Blogtrek

Rock Paper Scissors

Yesterday (2003 Oct 21) in Aaron Brown's Morning Papers, Aaron presented the Detroit Free Press with a story about the game Stone Paper Scissors. This I remembver well from childhood. Some other boy taught it to me. You put out your hand in the form of a stone, paper, or scissors, and the rule is that stone smashes scissors cuts paper covers stone. I played it a few times but did not think much of it.

Later on, when I was a teenager reading about game theory, I encountered the game in the classic book The Compleat Strategyst, by JD Williams. In the book JD shows how the "three active strategies" method leads to the optimal strategy of selecting each turn completely at random, with 1/3 odds on each of scissors, paper, and stone.

JD also showed some other games that are from the scissors-paper-stone family. These games are characterized by each of two players selecting an object from a set S (with replacement), at the same time without knowledge of what the other player is doing. If they select the same element, the game is a draw. If they select different elements, there is a relation > such that for each pair of elements from S, s and t, either s > t or t > s; one of them "beats" the other. JD describes another form of scissors-paper-stone by adding glass and water to S, and they describe a 7-element example in the form of two medieval damsels who each choose a knight from a set of knights and have these two knights joust with each other.

What is unusual about the article is that it describes the Rock-Paper-Scissors society, which had a recent meeting in the Detroit area. Another article appeared in a Fort Worth newspaper. (this link may go dead in a few days). I can't see how so many people can form an organization about a game whose solution was solved decades ago. How many ways are there of choosing at random an element out of a 3-element set? What I would want to do is to form a theory about all such games. I did some research on some of these, such as variations of the Stratego game (1 beats 2 beats… beats 9 beats spy beats 1, a part of a board game Stratego that was popular a number of years ago) and simple extensions of Rock-paper-scissors, for example, add a hammer and say that scissors cuts a hammer's head off, paper covers a hammer but a hammer smashes a stone to bits. If you solve the game, you get exactly the same strategy as with scissors-paper-stone: pick one of these three randomly and ignore the hammer.

Despite the devotion of some of these people to a game that has been settled years ago, at least in a game-theoretic sense, it was good to be reminded of an old childhood friend.