Steve Eppley SEppley at alumni.caltech.edu
Mon Dec 3 13:55:38 PST 2007

```Peter Barath wrote:
> Sorry if the thing already has a name.
>
> Let's suppose there is a vote, where the voters are
> to chose from a number of numbers. For example,
> the membership fee of the club, the minimum age
> of application, the size of the office, anything.
>
> Something where we can suppose: If a person prefers
> number a over number b, and a > b > c then she will
> prefer b over c (because c is even further from a).
>
> Also, if she prefers a over b and a < b < c then she
> will prefer b over c (because c is even further from a).
>
> Scenario one. They vote by everyone giving her first
> preference then they search the median value. For example:
>
> Let the membership fee be:
>
> ------------------------------- (-)
> ------------------------------- (-)
> ------------------------------- (+)
> ------------------------------- (+)
> ------------------------------- (+)
>
> In this ladder-like scheme I put a sign at the end of each
> separating line: the sign is - if there are more votes under
> the line, and is + if there are mor votes above the line.
> The winner is the 150\$ because there is the change, so more
> than half of the voters wants the fee be 150\$ or higher, and
> more than half wants it to be 150\$ or lower.
>
> Scenario two. They vote Condorcet, while they keep the
> above mentioned convention: if somebody sees 180\$ as the
> best option she must prefer 140\$ over 70\$ etc. The
> convention says nothing about the preference between, say,
> 200\$ and 70\$ in this case.
>
> Theorem 1: If in scenario one there is a winner (no ties)
> then in scenario two there is a Condorcet-winner and is the
> same as the "ladder" winner in scenario one.
>
> Theorem 2: The ladder voting is strategy-free.
>
> I don't waste the space with proofs, they seem pretty obvious.
>
> The theoretical (well, maybe sometimes practical, yes, we
> used sometimes to vote about numbers) significance I see
> mainly in the second theorem. We don't have many strategy-
> free voting methods, only this, the Clarke-tax and the
> random methods mentioned usually with the Gibbard-Satterthwaite
> theorem: (random vote and runoff between two random candidates)
> for more than 2 choseable possibilities.
>
> Peter Barath
>
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>

Salvador Barbera taught a course on strategy-proofness when he was a
visiting professor at Caltech in 2000.  The Median Voter's Choice method
is indeed strategy-proof in some domains of alternatives and preferences.

Unfortunately, those are special cases.  It's hard to be sure when
choosing the voting method whether the voters' preferences will indeed
be single-peaked.  Also, other complications can easily arise, such as