Questions tagged [probability]
A branch of statistics that studies the likelihood of uncertain events occurring.
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Probability for Economics
I need to learn ASAP on the list of topics: Multivariate Distribution, discrete and continuous random variables; integration and expectation; law of large numbers and central limit theorem, confidence ...
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About The Bayesian Conditional-Probability Systems in Myerson's Game Theory: Analysis of Conflict
I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his Bayesian conditional-probability system. The Bayesian conditional-probability ...
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Best-responding to a stochastically higher distribution of bids
In Auction Theory, Krishna writes that:
a bidder who faces a stochastically higher distribution of bids–in the
sense of reverse hazard rate dominance–will bid higher
(This follows the proof of ...
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Joint distribution from differential equations
I have the following problem -
Z is a random variable which can take any real value in the range [0,1]
a and b are independent variables drawn from uniform distribution in the interval [0,1].
Z is a ...
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Understanding the notations in Bayesian game definition
I am having trouble understanding the definition of a Bayesian game based on the following definition from class. I would appreciate it if you could explain the notations and overall meaning for point ...
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transition probabilities from a AR(1) stochastic process
I have a stochastic volatility model for commodity price which follows an AR(1) process:
ln(pt ) − m = ρ (ln(pt−1) − m) + exp(σt)ut ut ∼ IID(0, 1)
σt − μ = ρσ(σt−1 − μ) + ηεt εt ∼ IID(0, 1)
...
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Deriving an equation in Banerjee "A Simple Model of Herd Behavior" (1992)
I am reading "A Simple Model of Herd Behavior" by Banerjee (1992). A short summary of the model is the following.
There is a probability $\alpha$ that each person receives a signal telling ...
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Third price auction from Auction Theory by Krishna, Order statistics
Notation:
$Y_1$: Highest order statistics of $(N-1)$ players' valuation.
$F_n^M:$ The distribution function of the highest $n$th order statistics of $M$ players.
$f_n^M:$ The density of the highest $n$...
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Estimating excess probability
I have three players A,B,C, where C ideally functions as a baseline. They play different games (not against each other) and I can observe their win probability, for example P(A wins in game I) = 90%, ...
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Resampling for a Probabilistic Model to Balance Outcomes
I wanted to construct a logit model for determining the probability a recession will be determined for any given month using the usual Macro indicators; however, I noticed that 90% of the months in my ...
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What does it mean when I say that CDF is bounded away from 1?
Suppose $\theta \in [\underline\theta, \bar\theta]$ is distributed with CDF F(.). What does it mean when I say that this F is bounded away from 1? Does it mean that F can never take the value 1 in ...
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Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?
In a Mixed Strategy Nash Equilibrium, each player constructs their own probability distribution over the set of their respective possible strategies.
In a Correlated Equilibrium or a Coarse Correlated ...
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What does this notation mean?
I am following an IO paper and, at some point, a function $h(\cdot)\in \mathbb{R}^2_+$ is defined as
$$h(t) = \cases{\mathbb{1}(t=k)*|\mathbb{N}(0,1)|\\\mathbb{1}(t<k)*|\mathbb{N}(0,1)|}$$ where $k\...
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Slight Uncertainty of Continuation in Repeated Prisoner's Dilemma
In a repeated prisoner's dilemma with some probability δ of continuing after each round, a Subgame Perfect Nash Equilibrium may be found which induces cooperation instead of defection in each round. ...
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Why is Stata omitting some of my variables and mfx not working?
I'm trying to do a probit regression with some categorical and continuous variables but Stata keeps omitting certain variables and even claiming that some can't be used to to collinearity problems (I ...
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How realistic is the conclusion that players do not change their mixing proportions in response to changes in their own payoffs?
A major lesson from game theory seems to be that in simultaneous move games, a player does not change mixing proportions in response to changes in their own payoffs. Rather, their opponents change ...
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Mechanisms of communication in game theory
In the spirit of the previous question that I have done, here considering the paper here I am trying to make the matching definition $2.2$ here.
I will give two definitions and I would like to clarify ...
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Maximizing profit with a simple probabilistic production function (basic practice problem)
A restaurant finds that less orders for their soup of the day are placed on warmer days so they discount the usual 7USD price to 5USD on warmer days. The cost of making the soup is given by
$$
C = 0.1{...
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Bayes’ rule in "The sources of capital misallocation"
I am reading a paper titled "The sources of capital misallocation". In the model, firms are facing incomplete information about their future productivities. In particular, the productivity ...
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A question about conditional expectation involving independence
If the vector $(u,v)$ is independent of the vector $x$, then I would like to show that
$$E(u|x,v)= E(u|v)$$
The only thing I can derive from the definitions is that if $(u,v)$ is independent of $x$, ...
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Intuition behind risk premium
In Lecture 20 of MIT's Microeconomics course, a situation is proposed where a 50/50 bet will either result in losing \$100 or gaining \$125 with a starting wealth of \$100. It is stated that a person ...
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What is the intuition behind Blackwell's Equivalence Theorem on Information Structures?
Let us suppose that we have a Bayesian game where the information structure is defined to be as $P^X=\{(X,\mathcal{X},P_\theta)\}_{\theta\in\Theta}$ where a signal generated by the information ...
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Finding the risk attitude parameter in a CPT Risk Elicitation Model
I'm working with this article by Bauermeister et al. that compares the risk attitude parameters found using two different risk elicitation models. The models each use a series of gambling options to ...
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Negative certainty equivalent
Let us consider an agent of initial wealth $w_0$ whose utility function is $u(x)=\sqrt{x}$. This individual faces a risk of loss $Z$ which occurs with probability $p$.
It is assumed that $w_0=60000$, $...
