  In this paper, we investigate the discount allocation problem in social
networks. It has been reported that 40\% of consumers will share an email offer
with their friend and 28\% of consumers will share deals via social media
platforms. What does this mean for a business? Essentially discounts should not
just be treated as short term solutions to attract individual customer,
instead, allocating discounts to a small fraction of users (called seed users)
may trigger a large cascade in a social network. This motivates us to study the
influence maximization discount allocation problem: given a social network and
budget, we need to decide to which initial set users should offer the
discounts, and how much should the discounts be worth. Our goal is to maximize
the number of customers who finally adopt the target product. We investigate
this problem under both non-adaptive and adaptive settings. In the first
setting, we have to commit the set of seed users and corresponding discounts
all at once in advance. In the latter case, the decision process is performed
in a sequential manner, and each seed user that is picked provides the feedback
on the discount, or, in other words, reveals whether or not she will adopt the
discount. We propose a simple greedy policy with an approximation ratio of
$\frac{1}{2}(1 - 1/e)$ in non-adaptive setting. For the significantly more
complex adaptive setting, we propose an adaptive greedy policy with bounded
approximation ratio in terms of expected utility.
