Exchange
Paradox
The
exchange paradox, also known as the two envelopes problem, is a puzzle in logic
and probability which originated as a variation of the necktie paradox
(Hausmann, 2001). The exchange paradox is presented by formulation of a
hypothetical challenge, which presents two contradicting situations taking the
following form:
Two
identical sealed envelopes contain money, where the sum in one envelop is twice
the sum in the other envelop. They are presented to two players for picking.
Whatever envelop a player chooses, he owns the amount of money in it. Before
looking into the amount of money in the envelop a player chooses, he is offered
an option of swapping the envelopes.
The
first player may reason as follows: If
his envelope has $2x and with a 50% probability of choosing either envelope,
the other envelope contains either ½($2x) or ½($4x). The expected value of the
amount of money in the other envelope is ½($2x) + ½($4x), resulting to $3x.
This amount is greater than his current envelope’s $2x. In such a scenario, he
decides to swap the envelopes. The second player reasons identically and the
swapping keeps on.
Some
authors argue that the flawed calculation of the expected value is caused by
the insufficient information about the amounts of money in the envelops. Before
the players do the calculation, information on prior distribution of money in
the envelope is necessary. For example, if a player knew that the amounts in
the envelopes were $2 and $4, and saw the $2 envelope, he would know the other
envelope had $4. He will then swop. This case does not present a paradox.
In
as much as many authors have presented solutions to this paradox, none has been
widely accepted to be definitive (Epstein, 2012).
References
Epstein, R. A. (2012).
The Theory of Gambling and Statistical Logic. San Diego: Elsevier Science.
Hausmann, W. (2001). On
the two envelope paradox. Friedberg: Fachhochschule Gießen- Friedberg.
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