The earliest steps of a touchdown polymerase chain reaction cycle have high annealing temperatures. The annealing temperature is decreased in increments for every subsequent set of cycles.  The primer will anneal at the highest temperature which is least-permissive of nonspecific binding that it is able to tolerate.  Thus, the first sequence amplified is the one between the regions of greatest primer specificity; it is most likely that this is the sequence of interest.  These fragments will be further amplified during subsequent rounds at lower temperatures, and will outcompete the nonspecific sequences to which the primers may bind at those lower temperatures.  If the primer initially (during the higher-temperature phases) binds to the sequence of interest, subsequent rounds of polymerase chain reaction can be performed upon the product to further amplify those fragments. Touchdown increases specificity of the reaction at higher temperatures and increases the efficiency towards the end by lowering the annealing temperature.
From a mathematical point of view products of annealing at smaller temperatures are disadvantaged by  for the first annealing in cycle  and the second one in cycle  for .