A Voting Cycle

Three families agree to pick a New Year's Day menu by majority vote. Each ranks the three entrees, and each puts a different one first, so no first-choice vote settles anything. Comparing the choices two at a time looks like a way out, and it is where the trouble starts.

Turkey beats roast beef two to one. Roast beef beats lasagna two to one. That seems to settle it, until lasagna beats turkey two to one as well. Every option beats one rival and loses to another, so the winner is decided by the order in which the votes are taken rather than by what the majority wants. The slider changes the Alexander family's ranking; most of the six orderings break the cycle and leave one entree that beats both others, and the panel says which.