Rock-Paper-Scissors? A Fair Game?
The majority of people have played the game “Rock-Paper-Scissors”. Rock-Paper-Scissors originated from China and is a hand game between two players. Each player has three strategies at their disposal: playing rock, playing paper, or playing scissors. The core idea behind this game is that rock beats scissors, scissors beat paper, and paper beats rock. So when both players raise their hands, they can either lose, win, or tie by using the same hand gesture. In theory, this game should be fair no matter the strategy chosen as there is no best response to any option.
To do this we can use our knowledge of nash equilibrium, let us assume that winning a match is equal to 1, tying a match is 0, and losing a match is -1. When testing player 1’s strategies against player 2’s, the outcome leads to there being no pure nash equilibrium. Now let’s test for mixed nash equilibrium.
Player 1:
(0)q(rock) + (-1)q(paper) + (1)q(scissors)
(1)q(rock) + (0)q(paper) + (-1)q(scissors)
(-1)q(rock) + (1)q(paper) + (0)q(scissors)
Knowing that the probability of all the strategies is equal to 1, you can determine that the mixed nash equilibrium is (⅓, ⅓, ⅓ ). This means that mathematically, there is no best option when it comes to playing rock-paper-scissor. No matter how often you play rock-paper-scissors, if you randomly choose a strategy, there is a ⅓ chance that you win.
However, when playing this game in actuality, there is a pattern in which a player can have the advantage. In a study done at China’s Zhejiang University by Zhijian Wang, 360 students were placed in groups of 6 and played 300 consecutive games of rock-paper-scissors. Each student was given prize money in proportion to how many games they won. It was found that players who win their game with a certain hand gesture tend to stick with that choice in the next round, this is called “conditional response”. Humans naturally want to follow a plan that has previously won before, so there becomes a higher probability they pick the same strategy, some call this the “win-stay lose-shift” strategy. This can also be said for the opposite, where when the player loses they naturally don’t want to stick to the same strategy. Wang suggests that two consecutive events are correlated to each other. Wang also suggests that when players do switch, “counter-clockwise cycling”, they move from rock to paper, from paper to scissor, and from scissor to rock. Wang says that following the “conditional response” rule gives a small advantage over nash equilibrium. While we are taught there are “best responses” to certain situations, nash equilibrium doesn’t always lead to the most optimal outcome.
https://www.nature.com/articles/srep05830