Let's step back. Recall that the preferential ballot that will be used would provide us all the data we need to choose a Condorcet winner, if there is one. The Condorcet winner is the flag that would beat all other flags in a head-to-head match-up. So if my preferred flag, monkey-butt, would beat each of the fern designs and the red peak design in a head-to-head match, monkey-butt would be the Condorcet winner.
It's dead simple to check for a Condorcet winner in the preferential ballot. For each of the pairings, you pretend that all the other options don't exist and see how many voters say A beats B and how many say B beats A. If a majority of voters rank A higher than B in their preference ordering, regardless of how they rank them relative to all the other options, then A beats B. You do that across all the potential options. If some flag would beat each other option, that's the right flag.
Unfortunately, a preferential ballot with sequential elimination of options that receive the fewest first choices can sometimes choose the wrong option. Suppose some flag is everybody's strong second choice but the first choice of few voters. It could be quickly eliminated from the preferential ballot. It isn't necessary that a Condorcet winner exists, or that a preferential elimination would fail to choose an existing Condorcet winner, but it is definitely possible and it is definitely easily avoided. How? Before running the preference ballot, check to see if any option beats all the others and declare it the winner if so.
Even more unfortunately, we'll never know whether we've chosen the wrong flag. The Electoral Commission replied to my request as follows:
Dear Mr Crampton, Thank you for your e-mail of 24 September 2015 regarding the results of the first flag referendum and a request to provide the number of votes that conform to each of the potential preference orderings and publish the aggregates.
Section 38(1)(a) of the New Zealand Flag Referendums Act 2015 (‘the Act’), provides that in the case of the first flag referendum, once all the voting papers have been processed, the Returning Officer must:
(i) calculate the number of first preference votes received for each option; and(ii) count the votes in the manner described in Schedule 4; and
(iii) declare the result of the referendum by giving public notice of the following:(A) the absolute majority of votes determined at the first iteration; and(B) the number of first preference votes received for each option; and(C) the absolute majority of votes determined at each iteration at which an option was successful or excluded; and(D) the number of votes recorded for each option and the number of transferable votes at each iteration at which an option was excluded; and(E) the iteration number at which each option was excluded, where applicable; and(F) the number of informal voting papers; and(G) the outcome of the referendum; and
The results will be published on the Electoral Commission's website: www.elections.govt.nz in accordance with these requirements. The preliminary results will be released after 7pm on 11 December 2015 and the final result after midday on 15 December 2015.
Unfortunately, the Electoral Commission is unable to provide the additional result information that you would like because it is not permitted under the legislation.
I wonder if it's too late to get an emergency change to the referendum legislation.