Cycle analysis is basically drawing out a graph (nodes and edges) of what beats what. For standard Rock-paper-scissors you get a graph that looks like this:

ROCK -------> SCISSORS -------> PAPER

^-----------------------------------------------|

In systems that aren't balanced like the sword and armor problem here you can use it to decide what choice to make by giving each node a probability value based on how many people in game use that and then the best choice would be the node with the greatest (sum of probabilities on outgoing nodes) - (sum of probabilities on incoming nodes).