arXiv subjects

Peter J. Hammond

Publications and source records attributed to Peter J. Hammond.

2 recordsLinked to original sources

arXiv2026

Bounded Rationality with Subjective Evaluations in Enlivened but Truncated Decision Trees

In normative models a decision-maker is usually assumed to be Bayesian rational, and so to maximize subjective expected utility, within a complete and correctly specified decision model. Following the discussion in Hammond (2007) of Schumpeter's (1911, 1934) concept of entrepreneurship, as well as Shackle's (1953) concept of potential surprise, we consider enlivened decision trees whose growth over time cannot be accurately modelled in full detail. An enlivened decision tree involves more severe limitations than a mis-specified model, unforeseen contingencies, or unawareness, all of which are typically modelled with reference to a universal state space large enough to encompass any decision model that an agent may consider. We consider a motivating example based on Homer's classic tale of Odysseus and the Sirens. Though our novel framework transcends standard notions of risk or uncertainty, for finite decision trees that may be truncated because of bounded rationality, an extended and refined form of Bayesian rationality is still possible, with real-valued subjective evaluations instead of consequences attached to terminal nodes where truncations occur. Moreover, these subjective evaluations underlie, for example, the kind of Monte Carlo tree search algorithm used by recent chess-playing software packages. They may also help rationalize the contentious precautionary principle.

econ.TH↗

arXiv2025

Quantum Measurement Trees, II: Quantum Observables as Ortho-Measurable Functions and Density Matrices as Ortho-Probability Measures

Given a quantum state in the finite-dimensional Hilbert space $ \C^n $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with: (i) an ``ortho-measurable'' function defined on the Boolean ``ortho-algebra'' generated by the eigenspaces that form an orthogonal decomposition of $ \C^n $; (ii) a ``numerically identified'' orthogonal decomposition of $ \C^n $. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on $ \C^n $ is identified with a Bayesian prior ``ortho-probability'' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule.

quant-ph↗