The St. Petersburg Paradox

Imagine a simple coin-tossing game where you get paid $1 if you toss heads on the first toss, $2 if you get heads on the second toss, $4 on the third toss etc. The mathematical expectancy of this game is:

1/2 times $1 + 1/4 times $2 + 1/8 times $4 etc….

This means the game has a minimum payout of $1 and an infinite expectancy. An interesting paradox arises because if the game has infinite expectancy then it would seem to be reasonable to pay any amount to play the game. In reality, this is not a good idea so the question that requires an answer is:

How much should one pay to play each turn of the game?

This simple game is known as the St. Petersburg paradox as first devised by Nicholas Bernoulli. It is discussed in William Poundstone’s Fortune’s Formula (which is #12 on our ‘Trader Must Read Top List’ by the way) and addresses the question by assuming that the game is impractical because no-one can actually offer an infinite payout. If one changes the game to be capped at say, $1 billion, then the expectancy is reduced to just under $16. To me this seems like a poor answer to the dilemma that ‘cheats’ by adjusting the rules of the game, and gives an answer that is still too high a price to pay for playing (based on common sense and ‘gut feel’).

Note that none of this is directly related to position-sizing since the game has a minimum $1 payout and no maximum payout – there is no chance of loss and the payout does not increase with the amount wagered. This is a case where position-sizing algorithms do not have a direct bearing on the problem since the ‘correct’ amount to pay for a turn is fixed.

I have found that many times in trading (and other games of chance or gambling), there is a gulf between mathematical theory and practical reality that can be effectively bridged by simulation. In this case if one simulates the game as described a large number of turns (10,000 for example) and takes the total amount won divided by the number of turns, this average win per turn will be a good indication of the practical true value of a turn.

The results I got from my simulations were that a turn was worth between $5 and $9, so if a casino offered this game I would play if it cost less than 5 times the minimum win of $1. If you used a geometric mean instead of arithmetic mean you would get an even more conservative (lower) estimated value of a turn.

So, how does this relate to trading? Well, one can think of the future price of any equity in terms of mathematical expectancy. If you estimate that an equity has a 50% chance of increasing 100% in value, but only a 25% chance of decreasing 40% in value, then if it is currently selling for $100, the future expected price is:

$100 + (0.5 100) – (0.25 40) = $140

Therefore one could buy the equity at the current price of $100 and expect to make a profit of $40. Of course, these calculations are always completely dependent on accurate estimation of the probability of future price increases or decreases, which is not an exact science like the probability of a coin toss coming up heads or tails.

In my opinion, making money at trading partly relies on a more accurate estimation of these kinds of probabilities than other market participants in order to find ‘inefficiently-priced’ equities to trade.

Paul King
PMKing Trading LLC
www.pmkingtrading.com