The essence of St Petersburg paradox

Recall the St Petersburg paradox from an earlier blog entry. Consider the following variant of St Petersburg game. In the new game, the payoffs are

1 dollar with probability 1/2,

2 dollars with probability 1/4,

4 dollars with probability 1/8,

100 times (2 to power 1,000,000,000,000,000,000,000,000)

dollars with probability

2 to power -1,000,000,000,000,000,000,000,000,

and with the remaining probability, the reward is 0. The expected value of the reward is

1 times 1/2 plus 2 times 1/4 plus 4 times 1/8 plus

100 times (2 to power 1,000,000,000,000,000,000,000,000)

times

2 to power -1,000,000,000,000,000,000,000,000

= 1/2 + 1/2 + 1/2 + 100 = 101.5.

In my personal opinion, the "true" value of the payoff is

1 times 1/2 plus 2 times 1/4 plus 4 times 1/8 = 1.5.

In other words, I completely ignore the enormous payoff that has extremely small probability. My guess is that a number of other people would feel the same. My justification is that the probabilities in the range from 1/2 to 1/8 are common in everyday life. It makes sense to maximize the expected payoff corresponding to such probabilities because of the Law of Large Numbers. Maximizing the expected value in many decision situations with moderate payoffs and moderate probabilities results in the best possible long term trend.

On the other hand, the probability of the enormous payoff in the above game is so small that it should be ignored. Even if we add probabilities of all events relevant to the life of an individual which have similarly small magnitude, we will obtain a very small number. It is safe to assume that none of these events will ever happen.

I do not feel that the above game presents much of a practical or philosophical challenge. The essence of St Petersburg paradox is that it involves rewards and probabilities on all scales. Some of these probabilities are too small to hope that the Law of Large Numbers will translate them into a clear trend in the life of a single individual. But these probabilities are too big to be negligible.

Intermediate probabilities and rewards do occur in real life. Buying a house or choosing a college can result in substantial gains or losses. The Law of Large Numbers does not apply in such cases to decisions made by a single individual because only a small number of decisions with similarly large rewards occur in a human lifetime. On the other hand, the probabilities involved in these decision problems are too large to be neglected. A possible philosophical choice for an individual is to identify himself with a group of people, for example, all citizens of a country. This allows one to invoke the Law of Large Numbers but the voluntary identification of one's welfare with that of a group is far from the universal choice.