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The Agreement Theorem Aumann (1976 Ann. Stat. 4, 1236–1239. (doi:10.1214/aos/1176343654)) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot ‘agree to disagree’. This result applies in the classical domain where classical probability theory applies. But in non-classical domains, such as the quantum world, classical probability theory does not apply. Inspired principally by their use in quantum mechanics, we employ signed probabilities to investigate the epistemics of the non-classical world. We find that here, too, it cannot be common knowledge that two agents assign different probabilities to an event of interest. However, in a non-classical domain, unlike the classical case, it can be common certainty that two agents assign different probabilities to an event of interest. Finally, in a non-classical domain, it cannot be common certainty that two agents assign different probabilities, if communication of their common certainty is possible—even if communication does not take place.
This article is part of the theme issue ‘Quantum contextuality, causality and freedom of choice’.

We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. In contrast to other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players’ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, such stable coalition structures support Gamson’s theory of coalition formation (Gamson, 1961).

Gamson-Shapley Laws
(2023)

We consider a set of empirical assumptions formulated by Gamson (1961), namely, Gamson’s Laws, which remain at the heart of government formation forecast in parliamentary systems. While the critical resource postulated in Gamson’s approach is the proportion of votes received by each party, other versions of Gamson’s Laws can be defined by a different choice of critical resource. We model coalition formation as a cooperative game, and provide axiomatic foundations for a version of Gamson’s Laws in which the critical resource is identified with strategic influence, as measured by the Shapley value. We compare the empirical accuracy of the resulting Gamson–Shapley theory against the original Gamson’s Laws in a panel of 33 parliamentary elections, and find that it leads to significantly more accurate predictions of both coalition structure and power distribution. Finally, we propose an extension of the Gamson–Shapley approach which also incorporates information about policy distance among coalition partners. In particular, we discuss the advantages of the extended approach in the context of the German elections in 1987 and 2017.

The states of the qubit, the basic unit of quantum information, are 2×2 positive semi-definite Hermitian matrices with trace 1. We contribute to the program to axiomatize quantum mechanics by characterizing these states in terms of an entropic uncertainty principle formulated on an eight-point phase space. We do this by employing Rényi entropy (a generalization of Shannon entropy) suitably defined for the signed phase-space probability distributions that arise in representing quantum states.

Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the postquantum theories that are consistent with experimentally observed results. We explore this question in the context of epistemics, and ask whether agreement between observers can serve as a physical principle that must hold for any theory of the world. Aumann’s seminal Agreement Theorem states that two observers (of classical systems) cannot agree to disagree. We propose an extension of this theorem to no-signaling settings. In particular, we establish an Agreement Theorem for observers of quantum systems, while we construct examples of (postquantum) no-signaling boxes where observers can agree to disagree. The PR box is an extremal instance of this phenomenon. These results make it plausible that agreement between observers might be a physical principle, while they also establish links between the fields of epistemics and quantum information that seem worthy of further exploration.

Is the world quantum? An active research line in quantum foundations is devoted to exploring what constraints can rule out the post-quantum theories that are consistent with experimentally observed results. We explore this question in the context of epistemics, and ask whether agreement between observers can serve as a physical principle that must hold for any theory of the world. The seminal Agreement Theorem by Aumann (Annals of Statistics, 1976) states that two (classical) agents cannot agree to disagree. We examine the extension of this theorem to no-signalling settings. In particular, we establish an Agreement Theorem for quantum agents. We also construct examples of (post-quantum) no-signalling boxes where agents can agree to disagree. The PR box is an extremal instance of this phenomenon. These results make it plausible that agreement might be a physical principle, while they also establish links between the fields of epistemics and quantum information that seem worthy of further exploration.

We suggest a new component efficient solution for monotonic TU games with a coalition structure, the conditional Shapley value. Other than other such solutions, it satisfies the null player property. Nevertheless, it accounts for the players‘ outside options in productive components of coalition structures. For all monotonic games, there exist coalition structures that are stable under the conditional Shapley value. For voting games, the stability of coalition structures under the conditional Shapley value supports Gamson‘s theory of coalition formation (Gamson, Am Sociol Rev 26, 1961, 373-382).

Contextual mechanism design
(2019)

We show how quantum entanglement may be able to improve the joint performance of a system of telescopes, cameras, or other sensors which are widely separated in space. The improvement is relative to any observation strategy that uses only classical coordinating devices. Potential application domains include space-based observatories and multi-frequency interferometry.

