The St.. Petersburg paradox

The St. Petersburg lottery:

... is a game of chance which costs c coins to play and pays back to the player exactly one of the following prizes: 1 coin with probability 1/2, 2 coins with probability 1/4, 4 coins with probability 1/8, 8 coins with probability 1/16, etc.  The sequence of prizes 1,2,4,8,16,32 lists the powers of two.  The sequence of probabilities 1/2, 1/4, 1/8, 1/16 is chosen so that the sum of all probabilities is 1.  I discuss the importance of this -- that the prizes are exclusive, and one prize is guaranteed extensively in my next post.  Without this condition, I believe that the paradox can be resolved.

The mechanism of the lottery could be as follows: a fair coin is flipped.  If it lands on one side, the pot is doubled.  If it lands on the other side, the pot is paid to the player.  The St. Petersburg paradox is that there seems to be no good answer to the questions: What cost c will attract players to the lottery and what costs c will attract a casino to offer this game?

The paradox:

Informally, the paradox of the St. Petersburg lottery is that no one wants to play it -- neither as the casino nor as the player.  Formally, the paradox is that the expected payoff to the player is infinite, so the player should be willing to pay an infinite sum, in order to be able to play the game.  On the other hand, the player will surely receive only a finite payoff on each game; a player who paid an infinite sum cannot possibly win it back or ever profit from the game.

The game violates the notions that you can put a price on anything and that a game is favorable to one player or to the other -- this game scares off both players.  For many games, we can vary the cost, and we find that for high values of c, a casino would happily offer the game and win consistently; for low values of c the player would have the advantage and win consistently, and that for many values the casino is happy to offer the game and win consistently, while some players are happy to play.

The "winnings - odds lemma" :

Consider a simple gamble in which a player stands to lose a or win b with odds p:q.  The "lemma" is that the game will find players if q*b > p*a.

For a hypothetical player H who wins q times and loses p times, the odds equal the ratio of losses to wins; this player would win q * b and incur losses of p * a.  When p and q are small natural numbers, a real-life player may find that his experiences amount to the repetition of H's experiences, plus a random variation, the monetary value of which grows more slowly than the value of repeating H's experiences (more or less as the square root).  When p and q are small, the game will have players when q*b > p*a.

Utility and lotteries:

What happens as the odds against winning become extremely large?  Present-day lotteries find players for games for which q*b = 0.5 * p*a.  For instance, the lottery may offer million-to-one odds of winning half a million.  We could call this the Lottery Paradox.  It is a cliché that such a lottery imposes a "tax on bad math skills."  But economists insist of believing that people in society are in fact playing winning strategies, and that when the strategy seems senseless, then we should first attempt to find an explanation for the behavior, before giving up the assumption that people act rationally.  To play million-to-one odds of winning half a million could be rational if money had increasing utility, or if people viewed small possibilities in a "hopeful" way.  On the other hand, the correspondence of early mathematicians suggests that they expected money to have a decreasing utility, or that people would view small possibilities with distrust (http://www.cs.xu.edu/math/Sources/Montmort/stpetersburg.pdf).

The "expected value" lemma :

A player should play a complicated game if his expected winnings exceed the expected losses.  A special case of this lemma is the winnings - odds lemma stated above. 

Correspondence (my selection and paraphrases from the source cited above):

Montmort (Paris): [Despite the paradox] I cannot resolve to abandon our lemma, which must be generally be true.  (The lemma being that when expected winnings exceed expected losses, the game is favorable to the player.)

Cramer: Utility is finite.  Wealth above 2^24, or 16,777,216 coins is all the same to a player; this alone shows the game is not worth 13 coins.  And if the utility of wealth varies as the square root of that wealth, then the game is worth no more than 3 coins.

N Bernoulli: Perhaps gamblers ignore, and should ignore, very small probabilities.  If players all suppose that odds such as 31:1 are as good as 32:0, then they would play the game for 2 and a half coin, and no more.

