Rock, Paper, Scissors
Pure game theory is debatably social science, so here goes.
Inspired by various 5-, 7-, and even 25-move extensions to Rock, Paper, Scissors, my friends and I were wondering whether it's possible to have an even-move version. It doesn't seem possible, at least for 4-RPS. I conjecture that all even-RPS's might be similarly impossible. Here's a very! rough writeup.