Boy, this discussion is really heating up!  (Groan!)

  I trust my previous summation, to which I have privately received a
favourable confirmation by someone with practical experience, fully
clears up the matter of PWM including the corollary that grounding the
device during an "off" period would not only prevent the cooler from
working, but result in its rapid demise.

  The matter of stacking coolers is fascinating.  They must be stacked
in a "pyramid" with heat-distributing plena between ("heat pipes" come
to mind for this application), where successive lower layers have a
multiple of units inversely proportional to the efficiency of the units.

  If for example, the efficiency of the device is 25%, then you will
require four devices in the second layer; 16 in the third etc.  At this
rate, overall efficiency drops precipitously (exponentially to the
number of layers) and it is easy to see why cascaded Peltiers are rarely
used and *not* for primary cryo-cooling!

  Even in a cascade however, the optimal performance and temperature
differential presumably relate directly to current, and for the same
device, each of the units in the same layer as well as each layer would
best be electrically wired, all in series.

  In most applications, Peltiers are used purely for the convenience of
servo-control of temperature.  For this purpose, it is often convenient
to use another system (conventional evaporative/ condensation
refrigeration cycle using mechanical compressor) as the primary source
of "cool" and the Peltier as the vernier.

  Whether or not there is room for improvement of efficiency as was
hoped some 20 or 30 years ago to the extent that they could rival the
refrigerant compressor, at present they appear to be neither efficient
nor cheap.
--
  Cheers,
        Paul B.