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Calibrated Learning and Correlated Equilibrium
 Games and Economic Behavior
, 1996
"... Suppose two players meet each other in a repeated game where: 1. each uses a learning rule with the property that it is a calibrated forecast of the others plays, and 2. each plays a best response to this forecast distribution. ..."
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Cited by 86 (5 self)
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Suppose two players meet each other in a repeated game where: 1. each uses a learning rule with the property that it is a calibrated forecast of the others plays, and 2. each plays a best response to this forecast distribution.
Asymptotic calibration
, 1998
"... Can we forecast the probability of an arbitrary sequence of events happening so that the stated probability of an event happening is close to its empirical probability? We can view this prediction problem as a game played against Nature, where at the beginning of the game Nature picks a data sequenc ..."
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Cited by 68 (4 self)
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Can we forecast the probability of an arbitrary sequence of events happening so that the stated probability of an event happening is close to its empirical probability? We can view this prediction problem as a game played against Nature, where at the beginning of the game Nature picks a data sequence and the forecaster picks a forecasting algorithm. If the forecaster is not allowed to randomise, then Nature wins; there will always be data for which the forecaster does poorly. This paper shows that, if the forecaster can randomise, the forecaster wins in the sense that the forecasted probabilities and the empirical probabilities can be made arbitrarily close to each other.
Deterministic calibration and Nash equilibrium
 Proceedings of the Seventeenth Annual Conference on Learning Theory, volume 3120 of Lecture Notes in Computer Science
, 2004
"... Abstract. We provide a natural learning process in which the joint frequency of empirical play converges into the set of convex combinations of Nash equilibria. In this process, all players rationally choose their actions using a public prediction made by a deterministic, weakly calibrated algorithm ..."
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Cited by 39 (2 self)
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Abstract. We provide a natural learning process in which the joint frequency of empirical play converges into the set of convex combinations of Nash equilibria. In this process, all players rationally choose their actions using a public prediction made by a deterministic, weakly calibrated algorithm. Furthermore, the public predictions used in any given round of play are frequently close to some Nash equilibrium of the game. 1
Probabilistic forecasts, calibration and sharpness
 Journal of the Royal Statistical Society Series B
, 2007
"... Summary. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive dis ..."
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Cited by 38 (15 self)
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Summary. Probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions. We propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration. Calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize. Sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only. A simple theoretical framework allows us to distinguish between probabilistic calibration, exceedance calibration and marginal calibration. We propose and study tools for checking calibration and sharpness, among them the probability integral transform histogram, marginal calibration plots, the sharpness diagram and proper scoring rules. The diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the Stateline wind energy centre in the US Pacific Northwest. In combination with crossvalidation or in the time series context, our proposal provides very general, nonparametric alternatives to the use of information criteria for model diagnostics and model selection.
Conditional Universal Consistency
, 1997
"... Each period, a player must choose an action without knowing the outcome that will be chosen by "Nature," according to an unknown and possibly historydependent stochastic rule. We discuss have a class of procedures that assign observations to categories, and prescribe a simple randomized variation o ..."
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Cited by 32 (0 self)
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Each period, a player must choose an action without knowing the outcome that will be chosen by "Nature," according to an unknown and possibly historydependent stochastic rule. We discuss have a class of procedures that assign observations to categories, and prescribe a simple randomized variation of fictitious play within each category. These procedures are "conditionally consistent," in the sense of yielding almost as high a timeaverage payoff as could be obtained if the player chose knowing the conditional distributions of actions given categories. Moreover given any alternative procedure, there is a conditionally consistent procedure whose performance is no more than epsilon worse regardless of the discount factor. Cycles can persist if all players classify histories in the same way; however in an example, where players classify histories differently, the system converges to a Nash equilibrium. We also argue that in the long run the timeaverage of play should resemble a correlated equilibrium.
