By Warren D. Smith, 14 Aug 2010
Here's a simple IRV election with 57 voters and 3 candidates named "A", "B", and "C".
#voters  their vote 

18  A>B>C 
24  B>C>A 
15  C>A>B 
Instant runoff eliminates C, then A wins by 33 to 24 over B.
Notation: A vote "Jim>Amy>Bob" means that voter prefers Jim more than Amy more than Bob, and says so as her vote.
FAILURE OF THE SNIFF TEST: First of all, without any analysis at all, who do you think ought to win this election? It sure looks to me like B is the "most correct" and "most democratic" winner. (B has the most top and fewest bottomrankings.) But IRV elects A.
Score voting considers this election an easy call. It would elect B if all voters gave score X to their first choice, Y to their second, and Z to their third, for any scores X≥Y≥Z, not all equal.
IF WE REVERSE all ballots: then A still wins (but now after B eliminated). So the uniquely "best" candidate according to IRV is the same as the uniquely "worst" candidate according to IRV! (FairVote, the proIRV advocacy group, describes this as "IRV elects a majority winner.")
To "reverse" the ballot "Jim>Amy>Bob" we would change it to "Bob>Amy>Jim," i.e. now ranking Jim bottom instead of top.
LESS IS MORE: If 45 BCA voters lower the current loser B, that makes B win under IRV rules (IRV begins by eliminating A).
To "lower" Jim in the ballot "Jim>Amy>Bob" we would change it to "Amy>Jim>Bob," or (to lower Jim even further, all the way to rock bottom) we could change it to "Amy>Bob>Jim."
NO, MORE IS LESS: Raising the current winner A from bottom to top in 10 BCA votes stops A winning (causes C, the new bottomranked guy in all these votes, to win).
YOU POOR SUCKERS VOTED AGAINST YOUR ENEMY? BIG MISTAKE: If 10 BCA voters instead of voting maximallyagainst the hated A, dishonestly vote maximallyfor him (or just refuse to vote at all), that would have prevented A from winning and made their lesserevil C win. (Same thing as the last one, just different viewpoint. Point is, IRV in this election incentivized voters to lie in their votes and made voting honestly be stupid. Rob Richie, head of FairVote, the proIRV advocacy group, described this as "A[n IRV] voter's best strategy is to sincerely rank the candidates.")
IT SURE WASN'T WISE TO VOTE ABOTTOM, YOU CHUMPS SHOULD HAVE STAYED HOME: If we delete 1020 BCA voters, then the old winner A loses (and C wins). So it was very foolish of them to vote Abottom and honestly say they regarded A as the worst choice – that made A win – they should either have refused to vote, or lied in their vote. (FairVote, the proIRV advocacy group, describes this as "IRV eliminates the 'wasted vote' problem.")
IT SURE WAS CLEVER OF THOSE BBOTTOM VOTERS TO STAY HOME: If 426 more CAB voters came, that'd make B win. Good job for them they stayed home and didn't let anybody know they hated B.
YOU PATSIES VOTED FOR YOUR FAVORITE? BIG MISTAKE: 324 BCA voters could have prevented their greatest evil A from winning by dishonestly lowering their favorite B (including in some cases thus making their favorite B win!) .
WHOOWHEE, I'M A KINGMAKER (also known as a SPOILER): After the voters had voted, it was belatedly discovered that B was a convicted criminal and hence ineligible to run. Fortunately, B had not (embarrassingly) won; and since all voters provided their full preference orders for all candidates, this didn't matter – we can now just erase B from all ballots and redo the instant runoff count, which will of course still verify that A won and C finished in last place. Right? Wrong... now C wins. (FairVote, the proIRV advocacy group, describes this as "IRV eliminates the 'spoiler' problem.")
SOME OTHER WAYS TO LOOK AT THAT LAST ONE: B is a spoiler since by the act of running, he altered the winner from C to A; while if B drops out of the race at the last minute, he can alter the winner from A back to C. Also, this can be viewed as a tactical opportunity for Bvoters: instead of honestly voting for their favorite B, they could dishonestly vote for their second choice C. The latter move works out better for them. (And here's another IRV spoiler example.)
PRECINCT TOTALS? DON'T MAKE ME LAUGH: If we split up the country into 3 districts in various ways, e.g.
East:  North:  South:  




Then:
So "obviously" B wins the whole country, right? Wrong – A wins.
You cannot count IRV elections in precincts for this reason, you have to count the whole election centrally. No such thing as a precinct "subtotal." (More discussion.) That is a big change in counting procedures in many places and makes the election more susceptible to a centrally organized fraud.
YOU WANT COMPLEXITY? That was only a 3candidate election with only 3 kinds of voters. Simple. (FairVote describes this as "as simple as 123.") If you want something more complicated, think about how IRV behaves with 4, 5 or 6 candidates and a million voters.
If, on the other hand, you want a bit more simplicity: if you replace the numbers (18,24,15) in the election table we started with by (12,17,11) or (9,12,8) then you should still get all the same paradoxes (with the numbers appropriately changed) but with only 40 or 29 voters instead of 57. Readers may enjoy working those elections out for themselves.