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A fundamental structure of strategy-proof social choice correspondences with restricted preferences over alternatives

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Abstract

I prove that under each strategy-proof and unanimous social choice correspondence, there is at least one agent who is decisive. Because the result is established on a weak requirement on preferences over sets, the existence of a decisive agent is an underlying feature of most strategy-proof and unanimous social choice correspondences. Moreover, I consider a restriction on the space of preferences over alternatives. I prove that circular sets of preferences over alternatives are sufficient for the existence of a decisive agent.

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Notes

  1. We now have an excellent survey, Barberà (2010).

  2. See Taylor (2005) and Sect. 8 of Barberà (2010).

  3. Randy Johnson and Curt Schilling shared the MVP of the 2001 World Series.

  4. Some problems are out of the scope of this paper. An example is the problem of hiring exactly two new faculty members. See Özyurt and Sanver (2008) for an analysis of such a case.

  5. Anonymity requires a symmetric treatment of the agents, neutrality requires a symmetric treatment of the alternatives, Pareto efficiency requires that an alternative \(x\) does not belong to the social outcome if there exists an alternative \(y\) such that each agent prefers \(y\) to \(x\).

  6. See Taylor (2005) and Sect. 8 of Barberà (2010).

  7. Near unanimity is a property of a social choice correspondence. It requires that when all but one agent have a common most preferred alternative, then that single alternative must be the social outcome at the preference profile.

  8. I do not mean that leximin preferences should be ruled out from the analysis. If we do not have much information on preferences over sets, then such “extreme preferences” should be included in the analysis because we do not have a rationale to exclude them. However, once we get enough information about preferences over sets to exclude “extreme preferences” from the analysis, each analysis which critically depends on such preferences becomes ineffective. Benoît (2002)’s analysis is robust in this aspect. Barberà et al. (2001)’s analysis is also robust in this aspect. The results by Barberà et al. (2001) are independent of those by Benoît (2002).

  9. Precisely speaking, I employ the idea by Benoît (2002) as a way of extending preferences over alternatives to preferences over sets. The set of admissible preferences over sets considered by Benoît (2002) and those considered in this paper are not the same. I will discuss this point in Sect. 2. Because an essence of the idea is due to Benoît (2002), I will say “Benoît’s formulation”.

  10. I give the definition of decisiveness informally. Agent \(i\) is decisive for an alternative \(x\) if \(x\) belongs to the social outcome under a situation where there is an alternative \(y\) such that for agent \(i, x\) is the best alternative and \(y\) is the second alternative, and for the other agents, \(y\) is the best alternative and \(x\) is the worst alternative. Agent \(i\) is decisive if he is decisive for each alternative.

  11. The Condorcet loser is defeated by each other alternative in a pairwise comparison.

  12. The concept of circular sets is introduced by Sato (2010). Sato (2010) proves that on each circular set of preferences over alternatives, each strategy-proof and unanimous single-valued social choice rule is dictatorial.

  13. To the best of my knowledge, Nehring (2000) and Ching and Zhou (2002) are the only exceptions. Nehring (2000) considers “comprehensive domains”, and deals with the weakest notion of nonmanipulability in the literature. Ching and Zhou (2002) consider continuous preferences over alternatives. Some of the other existing results do not need the universal set of preferences over alternatives. Benoît (2002)’s theorem is an example of such a result. See arguments after Corollary 3.2 in this paper.

  14. Let \(S\) be a set. Each subset of \(S\times S\) is a binary relation on \(S\). A binary relation \(R\) on \(S\) is complete if for each pair \(x, y\in S\), either \(x R y\) or \(y R x\), transitive if for each triple \( x, y, z\in S, [x R y\; \& \; yR z]\) implies \(x R z\), antisymmetric if for each pair \( x, y\in S, [x R y \; \& \; yR x]\) implies \(x = y\). A binary relation is a weak order if it is complete and transitive, a linear order if it is complete, transitive, and antisymmetric.

  15. I will use \(r_k(\mathcal R )\) only when it can be defined unambiguously.

  16. In addition to these axioms, we could require that \(\mathcal{R }_i\) preserves the rankings of singletons: for each \(R_i\in D\), each pair \(x, y\in X\), and each \(\mathcal{R }_i\in E(R_i), x R_i y \iff x \mathcal R _i y\). I do not include this in the definition of an extension rule, just because it is not necessary for our results. The reader can assume this preservation condition if it is more comfortable.

