  In this paper, we study belief elicitation about an uncertain future event,
where the reports will affect a principal's decision. We study two problems
that can arise in this setting: (1) Agents may have an interest in the outcome
of the principal's decision. We show that with intrinsic competing incentives
(an interest in a decision that is internal to an agent) truthfulness cannot be
guaranteed and there is a fundamental tradeoff between how much the principal
allows reports to influence the decision, how much budget the principal has,
and the degree to which a mechanism can be manipulated. Furthermore, we show
that the Quadratic Scoring Rule is worst-case optimal in minimizing the degree
of manipulation. In contrast, we obtain positive results and truthful
mechanisms in a setting where the competing incentives stem instead from a
rational briber who wants to promote a particular decision. We show that the
budget required to achieve this robustness scales with the sum of squares of
the degree to which agent reports can influence the decision. (2) We study the
setting where the future event is only observed conditionally on the decision
taken. We give a category of mechanisms that are truthful when agent beliefs
are independent but fails with dependent beliefs, and show how to resolve this
through a decoupling method.
