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Quantum Information Elicitation

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  • Rafael Frongillo

Abstract

In the classic scoring rule setting, a principal incentivizes an agent to truthfully report their probabilistic belief about some future outcome. This paper addresses the situation when this private belief, rather than a classical probability distribution, is instead a quantum mixed state. In the resulting quantum scoring rule setting, the principal chooses both a scoring function and a measurement function, and the agent responds with their reported density matrix. Several characterizations of quantum scoring rules are presented, which reveal a familiar structure based on convex analysis. Spectral scores, where the measurement function is given by the spectral decomposition of the reported density matrix, have particularly elegant structure and connect to quantum information theory. Turning to property elicitation, eigenvectors of the belief are elicitable, whereas eigenvalues and entropy have maximal elicitation complexity. The paper concludes with a discussion of other quantum information elicitation settings and connections to the literature.

Suggested Citation

  • Rafael Frongillo, 2022. "Quantum Information Elicitation," Papers 2203.07469, arXiv.org.
    • Handle: RePEc:arx:papers:2203.07469
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    References listed on IDEAS

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    1. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    2. Rafael Frongillo & Ian A. Kash, 2015. "Elicitation Complexity of Statistical Properties," Papers 1506.07212, arXiv.org, revised Aug 2020.
    3. Robin Hanson, 2007. "Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation," Journal of Prediction Markets, University of Buckingham Press, vol. 1(1), pages 3-15, February.
    4. Lambert, Nicolas S. & Langford, John & Wortman Vaughan, Jennifer & Chen, Yiling & Reeves, Daniel M. & Shoham, Yoav & Pennock, David M., 2015. "An axiomatic characterization of wagering mechanisms," Journal of Economic Theory, Elsevier, vol. 156(C), pages 389-416.
    5. Frongillo, Rafael M. & Kash, Ian A., 2021. "General truthfulness characterizations via convex analysis," Games and Economic Behavior, Elsevier, vol. 130(C), pages 636-662.
    6. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    7. Robin Hanson, 2003. "Combinatorial Information Market Design," Information Systems Frontiers, Springer, vol. 5(1), pages 107-119, January.
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