  17. Example 2.1 is the same as Example 3 by Benoît (2002). Example 2.2 is a special case of Example 2 by Benoît (2002).

  18. The superscript “\(wb\)” stands for “Worst and Best”.

  19. For example, the plurality rule, the Borda rule, each Condorcet consistent rule, and so on.

  20. This argument assumes \(n\ge 3\).

  21. Later in this section, I will show that a “strongly” decisive agent does not necessarily exist. Thus, my result might be near the limit of what we can derive from the assumptions in the theorem.

  22. For example, when the agents have the leximin preferences over sets, the rule \(f\) defined by \(f({\varvec{\mathcal{R }}}) = \bigcup _{i\in N} r_1({\mathcal{R }}_i)\) is strategy-proof and unanimous. (This fact is pointed out by Özyurt and Sanver (2009)). Each agent is decisive under this rule.

  23. A circular set is minimal in terms of the cardinality when it consists of \(2m\) preference relations.

  24. Because \(D\) is circular, the elements of \(X\) are indexed \(x_1, \dots , x_m\).

  25. Leximax preferences are the counterpart of leximin preferences.

  26. The ideas of circular sets and linked sets are independent from each other.

  27. This approach first appears in Barberà et al. (2001).

  28. Strictly speaking, this statement assumes that for each \(x, y\in X, x R y\) iff \(\{x\} \mathcal R \{y\}\) for each \(\mathcal{R }\in E(R)\) and each \(R\in D\). This is an uncontroversial assumption. It is not assumed in this paper, just because I do not need it.

  29. I give a note on the description of preferences. I write only rankings over the singletons from which \(\mathcal R _1, \dots , \mathcal R _n\) are derived. The first column shows that \(\mathcal R _{i^*}\in E(R_{i^*})\) with \(r_1(R_{i^*})=x_{k-1}, r_2(R_{i^*})=x_{k-2}\), and \(r_m(R_{i^*})=x_k\). Because \(\mathcal R _{i^*}\) is \(r_2\)-favoring, the second ranked set is \(\{x_{k-1}, x_{k-2}\}\), and \(\{x_{k-2}\}\) is the third ranked set.

  30. I use the assumption that \(\mathcal R _{i^*}\) is \(r_2\)-favoring. Only \(x_{k-1}\) and \(\{x_{k-1}, x_{k-2}\}\) are preferred to \(x_{k-2}\).

  31. Let \(Q(0)=Q(m)\).

  32. See Definition 2.2

  33. Formally, \(r_1({\mathcal{R }}_i)=\{r_1(R_i)\}\).

References

  • Aswal N, Chatterji S, Sen A (2003) Dictatorial domains. Econ Theory 22:45–62

    Article  Google Scholar 

  • Barberà (2010) Strategyproof social choice. In: Arrow KJ, Sen A, Suzumura K (eds) . Handbook of social choice and welfare, vol 2, chapter 25. North-Holland, Amsterdam

  • Barberà S, Dutta B, Sen A (2001) Strategy-proof social choice correspondences. J Econ Theory 101:374–394

    Article  Google Scholar 

  • Benoît J-P (2002) Strategic manipulation in voting games when lotteries and ties are permitted. J Econ Theory 102:421–436

    Article  Google Scholar 

  • Ching S, Zhou L (2002) Multi-valued strategy-proof social choice rules. Soc Choice Welf 19:569–580

    Article  Google Scholar 

  • Feldman A (1979) Manipulation and the Pareto rule. J Econ Theory 21:473–482

    Article  Google Scholar 

  • Gibbard A (1973) Manipulation of voting scheme: a general result. Econometrica 41:587–601

    Article  Google Scholar 

  • Nehring K (2000) Monotonicity implies generalized strategy-proofness for correspondences. Soc Choice Welf 17:367–375

    Article  Google Scholar 

  • Özyurt S, Sanver MR (2008) Strategy-proof resolute social choice correspondences. Soc Choice Welf 30:89–101

    Article  Google Scholar 

  • Özyurt S, Sanver MR (2009) A general impossibility result on strategy-proof social choice hyperfunctions. Games Econ Behav 66:880–892

