We know that, in a pinch, you can use someone else's wand. But can you use nobody's wand?

Maybe the wands are intelligent enough that they don't fully activate (although they will still do testing waves) until after they choose a wizard, but stupid enough that they'll let anybody use them once they've been activated.

We also need a rule to deactivate the wand after its wizard dies, although a tradition of burying a wizard with their wand might be enough. There could be a black market in used wands, but as long as it's small, most people will still only have one.