# Find the Run-off Flaw

Steve Eppley seppley at alumni.caltech.edu
Tue Nov 19 15:59:16 PST 1996

```Donald wrote:
>     This is your assignment - "if you care to accept" - find the
>claim one flaw in the following run-off example:
-snip-
>             Ballot One    27A   26B   24C   23D
>             Ballot Two    36A   34B   30C
>             Ballot Three  51A   49B
>
>Show the claim one flaw - prove that this election did not elect
>the majority wish of the voters.
-snip-

Here's another assignment, closely related to Donald's:
The Totalitarian Club is electing its Dictator.  Everyone is
required to vote or be fined.
Only one candidate A is running.  B was denied ballot access
because the previous Dictator judged B is weak and therefore
it's not worth tallying an A vs B pairing.
One voter--guess who--supports A.
The other 99 members detest A and would have voted for B, but
since B is not running they vote for A to avoid being fined.
100: A
Show the flaw: prove from the votes cast that this election did
not elect the majority wish of the voters.

>     The reason I picked this type of run-off is because I want to
>avoid the ranking of candidates so that you will not slip and use
>Condorcet - remember, for this example Condorcet does not exist.

You may change the new assignment's numbers, as long as only one
candidate runs and gets most of the votes.  We wouldn't want you to
slip and allow true democracy -- remember, for this example only one
candidate exists...

Sorry, without providing the voters' preference orderings your
example is incomplete.  It's not a "slip" to analyze the voters'
orderings.   On the contrary, it's a slip not to.  It's your
recurrent hippo logic of ignoring voters' preferences, which you
refuse to explain.  That's the runoff flaw, and I've found it
as you requested.

You can tell in your example that in a race between A and B,
A would get 51 votes and B would get 49.  That's the pairwise logic
by which you and we agree that A is a better winner than B, all else
being equal.

But it would be consistent with your example's numbers that in a
runoff between A and D, D could get 73 votes.  And it doesn't take
4 candidates for Elimination (any runoff method, including IR)
to misfire; 3 is enough: in a runoff between A and C, C could get
these "vote-sums":
26BDCA
12CBDA
9DACB
8DBCA
3DCAB
3DCBA
If you check, you'll see that these voter preferences are consistent
with the Ballot One, Ballot Two, and Ballot Three numbers in
your example, assuming the voters vote in each round for their
favorite of the candidates on the ballot.

Guess what?  D would trounce any of the other candidates in a
2-candidate runoff.  And C would trounce anyone but D.  Here's the
runoff table which shows *all* the runoff results, not just the
"A vs B" pairing to which hippo logic has you straitjacketed:
A    B    C    D
A        51L  36   27
B   49        34   38
C   64L  66L       24
D   73L  62L  76L
(Each number shows how many votes the candidate at the left would
receive vs the candidate above.  The 'L' suffix marks a runoff loss
for the candidate above.)

D is the landslide favorite, yet is eliminated first by Instant Runoff.
Why?  Because to save labor, Instant Runoff doesn't tally info which
can be crucial.

You keep ducking the question which goes to the heart of the IR vs
pairwise debate: why do you think voters' preferences should be
ignored?  To save computer cpu time?
35A
32B
13CAB
20CBA    <--- why ignore these preferences for B over A?

Here's your mission, Donald, if you have the guts to accept it.
Answer this:  Which candidate ought to be elected here, and why?
35ABC
16BA
16BC
33CBA

C decides not to run because he doesn't want A to be elected.
After removing himself from the ballot, the voters above would
35AB
16BA
16B
33BA
B wins in a landslide.

The less things change, the more they remain the same.  By
counting only one vote at a time, ignoring the rest of the voters'
preferences, Instant Runoff still keeps the spoiler dilemma that
reduces the field of candidates.  This would keep the two "big tent"
parties from splintering and maintain the two-party system.

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)

```