    Article  Google Scholar 

  • Satterthwaite MA (1975) Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J Econ Theory 10:187–217

    Article  Google Scholar 

  • Sato S (2010) Circular domains. Rev Econ Des 14:331–342

    Google Scholar 

  • Taylor AD (2005) Social choice and the mathematics of manipulation. Cambridge University Press, Cambridge

    Book  Google Scholar 

Download references

Acknowledgments

I am grateful to Masashi Umezawa and seminar participants at Keio and Bilgi for their helpful comments. I also grateful to the Associate Editor and an anonymous reviewer for their suggestions and comments.

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Correspondence to Shin Sato.

Appendix

Appendix

A proof of Theorem 3.1

The idea of a main part of the proof is the combination of that of Benoît (2002) and Sato (2010), and indirectly of their predecessors.

Let \(D\subset L\) be circular and \(f\) be a unanimous and strategy-proof rule on \(\mathbb D (D, E)\). In the following, I freely use the following simple implication of strategy-proofness. A set \(A\) is isolated in \(\mathcal R _i\in \mathbb D (D, E)\) if for each \(B\in \mathcal X , A \mathcal R _i B\) and \(B \mathcal R _i A\) imply \(A=B\). In other words, \(A\) is isolated in \(\mathcal R _i\) if no set is indifferent to \(A\) in \(\mathcal R _i\).

Maskin monotonicity: Let \({\varvec{\mathcal{R }}}, {\varvec{\mathcal{R ^{\prime }}}}\in \mathbb D (D, E)^N\). If for each \(i\in N, f({\varvec{\mathcal{R }}})\) is isolated in either \(\mathcal R _i\) or \(\mathcal R _i^{\prime }, \{A\in \mathcal X \mid f({\varvec{\mathcal{R }}}) \mathcal P _i A\} \subset \{A\in \mathcal X \mid f({\varvec{\mathcal{R }}}) \mathcal P _i^{\prime } A\}\), and \(\{A\in \mathcal X \mid f({\varvec{\mathcal{R }}}) \mathcal R _i A\} \subset \{A\in \mathcal X \mid f({\varvec{\mathcal{R }}}) \mathcal R _i^{\prime } A\}\), then \(f({\varvec{\mathcal{R }}})= f({\varvec{\mathcal{R ^{\prime }}}})\).

Maskin monotonicity says that when \(f({\varvec{\mathcal{R }}})\) is isolated in either \(\mathcal R _i\) or \(\mathcal R _i^{\prime }\), and the strict lower contour set and the weak lower contour set of \(f({\varvec{\mathcal{R }}})\) expand in a weak sense for each agent \(i\) in the passage from \(\mathcal R _i\) to \({\mathcal{R }}_i^{\prime }\), then the social outcome does not change. To see this, consider that agent \(1\) changes his preference from \(\mathcal R _1\) to \(\mathcal R _1^{\prime }\). Then, the preference profile changes from \({\varvec{\mathcal{R }}}\) to \((\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). Suppose \(f({\varvec{\mathcal{R }}})\ne f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). Because \(\mathcal R _1\) is a weak order, either \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1}) \mathcal P _1 f({\varvec{\mathcal{R }}})\) or \(f({\varvec{\mathcal{R }}}) \mathcal R _1 f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). The former case is a contradiction to strategy-proofness. Consider the latter case. If \(f({\varvec{\mathcal{R }}})\) is isolated in \({\mathcal{R }}_1\), then \(f({\varvec{\mathcal{R }}}) {\mathcal{P }}_1 f({\mathcal{R }}_1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). By the assumption, \({f(\varvec{\mathcal{R )}}} {\mathcal{P ^{\prime }}}_1 f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). This is a contradiction to strategy-proofness. If \(f({\varvec{\mathcal{R }}})\) is isolated in \(\mathcal R _1^{\prime }\), then either \({f(\varvec{\mathcal{R )}}} \mathcal P _1^{\prime } f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) or \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1}) \mathcal P _1^{\prime } f({\varvec{\mathcal{R }}})\). The former case is a contradiction to strategy-proofness. In the latter case, it should be \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1}) \mathcal P _1 f({\varvec{\mathcal{R }}})\). (If not, \(f({\varvec{\mathcal{R }}}) \mathcal{R }_1 f(\mathcal{R }_1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). By the assumption, \(f({\varvec{\mathcal{R }}}) \mathcal R _1^{\prime } f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\), which is a contradiction to \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1}) \mathcal P _1^{\prime } f({\varvec{\mathcal{R }}})\).) However, this is a contradiction to strategy-proofness. Therefore, \(f({\varvec{\mathcal{R }}}) = f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). By repeating the arguments, we have \(f({\varvec{\mathcal{R }}}) = f({\varvec{\mathcal{R ^{\prime }}}})\).

