  Internet advertising is a sophisticated game in which the many advertisers
"play" to optimize their return on investment. There are many "targets" for the
advertisements, and each "target" has a collection of games with a potentially
different set of players involved. In this paper, we study the problem of how
advertisers allocate their budget across these "targets". In particular, we
focus on formulating their best response strategy as an optimization problem.
Advertisers have a set of keywords ("targets") and some stochastic information
about the future, namely a probability distribution over scenarios of cost vs
click combinations. This summarizes the potential states of the world assuming
that the strategies of other players are fixed. Then, the best response can be
abstracted as stochastic budget optimization problems to figure out how to
spread a given budget across these keywords to maximize the expected number of
clicks.
  We present the first known non-trivial poly-logarithmic approximation for
these problems as well as the first known hardness results of getting better
than logarithmic approximation ratios in the various parameters involved. We
also identify several special cases of these problems of practical interest,
such as with fixed number of scenarios or with polynomial-sized parameters
related to cost, which are solvable either in polynomial time or with improved
approximation ratios. Stochastic budget optimization with scenarios has
sophisticated technical structure. Our approximation and hardness results come
from relating these problems to a special type of (0/1, bipartite) quadratic
programs inherent in them. Our research answers some open problems raised by
the authors in (Stochastic Models for Budget Optimization in Search-Based
Advertising, Algorithmica, 58 (4), 1022-1044, 2010).
