I wish I could superpose votes on Election Day.

However much I agree with Candidate A about social issues, I dislike his running mate. I lean toward Candidate B’s economic plans and C’s science-funding record, but nobody’s foreign policy impresses me. Must I settle on one candidate? May I not vote

Now you can—at least in theory. Caltech postdoc Ning Bao and I concocted quantum elections in which voters can superpose, entangle, and create probabilistic mixtures of votes.

Previous quantum-voting work has focused on privacy and cryptography. Ning and I channeled quantum game theory. Quantum game theorists ask what happens if players in classical games, such as the Prisoner’s Dilemma, could superpose strategies and share entanglement. Quantization can change the landscape of possible outcomes.

The Prisoner’s Dilemma, for example, concerns two thugs whom the police have arrested and have isolated in separate cells. Each prisoner must decide whether to rat out the other. How much time each serves depends on who, if anyone, confesses. Since neither prisoner knows the other’s decision, each should rat to minimize his or her jail time. But both would serve less time if neither confessed. The prisoners can escape this dilemma using quantum resources.

Introducing superpositions and entanglement into games helps us understand the power of quantum mechanics. Elections involve gameplay; pundits have been feeding off Hilary Clinton’s for months. So superpositions and entanglement merit introduction into elections.

How can you model elections with quantum systems? Though multiple options exist, Ning and I followed two principles: (1) A general quantum process—a preparation procedure, an evolution, and a measurement—should model a quantum election. (2) Quantum elections should remain as true as possible to classical.

Given our quantum voting system, one can violate a quantum analogue of Arrow’s Impossibility Theorem. Arrow’s Theorem, developed by the Nobel-winning economist Kenneth Arrow during the mid-20^{th} century, is a no-go theorem about elections: If a constitution has three innocuous-seeming properties, it’s a dictatorship. Ning and I translated the theorem as faithfully as we knew how into our quantum voting scheme. The result, dubbed the Quantum Arrow Conjecture, rang false.

Superposing (and probabilistically mixing) votes entices me for a reason that science does: I feel ignorant. I read articles and interview political junkies about national defense; but I miss out on evidence and subtleties. I read quantum-physics books and work through papers; but I miss out on known mathematical tools and physical interpretations. Not to mention tools and interpretations that humans haven’t discovered.

Science involves identifying (and diminishing) what humanity doesn’t know. Science frees me to acknowledge my ignorance. I can’t throw all my weight behind Candidate A’s defense policy because I haven’t weighed all the arguments about defense, because I don’t know all the arguments. Believing that I do would violate my job description. How could I not vote for elections that accommodate superpositions?

Though Ning and I identified applications of superpositions and entanglement, more quantum strategies might await discovery. Monogamy of entanglement, discussed elsewhere on this blog, might limit the influence voters exert on each other. Also, we quantized ordinal voting systems (in which each voter ranks candidates, as in “A above C above B”). The quantization of cardinal voting (in which each voter grades the candidates, as in “5 points to A, 3 points to C, 2 points to B”) or another voting scheme might yield more insights.

If you have such insights, drop us a line. Ideally before the presidential smack-down of 2016.

I don’t believe in elections at all, no matter classical or quantum they are.

Please don’t take this criticism personally.

I just don’t think that this mechanism is worth studying at so much higher level.

Congratulations on your new paper! I like the idea of quantum voting and quantum game theory. If it is realized, discrete game theory can be even more interesting – now the number of choices of policy changes from finite to infinite. This will dramatically enrich the content of the discrete game theory and change the choices of optimal policy, which means the presidential candidates have to apply new campaign strategies, and the Wall Street bankers have to reformulate their investment and hedging strategies, etc. Well, my naive understanding … As a summary, it is an interesting idea and I would like to see more of your work in this subject. Also, hope on 2016 you can quantumly vote for your president with the greatest satisfaction on your voting scheme 🙂

Thanks, and I’m glad you liked the approach! We hope that the system admits of extensions and further analysis.

In the first paragraph of the body of the paper, you state that \mathcal{L}(\mathcal{H}) denotes the space of linear positive semidefinite operators defined on the Hilbert space \mathcal{H}. In the following line, you say that Society is associated with a joint state σsoc ∈ \mathcal{P}(\mathcal{H}_1) × . . . × \mathcal{P}(\mathcal{H}_N ) that represents the votes. What is \mathcal{P}? Is it a typo for \mathcal{L}?

Yep. Thanks for letting us know! The paper also contains a typo that involves the word “dictatorship.”

These are very interesting results. There are some classic results (by Barbera if I remember correctly) that gives probabilistic analogs of Arrow’s theorem. When the number of candidates is not too small “random dictatorship” is the only possibility. Maybe this can serve as a good starting point to the quantum case.

Thanks very much for your interest and for the reference! I hadn’t been aware of a Barbera approach to a probabilistic analogue of Arrow’s Theorem, which I could easily have missed. The probabilistic analysis of Arrow’s Theorem on which we relied (Ref. 3, by Mossel) does, however, cite Barbera’s “pivotal voter” proof of Arrow’s Theorem:

Barbera. Pivotal voters: A new proof of arrow’s theorem. Economics Letter, 6:13–16, 1980.

Thanks again for your comments!

Congratulations on the interesting paper. I would just like to mention that there are other “quantum voting” works, and there is a burgeoning field called “quantum interactions” whose goal is to apply the formalism of quantum mechanics to the social sciences. The rationale behind it is that such formalism is very suitable to treat highly contextual systems, which is true both of quantum and social systems. Of particular interest to your paper are the works of Polina Khrennikova, Emmanuel Haven, and Andrei Khrennikov. Khrennikova, for instance, just published a paper on Physica Scripta about describing actual voter preferences in terms of superpositions. Hopefully you find this useful.

Thank you for the references and insights, Acacio!

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