Saint Petersburg Paradox

Consider the following game:

  • a "banker" $B$ plays with a "player" $P$
  • $P$ must pay some fixed sum to enter the game
  • $B$ flips a coin until TAILS appear - then the game stops
  • $B$ pays $2^n$ euro to $P$ if the game stops on $n$th toss

How much a rational should be willing to pay to play the game?

  • Expected Value could be a good model to estimate the costs for entering the game
Seq Gain Prob
$T$ $2^1$ $0.5$
$HT$ $2^1$ $0.5^2$
$HHT$ $2^2$ $0.5^3$
... ... ...
$\underbrace{H..H}_{n}T$ $2^n$ $0.5^n$

Let's calculate the expected value

  • $EV = 2 \cdot \cfrac{1}{2} + 2^2 \cdot \cfrac{1}{2^2} + 2^3 \cdot \cfrac{1}{2^3} + ... = \infty$
  • so the expected value is $\infty$ with just 50% chance to win only 2 euro
  • clearly this is not good enough to model such situation


Sources