12 Decision Theory

The field of decision theory is vast and there are many examples of real-world
problems to choose from. We present an example from viral marketing, a small
snippet from the epinions.com database. Viral marketing is
based on the premise that members of a social network
influence each otherâ€™s purchasing decisions. The
goal is then to select the best set of people to market
to, such that the overall profit is maximized by propagation
of influence through the network. We modeled
this problem using the state predicates `Buys(x)` and
`Trusts(x1, x2)`, and the action predicate `MarketTo(x)`.
The utility function is represented by the unit clauses
`Buys(x)` (with positive utility, representing profits from
sales) and `MarketTo(x)` (with negative utility, representing
the cost of marketing). The topology of the
social network is specified by an evidence database of
`Trusts(x1, x2)` atoms.
The core of the model consists of two formulas:

0.6 Buys(x1) ^ Trusts(x2, x1) => Buys(x2) 0.8 MarketTo(x) => Buys(x)

The weight of Formula 1 represents how strongly x1 influences x2, and the weight of Formula 2 represents how strongly users are influenced by marketing. In addition, the model includes the unit clause:

-2:20 Buys(x)

with a utility of 20 (see above). The negative weight represents the fact that most users do not buy most products. In addition, we include the action predicate:

:-1 MarketTo(x)

We perform maximum expected utility (MEU) inference with the following command:

infer -i marketing.mln -e trust.db -r marketing.result -q MarketTo -ow Buys -decision

The output contains the maximum utility achieved and the optimal assignment of action atoms.