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Strictly Proper Scoring Rules, Prediction, and Estimation
, 2007
"... Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distribution F if he ..."
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Cited by 143 (17 self)
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Scoring rules assess the quality of probabilistic forecasts, by assigning a numerical score based on the predictive distribution and on the event or value that materializes. A scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distribution F if he or she issues the probabilistic forecast F, rather than G ̸ = F. It is strictly proper if the maximum is unique. In prediction problems, proper scoring rules encourage the forecaster to make careful assessments and to be honest. In estimation problems, strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand. This article reviews and develops the theory of proper scoring rules on general probability spaces, and proposes and discusses examples thereof. Proper scoring rules derive from convex functions and relate to information measures, entropy functions, and Bregman divergences. In the case of categorical variables, we prove a rigorous version of the Savage representation. Examples of scoring rules for probabilistic forecasts in the form of predictive densities include the logarithmic, spherical, pseudospherical, and quadratic scores. The continuous ranked probability score applies to probabilistic forecasts that take the form of predictive cumulative distribution functions. It generalizes the absolute error and forms a special case of a new and very general type of score, the energy score. Like many other scoring rules, the energy score admits a kernel representation in terms of negative definite functions, with links to inequalities of Hoeffding type, in both univariate and multivariate settings. Proper scoring rules for quantile and interval forecasts are also discussed. We relate proper scoring rules to Bayes factors and to crossvalidation, and propose a novel form of crossvalidation known as randomfold crossvalidation. A case study on probabilistic weather forecasts in the North American Pacific Northwest illustrates the importance of propriety. We note optimum score approaches to point and quantile
Eliciting Informative Feedback: The PeerPrediction Method
 Management Science
, 2005
"... informs ® doi 10.1287/mnsc.1050.0379 ..."
The Good NewsBad News Effect: Asymmetric Processing of Objective Information about Yourself
 American Economic Journal: Microeconomics
"... We study processing and acquisition of objective information regarding qualities that people care about, intelligence and beauty. Subjects receiving negative feedback did not respect the strength of these signals, were far less predictable in their updating behavior and exhibited an aversion to new ..."
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Cited by 12 (1 self)
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We study processing and acquisition of objective information regarding qualities that people care about, intelligence and beauty. Subjects receiving negative feedback did not respect the strength of these signals, were far less predictable in their updating behavior and exhibited an aversion to new information. In response to good news, inference conformed more closely to Bayes Rule, both in accuracy and precision. Signal direction did not affect updating or acquisition in our neutral control. Unlike past work, our design varied direction and agreement with priors independently. The results indicate that confirmation bias is driven by direction; confirmation alone had no effect.
Spatial Competition with Three Firms: An Experimental Study
 Economic Inquiry
, 1999
"... The paper reports the results of an experimental study of the three firm location problem. We compare the subjects' behavior in the experiments with the symmetric mixed strategy Nash equilibrium calculated by Shaked (1982). Overall, the findings are consistent with the equilibrium prediction. Howeve ..."
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Cited by 4 (0 self)
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The paper reports the results of an experimental study of the three firm location problem. We compare the subjects' behavior in the experiments with the symmetric mixed strategy Nash equilibrium calculated by Shaked (1982). Overall, the findings are consistent with the equilibrium prediction. However, the subjects' locations were significantly more dispersed than predicted by the theory. Three alternative explanations of this phenomenon  inexperience, approximate equilibrium behavior and risk aversion  are suggested and evaluated for their predictive power. Special attention is paid to risk aversion.
Comparing Density Forecasts Using Threshold and Quantile Weighted Scoring Rules
, 2008
"... We propose a method for comparing density forecasts that is based on weighted versions of the continuous ranked probability score. The weighting emphasizes regions of interest, such as the tails or the center of a variable’s range, while retaining propriety, as opposed to a recently developed weight ..."
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Cited by 3 (0 self)
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We propose a method for comparing density forecasts that is based on weighted versions of the continuous ranked probability score. The weighting emphasizes regions of interest, such as the tails or the center of a variable’s range, while retaining propriety, as opposed to a recently developed weighted likelihood ratio test, which can be hedged. Threshold and quantile based decompositions of the continuous ranked probability score can be illustrated graphically and prompt insights into the strengths and deficiencies of a forecasting method. We illustrate the use of the test and graphical tools in case studies on the Bank of England’s density forecasts of quarterly inflation rates in the United Kingdom, and probabilistic predictions of wind resources in the
Combining Probability Forecasts
, 2008
"... Linear pooling is by the far the most popular method for combining probability forecasts. However, any nontrivial weighted average of two or more distinct, calibrated probability forecasts is necessarily uncalibrated and lacks sharpness. In view of this, linear pooling requires recalibration, even i ..."
