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This paper uses laboratory experiments to study subjects' assessment of uncertainty resulting from strategic and non-strategic decisions of other players. Non-strategic events are defined by the colors of balls drawn from urns, whereas strategic events are defined by the action choice in Stag Hunt (SH) and Prisoners' Dilemma (PD) games. We elicit subjects' matching probabilities and examine if they satisfy the law of probability including monotonicity and additivity. Violations from the law are observed for both uncertainty sources, but are more substantial for strategic uncertainty. In particular, we observe a coordination fallacy, a violation of monotonicity whereby the probability weight placed on a symmetric coordination profile of the games exceeds that placed on the corresponding action choice. The violation is found to be severer for an efficient coordination profile.

The probability of nuclear war is a major factor in many important policy questions, but it has gotten little scholarly attention. This paper presents a model for calculating the total probability of nuclear war. The model is based on 14 interrelated scenarios for how nuclear war can break out, covering perhaps the entire range of nuclear war scenarios. Scenarios vary based on factors including whether a state intends to make a first strike attack, whether the nuclear attack is preceded by a conventional war or a non-war crisis, whether escalation is intentional or inadvertent, the presence of false alarms of various types, and the presence of non-war nuclear detonations such as nuclear terrorism. As a first step towards quantifying the probability of nuclear war using the model, the paper also includes a dataset of historical incidents that might have threatened to turn into nuclear war. 60 historical incidents are included, making it perhaps the largest such dataset currently available. The paper also includes background information about probabilistic analysis and modeling to help readers understand how to think about the probability of nuclear war, including new theory for the decision to initiate nuclear war.

Richard von Mises set forth a Limiting Frequency approach to probability for use primarily in physics and some areas of biology, chemistry and engineering. Emphasis was placed on the existence of series or sequences of events that resulted from a known generating mechanism. The series consisted of repeated, homogeneous events that were identical to each other, except for their particular place selection in an infinite series. This conception of probability is extremely limited and can play only a very limited role in liberal arts, social science, behavioral science, and especially in economics, business, and finance.Ludwig von Mises conception was that there were two different concepts of probability. First, there was a class concept of probability that was basically a version of a limiting frequency interpretation of probability. He had a second conception that he called case probability. Case probability was supposed to deal with unique events. It essentially is a subjective degree of belief approach that allows one to calculate no more than an ordinal relationship. Ludwig Von Mises is thus not a Bayesian Subjectivist. His dual approach to probability is reminiscent of Rudolf Carnap, although it is greatly underdeveloped technically when compared to Carnap.J. M. Keynes’s conception of probability is a logical theory of probability built on G. Boole’s earlier logical theory of probability. Keynes’s theory deals with the degree of rational belief one should believe and act on. It is not “a” or “the” degree of belief used by Subjectivists. It was a relative concept, which means that all probabilities are conditional probabilities. There is an objective probability relation that holds between a given body of evidence, the premises, E, and a conclusion or hypothesis, H. Thus, P (H/E) =α. The objective probability relation measures the degree of similarity (dissimilarity) that exists between the evidence, E (premises), and the conclusion or Hypothesis, H.There are many severe and numerous errors in Richard Von Mises book, Probability, Statistics and Truth, regarding Keynes’s logical theory of probability that will be identified and corrected in this paper. Neither of the Von Mises brothers understood Keynes’s interval valued approach to probability or had any clear cut understanding of the concept of weight or decision weights like Keynes’s conventional coefficient in chapter 26 of the A Treatise on Probability, although such an approach appears out of nowhere in Probability, Statistics and Truth in Lecture 4.Ludwig von Mises’s conception of case probability has no developed concept of the weight of the evidence underlying it. Its identification of probability as being ordinal probability would be too weak from Keynes’s point of view to help an entrepreneur in making decisions related to his revenues and costs and the question of whether he should invest in fixed capital now or wait to do so at some time in the future (liquidity preference and user cost of capital issues).

A patient player privately observes a persistent state and interacts with an infinite sequence of myopic uninformed players. The patient player is either a strategic type who maximizes his payoff or one of several commitment types who mechanically play the same action in every period. I focus on situations in which the uninformed player's best reply to a commitment action depends on the state and where the total probability of commitment types is sufficiently small. I show that the patient player's equilibrium payoff is bounded below his commitment payoff in some equilibria under some of his payoff functions. This is because he faces a trade‐off between building his reputation for commitment and signaling favorable information about the state. When players' stage‐game payoff functions are monotone‐supermodular, the patient player receives high payoffs in all states and in all equilibria. Under an additional condition on the state distribution, my reputation model yields a unique prediction on the patient player's equilibrium payoff and on‐path behavior.

We establish novel necessary and sufficient conditions for a particularly tractable concept of monotonic dependence of an exponential conditional probability model on its explanatory variables. We introduce convex programming methods to calibrate such models (which are monotone in conditional mean, conditional median and conditional mode). Armed with such methods, the model builder can incorporate prior knowledge, with respect to monotone dependence on a subset of the explanatory variables, into a broad class of nonlinear conditional probability models (including two-state models, as well as models with more than two states). Our method amounts to a novel form of regularization that, as our benchmarking experiments in the credit modeling context show, allows us to calibrate models that perform better than linear models, while maintaining the monotonic behavior that many users find reassuring.

