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