Blackwell-Optimal Strategies in Priority Mean-Payoff Games

We examine perfect information stochastic mean-payoff games - a class of games containing as special sub-classes the usual mean-payoff games and parity games. We show that deterministic memoryless strategies that are optimal for discounted games with state-dependent discount factors close to 1 are optimal for priority mean-payoff games establishing a strong link between these two classes

Dynamic Non-Bayesian Decision Making

The model of a non-Bayesian agent who faces a repeated game with incomplete information against Nature is an appropriate tool for modeling general agent-environment interactions. In such a model the environment state (controlled by Nature) may change arbitrarily, and the feedback/reward function is initially unknown. The agent is not Bayesian, that is he does not form a prior probability neither on the state selection strategy of Nature, nor on his reward function. A policy for the agent is a function which assigns an action to every history of observations and actions. Two basic feedback structures are considered. In one of them -- the perfect monitoring case -- the agent is able to observe the previous environment state as part of his feedback, while in the other -- the imperfect monitoring case -- all that is available to the agent is the reward obtained. Both of these settings refer to partially observable processes, where the current environment state is unknown. Our main result refers to the competitive ratio criterion in the perfect monitoring case. We prove the existence of an efficient stochastic policy that ensures that the competitive ratio is obtained at almost all stages with an arbitrarily high probability, where efficiency is measured in terms of rate of convergence. It is further shown that such an optimal policy does not exist in the imperfect monitoring case. Moreover, it is proved that in the perfect monitoring case there does not exist a deterministic policy that satisfies our long run optimality criterion. In addition, we discuss the maxmin criterion and prove that a deterministic efficient optimal strategy does exist in the imperfect monitoring case under this criterion. Finally we show that our approach to long-run optimality can be viewed as qualitative, which distinguishes it from previous work in this area.Comment: See http://www.jair.org/ for any accompanying file

Optimal and Myopic Information Acquisition

We consider the problem of optimal dynamic information acquisition from many correlated information sources. Each period, the decision-maker jointly takes an action and allocates a fixed number of observations across the available sources. His payoff depends on the actions taken and on an unknown state. In the canonical setting of jointly normal information sources, we show that the optimal dynamic information acquisition rule proceeds myopically after finitely many periods. If signals are acquired in large blocks each period, then the optimal rule turns out to be myopic from period 1. These results demonstrate the possibility of robust and "simple" optimal information acquisition, and simplify the analysis of dynamic information acquisition in a widely used informational environment

Towards Machine Wald

The past century has seen a steady increase in the need of estimating and predicting complex systems and making (possibly critical) decisions with limited information. Although computers have made possible the numerical evaluation of sophisticated statistical models, these models are still designed \emph{by humans} because there is currently no known recipe or algorithm for dividing the design of a statistical model into a sequence of arithmetic operations. Indeed enabling computers to \emph{think} as \emph{humans} have the ability to do when faced with uncertainty is challenging in several major ways: (1) Finding optimal statistical models remains to be formulated as a well posed problem when information on the system of interest is incomplete and comes in the form of a complex combination of sample data, partial knowledge of constitutive relations and a limited description of the distribution of input random variables. (2) The space of admissible scenarios along with the space of relevant information, assumptions, and/or beliefs, tend to be infinite dimensional, whereas calculus on a computer is necessarily discrete and finite. With this purpose, this paper explores the foundations of a rigorous framework for the scientific computation of optimal statistical estimators/models and reviews their connections with Decision Theory, Machine Learning, Bayesian Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty Quantification and Information Based Complexity.Comment: 37 page