Testing which of two distributions deviates more from the uniform distribution I'm running an experiment, and am unsure what statistic I should be using for my key test.
The design:
People are instructed to privately use a random number generator to generate a random number (1-6, 1-10, could be whatever) and then report it, earning a higher monetary payoff for higher random numbers. The random numbers are drawn from a uniform distribution, but you expect that some people will lie and report a higher number than they actually got. I've got a control and a treatment condition where I do a manipulation, and I'd like to see if it influences the level of lying.
Testing if people lie within a condition is super straightforward, and I can just do a Kolmogorov Smirnov test for that.
But, I'm not sure how I should test for the between condition differences. I was considering just testing for a difference between the reported numbers using something like a Mann Whitney U, but I'm not sure if this is most appropriate.  Is there a test geared more specifically to seeing which of two distributions deviates more from the uniform (or whichever) distribution, and would it make sense to use that here instead?
Any help would be appreciated- I feel like I might be other thinking this.
 A: I will assume a standard uniform distribution on the unit interval. You can transform to whatever interval used in the experiment.
A simple idea is to have a simple, one-parameter alternative model, estimate that parameter from your data and use that estimate as a measure of a possible cheat factor. The standard uniform distribution has cdf (cumulative distribution function)
$$ F(u)= u, \quad 0\le u \le 1 $$ introducing a positive parameter $\alpha$ define the alternative$^\S$
$$F_\alpha(u)=u^\alpha, \quad 0 \le u \le 1 $$
Note that $F=F_1$. This family is stochastically increasing in $\alpha$, and it can be recognized as a special case of the beta distribution.
Now use the data to estimate $\alpha$, you can also test the null hypothesis $H_0\colon \alpha=1$, which will be more powerful than a nonparametric test against a general alternative.
$^\S$ We have used the fact that for any cdf $F$, $F^\alpha$ is also a cdf.
A: I am unclear whether you are looking for a test or a measure: your title says "test" but then says "deviates more". One distribution being non-uniform in no way makes a different distribution either uniform or not-uniform, so I proceed thinking you are actually looking for a measure of deviance from uniformity.
You can use a one-sample $\chi^2$ statistic as a measure of deviation of observed discrete counts* from expected counts assuming the distribution is discrete uniform. Expected counts under such a scenario would be obtained by multiplying the total sample size by the probability for each value (which probability is, if you have $k$ discrete values, $\frac{1}{k}$ so expected counts for each value would be $\frac{n}{k}$).
The sample with the larger $\chi^{2}$ statistic has a larger standardized squared deviation of observed from expected uniform values. Of course, you can conduct a $\chi^2$ test if you want to provide evidence that one sample is drawn from a distribution that is not discrete uniform, and you can do this for both samples.
* I am assuming you are talking about discrete uniform variables, since you are giving integer ranges.
