The No Trade Theorem says that two traders with common prior beliefs will not find a mutally beneficial speculative trade provided they began with a Pareto efficient allocation.  There is in fact a converse.  If the traders do not share a common prior, then they can always find such a trade.

My kids demonstrated this experimentally today in the car coming home from Evanston’s Dixie Kitchen and Bait Shop (Recommended by Barack Obama!)  Two kids have identical rubber alligator swag from the restaurant.  3 year old believes that 6 year old has his alligator and demands a swap.  6 year old insists that all gators are with their rightful owners.  There is common knowledge that they disagree about this and therefore by Aumann’s famous theorem they do not share a common prior.

Dad takes temporary posession of both rubber reptiles.  In plain view of the 6 year old, Dad pretends to switch but doesn’t.  Sleight of hand deceives 3 year old.  Alligators returned to original owners.  Viola, Pareto improvement.

I forgot to get my commission.