We describe a general approach to modeling rational decision-making agents who adopt either quantum or classical mechanics based on the Quantum Bayesian (QBist) approach to quantum theory. With the additional ingredient of a scheme by which the properties of one agent may influence another, we arrive at a flexible framework for treating multiple interacting quantum and classical Bayesian agents. We present simulations in several settings to illustrate our construction: quantum and classical agents receiving signals from an exogenous source, two interacting classical agents, two interacting quantum agents, and interactions between classical and quantum agents. A consistent treatment of multiple interacting users of quantum theory may allow us to properly interpret existing multi-agent protocols and could suggest new approaches in other areas such as quantum algorithm design.
 Daniela Frauchigerand Renato Renner ``Quantum theory cannot consistently describe the use of itself'' Nature Communications 9, 3711 (2018).
 Veronika Baumannand Časlav Brukner ``Wigner’s friend as a rational agent'' Quantum, probability, logic: the work and influence of Itamar Pitowsky (2020).
 Alexei Yu Kitaev, Alexander H. Shen, and Mikhail N. Vyalyi, ``Classical and Quantum Computation'' American Mathematical Society (2002).
 Myrto Arapinis, Nikolaos Lamprou, Elham Kashefi, and Anna Pappa, ``Definitions and Security of Quantum Electronic Voting'' ACM Transactions on Quantum Computing 2, 1–33 (2021).
 Christopher A. Fuchsand Blake C. Stacey ``QBism: Quantum Theory as a Hero's Handbook'' Proceedings of the International School of Physics "Enrico Fermi": Course 197, Foundations of Quantum Theory 133–202 (2019).
 John B. DeBrota, Christopher A. Fuchs, and Rüdiger Schack, ``Respecting One’s Fellow: QBism’s Analysis of Wigner’s Friend'' Foundations of Physics (2020).
 Bruno de Finetti ``La prévision: ses lois logiques, ses sources subjectives'' Annales de l'Institut Henri Poincaré 7, 1–68 (1937) Reprinted as `Foresight: Its Logical Laws, Its Subjective Sources' in Breakthroughs in Statistics (S. Kotz and N. L. Johnson, eds.). New York: Springer, 134–174, 1992.
 Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack, ``Subjective probability and quantum certainty'' Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38, 255–274 (2007).
 Robin Blume-Kohoutand Patrick Hayden ``Accurate quantum state estimation via ``Keeping the experimentalist honest'''' (2006).
 R. Balescu ``Equilibrium and Nonequilibrium Statistical Mechanics'' Krieger Publishing Company (1991).
 John B. DeBrota, Christopher A. Fuchs, Jacques L. Pienaar, and Blake C. Stacey, ``Born's rule as a quantum extension of Bayesian coherence'' Phys. Rev. A 104, 022207 (2021).
 Christopher Ferrieand Joseph Emerson ``Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations'' Journal of Physics A: Mathematical and Theoretical 41, 352001 (2008).
 Christopher Ferrie, Ryan Morris, and Joseph Emerson, ``Necessity of negativity in quantum theory'' Physical Review A 82, 044103 (2010).
 John B. DeBrota, Christopher A. Fuchs, and Blake C. Stacey, ``The Varieties of Minimal Tomographically Complete Measurements'' International Journal of Quantum Information 9, 445–507 (2020).
 M. A. Nielsenand I. Chuang ``Quantum Computation and Quantum Information'' Cambridge University Press (2010).
 Mark Wilde ``Quantum information theory'' Cambridge University Press (2013).
 John B. DeBrotaand Blake C. Stacey ``Lüders channels and the existence of symmetric-informationally-complete measurements'' Physical Review A 100, 062327 (2019).
 John B. DeBrota, Christopher A. Fuchs, and Blake C. Stacey, ``Symmetric Informationally Complete measurements identify the irreducible difference between classical and quantum systems'' Physical Review Research 2, 013074 (2020).
 John B. DeBrotaand Blake C. Stacey ``Discrete Wigner functions from informationally complete quantum measurements'' Physical Review A 102, 032221 (2020).
 G. Zauner ``Quantendesigns. Grundzüge einer nichtkommutativen Design-theorie'' thesis (1999) Published in English translation: G. Zauner, "Quantum designs: foundations of a noncommutative design theory," International Journal of Quantum Information, vol. 9, pp. 445–508, 2011.
 J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, ``Symmetric Informationally Complete Quantum Measurements'' Journal of Mathematical Physics 45, 2171–2180 (2004).
 Howard Barnum ``Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases'' (2002).
 Markus Grassl (2021) In preparation.
 Robin Blume-Kohout ``Optimal, reliable estimation of quantum states'' New Journal of Physics 12, 043034 (2010).
 R. L. Hudsonand G. R. Moody ``Locally normal symmetric states and an analogue of de Finetti's theorem'' Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 33, 343–351 (1976).
 Carlton M. Caves, Christopher A. Fuchs, and Rüdiger Schack, ``Unknown quantum states: The quantum de Finetti representation'' Journal of Mathematical Physics 43, 4537–4559 (2002).
 Herbert A. Simon ``Administrative behavior: a study of decision-making processes in administrative organizations'' Free Press (1997).
 Richard H. Thalerand Cass R. Sunstein ``Nudge: improving decisions about health, wealth, and happiness'' Penguin Books (2009).
 Amos Tverskyand Daniel Kahneman ``Advances in prospect theory: Cumulative representation of uncertainty'' Journal of Risk and Uncertainty 5, 297–323 (1992).
 Michael Nielsenand Rush T. Stewart ``Persistent Disagreement and Polarization in a Bayesian Setting'' The British Journal for the Philosophy of Science 72, 51–78 (2021).
 V.I. Yukalovand D. Sornette ``Quantum decision theory as quantum theory of measurement'' Physics Letters A 372, 6867–6871 (2008).
 Andrei Khrennikov ``Quantum Bayesianism as the basis of general theory of decision-making'' Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 374, 20150245 (2016).
This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.