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Understanding the properties of extensive form games
In Heller et al, they use the Osborne and Rubinstein formal definition for the extensive form games with public information. To some point they refer to the following two properties
$P$ is a mapping ...
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Model the uncertain impact of a proposed policy by expected utility or other probabilistic approach
The impact of a proposed policy is often uncertain and subjected to randomness. As such, it seems natural to use probabilistic models. How to model the policy impact using the expected utility ...
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Risk neutral probability for each of 3 states
I need help to find the risk-neutral probability for states 1,2 and 3
I have two stocks: A and B.
The price of A today is 180 and in a year it will be worth 288 (S1), 180 (S2) or 120 (S3);
The ...
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Expected payoff calculation difficulty in the Bayesian environment of Bergemman and Morris
Suppose that we have two states of the world equally likely to occur, and say $\psi$ is the common prior of the state $\theta\in\Theta=\{G,B\}$. The types of the players are given by the following ...
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Bayesian update in the beliefs about the signals
Suppose that we have tow states of the world $\omega_1$ and $\omega_2$, where $p(\omega_1)=p(\omega_1)=1/2$ and there are three different signals, $s_H,s_M,s_L$ that are equally likely to occur in ...
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The most used probability distributions in moral hazard models?
I'm engaged in making a variation of the canonical moral hazard model, but I need some examples of probability distributions to make some simulations and graphs. What are the probability distributions ...
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How to compare investments with different risk and expected return?
Supposing I can choose to invest money in several different investments, each having
risk $\sigma_i$, for example, calculated as standard deviation
and expected return $r_i$
let's assume they have ...
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Difficulty in understanding the notation related from probability theory with game theory
The question that I have is a little technical and it has to do with the notation and the combination between some mathamatical properties in the probability theory of information economics.
Say $\...
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Olivier Gossner - Secure Protocols or How Communication Generates Correlation
The paper of Olivier Gossner in Security Protocols in 1998 has some definitions that confuse me too much. I will cite here these definitions and my questions and I hope someone is familiar with these ...
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How to use Girsanov theorem to prove $\hat{W_t}$ is $\hat{\mathbb P}$-Brownian motion?
Assumptions:
Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $...
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What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?
Related: Lewis' triviality result?
As I recall the one-step binomial model goes like this:
The time periods are now $t=0$ and later $t=1$.
We have
2.1. a stock that pays off $u$ for going up or $d$...
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What is the meaning of the support set in game theory?
What is the meaning of the support set in game theory? I have seen it, in many papers, however none there explains how did they find it or why did the define it in a specific way. I understand that ...
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In economics, what variables do we usually assume to follow an exponential distribution?
In economics, what variables do we usually assume to follow an exponential distribution? I would like as many examples as possible, and it would be great if you have a rationale(economic reason or ...
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Interpretation of multinominal logit regression (Stata)
I have a few questions about mlogit. I have a set of independent variables and a categorical, but ordered, dependent variable with three categories (Disagree, Neutral and Agree). The assumptions for ...
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Find all of the Pure and Mixed Strategy Nash Equilibria [closed]
When I do the basic calculations for mixed probability, I get that the Column player always plays B. However, I am getting a negative probability for the row. Any help is appreciated.
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Question on the choice of boundary in the CUSUM test when we make some resampling
Question on the choice of boundary in the CUSUM test when we make some resampling
We are considering to make a CUSUM test for some economical time series $𝑋=(𝑥_1,..,x_n)$. Suppose 𝑋 contains many ...
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Expected value of order statistics for uniform distribution
I have $X$ ~ $U(0,1)$ interval.
Let n=2, i.e. $X_1 < X_2$
I have to calculate the expected value of ${X_2}^{m/(1-m)}$.
Where, $0≤m≤1$
I want to confirm if I have calculated it correctly?
$$\int_0^1 ...
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Integration by parts with CDF
I am told that the following equality follows from integration by parts:
$$\int_{R-k}^{1}(\theta-R)dG(\theta)-G(R-k)k=\int_{R-k}^{1}(1-G(\theta))d\theta-k$$ Where $R>k>0$ and $G$ is the CDF of $\...
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Is it worth betting on this case?
Let's imagine a coin-flip game, which uses an unbiased coin.
Starting with X dollars, your total increases 50% every time you flip heads. But if the coin lands on tails, you lose 40% of your total. ...
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Show that conditional variance of error in linear probability model is heteroskedastic?
I have a problem that asks me the following:
" Consider the linear probability model, in which we specify the
regression equation to be linear in X,
E(Y |X = x) = Pr(Y = 1|X = x) = x'β
We can ...
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Average ability conditioning on having accepted an offer
There is a continuum of workers between 0 and 1. These have ability $\alpha\sim U[0,2]$. A firm offers them a salary $v$ and has profits
$$
\pi = (\rho \alpha-v) n(v)
$$
where $n(v)$ is the fraction ...
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How to average CDFs of one variable across years
I have wealth-to-income data for 10 years. I computed the cdf of this variable in each year.
Now I'm trying to average the cdfs across years. In each each, the number of observations is different.
...
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Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?
I found this picture in my stats book but I'm now confused to what 'positive' and 'negative' is referring to.
As seen in the table below, Type 1 error is the error that its H0 is actually true but ...
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Why is it possible to calibrate your subjective probabilities?
Humans tend to be overconfident in their predictions; when most people say that there's a 95% chance that something will happen, they're usually wrong far more than 5% of the time. Whereas what ought ...
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Expected Utility and Jensen's Inequality
Consider two random variables (costs and valuations) distributed $v\backsim G(.)$ and
$c \backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and
$v$ be $[x,y]$. Let $x<a=E(v)<b&...
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How to approach rigorous probability theory from an economics background?
I am attempting to read around the theory of probability theory from the ground up, coming from a background of economics I have little experience in set/measure theory, whilst I am not new to ...
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