Abstract

Proper scoring rules, particularly when used as the basis for a prediction market, are powerful tools for eliciting and aggregating beliefs about events such as the likely outcome of an election or sporting event. Such scoring rules incentivize a single agent to reveal her true beliefs about the event. Othman and Sandholm [16] introduced the idea of a decision rule to examine these problems in contexts where the information being elicited is conditional on some decision alternatives. For example, \What is the probability having ten million viewers if we choose to air new television show X? What if we choose Y?” Since only one show can actually air in a slot, only the results under the chosen alternative can ever be observed. Othman and Sandholm developed proper scoring rules (and thus decision markets) for a single, deterministic decision rule: always select the the action with the greatest probability of success. In this work we significantly generalize their results, developing scoring rules for other deterministic decision rules, randomized decision rules, and situations where there may be more than two outcomes (e.g. less than a million viewers, more than one but less than ten, or more than ten million).