22 problems

Consider Deutsch's theorem concerning the derivation of Born's law from assumptions about quantum measurements and rational decision-making. The assumptions include functional inva…

Let $\bm{S}$ be the sample scatter matrix and let $\widehat{\bm{\Sigma}}^U=n^{-1}\bm{S}$ be the unbiased estimator of the covariance matrix $\bm{\Sigma}$. For an orthogonally equiv…

Innateness of multi-step planning. An apparatus enabling multi-step planning is among humans' innate functions.

Let $F$ and $G$ be the continuous self-maps defining the model-based and robot value equations, with positive inverse-temperature parameters $\beta_h$ and $\beta_r$. Since $F$ and…

Savage's conjecture. Allowing partial orderings of acts would prove a blind alley, losing much in power and advancing little, if at all, in realism.

Soland's conjecture. The lemma asserting that every properly efficient decision admits a continuous, strictly increasing and strictly concave real-valued function $v$ on…

Representation conjecture. The three modified axioms should imply a corresponding representation theorem for variable lotteries.

Consider voting tables of type c, parametrized by $(\rho,\alpha)$, and suppose they are added to the set of positive tables as the competence level decreases. For each table, defin…

Bradt–Feldman conjecture. The myopic strategy is optimal when $\alpha+\beta\neq 1$.

For a bounded random variable $X$, let … be a preference satisfying the paper's continuity, risk-invariance, and first-order stochastic-dominance monotonicity axioms. Representatio…

First-order approximation conjecture. First-order approximation of equilibrium decision-rules based on the notion of materiality yields decision-rules using functions of the form

Multi-state tractability conjecture. A characterization in a general model with more than two states will not be tractable.

Deadline conjecture. If a firm deadline is introduced into the continuous-time model, the DM will shift her attention increasingly toward own-biased news sources as the deadline ap…

Let the decision maker acquire signals from a general signal distribution in a decision problem restricted, for example, to prediction of the payoff-relevant state. Eventual-myopia…

True Comparison Theory conjecture. Despite advances in the social sciences, an objective measuring stick for comparison—a so-called True Comparison Theory—has yet to be discovered…

Prediction–estimation link conjecture. There exists a link between the prediction problem and the estimation problem in the unknown-variance case under Kullback–Leibler risk, analo…

Intermediate-sample-size inadmissibility conjecture. The best equivariant predictive density is inadmissible for every integer sample size satisfying

Let $T$ be a generic transformation in the setting of decisions with partial information, where a strategy is called pathwise optimal if its time-average loss attains the optimal m…

Let $u_i,v_i>0$ and let the two arms of a finite-horizon Bernoulli bandit have independent ${\rm Beta}(u_i,v_i)$ priors, for $i=1,2$, under uniform discounting. Berry's prior-order…

Permutation-invariant extension conjecture. In Theorem 2, the condition that $Delta^n$ is simple symmetric for every $n$ may be replaced by the weaker condition that $Delta^n$ is p…

Let ${\cal P}$ be a set of distributions on ${\cal X}\times{\cal Y}$, and let $\delta_{\mathrm{prior}}$ denote the a priori minimax-optimal decision rule. A partition…

Let ${\cal P}$ be an arbitrary set of distributions on ${\cal X}\times{\cal Y}$, let $L$ be an arbitrary loss function on ${\cal Y}\times{\cal A}$, and let $\delta$ be the minimax…