What exactly is the paradox in St. Petersburg paradox?

Question

Currently I am reading St. Petersberg paradox.

However, I do not see the paradox.

Can someone explain to me why is this considered a paradox?

Answer 1

Mathematically, this is not a paradox, very much how mathematically, the birthday paradox is not a paradox.

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There are two kinds of paradoxes in mathematics. The first type is the logical paradox, in which a certain set of axioms turns out to allow proofs of false statements, i.e. statements like "$A$ and not $A$". Probably the most famous paradox in this category is Russel's paradox.

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The second kind of paradox is a meta-mathematical paradox, where something that is mathematically true doesn't make intuitive sense when it is applied to the real world. Examples of this include the St. Petersburg paradox, which shows that even though the game described looks like it should be fairly cheap to buy in, but in fact, mathematics shows us that no matter what the buy-in value is, the house will always lose.

That is, if you offer the game with a $1000$ dollar buy in to any one person, that person will most likely refuse to play. However, if you force everybody that has 1000 dollars to play your game, then you, as the house, will very probably lose a lot of money.

Answer 2

It is a paradox because, although it can be easily proved that anyone should play that game no matter how much that person would have to pay for that, in practice almost everyone would be unwilling to pay more than a small amount to do that.