For a fair game successive plays should be independent. It sounds like they are asking you to perform a test that consecutive results are uncorrelated.  You could do this by pairing the data let $R_1, R_2,...,R_{2n}$ be the first $2n$ results.  Then you can form $n$ distnct pairs
$(R_1,R_2)$, $(R3, R4),...,(R_{2n-1}, R_{2n})$.  Calculate the pearson correlation coefficient is different from zero (if the data is continuous or even a set of integers).  If the data are $0/1$ for lose/win  you can test for independence in the $2\times2$ table obtained by using the counts for $(0,0), (0,1),(1,0)$ and $(1,1)$.  In this case of $0/1$ the runs test of Wald and Wofowitz suggested above could also be used.  The way it is described in the rule it sounds like they want you to construct a confidence interval for the correlation with halfwidth equal to $3\sigma$.  You would pass if $0$ is contained in the interval.  These tests seem to be a little too easy to pass though.