- - - - - - - - - - - - - - - - - - - - - - - - Investigating the St . Petersburg Paradox with Maple

The Saint Petersburg Paradox is a well known illustration ofhow the psychology of risk and reward can diverge from the underlying mathematics. In the standard game a fair coin is tossed repeatedly until the first head occurs. The payoff is $1 ifthe game terminates in one flip, $2 ifin two flips, $4 if in three, and so on, doubling the payoff for each additional flip required to obtain the first head. The basic analysis of the game is a straightforward computation involving expected value, which turns out to be infmite (despite the fact that few individuals are willing to pay more than a few dollars to play). When one looks more closely at the game from the standpoint of the likelihood of making a profit after many repeated plays, however, more complex and interesting questions and computations arise. We use Maple to investigate these and other questions relating to the Saint Petersburg Paradox. This approach sheds some light on the nature ofthe paradox, because the human psychologyofrisk is often connected more closely to the probability ofloss, rather than the somewhat more involved notion of expected value. The computations associated with finding exact probabilities are intractable enough that the computational power ofMaple is a necessity. We also use Maple in our investigations to simulate repeated plays of the game, and check these against the theoretical probabilities. The mathematics and Maple used is at the level of an undergraduate special project.