What is the Gibbard–Satterthwaite theorem in simple terms?

In any election with three or more candidates, unless one person dictates the result, every voting method creates situations where someone does better by not voting their true preferences.

How does it differ from Arrow's impossibility theorem?

Arrow's theorem is about combining individual preferences into a consistent social ranking, while Gibbard–Satterthwaite is about whether people have an incentive to vote honestly. The two results are closely linked mathematically.

Gibbard–Satterthwaite Theorem

🏛️ Politics & Public Administration By CampusDown Wiki Editorial Team Last updated October 27, 2026

The Gibbard–Satterthwaite theorem, proved independently by the philosopher Allan Gibbard (1973) and the economist Mark Satterthwaite (1975), states that any deterministic voting rule with at least three possible outcomes, each of which can win, either is dictatorial or gives some voter an incentive to vote against their true preferences. In other words, no fair voting system is immune to strategic manipulation.

Contents

1. What the theorem says
2. Relation to Arrow's theorem
3. Exceptions and workarounds
4. FAQ
5. References

What the theorem says

A voting rule takes each voter's ranking of the options and selects a single winner. A rule is manipulable if some voter can obtain a better outcome by misreporting their true ranking. Gibbard (1973) and Satterthwaite (1975) showed that if there are at least three options and every option can win, a rule is strategy-proof only if it is a dictatorship, in which one person's preferences always decide the outcome. A familiar case of strategic voting is a plurality-rule voter who backs a strong second choice instead of a long-shot favorite. Dummett and Farquharson (1961) had conjectured such manipulability early on, and Farquharson (1969) systematized the theory of strategic voting.

Relation to Arrow's theorem

The theorem is closely tied to Arrow's impossibility theorem. Satterthwaite (1975) showed that strategy-proofness corresponds to Arrow's conditions, and Reny (2001) proved both theorems with a single short argument. Whereas Arrow's theorem says there is no consistent way to aggregate individual preferences into a social ranking, Gibbard–Satterthwaite says there is no fair way to make honest voting always optimal. Together with the concept of incentive compatibility introduced by Hurwicz (1972), it became a basic constraint in mechanism design. Duggan and Schwartz (2000) generalized the theorem to rules that allow ties or select sets of winners.

Exceptions and workarounds

Relaxing the assumptions allows strategy-proof rules. Moulin (1980) showed that with single-peaked preferences there are strategy-proof rules, such as choosing the median peak, while Gibbard (1977) proved that among rules mixing voting with chance, the only strategy-proof ones take very restricted forms such as random dictatorship. Barberà (2011) surveyed these results. In computer science, Bartholdi, Tovey and Trick (1989) showed that for some rules manipulation, though possible in principle, is computationally hard to find, and Faliszewski, Hemaspaandra and Hemaspaandra (2010) reviewed research on using complexity to protect elections.

FAQ

What is the Gibbard–Satterthwaite theorem in simple terms?: In any election with three or more candidates, unless one person dictates the result, every voting method creates situations where someone does better by not voting their true preferences.
How does it differ from Arrow's impossibility theorem?: Arrow's theorem is about combining individual preferences into a consistent social ranking, while Gibbard–Satterthwaite is about whether people have an incentive to vote honestly. The two results are closely linked mathematically.

Sources reviewed: 2026-10-26

References

Gibbard, A. (1973). Manipulation of voting schemes: A general result. Econometrica, 41(4), 587–601.
Satterthwaite, M. A. (1975). Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory, 10(2), 187–217.
Arrow, K. J. (1951). Social Choice and Individual Values. New York: Wiley.
Dummett, M., & Farquharson, R. (1961). Stability in voting. Econometrica, 29(1), 33–43.
Farquharson, R. (1969). Theory of Voting. New Haven, CT: Yale University Press.
Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455.
Barberà, S. (2011). Strategyproof social choice. In K. J. Arrow, A. Sen, & K. Suzumura (Eds.), Handbook of Social Choice and Welfare (Vol. 2, pp. 731–831). Amsterdam: North-Holland.
Bartholdi, J. J., Tovey, C. A., & Trick, M. A. (1989). The computational difficulty of manipulating an election. Social Choice and Welfare, 6(3), 227–241.
Gibbard, A. (1977). Manipulation of schemes that mix voting with chance. Econometrica, 45(3), 665–681.
Duggan, J., & Schwartz, T. (2000). Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized. Social Choice and Welfare, 17(1), 85–93.
Hurwicz, L. (1972). On informationally decentralized systems. In C. B. McGuire & R. Radner (Eds.), Decision and Organization (pp. 297–336). Amsterdam: North-Holland.
Faliszewski, P., Hemaspaandra, E., & Hemaspaandra, L. A. (2010). Using complexity to protect elections. Communications of the ACM, 53(8), 74–82.
Reny, P. J. (2001). Arrow's theorem and the Gibbard–Satterthwaite theorem: A unified approach. Economics Letters, 70(1), 99–105.

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This article was last updated on October 27, 2026. It is based on widely recognized original works and textbooks; when citing it in a paper, please check the original sources listed in the references.