It may seem like stating the obvious, but when we play Scissors, Rock, Paper (known by a hilarious variety of names in other languages, such as Schnick-Schnack-Schnuck in German), we are playing one of the fairest games ever invented. It is fair because the outcome is entirely determined at the moment of play, and any attempt to pervert that fairness is both easy to detect and almost impossible to achieve. In reality, the names of the three items do not matter. They could be called Fred, Bill, and Mary. As long as we have the system A beats B, B beats C, and C beats A, we have the game.
Game theory tells us that in nearly every game, playing the “long game” produces a distinctly different outcome to any individual round. Sometimes you lose the battle to win the war. But Schnick-Schnack-Schnuck defies being “gamed.” The chances of being certain of a win over several rounds diminish to negligible within three rounds and approach zero after ten rounds. Scissors–Rock–Paper, it seems, is impervious to corruption.
Why not just flip a coin? Certainly, this is an option. But coin flipping has an inherent problem: someone has to call, and the selection of that person is arbitrary. The other player must accept the opposite. You cannot both choose heads. With a coin, the only strategy available to influence the outcome is to change your call when it is your turn to call. If the nomination of the caller is manipulated, one player must always accept an outcome they did not control.
But Scissors–Rock–Paper, simply by adding one more element, introduces a level of fairness because both players choose simultaneously. Interestingly, if you add an extra element—say “Gun”—in order to maintain fairness, each element must defeat two other elements. Added complexity does not create fairness unless it is structured carefully.
Game theory shows that perfectly balanced, non-transitive expansions are easiest to achieve with an odd number of strategies (5, 7, 9, etc.). The most famous expansion is the Five-Element Game, which adds Lizard and Spock. In this system, every element defeats two elements and is defeated by two elements, guaranteeing a stable, balanced structure where the Nash equilibrium is once again a mixed strategy in which every player chooses all five options with equal probability (20% each). Rock beats Scissors and Lizard. Paper beats Rock and Spock. Scissors beat Paper and Lizard. Lizard beats Paper and Spock. Spock beats Rock and Scissors. This symmetry is necessary to keep the game theoretically fair and unpredictable.
Sadly, as the number of elements increases, maintaining fairness requires an increasing number of rounds, because the number of possible draws increases unless the number of defeats-per-element increases accordingly. The comparison matrix grows at a squared rate, so a nine-element game requires an 81-cell matrix to show all pairings.
Now, if you think this is all very interesting but not very useful, consider that in nature, this type of game is important for maintaining species diversity. It prevents one species from dominating and producing a monoculture. Three-element non-transitive arrangements are surprisingly common, even in higher-order mammals. Biodiversity increases with the complexity of the “game,” but because nature tends toward efficiency, the three-element structure often prevails.