Agreement and disagreement in a non-classical world.

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'.

Bad Temptation.

We study a static self-control model in which an agent's preference, temptation ranking, and cost of self-control drive her choices among a finite set of options. We show that it is without loss to assume that the agent's temptation ranking is the opposite of her preference. We characterize the model by relaxing the Weak Axiom of Revealed Preference (WARP), and exploit WARP violations to identify the model's parameters.

Sensitivity of reaction time to the magnitude of rewards reveals the cost-structure of time.

The Drift-Diffusion Model (DDM) is the prevalent computational model of the speed-accuracy trade-off in decision making. The DDM provides an explanation of behavior by optimally balancing reaction times and error rates. However, when applied to value-based decision making, the DDM makes the stark prediction that reaction times depend only on the relative utility difference between the options and not on absolute utility magnitudes. This prediction runs counter to evidence that reaction times decrease with higher utility magnitude. Here, we ask if and how it could be optimal for reaction times to show this observed pattern. We study an algorithmic framework that balances the cost of delaying rewards against the utility of obtained rewards. We find that the functional form of the cost of delay plays a key role, with the empirically observed pattern becoming optimal under multiplicative discounting. We add to the empirical literature by testing whether utility magnitude affects reaction times using a novel methodology that does not rely on functional form assumptions for the subjects' utilities. Our results advance the understanding of how and why reaction times are sensitive to the magnitude of rewards.

Axioms for the Boltzmann Distribution.

A fundamental postulate of statistical mechanics is that all microstates in an isolated system are equally probable. This postulate, which goes back to Boltzmann, has often been criticized for not having a clear physical foundation. In this note, we provide a derivation of the canonical (Boltzmann) distribution that avoids this postulate. In its place, we impose two axioms with physical interpretations. The first axiom (thermal equilibrium) ensures that, as our system of interest comes into contact with different heat baths, the ranking of states of the system by probability is unchanged. Physically, this axiom is a statement that in thermal equilibrium, population inversions do not arise. The second axiom (energy exchange) requires that, for any heat bath and any probability distribution on states, there is a universe consisting of a system and heat bath that can achieve this distribution. Physically, this axiom is a statement that energy flows between system and heat bath are unrestricted. We show that our two axioms identify the Boltzmann distribution.

Choice-theoretic foundations of the divisive normalization model.

Recent advances in neuroscience suggest that a utility-like calculation is involved in how the brain makes choices, and that this calculation may use a computation known as divisive normalization. While this tells us how the brain makes choices, it is not immediately evident why the brain uses this computation or exactly what behavior is consistent with it. In this paper, we address both of these questions by proving a three-way equivalence theorem between the normalization model, an information-processing model, and an axiomatic characterization. The information-processing model views behavior as optimally balancing the expected value of the chosen object against the entropic cost of reducing stochasticity in choice. This provides an optimality rationale for why the brain may have evolved to use normalization-type models. The axiomatic characterization gives a set of testable behavioral statements equivalent to the normalization model. This answers what behavior arises from normalization. Our equivalence result unifies these three models into a single theory that answers the "how", "why", and "what" of choice behavior.