Rock Paper Scissors
https://www.quantamagazine.org/the-game-theory-math-behind-rock-paper-scissors-20180402/
The game Rock Paper Scissors is played with two players who each have three strategies. They can either make a fist with their hand (rock), put up two fingers (scissors), or lay their hand out flat (paper). Each player’s strategy depends on what the other player will do, as Rock beats Scissors, Scissors beats Paper, and Paper beats Rock. If a player changes their strategy throughout the game (assuming they continue to play beyond just a few rounds), it will not be helpful unless the alternate player also changes their initial strategy.
The article explains why there cannot be a pure strategy for either player, because if they always choose rock, for example, their opponent will always choose paper. If this continues to happen, the player who is playing rock will switch to scissors to beat paper, and eventually their opponent will switch to rock to beat scissors. This creates an endless chase of strategies. As for a mixed strategy, this may help the score when someone changes their mixed strategy (for example, playing rock ¼ of the time, playing paper ½ of the time, and playing scissors ¼ of the time). If a player alters how often they play each option, their opponent will respond accordingly. In order to reach Nash equilibrium in Rock Paper Scissors, Player 1 switches from (⅓, ⅓, ⅓) to a strategy that is imbalanced, such as (¼, ½, ¼). Player 2 will follow Player 1’s lead, and also change their strategy. Once they both alter their strategies, the game will reach Nash equilibrium. The article then explains how these outcomes would be different if the scoring system was not public knowledge. It then ends by saying that some games are harder to naturally reach Nash equilibrium for than others, Rock Paper Scissors included, as it is very intricate.