The two-note play or also two-envelope problem optimizes the probability the larger from two numbers to to find, of which nothing admits is, except that it are different.

Problem

In the formulation set up by Cover (1987) it reads: Player 1 writes two arbitrary, different numbers on notes. Player 2 selects coincidentally one of it and regards the number. Player 2 must decide now whether the selected number is the larger. The probability is P = 1/2, seems better it not to go.

A practical example is that of the house selling with two prospective customers, whereby one cannot return on refusal of the offer no more on the prospective customers. To that extent the problem definition a little reminds of the secretary problem with the problem size n = 2. Further possibilities in the daily life are a special offer in the supermarket, a dwelling, working premises, the partner for the life, etc. always stand one before the dilemma for or against an alternative to decide to have itself, without knowing whether still another better opportunity does not come.

Strategy

Determine (arbitrary, coincidental and of the other two numbers independent) a third number of Z and select the well-known, first number of X (first offer), if X is larger than Z, otherwise select the unknown, second number of Y (second offer).

Solution

X is the number selected first (first offer), Y the unknown second number (second offer).

Now three cases can arise:

• A) Z is larger both and X and Y, or at the most equal in size. Thus X \ leq Z and Y \ leq Z.
• B) Z lies between X and Y (inclusively). Thus X \ leq Z \ leq Y or <math>Y \ leq Z < X</math>.
• C) Z is smaller both and X and Y. thus X > Z and Y > Z.

In the case A) the choice falls because of X \ leq Z on Y, in the case C) because of X > Z on X. in both cases is that in the long run a coincidence choice and the probability 1/2. In the case B) Z falls \ leq Y the choice on Y, for <math>Y \ leq Z < X</math> on X. with security one the larger number met for X \ leq.

Presupposed the events A), B) and C), with its probabilities P (A), P (B) and P (C), are independent of the choice of Z, arise the probability the largest number to meet P (G) out:

P (G) = \ frac {1} {2} P (A) +1*P (B) + \ frac {1} {2} P (C) = \ frac {1} {2} \ left (P (A) +P (B) +P (C) \ right) + \ frac {P (B)}{2}

There however one of the events A), B) or C) to occur must, is P (A) + P (B) + P (C) = 1. For the total probability follows

P (G) = \ frac {1} {2} + \ frac {P (B)}{2}

P (G) becomes now maximally, if P (B) maximum will to thus find the probability a Z which lies between X and Y. But no general rules can be indicated, a good choice depend on the concrete situation. In case of the house selling one should an idea of the market value have and Z "“somewhere"” in the proximity settle. In case of a distribution over all real numbers Z = 0 can be a good choice.

Applicability in practice

The range of application of the two-note play in practice is smaller, than one would assume first. Because the compelling premise, i.e. that the two numbers must be arbitrary in each case, might not be given usually. With the entrance example of the house buying for example there is a Spannbreite, within those the numbers appears meaningful, however are - with a normal house - for example numbers of smaller than 10,000 or of more largely than 10 million extremely improbable. Anyhow it is missing at a uniform distribution of the probabilities. Around such probabilities to measure however foreknowledge is necessary. The difference becomes clear with a purchase in the supermarket: If a housewife recognizes that a product is offered for a price, which was undercut never before, it becomes the probability the fact that this product in another shop is attractive offer, than very small estimates an application of the two-note play would be paradoxically. A customer however, who did not buy this product ever, and who has also otherwise no advance informations, can very probably according to the method of the two-note play proceed.

See also

Used topics, with which one can make the optimal decision of the remainder problem from partial information:

• Secretary problem
• Goat problem
• Odds strategy

Likewise related, to paradoxical solution:

• Conversion paradox

Literature

• F. Thomas Bruss: The uncertainty a Schnippchen strike. In: Spektrum der Wissenschaft. 06/2000. Spektrum der Wissenschaft publishing house company, P. 106
• Cover, T. (1987): Open of problem in Communication and Computation. Problem of 5.1: Pick the largest NUMBERs. Springer publishing house, New York. (English)

Related links

• http://www.wissenschaft-online.de/abo/spektrum/archiv/4947

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