I was reading the below statistics paper the other week and it occurred to me that part of their result might apply to the planning poker process, specifically since planning poker requires a unanimous decision.
Abstract:
http://arxiv.org/abs/1601.00900Preprint:
http://arxiv.org/pdf/1601.00900v1.pdf (the graph on pg3 is particularly interesting)
The application to planning poker comes about by considering it a two-part process: we're both trying to estimate what work needs to be done as part of a story, and also trying to estimate a points value for that work. There is some extra maths needed to make the concept work for multiple hypotheses (planning poker cards) and F-values (different ways to mis-estimate the work needed), but I'll spare you the details.
Their main point doesn't particularly apply to round 1 of planning poker because mis-estimating the work needed doesn't really affect single engineer's accuracy at estimating the points for that work. It does obviously affect what the resulting estimate is, but in our case this error can't be detected by looking at the amount of votes for the majority option as the paper discusses.
In round 2 onwards, there is a bias for minority voters to align with the majority, due to social pressure imposed by requiring a unanimous decision. This means that a unanimous vote has less predictive power in rounds after the first.
Instead of requiring unanimity, I think it would be better to concentrate on the planning poker principle of discussing dissenting opinions after each round.
This is useful in that it forces people to contribute their thoughts on the first part of the process (estimating the work needed) and sharing with the group.
So we've finished planning a task once nobody has any further discussion to add: hopefully this results in a naturally unanimous vote, but if not we could just take the majority.