No Arabic abstract
Cheng(2021) proposes and characterizes Relative Maximum Likelihood (RML) updating rule when the ambiguous beliefs are represented by a set of priors. Relatedly, this note proposes and characterizes Extended RML updating rule when the ambiguous beliefs are represented by a convex capacity. Two classical updating rules for convex capacities, Dempster-Shafer (Shafer, 1976) and Fagin-Halpern rules (Fagin and Halpern, 1990) are included as special cases of Extended RML.
After observing the outcome of a Blackwell experiment, a Bayesian decisionmaker can form (a) posterior beliefs over the state, as well as (b) posterior beliefs she would observe any given signal (assuming an independent draw from the same experiment). I call the latter her contingent hypothetical beliefs. I show geometrically how contingent hypothetical beliefs relate to information structures. Specifically, the information structure can (generically) be derived by regressing contingent hypothetical beliefs on posterior beliefs over the state. Her prior is the unit eigenvector of a matrix determined from her posterior beliefs over the state and her contingent hypothetical beliefs. Thus, all aspects of a decisionmakers information acquisition problem can be determined using ex-post data (i.e., beliefs after having received signals). I compare my results to similar ones obtained in cases where information is modeled deterministically; the focus on single-agent stochastic information distinguishes my work.
We examine the long-term behavior of a Bayesian agent who has a misspecified belief about the time lag between actions and feedback, and learns about the payoff consequences of his actions over time. Misspecified beliefs about time lags result in attribution errors, which have no long-term effect when the agents action converges, but can lead to arbitrarily large long-term inefficiencies when his action cycles. Our proof uses concentration inequalities to bound the frequency of action switches, which are useful to study learning problems with history dependence. We apply our methods to study a policy choice game between a policy-maker who has a correctly specified belief about the time lag and the public who has a misspecified belief.
We introduce a new updating rule, the conditional maximum likelihood rule (CML) for updating ambiguous information. The CML formula replaces the likelihood term in Bayes rule with the maximal likelihood of the given signal conditional on the state. We show that CML satisfies a new axiom, increased sensitivity after updating, while other updating rules do not. With CML, a decision makers posterior is unaffected by the order in which independent signals arrive. CML also accommodates recent experimental findings on updating signals of unknown accuracy and has simple predictions on learning with such signals. We show that an information designer can almost achieve her maximal payoff with a suitable ambiguous information structure whenever the agent updates according to CML.
We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems.
We propose an extended version of Gini index defined on the set of infinite utility streams, $X=Y^mathbb{N}$ where $Ysubset mathbb{R}$. For $Y$ containing at most finitely many elements, the index satisfies the generalized Pigou-Dalton transfer principles in addition to the anonymity axiom.