[EM] Generalizing "manipulability"

[Jonathan Lundell]

On Jan 8, 2009, at 4:45 PM, Abd ul-Rahman Lomax wrote:

> The whole concept of strategic voting is flawed when applied to  
> Range. Voters place vote strength where they think it will do the  
> most good -- if they think. Some don't. Approval is essentially, as  
> Brams claimed, "strategy-free,"  in the old meaning, and the only  
> way that it was at all possible to call it vulnerable was that  
> critics claimed that there was some absolute "approval" relation  
> between a voter and a candidate.

It would be useful to generalize the concept of strategic voting (and  
the related concepts of manipulation and sincerity) to other than  
linear ballots (that is, a ballot with an ordinal ranking of the  
voter's preferences). With linear ballots (and so Borda, IRV and  
various Condorcet methods) we define a "sincere" ballot as the one a  
voter would cast if the voter were a dictator, and manipulability as  
the ability of a voter to achieve a better result (where "better"  
means the election of a candidate ranked higher on that voter's  
sincere ballot) by voting "insincerely" or "strategically"--that is,  
by casting a ballot different from their sincere ballot. An election  
method that is not manipulable in this sense is defined to be  
"strategy-free".

A two-candidate plurality election is strategy-free. Most interesting  
elections are not.

With any practical election method using linear ballots, manipulation  
cannot succeed unless the voter has knowledge of how the other voters  
are voting. This knowledge need not be perfect. I propose (and I don't  
claim that this is original, though I don't recall seeing the  
definition) that we use this observation to generalize the idea of  
manipulability to election methods, such as Range and Approval, that  
do not use linear ballots, thus:

> An election method is manipulable if a voter has a rational  
> motivation to cast different ballots depending on the voter's  
> knowledge (or belief) of the ballots of other voters.

In such an election, a voter should vote strategically when the ballot  
that will produce the "best" outcome (for that voter) depends on the  
behavior of the other voters, the strategy consisting of determining,  
by some means depending on the method, which ballot that is.

For example, in an Approval election, with a preference of A>B>C, we  
will always vote for A, but whether we vote for B depends on how well  
we believe B and C are doing with other voters. If we believe that C  
cannot win, then we vote for A only, to improve our chance of electing  
A over B. If we believe that C is a serious threat, then we vote for A  
and B, to improve our chance of rejecting C.

The generalization of a "sincere" ballot then becomes the zero- 
knowledge (of other voters' behavior) ballot, although we might still  
want to talk about a "sincere ordering" (that is, the sincere linear  
ballot) in trying to determine a "best possible" outcome.


It seems to me that it's clearly desirable to be able to optimize the  
outcome by casting a sincere linear ballot. Such a ballot is  
reasonably expressive (that is, it contains more information about my  
preferences than, say, a plurality or approval ballot) without (in  
itself) requiring me to strategize. Unfortunately, no such election  
method exists, and many (most?) of the arguments on this list are over  
the tradeoffs implied by that sad fact. The best we can do is to find  
a method in which it's very unlikely that we can improve our outcome  
by voting other than sincerely.