The St Petersburg Game's Win-Loss Ratio

  It is interesting to look at the win-loss ratio for the n-toss St Petersburg game. We start by determining the value k for which the pot equals the wager or when 2^k-1 = w and then define κ as this value of k lowered to its nearest integer. The determination of the win-loss ratio is as follows.

The key formulas with 2^κ approximately equal to n are,

So as the number of tosses becomes indefinitely large as in the St Petersburg paradox the win-loss ratio approaches zero. The contradiction is that one can expect to win an infinitely large amount while at the same time have relatively no chance of winning. The win-loss ratio needs to be considered in addition to the expected value in order to determine a game's fairness. Games with higher win-loss ratios are more alluring to prospective players.