In the setting where we ask participants multiple similar possibly subjective multi-choice questions (e.g. Do you like Bulbasaur? Y/N; do you like Squirtle? Y/N), peer prediction aims to design mechanisms that encourage honest feedback without verification. A series of works have successfully designed multi-task peer prediction mechanisms where reporting truthfully is better than any other strategy (dominantly truthful), while they require an infinite number of tasks. A recent work proposes the first multi-task peer prediction mechanism, Determinant Mutual Information (DMI)-Mechanism, where not only is dominantly truthful but also works for a finite number of tasks (practical). However, few works consider how to optimize the multi-task peer prediction mechanisms. In addition to the definition of optimization goal, the biggest challenge is we do not have space for optimization since there is only a single practical and dominantly truthful mechanism. This work addresses this problem by proposing a tractable effort incentive optimization goal and generalizing DMI-Mechanism to a new family of practical, dominantly truthful mechanisms, Volume Mutual Information (VMI)-Mechanisms. We show that DMI-Mechanism may not be optimal. But we can construct a sequence of VMI-Mechanisms that are approximately optimal. The main technical tool is a novel family of mutual information measures, Volume Mutual Information, which generalizes Determinant Mutual Information. We construct VMI by a simple geometric idea: we measure how informative a distribution is by measuring the volume of distributions that is less informative than it (inappropriately, its similar to measuring how clever a person is by counting the number of people that are less clever than he/she).
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