Rock blunts scissors, scissors cut paper, and paper wraps rock. Both players show a hand at the same moment.
A win is worth 1, a loss minus 1, and the same hand is a draw worth 0.
You are Player A
Watch one round, or choose now.
Player A
oror
You
Player A
?
You
Player B
oror
The other player
Player B
?
The other player
The other player picks at random here, like a coin toss.
The whole game on one grid
Every pair of choices and what each side gets: yours first, then theirs. Each number is a payoff.
Step 0 of 3
The cards are face down. Turn them, then step through which squares are stable.
What it teaches
Best responses can run in a circle, so no choice is safe and the only stable play is to be equally unpredictable across three options rather than two.
The usual mistakeHearing that the equilibrium is uniform and concluding there is no skill in the game, when against human opponents who are not actually random there is a great deal of exploitable pattern.
The checkThe uniform mix is what you play when you have no read on the opponent. It is the floor you cannot be beaten below, not the ceiling of what can be won.
What people actually do
Frequency-dependent strategies cycle instead of settling
The theory says
In a population game with no evolutionarily stable pure strategy, the static analysis gives a mixed equilibrium, and a population at the equilibrium frequencies is at rest, with selection removing any deviation.
People actually
Male side-blotched lizards occur in three morphs standing in a strict rock-paper-scissors relation, and their frequencies in the wild oscillate over a cycle of several years rather than holding at the predicted mixture. The analogous laboratory and simulation result is that populations of adaptive agents orbit the equilibrium without converging to it.
Why is still argued over.
In the real world
Real world
Three throat colours in the side-blotched lizard
Male side-blotched lizards come in three throat colours, each with a different mating strategy, and each strategy beats one of the others and loses to the remaining one. Sinervo and Lively's field work found that the relative numbers of the three morphs cycle rather than settle, because whichever morph becomes common is the one its own predator-in-kind exploits. No morph is fit in the abstract, only relative to the current mix.
Three strategies whose best responses run in a circle, so there is no safe choice and no stable pure outcome, which is what makes it the three-way version rather than a two-sided guessing game.
Fiction: Kaiji
Restricted rock paper scissors on the Espoir
Debtors aboard a gambling ship are each given a hand of cards bearing the three symbols and a small number of stars, and must spend every card in one-off matches against other players, winning or losing a star each time. Within any single match the three symbols stand in the usual circle, so there is nothing to aim for and the only protection is that the opponent cannot tell which card is coming. What drives the plot is that the hands are finite and public knowledge of how many of each symbol remain gradually removes that protection, so the end of the night is no longer a game of chance.
Each round is three options with best responses running in a cycle, so no symbol is safe and the only stable play is to be unreadable across all three rather than across two.
Cyclic population gameA win is worth more than a loss costs, so the cycling converges instead of orbiting forever, and the mixed equilibrium is an evolutionarily stable strategy rather than merely a Nash equilibrium. Not on the wall yet.
A zero-sum game with a saddle pointA saddle point exists, so pure strategies suffice in a game of pure conflict. Not on the wall yet.