## Answer to puzzle #?? – "Naive-exaggeration" strategy in IRV

**Puzzle**

Consider this "*n*aive-*e*xaggeration"
voting *s*trategy (NES): rank the two frontrunners A&B (where you prefer A>B)
top & bottom, then rank anybody you consider better than A co-equal top (or if
the rules of the voting system forbid that, then just below A) and rank
anybody worse than B co-equal bottom (or if that forbidden
then just above B)
and finally rank anybody between A&B using honest normalized utility.

**A.**
Show that if >75% of voters use NES strategy, then it is mathematically
impossible, with instant runoff (IRV) voting, for anybody besides A & B to win.
However, if any amount *below* 75% of the voters employ NES strategy, then
it *is* mathemetically possible for an "underdog" C to win (albeit this may be unlikely).

**B.**
Compare this 75% threshhold with the corresponding threshholds for plain plurality
voting and for Condorcet and Bucklin voting.

### Answers

**A.**
In instant runoff voting, if A gets 49.99%, B gets 25%, and C gets 25.01% of the
top-preference votes, then C can win if all of B's votes transfer to C
as second preference. Here (49.99+25=74.99)% of the voters employed NES strategy.

However it is not possible for C to win if over 75% of the voters use NES.
Suppose without loss of generality that A gets at least as many top-preference votes as B.
If so, then either A gets over 50% (hence wins) or both A and B get over 25% each, while
meanwhile C gets below (100-75=25)%, hence C is eliminated and cannot win.

**B.**
With plurality voting the threshhold is 66.67% (more precisely 2/3).
If over 2/3 of plurality voters employ NES, then
one of {A,B} must get over 1/3 of the vote, while meanwhile any underdog C must get
below 1/3 and hence cannot win. But if only 66.66% of the voters use NES, then
it becomes mathematically possible for C to win,
because the votes could be A=33.33%, B=33.33%, C=33.34%.

With Bucklin and Condorcet voting, no such threshhold exists (or the "threshhold"
could be said to be 100%). That is because, even if 99.99% of the voters
employ NES, it is still mathematically possible for the "underdog" C to win
if there is an almost-exact tie between A and B. E.g. with either Bucklin or Condorcet,
if the top-preference votes are A=49.995%, B=49.995%, C=0.01%, and all A & B voters prefer
C as their 2nd choice, then C wins.

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