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Cited by 3 (0 self)
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Linear pooling is by the far the most popular method for combining probability forecasts. However, any nontrivial weighted average of two or more distinct, calibrated probability forecasts is necessarily uncalibrated and lacks sharpness. In view of this, linear pooling requires recalibration, even in the ideal case in which the individual forecasts are calibrated. Toward this end, we propose a beta transformed linear opinion pool (BLP) for the aggregation of probability forecasts from distinct, calibrated or uncalibrated sources. The BLP method fits an optimal nonlinearly recalibrated forecast combination, by compositing a beta transform and the traditional linear opinion pool. The technique is illustrated in a simulation example and in a case study on statistical and National Weather Service probability of precipitation forecasts.
Mechanism Design with Multidimensional, Continuous Types and Interdependent Valuations
, 2006
"... We consider the mechanism design problem when agents ’ types are multidimensional and continuous, and their valuations are interdependent. If there are at least three agents whose types satisfy a weak correlation condition, then for any decision rule and any ε>0 there exist balanced transfers that r ..."
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Cited by 2 (1 self)
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We consider the mechanism design problem when agents ’ types are multidimensional and continuous, and their valuations are interdependent. If there are at least three agents whose types satisfy a weak correlation condition, then for any decision rule and any ε>0 there exist balanced transfers that render truthful revelation a Bayesian εequilibrium. A slightly stronger correlation condition ensures that there exist balanced transfers that induce a Bayesian Nash equilibrium in which agents ’ strategies are nearly truthful.
Learning to Respond: The Use of Heuristics in Dynamic Games 1
, 2001
"... While many learning models have been proposed in the game theoretic literature to track individuals ’ behavior, surprisingly little research has focused on how well these models describe human adaptation in changing dynamic environments. Analysis of human behavior demonstrates that people are often ..."
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While many learning models have been proposed in the game theoretic literature to track individuals ’ behavior, surprisingly little research has focused on how well these models describe human adaptation in changing dynamic environments. Analysis of human behavior demonstrates that people are often remarkably responsive to changes in their environment, on time scales ranging from millennia (evolution) to milliseconds (reflex). The goal of this paper is to evaluate several prominent learning models in light of a laboratory experiment on responsiveness in a lowinformation dynamic game subject to changes in its underlying structure. While historydependent reinforcement learning models track convergence of play well in repeated games, it is shown that they are ill suited to these environments, in which sastisficing models accurately predict behavior. A further objective is to determine which heuristics, or “rules of thumb, ” when incorporated into learning models, are responsible for accurately capturing responsiveness. Reference points and a particular type of experimentation are found to be important in both describing and predicting play.
An Experiment
, 2008
"... We experimentally investigate the disposition of decision makers to use case—based reasoning as suggested by Case—Based Decision Theory (Gilboa and Schmeidler, 1995). Our subjects face a monopoly decision problem about which they have very limited information. Information is presented in a manner wh ..."
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We experimentally investigate the disposition of decision makers to use case—based reasoning as suggested by Case—Based Decision Theory (Gilboa and Schmeidler, 1995). Our subjects face a monopoly decision problem about which they have very limited information. Information is presented in a manner which makes similarity judgements according to the feature matching model of Tversky (1977) plausible. In a 2×2 between—subject design, we vary whether information about the current market is given, in addition to a “history, ” or not. Furthermore, we investigate the impact feedback has on behavior in several independent markets by either giving immediate feedback about obtained profits or not. The results provide some support for the predictions of Case—Based Decision Theory, particularly when no immediate feedback is provided. ∗We thank Gerd Gigerenzer and the audiences at the Max Planck Institute for Human
Minimizing Inaccuracy for Self Locating Beliefs
"... One’s inaccuracy for a proposition is defined as the squared difference between the truth value (1 or 0) of the proposition and the credence (or subjective probability, or degree of belief) assigned to the proposition. One should have the epistemic goal of minimizing the expected inaccuracies of one ..."
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One’s inaccuracy for a proposition is defined as the squared difference between the truth value (1 or 0) of the proposition and the credence (or subjective probability, or degree of belief) assigned to the proposition. One should have the epistemic goal of minimizing the expected inaccuracies of one’s credences. We show that the method of minimizing expected inaccuracy can be used to solve certain probability problems involving information loss and selflocating beliefs (where a selflocating belief of a temporal part of an individual is a belief about where or when that temporal part is located). We analyze the Sleeping Beauty problem, the duplication version of the Sleeping Beauty problem, and various related problems. 1.