Consider the following scenario: control group is 50% female while the participant sample is 60% female. Also, assume that this discrepancy is solely a matter of chance and that the effect of the experiment varies by gender. To estimate the effect of the experiment, one needs to adjust for the discrepancy, which can be done via matching, regression, etc.
If the effect of the experiment depends on the nature of the participant pool, such adjustments won’t be enough. Part of the effect of Deliberative Polls is a consequence of the pool of respondents. It is expected that the pool matters only in small group deliberation. Given people are randomly assigned to small groups, one can exploit the natural variation across groups to estimate how say proportion females in a group impacts attitudes (dependent variable of interest). If that relationship is minimal, no adjustments outside the usual are needed. If, however, there is a strong relationship, one may want to adjust as follows: predict attitudes under simulated groups from a weighted sample, with the probability of selection proportional to the weight. This will give us a distribution — which is correct— as women may be allocated in a variety of ways to small groups.
There are many caveats, beginning with limitations of data in estimating the impact of group characteristics on individual attitudes, especially if effects are heterogeneous. Where proportions of subgroups are somewhat small, inadequate variation across small groups can result.
This procedure can be generalized to a variety of cases where the effect is determined by the participant pool except where each participant interacts with the entire sample (or a large proportion of it). Reliability of the generalization will depend on getting good estimates.