Roshambo, math stuff

Rock, paper, scissors, or roshambo* is a really cool game, and a nice way to add some flavor to a conflict system. Everyone probably knows the game, but I'll describe it for completeness' sake. Two players each choose a "method of attack" independently without seeing the other's choice (usually by doing it quickly at the same time). There are three choices, if both players choose the same attack, it's a draw. otherwise,

Rock crushes Scissors,
Scissors cuts Paper,
Paper covers Rock.

*You can spell it Rochambeau if you want to be utterly pretentious. Or French.

This system of defeats is cool for several reasons. It's completely symmetrical (one other option will beat you, one can be beaten, for every choice) It's "non-transitive". That means that if A beats B and B beats C, A won't necessarily beat C. (Imagine the relationship "greater than". If A is greater than B and B is greater than C, then A is always greater than C. That's a transitive relationship.) Transitive relationships always have a neat hierarchy, but non-transitives can have loops.

These mathematical properties mean that there's no best choice. There is never an easy rule for winning. That's a good recipe for an interesting game.

Another much richer system with some similarities to Roshambo is the Chinese elements. There is a constructive and a destructive cycle which both use five elements:

Constructive Cycle:
Fire makes ashes which are Earth
Earth gives the ore to make Metal
Metal forms Water droplets through condensation
Water feeds Wood
Wood burns to create Fire

Destructive Cycle:
Fire melts Metal
Metal cuts Wood
Wood draws up Earth
Earth dams Water
Water quenches Fire

This system is also completely symmetric and non-transitive, and the mnemonics for the cycles are very suggestive. The five elements are rich with meaning from the traditions of Feng Shui. But it is a bit harder to remember than the three "element" roshambo.

Another thing I might mention, symmetry is not really an advantage. It tends to make the system boring for the users. Only by the symbolism of the elements is a symmetric system made interesting. A slightly unbalanced but still non-transitive system without any dominating element should be more interesting, at least in theory. How about this one? (Called "Undercut", introduced by my hero Douglas Hofstadter).

Undercut:
Two players each choose a number from one to five. The highest number wins the difference between the two choices, unless one player undercuts the other's number by exactly one. Then the undercutting player gets the sum of both numbers. Of course a tie gives no one any points.

This system has no dominating number (any number can be beaten by another number) but a strategy of playing each number 1/5 of the time isn't the best idea-certain numbers have a better payoff and should be played more frequently. Can you guess which number should be played the most?

The system of four card suits with one trump is also surprisingly interesting. The trump is usually restricted to only be played after nothing else can be played legally for a certain user. (the trump would beat any card, so it has to be restricted some way.)

Usually small wrinkles added to a symmetric system can add a surprisingly large amount of strategy, interest, and replay value.

As usual, I'm disappointed by the element magic systems used in most RPGs. it's usually not very meaningful in the game or useful only in a certain place. And the magic systems are usually very boring. Making a system that is interesting and simple enough to remember easily is a great challenge.