A fair coin will be tossed until a heads results. You will then be paid $2^{n-1}$ dollars where $n$ equals the number of flips. Now why is the expected pay out infinite? $$ \sum_{n \geq 1} (\frac{1}{2^n})2^{n-1} = \sum_{n \geq 1} \frac{2^{n-1}}{2^n} $$ Why does this give the payout? The payout shouldn't be infinite, it should be a potential infinity. For example, if I flip the coin once and it is a tails, then I will receive two dollars for certain. On the second flip there is a $25$ percent chance that I will receive a minimum of $4$ dollars. but a $75$ percent chance that I will receive only two dollars. And so on as the chances of receiving more money decreases and eventually approaches $0$.
Why then is it suggested that an investor pay any amount of money for such an opportunity?
Source: A Brief History of the Paradox, Philosophy and Labyrinths of the Mind
Obviously this appears false by intuition, but I'd like to know the argument for the paradox; why one would pay a trillion dollars to enter such a game. The argument just seems far too fallacious to me.
How many times would one have to play? After almost twenty thousand plays, we are still making less than 8 dollars. IF every twenty thousand plays we increased our winnings by 8 dollars, we would have to play hundreds of billions of times just to start to have an average earnings equal to one trillion… we'd run out of time in the universe it seems…
Edit I am curious, however, if the overall slope of the simulation data would remain consistent, if so why? If not, why? What accounts for the vertical jumps we see occurring less and less often? Why do they occur at all?
Best Answer
Suppose you pay a trillion dollars to enter the game. The following table contains some of the values of the net payoff you can possibly end up with and the corresponding probabilities:
\begin{align*} \begin{array}{rr} 1/2&-\,1\mathord{,}000\mathord{,}000\mathord{,}000\mathord{,}000\\ 1/4&-\,999\mathord{,}999\mathord{,}999\mathord{,}998\\ 1/8&-\,999\mathord{,}999\mathord{,}999\mathord{,}996\\ 1/16&-\,999\mathord{,}999\mathord{,}999\mathord{,}992\\ \vdots\\ 1/2^{10}&-\,999\mathord{,}999\mathord{,}999\mathord{,}488\\ \vdots\\ 1/2^{20}&-\,999\mathord{,}999\mathord{,}475\mathord{,}712\\ \vdots\\ 1/2^{30}&-\,999\mathord{,}463\mathord{,}129\mathord{,}088\\ \vdots\\ 1/2^{40}&-\,450\mathord{,}244\mathord{,}186\mathord{,}112\\ 1/2^{41}&99\mathord{,}511\mathord{,}627\mathord{,}776\\ 1/2^{42}&1\mathord{,}199\mathord{,}023\mathord{,}255\mathord{,}552\\ \vdots\\ 1/2^{50}&561\mathord{,}949\mathord{,}953\mathord{,}421\mathord{,}312\\ \vdots\\ 1/2^{100}&633\mathord{,}825\mathord{,}300\mathord{,}114\mathord{,}114\mathord{,}699\mathord{,}748\mathord{,}351\mathord{,}602\mathord{,}688\\ \vdots\\ 1/2^{200}&803\mathord{,}469\mathord{,}022\mathord{,}129\mathord{,}495\mathord{,}137\mathord{,}770\mathord{,}981\mathord{,}046\mathord{,}170\mathord{,}581\mathord{,}301\mathord{,}261\mathord{,}101\mathord{,}496\mathord{,}890\mathord{,}396\mathord{,}417\mathord{,}650\mathord{,}688\\ \vdots \end{array} \end{align*}
The paradox lies in the following observation. If you take out a loan of one trillion dollars to play this game, you will go bankrupt with a very large probability. However, once in a lifetime (not even of a human but of the universe) you win an unspeakably large amount of money, so large you can't even imagine.
In the light of this observation, a reasonable person would never play this game only once. It is worth playing only if you can play it indefinitely, while you have access to unlimited borrowing. What will happen is that you will keep playing for billions of years, accumulating an enormous debt using your infinite line of credit. But after a very long time, you will win so much money that is sufficient for you to pay off this large debt and still purchase the whole world. As @IanColey put it, this is because the chances of winning so much money are very, very tiny, but the payoffs associated with these very, very tiny probabilities are much, much, much more enormous than the probabilities are tiny.