[EM] Some unfortunately too strong Defensive Strategy Criterion

[Bart Ingles]

Jobst Heitzig wrote:
> 
> In the last days, I thought about some form of strategy-proofness like
> the following:
> 
> Criterion:
> Suppose that, with all voters voting sincerely, the method elects A, but
> some voter prefers B to A and can get B elected by voting insincerely.
> Then those voters not preferring B to A must have a way of voting which
> ensures that A or some option C gets elected which the first voter ranks
> *below* A, so that either the sincere result can be guaranteed or the
> incentive to vote insincerely is removed.
> 
> However, I then came up with the following, very simple 3-by-3-example
> which seems to render those thoughts ridiculous...
> 
> Problematic Example:
> 
> Sincere preferences
> Voter 1: A>B>C
> Voter 2: B>C=A
> Voter 3: C>A>B
> 
[...]
> 
> I sincerely hope I missed some essential point in that example... Can
> anyone tell what this would look like with Approval (I mean, what is a
> sincere Approval vote in the first place?)?

You would need information about utilities to predict how Approval
voters should behave, but if you assume that voters 1 and 3 have
utilities of 1.0, 0.5, and 0.0 for the three choices, and voter 2 has
1.0, 0.0, 0.0, and ties are resolved randomly, then:

The utility of bullet voting is:
Voter 1:  1/3 + .5 * 1/3 = 1/2
Voter 2:  1/3 + 0        = 1/3
Voter 3:  1/3 + .5 * 1/3 = 1/2

If only voter 1 approves of A & B, B wins so his utility is still 1/2
(but voter 2's is 1.0 and voter 3's is 0.0).

If only voter 2 approves of any two candidates, the 2nd candidate wins,
so his utility is 0.0 (but the beneficiary's is 1.0).

If only voter 3 approves of C & A, A wins so his utility is still 1/2
(but voter 1's is 1.0).

If both voters 1 & 3 approve of 2 candidates,the result is a tie between
A and B, so the utility to voter 1 is 0.75, and the utility to voter 3
is only 0.25.

It looks like voter 1 should approve of two candidates, since he won't
be worse off (and gets a more stable outcome, since least-favorite C is
guaranteed to lose).  In addition, this gives voter 3 strong incentive
to approve two candidates as well (to increase utility from 0.0 to
0.25).

I can't find any incentive for voter 2 to approve of a 2nd candidate.

So the likely outcome should be:

Voter 1:  AB
Voter 2:  B
Voter 3:  CA

Result:  
Tie between A and B.

Of course if the utility of a 2nd choice is something other than 0.5,
strategies could change.

Anyone else see anything different?
Bart