  AI systems often rely on two key components: a specified goal or reward
function and an optimization algorithm to compute the optimal behavior for that
goal. This approach is intended to provide value for a principal: the user on
whose behalf the agent acts. The objectives given to these agents often refer
to a partial specification of the principal's goals. We consider the cost of
this incompleteness by analyzing a model of a principal and an agent in a
resource constrained world where the $L$ attributes of the state correspond to
different sources of utility for the principal. We assume that the reward
function given to the agent only has support on $J < L$ attributes. The
contributions of our paper are as follows: 1) we propose a novel model of an
incomplete principal-agent problem from artificial intelligence; 2) we provide
necessary and sufficient conditions under which indefinitely optimizing for any
incomplete proxy objective leads to arbitrarily low overall utility; and 3) we
show how modifying the setup to allow reward functions that reference the full
state or allowing the principal to update the proxy objective over time can
lead to higher utility solutions. The results in this paper argue that we
should view the design of reward functions as an interactive and dynamic
process and identifies a theoretical scenario where some degree of
interactivity is desirable.
