dimecres, 15 d’agost de 2012

The St. Petersburg paradox

Extracted from wikipedia: http://en.wikipedia.org/wiki/St._Petersburg_paradox,
from the book El libro de las Matemáticas of C. Pickover: http://divulgamat2.ehu.es/divulgamat15/index.php?option=com_content&view=article&id=11733:el-libro-de-las-matematicas&catid=53:libros-de-divulgaciatemca&directory=67
and from the article The St. Petersburg Paradox: http://plato.stanford.edu/archives/fall2004/entries/paradox-stpetersburg/.

In 1713, Nicolas Bernoulli, a member of the most famous mathematical family of the history (the Bernoullis), proposed the following problem:

Imagine a game of chance for a single player in which at each stage a coin is tossed. The pot starts at 1 dollar and is doubled every time a head appears. The first time a tail appears, the game ends and the player wins whatever is in the pot. Thus, the player wins $1 if the tail appears in the first toss, $2 if the tail appears in the second toss, $4 if the tail appears in the third toss, and, in general, $2^(k-1) if the tail appears in the k-th toss.

The question is: what would be a fair price to pay for entering the game?

Rational people would enter the game if the expected win is bigger than the price paid to enter the game. In this case, the expected win is:

E = 1/2*1 + 1/4*2 + 1/8*4 + 1/16*8 + ... = 1/2 + 1/2 + 1/2 + 1/2 + ... = infinite

So, if we follow the usual treatment of this kins of problems we would have to play the game at any price if offered the opportunity!

However, some studies showed that many people would pay more than $20. The paradox here is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value suggested by the above analysis.

For more information follow the links in the top of this post, to the Wikipedia, to the book or to the article.

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