We describe three robot designs suitable for the unsupervised construction of habitable structures on Mars using locally sourced materials, and discuss the relative advantages and disadvantages associated to each of those designs. The first and most basic type of autonomous robot builder, ARBie, is a stone gatherer. Characterized by very modest energy requirements, ARBie can build modular, cone-shaped structures in dry masonry style out of gathered stones, gravel and clay material collected on location. The second type of robot builder, B-ARBie, is a brick-maker. Equipped with a static press, B-ARBie collects, filters and mixes locally sourced clay and gravel material, then vibro-presses it to produce dry bricks with interlocking profile. With moderate energy requirements, B-ARBie proceeds to build modular, vaulted structures with square or rectangular base. The third type, C-ARBie, is a stone-cutter. Equipped with a stone-cutting tool, C-ARBie has higher energy requirements relative to ARBie or B-ARBie, but can build massive vaulted structures with a variety of large, precisely shaped elements. The three robot builders share the same basic body plan: a six-legged frame with on-board PV cells and control electronics, actuated by compressed CO2 harvested from the Martian atmosphere and stored in liquid form as a rechargeable power source. As a power storage medium liquefied CO2 has energy density comparable to lead-acid batteries, but contrary to the latter can be cycled indefinitely, and is able to operate at Martian temperatures. An important advantage shared by the three robot designs is that their functionality and productivity can be easily tested and optimized on Earth. Over the course of its operational lifetime a small team of autonomous robot builders would have the potential to build a vast grid of interconnected, modular habitats out of locally sourced materials with no further need for external provisions or support.

We study team decision problems where communication is not possible, but coordination among team members can be realized via signals in a shared eironment. We consider a variety of decision problems that differ in what team members know about one another’s actions and knowledge. For each type of decision problem, we iestigate how different assumptions on the available signals affect team performance. Specifically, we consider the cases of perfectly correlated, i.i.d., and exchangeable classical signals, as well as the case of quantum signals. We find that, whereas in perfect-recall trees (Kuhn [15, 1950], [16, 1953]) no type of signal improves performance, in imperfect-recall trees quantum signals may bring an improvement. Isbell [13, 1957] proved that in non-Kuhn trees, classical i.i.d. signals may improve performance. We show that further improvement may be possible by use of classical exchangeable or quantum signals. We include an example of the effect of quantum signals in the context of high-frequency trading.

The Heisenberg uncertainty principle is one of the most famous features of quantum mechanics. However, the non-determinism implied by the Heisenberg uncertainty principle --- together with other prominent aspects of quantum mechanics such as superposition, entanglement, and nonlocality --- poses deep puzzles about the underlying physical reality, even while these same features are at the heart of exciting developments such as quantum cryptography, algorithms, and computing. These puzzles might be resolved if the mathematical structure of quantum mechanics were built up from physically interpretable axioms, but it is not. We propose three physically-based axioms which together characterize the simplest quantum system, namely the qubit. Our starting point is the class of all no-signaling theories. Each such theory can be regarded as a family of empirical models, and we proceed to associate entropies, i.e., measures of information, with these models. To do this, we move to phase space and impose the condition that entropies are real-valued. This requirement, which we call the Information Reality Principle, arises because in order to represent all no-signaling theories (including quantum mechanics itself) in phase space, it is necessary to allow negative probabilities (Wigner [1932]). Our second and third principles take two important features of quantum mechanics and turn them into deliberately chosen physical axioms. One axiom is an Uncertainty Principle, stated in terms of entropy. The other axiom is an Unbiasedness Principle, which requires that whenever there is complete certainty about the outcome of a measurement in one of three mutually orthogonal directions, there must be maximal uncertainty about the outcomes in each of the two other directions.

The Heisenberg uncertainty principle is one of the most famous features of quantum mechanics. However, the non-determinism implied by the Heisenberg uncertainty principle --- together with other prominent aspects of quantum mechanics such as superposition, entanglement, and nonlocality --- poses deep puzzles about the underlying physical reality, even while these same features are at the heart of exciting developments such as quantum cryptography, algorithms, and computing. These puzzles might be resolved if the mathematical structure of quantum mechanics were built up from physically interpretable axioms, but it is not. We propose three physically-based axioms which together characterize the simplest quantum system, namely the qubit. Our starting point is the class of all no-signaling theories. Each such theory can be regarded as a family of empirical models, and we proceed to associate entropies, i.e., measures of information, with these models. To do this, we move to phase space and impose the condition that entropies are real-valued. This requirement, which we call the Information Reality Principle, arises because in order to represent all no-signaling theories (including quantum mechanics itself) in phase space, it is necessary to allow negative probabilities (Wigner [1932]). Our second and third principles take two important features of quantum mechanics and turn them into deliberately chosen physical axioms. One axiom is an Uncertainty Principle, stated in terms of entropy. The other axiom is an Unbiasedness Principle, which requires that whenever there is complete certainty about the outcome of a measurement in one of three mutually orthogonal directions, there must be maximal uncertainty about the outcomes in each of the two other directions._x000D_ <link https://www.hhl.de/en/research/publications/detail-page/?tx_publikationen_pi1%5Buid%5D=5086 _blank external-link-new-window "Opens external link in new window">Updated version</link> (March 2015)

We discuss the possible nature and role of non-physical entanglement, and the classical vs. non-classical interface, in models of human decision-making. We also introduce an experimental setting designed after the double-slit experiment in physics, and discuss how it could be used to discriminate between classical and non-classical interference effects in human decisions.