomes, then every nondictatorial " voting scheme " (i.e. every non-dictatorial group decision procedure which yields exactly one outcome for every voting situation and every issue) leaves scope for strategic misrepresentation of preferences by individuals.' This has serious implications. For, the sanctity of social decisions can be upheld only in so far as these decisions are based on the true or sincere preferences of individuals in the society. Social decisions based on distorted versions of individual preferences lose this sanctity, whatever be the ethical appeal of the method of aggregating the individuals' expressed preferences so as to arrive at the social decision. Thus Gibbard's result is of great importance. However, the structure within which Gibbard proves his result is relatively simple in so far as it does not include the phenomenon of possible counter-coalitions when a coalition (of one or more than one individual) threatens to disrupt a voting situation by strategic manipulations. In actual life the stability of a given voting situation (i.e. the absence of any strategic manipulation by any coalition, which will disrupt the voting situation) is often the result of a counterbalancing of " threats " and " counter-threats ,2 (or coalitions and counter-coalitions). The purpose of this paper is to incorporate this element into the analysis and to explore the consequences of this wider framework for the theorem proved by Gibbard.3 It turns out that this negative result remains practically intact even within the more general framework. It is shown that even when we use a notion of stability which takes into account the possibility of counter-threats (and which is, to that extent, weaker than the stability notion underlying Gibbard's analysis), for every non-dictatorial voting scheme satisfying some very mild conditions, there exist sincere voting situations which are unstable because of strategic voting by single individuals. This essentially strengthens the negative impact of Gibbard's theorem since even the fear of a counter-coalition from the rest of the group does not rule out strategic or non-sincere voting by a single individual under, practically any non-dictatorial voting scheme in which one is likely to be interested.