Assign a number from \(1\) to \(m\) to each alternative so that it makes \(D\) circular. Let \(x_1, \dots , x_m\) be the indexed alternatives. Unless otherwise stated, in the following, each preference relation over sets is \(r_2\) favoring. The following two results are keys in proving Theorem 3.1.

Lemma 1

Let \(k\in \{1, \dots , m\}, i^*\in N\), and \({\varvec{\mathcal{R }}}\in \mathbb D (D, E)^N\) be such thatFootnote 29

figure d

where for each \(j\in N\setminus \{i^*\}, \mathcal R _j\) is not necessarily \(r_2\)-favoring. If \(x_{k-1}\not \in f({\varvec{\mathcal{R }}})\), then as long as the agents in \(N\setminus \{i^*\}\) rank \(x_{k-2}\) at the top, the social outcome is \(x_{k-2}\).

Proof of Lemma 1

Assume \(x_{k-1}\not \in f({\varvec{\mathcal{R }}})\). Then, the social outcome at \({\varvec{\mathcal{R }}}\) should be \(x_{k-2}\). (If not, then the social outcome is less preferred to \(x_{k-2}\) according to \(\mathcal R _{i^*}\).Footnote 30 Thus, agent \(i^*\) benefits from misreporting that \(x_{k-2}\) is the top ranked alternative. This is a contradiction to strategy-proofness.)

From \({\varvec{\mathcal{R }}}\), consider the successive change of preferences of the agents in \(N\setminus \{i^*\}\) to the following preference profile \({\varvec{\mathcal{R }}}^{\prime }\): (In this profile, for each \(j\in N\setminus \{i^*\}, \mathcal R _j^{\prime }\) is \(r_2\)-favoring.)

figure e

Maskin monotonicity and \(f({\varvec{\mathcal{R }}})=x_{k-2}\) imply \(f({\varvec{\mathcal{R ^{\prime }}}})=x_{k-2}\).

Next, agent \(i^*\) changes his preferences from \(\mathcal{R }_{i^*}^{\prime }\) to the one in the following preference profile \({\varvec{\mathcal{R ^{\prime \prime }}}}\):

figure f

By strategy-proofness, \(f({\varvec{\mathcal{R ^{\prime \prime }}}})\not \in \{\{x_{k-1}, x_{k-2}\}, x_{k-1}\)}. Then, the social outcome should be \(x_{k-2}\). (Suppose not. Then, for each \(j\in N\setminus \{i^*\}\), the social outcome is less preferred to \(x_{k-1}\) according to \(\mathcal R _j^{\prime \prime }\). When the agents in \(N\setminus \{i^*\}\) successively misreport that \(x_{k-1}\) is the top ranked alternative, by unanimity, the social outcome becomes \(x_{k-1}\) at some stage. This is a beneficial change for the agent whose change makes the social outcome \(x_{k-1}\). This is a contradiction to strategy-proofness.)

Maskin monotonicity and \(f({\varvec{\mathcal{R ^{\prime \prime }}}})=x_{k-2}\) imply that as long as the agents in \(N\setminus \{i^*\}\) rank \(x_{k-2}\) at the top, the social outcome is \(x_{k-2}\). \(\square \)

Lemma 2

Let \(k\in \{1, \dots , m\}\) and \(N^{\prime }\subset N\). Assume that for each \({\varvec{\mathcal{R }}}\in {\mathbb{D }}(D, E)^N\) such that \(r_1(\mathcal R _i)=x_{k-1}\) for each \(i\in N^{\prime }, f({\varvec{\mathcal{R }}})=x_{k-1}\). Let \(i^*\in N^{\prime }\). If \(f({\varvec{\mathcal{R ^{\prime }}}})\) is either \(x_k\) or \(\{x_{k-1}, x_k\}\) for some \({\varvec{\mathcal{R ^{\prime }}}}\in {\mathbb{D }}(D, E)^N\) such that

figure g

then, for each \({\varvec{\mathcal{R ^{\prime \prime }}}}\in {\mathbb{D }}(D, E)^N\) such that

figure h

we have \(f({\varvec{\mathcal{R ^{\prime \prime }}}})\ne x_{k-2}\).

