How do I express how precise the results of my simulation are? I am working on an UNO Simulator to measure the effectiveness of different strategies on an players win percentage.
I would like to know the proper way of expressing the results of my testing. For example, I ran the simulator using two identical random player strategies, with different sample sizes. For each sample size, I ran the simulator 20 times and calculated the Mean and Standard Deviation for each set of 10 thousand - 5 million sample games.
      |  Mean  |Std Dev |
      +--------+--------+
   10k|0.495390|0.027331|
   50k|0.502642|0.013702|
  100k|0.500470|0.005940|
  200k|0.501116|0.005152|
  500k|0.502088|0.003186|
1,000k|0.502720|0.001989|
5,000k|0.502542|0.000790|

I wanted to use these examples to express how confident I am that later tested strategies are precise to some fixed decimal Win:Loss ratio. 
Can I simply calculate the CI or margin of error (1.96 * Standard Deviation/SQRT(sample size))?
Would this mean that I am 95% confident that the true Win:Loss ratio is +/- (sigma/SQRT(n))?
Is this the proper way of expressing the precision of simulations results, or is their a more accurate way of expressing precision?
 A: How you are calculating confidence intervals is correct, and the interpretation of those intervals is that on repeated simulations, 95% of those simulations will fall into the range of the 95% confidence interval. That is subtly different from your interpretation.
Some notes of caution however, in regards to the reporting of large-scale simulation results like this:


*

*The notion of a confidence interval is built on the idea of theoretical random repetitions of an actual, real-world study. While very useful, for simulation models, you actually have those repetitions. I much prefer a "Simulation Interval", which is the 2.5th and 97.5th percentiles of your results - this gives you a true look at the spread of your results, and does allow you to say things like "95% of the time, the Win:Loss ratio falls into this range".

*For a single simulation study this is less of a problem, but keep in mind that for simulation studies, statistical power is a function of computational power and patience. This means if you're doing comparison-based testing, it's rather trivial to overpower a study, and detect differences statistically that are meaningless when applied to the real world.

