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Impossibility theorems are modified and unified

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Abstract

Arrow’s impossibility theorem and the Muller–Satterthwaite theorem are further interconnected by showing that translating one result into the other’s setting leads to similar results. This approach generated more impossibility theorems which are interesting on their own. Moreover, an implication of having these impossibility results on a unifying impossibility theorem in Eliaz (Soc Choice Welf 22:317–330, 2004) is also discussed.

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Notes

  1. Weak ranking is a complete and transitive binary relation, while strict ranking is a weak ranking which is also antisymmetric.

  2. An alternative definition is as follows. For all \(P\in P^{n}\) and all \(a\in A\), let \(MT(P,a)=\{P^{\prime }\in P^{n}: \forall i\in N, \forall c\in A, \text { if } a\succ _{P_{i}}c\text { then } a\succ _{P_{i}^{\prime }}c\}\), i.e., the set of monotonic transformations of \(P\in P^{n}\) at \(a\in A\). Then, \(F:P^{n}\rightarrow P^{*}\) is Monotonic (M) if whenever \(a,b\in A, P, P^{\prime }\in P^{n}\) are such that \(a\gtrsim _{F(P)}b\) and \(P^{\prime }\in MT(P,a)\), we have \(a\gtrsim _{F(P^{\prime })}b\).

  3. CI axiom is already appeared in Denicolò (1993, 2000). The former paper calls it as independence while the latter as Hansson pairwise independence.

  4. Existence of a version of the M–S theorem for SWFs is mentioned in the final section of Chap. 9 in Rubinstein (2012). But neither a formal statement of the result nor any reference, except Muller and Satterthwaite (1977), is given. Moreover, the monotonicity axiom described there seems to be weaker than SM. But Example 4 below shows that replacing SM with a weaker axiom allows SWFs which are not D.

  5. Thus, an individual is a weak dictator if society’s ranking never disagrees with her ranking, i.e., if she has veto power.

  6. Regarding to the reverse implication, in the light of Theorems 4 and 5 we can conclude that when \(m\ge 4\), PE and weak IIA imply M. But when \(m=3\), this implication is not necessarily true as the SWF constructed in Example 2 in Campbell and Kelly (2000) shows.

  7. A proof of the Gibbard–Satterthwaite (GS) theorem which is very similar to the proof of Theorem 6 can be found in Schmeidler and Sonnenschein (1974). Indeed, we believe that by combining these proofs one can come up with a line by line same proof, in the style of Reny (2001), of the GS theorem and Theorem 6.

  8. The obvious proof is omitted.

  9. Strictly speaking, the notion of ‘social aggregator’ already appeared in the earlier literature: see for instance, definitions of social decision function in Denicolò (1993) and social aggregating function in Sánchez and Peris (1999). In our opinion, these works anticipate, in many respects, more recent developments on unifying classic impossibility results in Eliaz (2004) and Man and Takayama (2013).

  10. Recently, Can and Storcken (2012, 2013) considered monotone social preference correspondences. However, the monotonicity property they studied (update monotonicity) is different from ours. In particular, the so called selective Kemeny rule is an update monotone SWF [see Can and Storcken (2013)]. But, one can show that it is not monotone in our sense.

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Acknowledgments

I am thankful to Susumu Cato and two anonymous referees for their helpful comments. The ARDI program coordinated by the World Intellectual Property Organization provided access to most of the recent publications and without this opportunity writing this paper would have been more difficult.

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  1. Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia

    Uuganbaatar Ninjbat

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Ninjbat, U. Impossibility theorems are modified and unified. Soc Choice Welf 45, 849–866 (2015). https://doi.org/10.1007/s00355-015-0887-2

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