Narrow Results

We describe a computationally efficient method to produce a specific Bayesian mixture of all the models in a finite set of feature-based models that assign a probability to the observed data set. Special attention is given to the bound on the regret of using the mixture instead of the best model in the set. It is proven theoretically and verified through synthetic data that this bound is relatively tight. Comparing the workload of the proposed method with the direct implementation of the Bayesian mixture shows an almost exponential improvement of computing time.

We introduce comfort decision modeling for decision problems in which an alternative is to be selected based on a measure of satisfaction we refer to as comfort. We define comfort as the difference between the payoff received by selecting a particular strategy and the worst payoff that could have been received under the manifestation of the same state-of-nature. We define the effective comfort associated with an alternative as the aggregation of an alternative’s comforts across all possible states-of-nature. We study several methods of aggregating an alternative’s individual comforts across the different states-of-nature, incorporating various types of information about the uncertainty associated with the states-of-nature. We provide a Comfort Decision Model to determine the value of alternatives utilizing attitudinal measures of the decision maker. We demonstrate a process of performing sensitivity of the resulting decision to a measure of the attitude of the decision maker. Lastly, we use an illustration to show the practicability and cogency of the new method.

We study the problem of determination of asset prices in an incomplete market proposing three different but related scenarios, based on utility pricing. One scenario uses a market game approach whereas the other two are based on risk sharing or regret minimizing considerations. Dynamical schemes modeling the convergence of the buyer and seller prices to a unique price are proposed. The case of exponential utilities is treated in detail, in the simplest possible example of an incomplete market, the trinomial model.

This paper provides an investigation of the effects of an investment’s return moments on drawdown-based measures of risk, including Maximum Drawdown (MDD), Conditional Drawdown (CDD), and Conditional Expected Drawdown (CED). Additionally, a new end-of-period drawdown measure is introduced, which incorporates a psychological aspect of risk perception that previous drawdown measures had been unable to capture. While simulation results indicate many similarities in the first and second moments, skewness and kurtosis affect different drawdown measures in radically different ways. Thus, users should assess whether their choice of drawdown measure accurately reflects the kind of risk they want to measure.

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: in the long run, the player is guaranteed an average payoff as large as the best-reply payoff to the empirical distribution of play of the other players; i.e., there is no “regret.” Smooth fictitious play (Fudenberg and Levine 1995) and regret matching (Hart and Mas-Colell 2000) are particular cases. The motivation and application of the current paper come from the study of procedures whose empirical distribution of play is, in the long run, (almost) a correlated equilibrium. For the analysis we first develop a generalization of Blackwell's (1956) approachability strategy for games with vector payoffs.

We exhibit a large class of simple rules of behavior, which we call adaptive heuristics, and show that they generate rational behavior in the long run. These adaptive heuristics are based on natural regret measures, and may be viewed as a bridge between rational and behavioral viewpoints. Taken together, the results presented here establish a solid connection between the dynamic approach of adaptive heuristics and the static approach of correlated equilibria.