Lesson 6: The fixed quantities paradox
Also known as the Two Envelope Paradox, and many other names, but the basic
problem that has puzzled and been the discussion of many mathematicians,
economists, philosophers, psychologists, etc. etc.
Basic setup: You are shown two indistinguishable envelopes, both of which holds a
positive amount of money. One envelope holds twice as the other. You may choose
one envelope and have whatever sum it holds. You choose an envelope at random
but right before you open the envelope you are offered the opportunity to take the
other envelope instead.
Here is the puzzle or argument in mathematical equations:
The exchanging argument: Now presume you justify as follows:
1. You represent by A the sum in your chosen envelope.
2. The chances that A is the smaller sum is 1/2, and that it is the greater sum is
also 1/2.
3. The opposite envelope may have either 2A or A/2.
4. If A is the lesser sum, then the opposite envelope holds 2A.
5. If A is the greater sum, then the opposite envelope holds A/2.
6. Therefore, the opposite envelope holds 2A with probability 1/2 and A/2 with
probability 1/2.
7. Thus, the expected value of the money in the opposite envelope is:
½(2A) +1/2(A/2) =5/4A
8. This is larger than A, so you profit on average by exchanging.
9. After the exchange, you then could gather that content by B and reason in
exactly the same way as above.
10. You can deduce that the most rational concept to do is to exchange back
again.
11. In being rational, you will perpetually end up exchanging envelopes forever.
12. It would seem more rational to open just any envelope than to exchange
forever, you have a paradox.
Within this paradox is the puzzle that compels some economist in the trading world,
so we must look at the puzzle itself a little more closely, and a few ways this puzzle
has been looked at being solved in over several ways depending on the field of
study. Let's take a closer look at
The puzzle: The puzzle is to discover the error in the very persuasive line of
deduction above. This involves concluding exactly why and by what
conditions which step is incorrect, it’s imperative not to make any errors in a more
complex situation where the miscalculation might not be so evident. In summary, the
puzzle is to decipher the paradox. Therefore, in specific, the puzzle isn’t deciphered
by the easy duty of discovering another way to evaluate the probabilities that doesn’t
direct to a contradiction.
The puzzle has many solutions depending on the why and what conditions. Those
two correlations determine the equations of the puzzle, but the solutions perpetually
are similar. Look at a few of the solutions especially ones through
economics/mathematician’s eyes and why this puzzle is important and studied in this
class.
Solutions
Numerous solutions to determining the paradox have been exhibited. The probability
theory triggering the problem is well identified, and some obvious paradox is
recognized when what is actually a conditional probability is treated as an
unconditional probability. A large diversity of comparable formulations of the paradox
are possible and have caused voluminous literature on the subject.
Forms of the problem have endured sparking attention in the areas
of philosophy and game theory; for example, here is a
Simple resolution
The total sum in both envelopes is a constant c=3x, with x in one envelope and 2x in
the opposing envelope.
If you choose the envelope with x first, you increase the sum x by exchanging. If you
choose the envelope with 2x first, you lose the sum x by exchanging, but you gain on
average
G=1/2(x) +1/2(-x) =1/2(x-x) =0 by exchanging.
Exchanging is no better than keeping the expected value
E= ½ 2x =1/2x = 3/2x is the same for each of the envelopes; therefore, there are no
contradictions.
Line of Reasoning
No matter how many times one looks at this puzzle, the puzzle in all forms of
psychology boils down to one person’s line of reasoning. What made a person
choose X or 2x in the first place or want to exchange. These are the variables that
are more important than the puzzle itself and when it comes to trading. How much
are you willing to risk? What is your root system variable? Do you know when to
enter and get out? These are the same variables that are in the paradox that
bludgeons mathematicians.
The line of Reasoning for Day Traders is a course of reasoning directed in
developing a methodical process of logical reasoning based on information given. In
the 2 envelope paradox, the line of reasoning says either envelope is willing to take
the risk on, you are going to win money in either case.