Non-transitive gaming elements, including a first, a second and a third one or more gaming elements, respectively, wherein the second provides an advantage represented by higher scores, on average, as compared with the scores of the first, the third provides an advantage represented by higher scores, on average, as compared with the scores of the second and the first provides an advantage represented by higher scores, on average, as compared with the scores of the third, and wherein each of the three respective advantages are all less than 3 percent and each advantage does not differ from the mean of the three advantages by more than 20 percent.