Proof of Lemma 2

Assume the premise of the lemma. Suppose \(f({\varvec{\mathcal{R ^{\prime \prime }}}}) = x_{k-2}\). Consider that the agents in \(N^{\prime }\setminus \{i^*\}\) successively change their preferences from \(\mathcal R _i^{\prime \prime }\) to \({\bar{\mathcal{R }}}_i\):

figure i

By strategy-proofness, the candidates for the social outcome are \(x_{k-2}, \{x_{k-1}, x_{k-2}\}\), and \(x_{k-1}\). If the outcome is either \(x_{k-2}\) or \(\{x_{k-1}, x_{k-2}\}\), then agent \(i^*\) benefits from misreporting that \(x_{k-1}\) is the top ranked alternative. (By the premise of the lemma, the social outcome becomes \(x_{k-1}\) as a result of the misrepresentation, and according to \(\bar{\mathcal{R }}_{i^*}, x_{k-1}\) is preferred to \(x_{k-2}\) and \(\{x_{k-1}, x_{k-2}\}\). This is a contradiction to strategy-proofness.) Assume that the social outcome is \(x_{k-1}\). Then, by Maskin monotonicity, \(f({\varvec{\mathcal{R }}}^{\prime })=x_{k-1}\), which is a contradiction to the premise of the lemma. \(\square \)

A proof strategy: We adopt the following strategy in proving Theorem 3.1. First, we find \(i^*\in N\) who has a “special power”. Next, for each \(k\in \{1, \dots , m\}\), let \(Q(k)\) denote the statement

\(x_k\in f({\varvec{\mathcal{R }}})\) for each \({\varvec{\mathcal{R }}}\in \mathbb D (D, E)^N\) such that

figure j

(for each \(j\in N\setminus \{i^*\}, \mathcal R _j\) is not necessarily \(r_2\)-favoring.)”

By induction, we prove that \(Q(k)\) is true for each \(k\in \{1, \dots , m\}\). Finally, we prove that \(i^*\) is decisive. In the following, there are many preference profiles. I use superscripts to distinguish them.

Finding \(i^*\): Following the proof strategy, I find \(i^*\in N\) having a “special power”.

Step 1: Consider \({\varvec{\mathcal{R }}}^1\in \mathbb{D }(D, E)^N\) having the structure in the following table:

figure k

Unanimity implies \(f({\varvec{\mathcal{R }}}^1) = x_{2}\).

Step 2: I find \(i^*\in N\) by the following way. For each \(i\in N\), let \(R_i^2\in D\) be such that \(r_1(R_i^2) = x_1, r_2(R_i^2) = x_{2}\), and \(r_m(R_i^2) = x_{m}\). Let \(\mathcal R _i^2\in E(R_i^2)\). Consider the following successive change of preferences. The starting point is \({\varvec{\mathcal{R }}}^1\). First, agent \(1\), next, agent \(2, \dots \), and finally, agent \(n\) change their preferences from \(\mathcal R _i^1\) to \(\mathcal R _i^2\). Throughout this process, by strategy-proofness, the social outcome is either \(x_1\) or \(x_2\) or \(\{x_1, x_2\}\). By unanimity, at the end of this process, the social outcome is \(x_1\). Then, there is \(i^*\in N\) whose change of preferences leads to the change of the social outcome from \(x_2\) or \(\{x_1, x_2\}\) to \(x_1\). In other words, there is \(i^*\in N\) such that

  • after agent \(i^*-1\) changes his preference, the social outcome is not \(x_1\):

    figure l

    and

  • after agent \(i^*\) changes his preference, the social outcome is \(x_1\):

    figure m

    By Maskin monotonicity, as long as agents \(1, \dots , i^*\) put \(x_1\) at the top, the social outcome is \(x_1\).