Under expected utility theory, unconditional expected utility can be decomposed into a weighted sum of conditional expected utilities where the weights are marginal probabilities. We derive necessary and sufficient conditions for a similar decomposition in the framework of Cumulative Prospect Theory (CPT). The conditions also ensure that a decision maker’s conditional preferences (given some event) remain within the CPT class. Our results are important for empirical analyses in which weighted marginal probabilities of events are used to explain a decision maker’s choices. The use of such marginal probabilities is a practical necessity in non-experimental settings.

Baye's Theorem states that the conditional probability of A given B can be found out by dividing the product of the conditional probability of B given A and the marginal probability of A by the sum of the product of the conditional probability of B given A and the marginal probability of A (Law of total probability).

The rule states that the marginal probability of an event is equal to the sum of the product of the conditional probability of some other event in the sample space (the probability that an event will occur given that some other event has occurred or will occur. Here the probability of B given the probability of A) and the marginal probability of that other event (Proof/Derivation).

The rule states that for two events A and B, the probability that any one of them will occur given that the other has occurred or will occur is equal to the probability that both the events will occur together divided by the probability of the event that has occurred or will occur. (Proof/Derivation)

In the existing literature on barrier options much effort has been exerted to ensure convergence through placing the barrier in close proximity to, or directly onto, the nodes of the tree lattice. For a variety of barrier option types we show that such a procedure may not be a necessary prerequisite to achieving accurate option price approximations. Using the Kamrad and Ritchken (1991) trinomial tree model we show that with a suitable transition probability adjustment our “probability adjusted” model exhibits convergence to the barrier option price. We study the convergence properties of several option types including exponential barrier options, single linear time-varying barrier options, double linear timevarying barriers options and Bermuda options. For options whose strike price is close to the barrier we are able to obtain numerical results where other models and techniques typically fail. Furthermore, we show that it is possible to calculate accurate option price approximations with minimal effort for options with complicated barriers that defeat standard techniques. In no single case does our method require a repositioning of the pricing lattice nodes.

This article derives key variables in the analysis of standards of proof in criminal law from basic conditional probabilities. The variables derived are the probability of correct and wrongful conviction, the expected sanction and society’s incarceration costs, while the basic conditional probabilities are the probability of observing (any given) evidence against individual i given that individual j committed the crime (for any j including j equal to i). The variables are derived from the conditional probabilities as a function of the standard of the proof using simple Bayesian updating.

The Capital Asset Pricing Model (CAPM) is based on the Expected Utility (EU) theory that assumes linearity in probabilities and models the assumption of risk-aversion via a concave utility function. Studies in psychology and economics, however, show that individuals systematically violate the assumption of linearity in probabilities and this affects their risk attitudes. I derive a measure of risk-premium that arises due to non-linearity in probabilities, captured by a non-linear probability weighting function, and arrive at a measure of conditional risk-aversion. I show that properties of the weighting function implies that the measure of conditional risk-aversion is decreasing in the outcome, in the case of a gamble with a uniform distribution. For the case of a normal distribution, I describe sufficient conditions on the distribution under which the measure of conditional risk-aversion is decreasing in the outcome. Further, I make a set of plausible assumptions on the weighting function and the measure of conditional risk-aversion, to reflect an individual's probability beliefs about a gamble; and characterize the shape of implied marginal utility function in the EU framework. In an asset-pricing model, such probability beliefs implies that the representative agent behaves as if risk-seeking in some regions of outcome, in particular, the small decumulative probability region of positive excess market returns (corresponding to high market gains). Moreover, the stochastic discount factor (SDF) is increasing in, among possibly other regions, the small decumulative probability region of positive excess market returns. This is in contrast to a decreasing SDF implied by the CAPM and its extensions. The model's prediction for the SDF, and consequently, its implications for the cross-section of expected returns, provides a plausible explanation for the weak relationship between a stock's average return and its CAPM beta.

For a number of statistical applications, experts' subjective estimates are needed for distributional parameters, like mean and variance; sometimes even the whole density is wanted. The literature agrees that it is wise behaviour to ask the expert only quantile estimates; from these, the desired parameter or density is derived. Many methods, using different sets of quantiles, have been suggested. A new method is proposed here. Experts are asked the relatively simple task of estimating the seven values that divide the total probability mass into eight equal parts. These seven quantiles uniquely determine one distribution within both the Pearson and the Johnson system of distributions. Comparisons between different methods were, in the past, generally based on theoretical arguments. A more practice-oriented approach is followed here: together with three often-used methods, our new method is tested experimentally: 56 students with some expert knowledge estimated quantiles of four distributions. The accuracy of the four methods in estimating both the quantiles and the distributional parameters is calculated and discussed.