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## Computational & Applied Mathematics

*Print version* ISSN 2238-3603*On-line version* ISSN 1807-0302

#### Abstract

TANAKA, Yasuhito. **Undecidability of the existence of dictator for strongly candidate stable voting procedures in an infinite society and Cantor 's diagonal argument**.

*Comput. Appl. Math.*[online]. 2008, vol.27, n.3, pp.251-268. ISSN 2238-3603.

The strong candidate stability theorem by Dutta et al. [ 4], one of the major theoremsof social choice theory, states that, with a finite number of voters, there exists a dictator for any voting procedure which satisfies *strong candidate stability, strong unanimity* and *independence of irrelevant alternatives (IIA)*. This paper investigates a decidability problem of voting procedures in a society with an infinite number of individuals (infinite society) using Cantor*'*s diagonal argument presented by Yanofsky [ 19] which is based on Lawvere [ 10]. We will show the following result. The problem whether a strongly candidate stable voting procedure has a dictator or has no dictator in an infinite society is undecidable. It is proved using the arguments similar to those used to prove an extended version of Cantor*'*s theorem that there cannot be an onto function from (the set of natural numbers) to its power set (). This undecidability means that for any strongly candidate stable voting procedure we can not decide whether or not it has a dictator *in finite steps* by some program. A dictator of a voting procedure is a voter such that if he strictly prefers a candidate (denoted by *x*) to another candidate (denoted by *y*), then the voting procedure does not choose *y*. Strong candidate stability requires that there be no change in the outcome of an election if a candidate withdraws who would lose ifevery candidate stood for office.

**Keywords
:
**strongly candidate stable voting procedure; Cantor*'*s diagonal argument.