The following Steps 3, 4, and 5 establish the Induction Base, i.e., \(Q(1)\).

A proof of the induction base \(Q(1)\):

Step 3: Consider the first preference profile in Step 2. When agent \(i^*\) changes his preference relation so that we have the following profile, by Maskin monotonicity, the social outcome does not change.

figure n

Step 4: Consider \({\varvec{\mathcal{R }}}^3\in \mathbb D (D, E)^N\) having the following structure, where for each \(j\in N\setminus \{i^*\}, \mathcal R _j^3\) is not necessarily \(r_2\)-favoring. (\(\mathcal R _{i^*}^3\) is \(r_2\)-favoring.)

figure o

I claim \(x_1\in f({\varvec{\mathcal{R }}}^3)\). (If we establish this claim, we establish the Induction Base.) Suppose not. By Lemma 1, as long as the agents in \(N\setminus \{i^*\}\) rank \(x_{m}\) at the top, the social outcome is \(x_{m}\).

Step 5: We apply Lemma 2. Let \(N^{\prime }\) in Lemma 2 be \(\{1, \dots , i^*\}, x_{k-2} = x_m, x_{k-1} = x_1, x_k = x_2\), and \(x_{k+1} = x_3\). Because the social outcome at the profile in Step 3 is either \(x_2\) or \(\{x_1, x_2\}\), we can apply Lemma 2. Then, the conclusion of Lemma 2 contradicts the last statement of Step 4. Thus, the supposition that \(x_1\not \in f({\varvec{\mathcal{R }}}^3)\) should be wrong, and we complete the proof of the Induction Base.

Next, let \(k\in \{1, \dots , m\}\). Assume the Induction Hypothesis \(Q(k)\). Steps 6 through 9 establish the Induction Step \(Q(k-1)\).Footnote 31

A proof of the induction step \(Q(k-1)\):

Step 6: Consider \({\varvec{\mathcal{R }}}^4\in \mathbb D (D, E)^N\) such that

figure p

Unanimity implies \(f({\varvec{\mathcal{R }}}^4) = x_{k}\). Consider the successive change of preferences of the agents in \(N\setminus \{i^*\}\) so that we have

figure q

At \({\varvec{\mathcal{R }}}^5\), by strategy-proofness, the candidates for the social outcome are \(x_{k-1}, \{x_{k-1}, x_k\}\), and \(x_k\). The social outcome at \({\varvec{\mathcal{R }}}^5\) is not \(x_{k-1}\). (If it is, by Maskin monotonicity, \(f({\varvec{\mathcal{R }}}) = x_{k-1}\), where \({\varvec{\mathcal{R }}}\) is the one in \(Q(k)\). This is a contradiction to \(Q(k)\).) Thus, the social outcome at \({\varvec{\mathcal{R }}}^5\) is either \(x_k\) or \(\{x_{k-1}, x_k\}\).

Step 7: When agent \(i^*\) changes his preferences so that we have the following profile, by Maskin monotonicity, the social outcome does not change.

figure r

Step 8: Consider \({\varvec{\mathcal{R }}}^{6}\in \mathbb D (D, E)^N\) having the following structure, where for each \(j\in N\setminus \{i^*\}, \mathcal R ^{6}_j\) is not necessarily \(r_2\)-favoring.

figure s

I claim \(x_{k-1}\in f({\varvec{\mathcal{R }}}^{6})\). (This is the conclusion of the Induction Step.) Suppose not. Lemma 1 implies that as long as the agents in \(N\setminus \{i^*\}\) rank \(x_{k-2}\) at the top, the social outcome is \(x_{k-2}\).

Step 9: We apply Lemma 2. Let \(N^{\prime }\) in Lemma 2 be \(N\). Because the social outcome at the profile in Step 7 is either \(x_k\) or \(\{x_{k-1}, x_k\}\), we can apply Lemma 2. Then, the conclusion of Lemma 2 contradicts the last statement of Step 8. Thus, the supposition that \(x_{k-1}\not \in f({\varvec{\mathcal{R }}}^{6})\) should be wrong, and we complete the proof of the Induction Step.

