  We study the problem of selling information to a data-buyer who faces a
decision problem under uncertainty. We consider the classic Bayesian
decision-theoretic model pioneered by [Blackwell, 1951, 1953]. Initially, the
data buyer has only partial information about the payoff-relevant state of the
world. A data seller offers additional information about the state of the
world. The information is revealed through signaling schemes, also referred to
as experiments. In the single-agent setting, any mechanism can be represented
as a menu of experiments. [Bergemann et al., 2018] present a complete
characterization of the revenue-optimal mechanism in a binary state and binary
action environment. By contrast, no characterization is known for the case with
more actions. In this paper, we consider more general environments and study
arguably the simplest mechanism, which only sells the fully informative
experiment. In the environment with binary state and $m\geq 3$ actions, we
provide an $O(m)$-approximation to the optimal revenue by selling only the
fully informative experiment and show that the approximation ratio is tight up
to an absolute constant factor. An important corollary of our lower bound is
that the size of the optimal menu must grow at least linearly in the number of
available actions, so no universal upper bound exists for the size of the
optimal menu in the general single-dimensional setting.
  For multi-dimensional environments, we prove that even in arguably the
simplest matching utility environment with 3 states and 3 actions, the ratio
between the optimal revenue and the revenue by selling only the fully
informative experiment can grow immediately to a polynomial of the number of
agent types. Nonetheless, if the distribution is uniform, we show that selling
only the fully informative experiment is indeed the optimal mechanism.
