Understanding 'Absolutely Fair' in Statistics & Probability

[Biosyn]

What does the term 'absolutely fair' in statistics & probability mean?

Does it mean that the probability for each variable* is 0.
And that the expected value is 0. I did a search on Google and the only thing I came up with was this:
http://mathworld.wolfram.com/AbsolutelyFair.html

 

[Stephen Tashi]

Does it mean that the probability for each variable* is 0.

It doesn't mean that because it doesn't make sense to say "the probability for each variable is 0". Random variables don't have probabilities. It is their possible values that have probabilities.

 

[chiro]

Hey Biosyn.

To add to Stephen Tashi's comments, I think you should consider the context of what the random variable is in.

One way of assessing "fair" is that probabilities are equal in a random variable. If this is discrete uniform with N states then P(X = x) = 1/N can be considered "fair" as can a continuous uniform with P(X = x) = 1/N in the interval [a,a+N].

In finance, we consider fair to be one where there is no risk of arbitrage or a "free lunch" under specific assumptions (which may not be right in practice).

We also consider fairness in the context of unpredictability or pure-randomness where no information at all helps predict the outcome and this is a case of maximum entropy which when used in the proper context (i.e. for discrete random variables with a finite number of outcomes) yields the uniform distribution in discrete state-space.

 

[lavinia]

In finance one has the idea of a fair game. The naive model is a random walk or a Brownian motion but more generally the model is called a martingale. This is a game where the current value of a random variable(e.g. the price of a security) gives you the best estimate of its future value after you have accounted for all of the relevant information about it.

This means that any change in value e.g. change in the price of a stock is completely unpredicatable. The game would be unfair if some players had special information that was not generally available that enabled them to get an edge in the betting.For instance insider trading or knowledge of price patterns that were difficult to find and not generally seen by the marketplace.

 

[Stephen Tashi]

I did a search on Google and the only thing I came up with was this:
http://mathworld.wolfram.com/AbsolutelyFair.html

which says:

Absolutely Fair

A sequence of random variates $X_0, X_1, ...$ is called absolutely fair if for $n=1, 2, ...,$
$<X_1>=0$

and
$<X_{n+1}|X_1,...,X_n>=0$

(Feller 1971, p. 210).

That definition is expressed in terms of "expectations" of random variables rather than probabilities. It has a misprint since the first condition should say $<X_n> = 0$ instead of $<X_1> = 0$. The notation $<X_n>$ refers to the expected value of $X_n$.

This definition differs from the definition of a martingale.

 

[lavinia]

which says:



That definition is expressed in terms of "expectations" of random variables rather than probabilities. It has a misprint since the first condition should say $<X_n> = 0$ instead of $<X_1> = 0$. The notation $<X_n>$ refers to the expected value of $X_n$.

This definition differs from the definition of a martingale.

Steve it seems to me that this definition is the same as a type of Martingale. If one takes the new process which is the sum of the X's then the expectation of the sum in the next period is its current value.

 

[lavinia]

If a betting game is fair them your current winnings should be your expected future winnings.