Manipulation of social choice correspondences under incomplete information

We study the manipulability of social choice correspondences in situations where individuals have incomplete information about others’ preferences. We propose a general concept of manipulability that depends on the extension rule used to derive preferences over sets of alternatives from preferences over alternatives, as well as on individuals’ level of information. We then focus on the manipulability of social choice correspondences when the Kelly extension rule is used, and individuals are assumed to have the capability to anticipate the outcome of the collective decision. Under these assumptions, we introduce some monotonicity and sensitivity properties for social choice correspondences that combined imply manipulability. Then we prove a result of manipulability for unanimous positional social choice correspondences, and present a detailed analysis of the manipulability properties for the Borda, the plurality, the negative plurality and the Copeland social choice correspondences.

💡 Research Summary

The paper investigates the manipulability of social choice correspondences (SCCs) when voters have only incomplete information about the preferences of others. The authors first formalize a general notion of manipulability that depends on two parameters: an extension rule that lifts individual preferences over alternatives to preferences over sets of alternatives, and an information function profile that captures the level of knowledge each voter possesses about the others’ preferences.

Among many possible extension rules, the study focuses on the Kelly extension (Kelly 1977). Under this rule a voter prefers set B to set C if every element of B is at least as good as every element of C according to her underlying linear order. The Kelly rule is very conservative, producing a partial order on the power set and making manipulation difficult in principle.

The authors then introduce the “winner information function” profile, which assumes that each voter can anticipate the set of alternatives that the SCC will select. In other words, a voter knows that the other voters’ true preference profile belongs to a certain information set that, when combined with her own reported preferences, yields the observed outcome. This models a realistic situation where voters do not know the exact rankings of others but have enough information to predict the winner(s).

Two structural properties of SCCs are defined:

1. Monotonicity (a weakened set‑monotonicity) – if an alternative x belongs to the chosen set under a profile, then raising x in a voter’s ranking cannot cause x to be dropped from the chosen set.

2. Sensitivity – there exists a voter and a non‑chosen alternative y such that raising y in that voter’s ranking (while leaving everything else unchanged) actually changes the outcome.

Theorem 14 shows that any SCC satisfying both properties is manipulable under the Kelly extension together with the winner‑information profile. The proof exploits the fact that the voter can foresee the outcome and can profitably misreport only when a small change in her ranking can affect the chosen set.

Applying this general result, the paper proves that every unanimous positional SCC (i.e., SCCs defined by a scoring vector w with w₁ > w₂) is manipulable whenever there are at least three alternatives and at least four voters (Theorem 15). This covers a large class of voting rules, including Borda, plurality, negative plurality, and many others.

The authors then conduct a detailed case‑by‑case analysis:

• Borda SCC – Using the Borda scoring vector (|A|‑1, |A|‑2, …, 0), Theorem 16 shows that manipulation is possible for almost all numbers of voters and alternatives, except for a few degenerate cases (e.g., an even number of voters with exactly three alternatives).

• Plurality SCC – With the scoring vector (1,0,…,0), Theorem 17 demonstrates that manipulation is generally possible, but there exist special configurations (even number of voters, three alternatives) where the rule is strategy‑proof under the Kelly‑winner information model.

• Negative Plurality SCC – Defined by the vector (1,1,…,1,0), Theorem 18 finds that with three alternatives the rule is immune to manipulation, while with four or more alternatives manipulation becomes easy, regardless of the number of voters.

• Copeland SCC – Based on pairwise majority defeats, Theorem 19 shows that whenever the underlying majority graph contains a cycle (which is typical when the number of voters is odd), a voter can manipulate by raising a non‑chosen alternative that participates in the cycle.

Each theorem is proved by constructing explicit preference profiles that satisfy the monotonicity and sensitivity conditions, thereby invoking Theorem 14. The analysis highlights how the manipulability of a rule depends not only on its scoring structure but also on the interaction between the number of alternatives and the number of voters.

In the concluding discussion, the authors contrast their findings with the classic Gibbard‑Satterthwaite theorem. While the latter states that every non‑dictatorial resolute rule with at least three alternatives is manipulable under complete information, the present work shows that even under incomplete information, many non‑resolute rules remain vulnerable, provided the information required for a profitable deviation is modest. The Kelly extension together with the winner‑information profile offers a concrete framework for measuring that information requirement.

The paper’s contribution is twofold: it extends the theory of strategy‑proofness to settings with limited knowledge, and it provides concrete criteria (monotonicity + sensitivity) that can be checked for any SCC. Practitioners designing voting mechanisms can therefore assess the robustness of a rule against strategic behavior in realistic environments where voters only have partial insight into others’ preferences. The results suggest that to achieve genuine strategy‑proofness one may need to either restrict the information available to voters further or adopt rules that deliberately violate either monotonicity or sensitivity, a direction left open for future research.