# Tag Info

Accepted

### Type - I Error & Type - II Error: Pregnancy test analogy - is it legit?

Presumably here the null hypothesis is $H_0:$ You are not pregnant the alternative hypothesis is $H_1:$ You are pregnant so being pregnant would be the positive result. You take a pregnancy test ...
• 4,765
Accepted

### Is First Order Stochastic Dominance (FOSD) relation convex?

The first order stochastic dominance relation is convex. An easy way to prove this is to use the property that a cdf $F$ FOSD another cdf $G$ if and only if $F(x)\le G(x)$ for all $x$. That is, $F$ ...
• 15.5k
Accepted

### second order stochastic dominance without the same mean

Let $u(x) = x$ which is increasing and concave. Then the defining condition of SOSD reads $$\int x\mathrm dF(x)\ge \int x\mathrm dG(x) \implies E_F(X) \geq E_G(X) \tag{1}$$ ..which would contradict ...
• 33.9k
Accepted

### Difficulty in understanding the notation related from probability theory with game theory

We have that ${\cal I} = ((X^i)_i, \mu)$​ and ${\cal J} = ((Y^i)_i, \nu)$​ are two information structures. An Interpretation mapping for player $i$​​ is a mapping $\phi^i: X^i \to \Delta(Y^i)$​ so it ...
• 12.4k
Accepted

### A question about conditional expectation involving independence

Well assuming the random variables are absolutely continuous, you can use densities: $$f(u|x,v)=\frac{f(u,v,x)}{f(x,v)}=\frac{f(u,v)f(x)}{f(x,v)}=\frac{f(u|v)f(v)f(x)}{f(x,v)}=f(u|v),$$ where the ...
• 243

### What does it mean when I say that CDF is bounded away from 1?

This would mean that $\sup\{F(\theta):\,\underline\theta\le\theta\le\bar\theta\}<1$, which makes no sense as $F(\bar\theta)=1$.
• 6,994
Accepted

### understanding the proof of stochastic dominance.

That's just a question of notation. Note that $$\mathcal{d}F(t) = f(t)\mathcal{d}t$$ Then the proof should be easy to understand.
• 181

### How to average CDFs of one variable across years

Why don't you just take a weighted average? Suppose you have ten years $t \in \{1,...,10\}$ and year $t$ has $N_t$ observations such that in total you have $\sum_t N_t=N$ observations. Let the year-$t$...
• 5,291
Accepted

### Understanding the properties of extensive form games

So $h$ is just some history of the game. Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities ...
• 1,899
Accepted

### What is the intuition behind Blackwell's Equivalence Theorem on Information Structures?

To answer the first part of your question, we do not need any more assumptions for the comparison of experiments (besides some measurability issues). Before going on, I'll fix some notations to ones ...
• 66
Accepted

• 29.3k
Accepted

### Bayes’ rule in "The sources of capital misallocation"

It follows immediately from Bayesian Updating using Bayes Rule for normal random variables. You can find derivations in e.g. Baley/Veldkamp (2021): Bayesian Learning See also the related answer at ...
• 529

### Resampling for a Probabilistic Model to Balance Outcomes

Imbalanced classes present minimal problem to proper statistical methods. A standard criticism of class imbalance is that it can result in models always or often classifying as the majority class. ...
• 410
Accepted

### How realistic is the conclusion that players do not change their mixing proportions in response to changes in their own payoffs?

I don't think the major lesson is quite that. If an expected payoff maximizing player mixes between several pure strategies, they must be indifferent between playing all of them. This has the curious ...
• 13.3k
Accepted

### Slight Uncertainty of Continuation in Repeated Prisoner's Dilemma

Imagine you have played the first 19 rounds. Now a chance event decides on whether there will be another, final, round. What's your optimal action in this last round, in case it actually occurs? ...
• 6,994

### Why are Mixed Strategy Nash Equilibria special cases of Correlated Equilibria and Coarse Correlated Equilibria?

Formally, a NE is indeed a different mathematical object than a CE. But, roughly speaking, the probability distribution on outcomes generated by a NE is a CE. This is what is really meant when saying ...
• 6,994
Accepted

### Third price auction from Auction Theory by Krishna, Order statistics

For questions like this, it can help to first consider the CDF. We can then take the derivative and have our density. I'll give you some hints and you need to fill in the gaps. Ask yourself: What is ...
• 5,291