Daniel Bernoulli: The "expected value" lemma ignores Risk.  The masters of the Bernoulli clan lost all to the bankruptcy of Mr. Muller.  Chasing the best expected return would not have prevented this.  I have written a monograph on Risk and I am sending it to you.  This monograph explains how to avoid disaster when investing.

Nicolas Bernoulli to Daniel Bernoulli: Thanks for sending me your manuscript.  Cramer wrote to say that a millionaire gains little from an additional thousand.  You point out that a family accustomed to millions might lightheartedly win or lose thosands, but should be less light-hearted about losing "all."  Yes, the expected value of 1000 coins, invested with 9:1 odds against bankruptcy, is 900 coins; the expected value of those coins, invested in two businesses, each with 9:1 odds agaisnt bankruptcy is the same.  It is very true and we know it without paying attention to your principle, that one does nonetheless better to place 500 coins in 2 places, than 1000 coins in a single place, since the chance of losing all is then 1 in 100, rather than 1 in 10.  But you would not have done better if you had been in charge of investing our money, seeking to earn interest on it, and without the chance to divide it into small pieces.

The limits of real money:

A player could play with money borrowed from many investors.  Hundreds of millions of investors can share ten trillion (ten million million) coins as dividends of a hundred thousand coins each.  Such dividends are low enough that individuals can easily comprehend how to spend them and get utility.  Windfalls like this occur in my lifetime to the citizens of small nations with stable democracies when tremendous natural resources are discovered there.  Even the trickle-down effect of a successful banking enterprise in a small country can be worth far more to its citizens than a single windfall of a hundred thousand coins.  In this way, each additional coin is enjoyed by one of the investors in such a way that the coin can have high utility -- it is no one person's ten-trillionth coin.  Of course, Cramer could have imagined this.  In 1713 the population of the globe exceeded half a billion people.  Personal investment and massive transfers of wealth between nations occurred (less than 100 years later, the Louisiana Purchase was valued at tens of millions of dollars or francs).  Utility does not so much diminish as become complicated, involving more people, so long as the winnings and losses in question do not exceed the net aspirations and wealth of the poor of the earth.  We can imagine sums of money comparable to the world's net wealth.  Europe could trade wealth with China -- we can imagine transfers of this magnitude.  Should China play odds of a trillion to one to win 1.1 trillion from the USA?  If the payouts were capped at a trillion, then we've raised the value of the truncated game slightly -- from 12 to 20 coin-- where the truncated game offers all the prizes from 1 to 2^n, with the same probabilities, but it offers none of the larger prizes.

Simple winnings and odds, for nations : 

Consider this simple gamble in which the player stands to lose 1 coin or win 1.1 trillion coin, with odds of a trillion to one.  As this is a finite game, our prejudices become theorems: whatever the relationship between utility and money, and however we hopefully overestimate or underestimate probability, there is a certain price at which this game is fair.  Below that price, it is to the player's advantage, and above that price it is to the casino's advantage.  As ambassador, I probably would accept advantageous odds to spend pocket change and possibly win part of the global GDP for your nation. 

Dream big.

With the right bet, you have a chance to satisfy a child's rhetorical request for a Christmas present such as "let everyone eat enough food this year" -- if you won the right to manage this year's global GDP, you could scrap every piece of military hardware produced this year and lay them as the foundation for an awesome scuba-diving playland.  That's probably not your child's dream, but if you gambled and won "the rights to everything produced this year in the world" you could make a very strange dreams come true.  We all hope you tread carefully on the economy during your year of absolute control!

Tell me the way to the St. Petersburg Casino!

I would definitely pay 40 coin to play at the same time each of the lotteries "2^n : 1 odds of winning 2^n" for n from 1 to 40.  I would also pay 40 coin, the fair market price, to play the finite St. Petersburg lottery with 40 coins.  I would like to buy many of these and sell the separate lotteries to different users.  The small lotteries I would roll back into the game.  The million-coin lotteries, on the scale of a massive government-protected jackpots I would re-sell at a profit.  The larger lotteries make wonderful Christmas gifts to people with big dreams; I'd attach a few coins to some of my own dreams.