By Clay Shentrup & Warren D. Smith

Imagine that my sincere election utilities are

Suppose I believe that, if I alter my vote to "bury the rivals B and C" (as opposed to voting A>B>C>D honestly), that can either

**X.**- Have no winner-altering effect. (The most likely possibility, by far.)
**Y.**-
If I choose to "bury the rivals" that unfortunately might
cause D to win, whereas someone else
(whose expected utility is [10+2+1]/3 = 13/3 = 4.3
assuming equal chances for each of {A,B,C})
would have otherwise won.
My utility loss in this case is
**–4.3**. **Z.**-
If I choose to "bury the rivals" that might work and
cause A to win, whereas someone else (whose expected utility is [2+1+0]/3 = 1
if all three among {B,C,D} are equally likely; but no matter what the likelihoods
the expected utility is at most 2)
would have otherwise won.
My utility gain in this case is somewhere between
**+8**and**+10**.

The expected alteration in value for me
*if I choose to bury*
is
≥8×P(Z) - 4.3×P(Y).
If this is *positive*, then it is strategically wise for me to bury.
If P(Z) and P(Y) are approximately equal – or if I consider Z to be more likely
than Y (or even if I consider Y more likely by a factor of 186%) –
then the correct strategy for me is
to bury the rivals.

Most people who object to DH3 as a criticism of Condorcet voting methods,
claim event Z is
very unlikely because it requires a lot of voters to strategize.
(For example, if 1000 out of 2000 voters needed to strategize to make Z happen
and each one did so with
independent probability≤33.3%, then
the likelihood of Z would be below 10^{-51}.)
Because Z is unlikely (the objectors continue) strategizing is not worth it, so we do not
have to worry about it.

However, Y tends
to require even *more* voters to strategize,
hence by their reasoning is even *less* likely.
(For example, if 1**1**00 out of 2000 voters need to strategize to make Y happen,
and each one does so with
independent probability≤33.3%, then
the likelihood of Y would be below 10^{-30} times the likelihood of Z!)
If so, then 8×P(Z) - 4.3×P(Y) is positive.
Then burying will rationally happen. Then the DH3 pathology
will happen. Note that this happens even if the DH3-objectors were exactly correct.
Indeed, the more-correct they are that strategic Condorcet voting is unlikely, the
*more* justified it becomes for any given voter to strategize.

The most severe form of the DH3 pathology supposes A,B,C are nearly equal in utility to all voters, while D is a lot worse. (What we just analysed was a less-severe form of the DH3 pathology, with numbers altered to make it less severe but more-clearly likely to happen.) Let's now examine this most-severe form.

Imagine that my sincere election utilities are

**X.**- Have no winner-altering effect. (The most likely possibility, by far.)
**Y.**-
If I choose to "bury the rivals" that unfortunately might
cause D to win, whereas someone else
(whose expected utility is [10+9+8]/3 = 27/3 = 9
assuming equal chances for each of {A,B,C})
would have otherwise won.
My utility loss in this case is
**–9**. **Z.**-
If I choose to "bury the rivals" that might work and
cause A to win, whereas someone else (whose expected utility is [9+8+0]/3 = 17/3 = 5.7
if all three among {B,C,D} are equally likely; but no matter what the likelihoods
the expected utility is at most 9)
would have otherwise won.
My utility
*gain*in this case is somewhere between**+1**and**+10**.

The expected alteration in value for me
*got by choosing to bury*
is
≥1×P(Z) - 9×P(Y).
If this is *positive*, then it is strategically wise for me to bury.
If Z is viewed as lots more likely than Y
then burial is a good idea.
(Burying is always a good idea if Z at least 9 times more likely.
Burying is
never a good idea if P(Y)≥1.25×P(Z).
If 0.8×P(Y)≤P(Z)≤9×P(Y)
then burying *might* be a good idea.)