To insure against the costs of failure, research managers often initiate se
veral more-or-less independent research projects with the same target. But
then they face the problem of how many parallel "teams'' to fund in order t
o maximize the probability of a timely breakthrough. If there are too many
teams, resources are stretched too thin, but if there are too few teams, an
opportunity for exceptional achievement may be missed. Several possible pr
oject objectives are identified, including maximizing the achievement of th
e best team and maximizing the probability that at least one team will atta
in a threshold. General optimization problems based on these goals are then
formulated in fixed, uncertain, and competitive environments. These proble
ms are solved analytically and numerically for achievement distributions in
a family based on the exponential distribution. The applicability of these
solutions to funding R & D, and to other problems such as supporting athle
tes, is then discussed.