What Would Be A Fair Price To Pay The Casino For Entering The Game

What would be a fair price  to pay the casino for entering the game (accordingly Bernoulli)?

 

1 " 4 1 8 1 16 The St. Petersburg paradox. A casino offers a game of chance for a single player in which a fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time heads appears. The first time tails appears, the game ends and the player wins whatever is in the pot. Thus, the player wins 2 dollars if tails appears on the first toss, 4 dollars if heads appears on the first toss and tails on the second, 8 dollars if heads appears on the first two tosses and tails on the third, and so on. Mathematically, the player wins 2k dollars, where k equals number of sequential tosses when heads appears. The expected value of the pot is E = 1 2. 2+ 4+ .8 + 16+...= 1 + 1 + 1 + 1+... = o. Assuming the game can continue if the coin toss results in heads and that the casino has unlimited resources, this sum grows without bound and so the expected win for repeated play is an infinite amount of money. Considering nothing but the expected value of the net change in one's monetary wealth, one should therefore play the game at any price to pay the casino for entering the game if offered the opportunity. Yet, in published descriptions of the game, people expressed disbelief in the result and never agreed to pay more than a few dollars to enter such a game the paradox is the discrepancy between what people seem willing to pay to enter the game and the infinite expected value of the pot. To resolve the paradox and explain the human behavior, Daniel Bernoulli suggested (in 1738) that it's important for people to increase not by a certain amount of money, but by a certain number of times. Given that the price for entering the game is Xo, he suggested that the proper expected value (he named it the utility function") describing the human behavior in the game should be E(X) = where X = 2 is the amount the player would win and is the probability of 1100 X - 1 2k such a win.