Decisiveness of \(i^*\): Step 10: By induction, for each \(x_k\in \{x_1, \dots , x_m\}\), we have \(x_k\in f({\varvec{\mathcal{R }}})\) for each \(\varvec{\mathcal{R }}\in \mathbb{D }(D, E)^N\) such that \(\mathcal R _{i^*}\) is \(r_2\)-favoring and

figure t

(For each \(j\in N\setminus \{i^*\}, \mathcal R _j\) is not necessarily \(r_2\)-favoring.)

Let \(x_k\in \{x_1, \dots , x_m\}\), and let \({\varvec{\mathcal{R }}}\) denote the profile in the previous paragraph. Let \({\varvec{\mathcal{R }}}^{\prime }\in \mathbb D (D, E)^N\) be such that \(\mathcal R _{i^*}^{\prime }\) is \(r_2\)-favoring and

figure u

(For each \(j\in N\setminus \{i^*\}, \mathcal R _j^{\prime }\) is not necessarily \(r_2\)-favoring.) By unanimity and strategy-proofness, \(f({\varvec{\mathcal{R }}}^{\prime })\in \{x_{k-1}, \{x_{k-1}, x_k\}, x_k\}\). (If not, agent \(i^*\) benefits from misrepresentation.) If \(f({\varvec{\mathcal{R }}}^{\prime })=x_{k-1}\), then by Maskin monotonicity, \(f({\varvec{\mathcal{R }}})=x_{k-1}\), which is a contradiction to the statement in the previous paragraph. Thus, \(f({\varvec{\mathcal{R }}}^{\prime })\) is either \(x_k\) or \(\{x_{k-1}, x_k\}\). Therefore, agent \(i^*\) is decisive for \(x_k\). (The alternative \(x_{k-1}\) plays the role of \(y\) in the definition of decisiveness.Footnote 32) Because \(x_k\) was arbitrary in \(\{x_1, \dots , x_m\}\), agent \(i^*\) is decisive. \(\square \)

A proof of Proposition 3.1

Assume \(m\ge 4\), and let \(D\) be a minimal circular set of preferences over alternatives. Let \(E\) and \(f\) be the extension rule and the rule, respectively, defined after Proposition 3.1. For each \({\mathcal{R }}_i\in {\mathbb{D }}(D, E)\), let \(R_i\) denote the element of \(D\) such that \(E(R_i)=\{{\mathcal{R }}_i\}\). (By the definition of \(E\), there is a one-to-one correspondence between a preference relation over sets and a preference relation over alternatives.) Thus, \(r_1(\mathcal R _i) = r_1(R_i)\).Footnote 33

Step 1: I claim that \(f\) is strategy-proof.

Let \({\varvec{\mathcal{R }}}\in \mathbb D (D, E)^N\). Each agent in \(N\setminus \{1, 2\}\) has no incentive to lie because he has no influence over social choice. I prove that agent \(1\) has no incentive to lie. (By similar arguments, it can be seen that agent \(2\) has no incentive to lie.)

Case 1: \(r_1(\mathcal R _1) = r_1(\mathcal R _2)\).

In this case, \(f({\varvec{\mathcal{R }}})=r_1(\mathcal R _1)\). Therefore, agent \(1\) has no incentive to lie.

Case 2: \(r_1(\mathcal R _1)\) and \(r_1(\mathcal R _2)\) are adjacent.

Because \(D\) is a minimal circular set, at \(R_1, r_1(\mathcal R _2)\) is either second preferred or least preferred.

Subcase 2.1: \(r_1(\mathcal R _2) = r_2(R_1)\), i.e., the top ranked alternative by agent \(2\) is the second ranked alternative according to \(R_1\).

In this case, \(f({\varvec{\mathcal{R }}}) = \{r_1(R_1), r_2(R_1)\}\), which is the second ranked set according to \(\mathcal R _1\). According to \(\mathcal R _1, r_1(R_1)\) is the only set preferred to \(\{r_1(R_1), r_2(R_1)\}\). For each \(\mathcal R _1^{\prime }\in \mathbb D (D, E), f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\ne r_1(R_1)\). Thus, agent \(1\) has no incentive to lie.

Subcase 2.2: \(r_1(\mathcal R _2) = r_m(R_1)\), i.e., the top ranked alternative by agent \(1\) is the worst alternative for agent \(1\).

