Why is a shortlist of six harder to settle than a list of fifty?
Because they are different problems. Fifty places is a browsing problem: you are still finding out what exists, and every hour spent adds information. Six places you have all already accepted is a commitment problem, and more discussion adds nothing — the information ran out weeks ago, and what is left is that nobody wants to be the one who chose.
That is how the same group researches for a month and then stalls for another one. The instrument that solves browsing — read more, compare more — is exactly the wrong instrument for committing, and it is the one people keep reaching for because it is the one that worked last.
What belongs on the wheel — countries, cities or trip types?
Whatever the decision is actually about. If the argument is between two weeks of walking and a week of museums, a wheel of cities answers a question nobody asked. Put trip types on it, spin, and the cities follow from the answer in an afternoon.
Where the shape of the trip is settled and only the place is open, cities are the right unit, and this is where the photographs earn their place: a coastline or a skyline you recognise on sight makes the case for a destination in a way its name cannot. Countries are right when flights and season are the real constraint and the city can be chosen later.
How do you build a wheel you would be happy to lose to?
One rule: nothing goes on the wheel that you would argue against afterwards. If Reykjavik is there because somebody did not want to seem difficult, the wheel is not settling the decision — it is postponing the argument to the day the flights are booked.
So the list is the work, and it is worth doing out loud. Say the six, let anyone strike one without explaining why, and spin what survives. Six acceptable places make a wheel whose every outcome is a good trip; six negotiated ones make a wheel that only decides who has to be disappointed.
What else is this wheel good for?
It is the plain picker wheel with a photograph on each segment. If you would rather the pictures did the whole job and the names came second, the picture wheel is built that way round.
Anything else the group cannot agree on works the same way on the decision wheel. An afternoon with nothing in it is the small version of the same problem, and that is the what to do when bored wheel.