Friday, June 10, 2011

The St Petersburg paradox; a resolution in favor of gambling

Utility companies buy and sell energy.

When money runs out, we can ask to be paid in joules of energy, or Kilowatt-hours.  If we are paid more energy than we can use, we can use a few joules to build machines which could use the total sum.  Is there utility in smashing protons?  Utility in interstellar travel?  G8 countries consider these things to have positive utility.  Engineers and physicists can propose ways to use any amount of energy.  Imagine if the public could fund big science via a fixed-price lottery with infinite expected return on investment!  I would vote to buy that investment. I conclude that infinite and finite versions of the St. Petersburg lotteries will attract players who want to spend some part of their current utility on a chance to visit Alpha Centauri (e.g., via Project Orion).  Thus, like any lottery with giant payouts:

The St. Petersburg lottery will have players at any price which players can afford.

Combinatorics matters:

Consider two games: The Petersburg lottery "independent-style" and the Petersburg lottery "guaranteed".  Game PG ends when you win any prize, and you are guaranteed to eventually win some prize.  Game PI never ends.  Winning one prize does not stop the game.  Suppose one prize is revealed each year.  Game PI is, clearly, worth one coin per year to play... for, each year, the expected winnings is 1 coin.  With game PG, in which the prizes are disjoint, exclusive, and guaranteed, has the same expected value, much less risk, and is guaranteed to leave you feeling an infinite regret.

Infinite regret:

The independent lottery is manifestly fair: each year it pays out an expected one coin, equal to your one-coin investment.  The Bernoulli "guaranteed" lottery is different: each year you anticipate an even bigger reward.  But you know in advance that one year you will win.  After n years of paying 1 coin per year, you will then earn 2^n coin.  You could spend half of your earnings every year.  n years later, you would be bust.  You would then continue to pay forever, 1 coin per year, for the pleasure of having played, and won, the Bernoulli lottery.  This is why Bernoulli's lottery is a paradox: the independent lottery is fair.  Bernoulli's lottery has the same expected payoff, lower risk, and yet you are sure to regret having played it.  If we consider a payment scheme other than "one coin per prize," we find that if the sum of the payment scheme (over all time) is finite, the casino's expected loss to the player is infinite; if the payment scheme is infinite, the player's expected loss is infinite. 

On the other hand, if the player could invest 2^n coins and pay the infinite cost of the game from the interest earned, then things are different.  Of course, the value of a lottery depends on what other investments are available -- competing for the price of the lottery, and available for investing the winnings.

Bernoulli's paradox:

The St. Petersburg lottery is completed-infinite, and you are guaranteed to win one prize.  Reasoning rationally, the kind of person who is happy to play a lottery for tremendous payoffs in joules would much prefer to play PI than PG.  But PG is offered.  The player who will sign the infinite contract -- to pay 1 coin per year for the pleasure of having played -- will end up with n joules of investment and 2^n joules of largesse, followed by infinite regret.

Bernoulli succeeded in showing that there is another variable to be considered when comparing games... not only expected value, and not only risk, but something else, which can be coded into the combinatorics of the game.  By combinatorics, in this case, I mean the condition that you can't win more than one prize -- the condition that the prizes are disjoint.

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