Calibrated Forecasting and Merging
, 1996
"... Consider a general finitestate stochastic process governed by an unknown objective probability distribution. Observing the system, a forecaster assigns subjective probabilities to future states. The resulting subjective forecast merges to the objective distribution if, with time, the forecasted pro ..."
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Cited by 20 (4 self)
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Consider a general finitestate stochastic process governed by an unknown objective probability distribution. Observing the system, a forecaster assigns subjective probabilities to future states. The resulting subjective forecast merges to the objective distribution if, with time, the forecasted probabilities converge to the correct (but unknown) probabilities. The forecast is calibrated if observed longrun empirical distributions coincide with the forecasted probabilities. This paper links the unobserved reliability of forecasts to their observed empirical performance by demonstrating full equivalence between notions of merging and of calibration. It also indicates some implications of this equivalence for the literatures of forecasting and learning.
An Easier Way to Calibrate
, 1995
"... This document is copyrighted by the authors. You may freely reproduce and distribute it electronically or in print, provided it is distributed in its entirety, including this copyright notice. 1 The authors are grateful for financial support from NSF grants SBR9223320, SBR9223175, SBR9409180 and ..."
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Cited by 18 (0 self)
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This document is copyrighted by the authors. You may freely reproduce and distribute it electronically or in print, provided it is distributed in its entirety, including this copyright notice. 1 The authors are grateful for financial support from NSF grants SBR9223320, SBR9223175, SBR9409180 and the UCLA Academic Senate. This paper benefited from conversations with Glen Ellison and Dean Foster
Any inspection is manipulable
 Econometrica
, 2001
"... Abstract. A forecaster provides a probabilistic prediction regarding the following day’s state of nature. To examine the forecaster, an inspector employs calibration tests that compare the average prediction and the empirical frequency of prespecified events. This paper shows that any mixed test ca ..."
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Cited by 16 (1 self)
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Abstract. A forecaster provides a probabilistic prediction regarding the following day’s state of nature. To examine the forecaster, an inspector employs calibration tests that compare the average prediction and the empirical frequency of prespecified events. This paper shows that any mixed test can be manipulated in the sense that, independently of the state realizations, the difference between the average prediction and the past empirical frequency that corresponds to almost any test employed diminishes to zero. In other words, a forecaster has a prediction scheme that passes almost any test. In particular, a forecaster can pass all the tests in a countable set simultaneously. I am grateful to Rann Smorodinsky, Sylvain Sorin and two anonymous referees for their helpful
Learning to Play Network Games
, 1997
"... this paper, the common knowledge assumptions are dropped in varying capacities, thereby limiting the credibility of the Nash equilibrium solution concept in its usual form. This research focuses on alternative forms of equilibria which arise as a result of learning in repeated games in the absence o ..."
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Cited by 10 (8 self)
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this paper, the common knowledge assumptions are dropped in varying capacities, thereby limiting the credibility of the Nash equilibrium solution concept in its usual form. This research focuses on alternative forms of equilibria which arise as a result of learning in repeated games in the absence of common knowledge. Without the assumptions of common knowledge of rationality and payoff structure, games are not conducive to deductive solutions, such as Nash equilibrium. One of the focal points of modern game theory is inductive reasoning in repeated games, which is described as follows in Arthur [1]: Each agent keeps track of the performance of a private collection of beliefmodels. When it comes time to make choices, he acts upon his currently most credible (or possibly most profitable) one. The others he keeps at the back of his mind, so to speak. Alternatively, he may act upon a combination of several...Once actions are taken, agents update the track record of all their hypotheses. This type of reasoning is known as beliefbased learning. Examples of beliefbaser learning algorithms include Bayesian updating and calibration. Under certain conditions, Bayesian learning converges to Nash equilibrium, while calibrated learning always converges to a generalization of Nash equilibrium known as correlated equilibrium. The intent of this thesis research is to develop efficient learning algorithms which quickly converge to reasonable approximations of equilibria in gametheoretic models of network routing and congestion problems. 2 Nash Equilibrium