Paradoxical winning strategy in guessing numbers

In 1987, Thomas M. Cover posed the following astounding question in "Open Problems in Communication and Computation": Player \(X\) writes two different and randomly chosen natural numbers \(A\) and \(B\) on two different pieces of paper and places them face down on a table. Player \(Y\) now randomly chooses one of these pieces of paper, sees the number and must now decide whether this number is smaller or larger than the other number that is still face down on the table.

Player \(Y\) may not turn the face down card. He first lets the coin decide and has thus found a strategy with a winning probability of \(50\%\) . Is there another strategy with a higher probability?

Before player \(Y\) randomly selects one of the two pieces of paper, he determines an arbitrary natural number \(C\) . Then he turns over one of the two pieces of paper at random. Now he decides as follows: If the inverted number is \( \leq C \) , he selects the number on the other piece of paper as the larger one; if the inverted number is \( > C\) , he selects the number that has just been inverted as the larger one. Amazingly, the winning probability is now \( > 50\% \) .

We first set the designation of the two numbers to \(A < B\) and then, immediately after choosing \(C\ ), exactly one of the following three cases occurs:

  • 1st case: \( C \leq A < B \): Then the probability of winning is \( 50\%\), because there is no knowledge of \(A\ ) and \(B\ ).
  • 2nd case: \( A < B \leq C \) : Then the probability of winning is \(50\%\) , since there is no knowledge about \(A\) and \(B\) .
  • 3rd case: \( A < C < B \) : Then the probability of winning is \(100\%\) , because if \( B \) turned first, one stays with \( B \) and if \(A\) turned around first, you switch to \(B\) , so you always choose the larger number.

Surprisingly, this strategy is also used in everyday life: If, for example, when shopping, you have to decide directly for or against the purchase of a product without being able to obtain a comparative offer, you set a financial limit in advance. If this limit is met by the actual price, the purchase is made - otherwise not.