In this case, \(f({\varvec{\mathcal{R }}}) = \{r_1(R_1), r_m(R_1)\}\), which is a fourth ranked set according to \(\mathcal R _1\). To achieve a better social choice than \(f({\varvec{\mathcal{R }}})\) according to \(\mathcal R _1, r_m(R_1)\) should be excluded from the social choice. To exclude \(r_m(R_1)\) from the social choice, agent \(1\) should reports \(\mathcal R _1^{\prime }\) such that \(r_1(\mathcal R _1^{\prime })\ne r_1(\mathcal R _2)\) and \(r_1(\mathcal R _1^{\prime })\) and \(r_1(\mathcal R _2)\) are not adjacent. However, in that case, either \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) contains more than two alternatives or \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) consists of two alternatives which are not adjacent. (The latter case occurs only when \(m=4\).) In each case, \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) is also a fourth ranked set according to \(\mathcal R _1\). Thus, agent \(1\) has no incentive to lie.

Case 3: \(r_1(\mathcal R _1)\ne r_1(\mathcal R _2)\), and \(r_1(\mathcal R _1)\) and \(r_1(\mathcal R _2)\) are not adjacent.

In this case, \(f({\varvec{\mathcal{R }}})=X \setminus \{r_1(R_1), r_1(R_2)\}\). Then, either \(f({\varvec{\mathcal{R }}})\) contains more than two alternatives or \(f({\varvec{\mathcal{R }}})\) consists of two alternatives which are not adjacent. Thus, \(f({\varvec{\mathcal{R }}})\) is a fourth ranked set according to \(\mathcal R _1\). Let \(\mathcal R _1^{\prime }\in \mathbb D (D, E)\).

If \(r_1(\mathcal R _1^{\prime })\ne r_1(\mathcal R _2)\), and \(r_1(\mathcal R _1^{\prime })\) and \(r_1(\mathcal R _2)\) are not adjacent, \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1}) = X\setminus \{r_1(R_1^{\prime }), r_1(R_2)\}\). Then, either \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) contains more than two alternatives or \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) consists of two alternatives which are not adjacent. In each case, \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) is also a fourth ranked set according to \(\mathcal R _1\). Thus, agent \(1\) has no incentive to lie.

If \(r_1(\mathcal R _1^{\prime }) = r_1(\mathcal R _2)\), or \(r_1(\mathcal R _1^{\prime })\) and \(r_1(\mathcal R _2)\) are adjacent, \(r_1(\mathcal R _2)\) belongs to \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\). Because \(r_1(\mathcal R _1)\ne r_1(\mathcal R _2)\) and \(r_1(\mathcal R _1)\) and \(r_1(\mathcal R _2)\) are not adjacent, \(r_1(R_2)\not \in \{r_1(R_1), r_2(R_1)\}\). (In words, \(r_1(R_2)\), which belongs to \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\), is neither the first ranked nor the second ranked alternative according to \(R_1\).) Thus, \(f(\mathcal R _1^{\prime }, {\varvec{\mathcal{R }}}_{-1})\) is at most a fourth ranked set according to \(\mathcal R _1\). Thus, agent \(1\) has no incentive to lie.

Step 2: I claim that agents \(1\) and \(2\) are decisive.

I show that agent \(1\) is decisive. Let \(x_k\in X\). Then, for each \({\varvec{\mathcal{R }}}\in \mathbb D (D, E)^N\) such that

figure v

we have \(f({\varvec{\mathcal{R }}})=\{x_k, x_{k+1}\}\). Thus, agent \(1\) is decisive for \(x_k\). Because \(x_k\) was arbitrary, agent \(1\) is decisive.

By similar arguments, agent \(2\) is also decisive.

Step 3: I claim that no agent is strongly decisive.

Let \(x_k\in X\), and consider \({\varvec{\mathcal{R }}}\in \mathbb D (D, E)^N\) such that

figure w

Then, \(f({\varvec{\mathcal{R }}}) = X\setminus \{x_k, x_{k+2}\}\). Thus, no agent is strongly decisive.

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Sato, S. A fundamental structure of strategy-proof social choice correspondences with restricted preferences over alternatives. Soc Choice Welf 42, 831–851 (2014). https://doi.org/10.1007/s00355-013-0755-x

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