Monday, December 20, 2010

St. Petersburg, Part 2

This post is the second in a planned series on the St. Petersburg Paradox. Start reading from the first post.

In the first post I asked, "How much would you pay for a 1 in a billion chance to earn $1 billion?"

In this post, I explore the question, how much would you charge for a 1 in a billion chance to owe $1 billion?

Details below the line:

This is the flip side of the St. Petersburg paradox. Ian Hacking has suggested (see citations in the link) that the maximum someone might be willing to pay for a St. Petersburg gamble is about $25. Would anyone be willing to sell such a gamble at this price? Again, I'm considering a bounded variation of the paradox. In this case, I'll set the bound at 30 iterations, roughly $1 billion at risk.

Suppose a billionaire found 100,000 people willing to pay $25 for a St.Petersburg gamble.

There is 5/6 chance that the billionaire would owe less than the 2.5 million he collected in wagers. His average rake in those scenarios would be $800,000, or 32% of the wager.

The chance that the total winning would exceed double the total wagers is only 4%. The chance that a billionaire would go bankrupt selling 100,000 St. Petersburg wagers at a price of $25 is well under 0.1%.

This should bring us closer to an understanding of the paradox. From the perspective of both the buyer and the seller, this wager looks like a bad deal. If the seller underprices the wager by 17%, the buyer still sees a gamble that almost always favors the seller, even after a very large number of wagers. On the other hand, the amount at risk for the seller is enormous, no matter how much he is able to charge. The seller has to put over $1 billion at risk for winnings that are usually under $1 million.

I consider this a settled question. If the risks are fully disclosed, a St. Petersburg wager is unlikely to find both a willing buyer and a willing seller. The asymmetries of the gamble are too severe for both parties.

The more interesting question is, what happens if a real risk resembles a St. Petersburg wager, but neither the buyer nor the seller are aware of the risk? I'll save that for another post.

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