determining a lower bound price of {TAIL ~ a group} of {HEAD ~ bids} of a subtree of the search tree 
determining a lower bound price of { TAIL ~ a group } of { HEAD ~ bids } of a subtree of the search tree
OTH OTH OTH OTH OTH OTH TAIL TAIL TAIL TAIL TAIL TAIL OTH HEAD HEAD HEAD HEAD HEAD OTH OTH OTH OTH OTH OTH OTH