A Guide to the Risk-Averse Gambler and Resolving the St. Petersburg Paradox Once and For All

A Guide to the Risk-Averse Gambler and Resolving the St. Petersburg Paradox Once and For All
By Lucy Martinez and Doron Zeilberger
.tex
.pdf
Journal version .pdf
[Published on-line before print in the College Mathematics Journal, volume and page number tbd]

We use three kinds of computations: Simulation, Numeric, and Symbolic, to guide risk-averse gamblers in general, and offer particular advice how to resolve the famous St. Petersburg paradox.

Written: July 2023

Maple package

StPete.txt, a Maple package to advise the risk-averse gambler with particular applications for the St. Petersburg paradox.
You also need to put in the same directory in your computer the following file:
StPeteData.txt,

Sample Input and Output for StPete.txt

If you want to see advise for the risk-averse gambler on how many times to insist on playing the gamble : You lose one dollar with probability (i-1)/i and win i dollars with probability 1/i (or equivalently: You lose i dollar with probability (i-1)/i and win i² dollars with probability 1/i for i from 2 to 15 (without the very complicated recurrences, that are distracting)
the input file yields the output file

If you want to see the above information with the very complicated recurrences,
the input file yields the output file

If you want to see advise for the risk-averse gambler how many times to insist on playing various finite forms of the St. Petersburg gamble:
the input file yields the output file

If you want to see more succinctly, advise for playing "You lose one dollar with probability (i-1)/i and win i dollars with probability 1/i" for risk-averseness, for i from 2 to 15 1/10,1/100, 1/1000, 1/10000,
the input file yields the output file

If you want to see advise for playing "You lose one dollar with probability (i-1)/i and win i dollars with probability 1/i" , for i from 2 to 20, for risk-averseness, but using the Central Limit Theorem Approximation (so you get cutoffs slightly bigger than necessary) for risk averseness 1/10,1/100, 1/1000, 1/10000,
the input file yields the output file

If you want to see advise risk-averseness cutoffs for the finite versions of the St. Petersburg game, with i rounds, for i from 3 to 20, using the Central Limit Theorem approximation (so you get cutoffs slightly bigger than necessary) for risk averseness 1/10,1/100, 1/1000, 1/10000,
the input file yields the output file

Pictures

Here is the risk-averseness graph for the gamble [[-1,1/2],[2,1/2]]

Here is the risk-averseness graph for the gamble [[-1,2/3],[3,1/3]]

Here is the risk-averseness graph for the gamble [[-1,3/4],[4,1/4]]

Here is the risk-averseness graph for the gamble [[-1,4/5],[5,1/5]]

Here is the risk-averseness graph for the gamble [[-1,5/6],[6,1/6]]

Here is the risk-averseness graph for the gamble [[-1,6/7],[7,1/7]]

Here is the risk-averseness graph for the gamble [[-1,7/8],[8,1/8]]

Here is the risk-averseness graph for the gamble [[-1,8/9],[9,1/9]]

Here is the averseness graph for the gamble [[-1,9/10],[10,1/10]]

Here is the EXACT averseness graph for the 7-round St. Petersburg Gamble with Entrance Fee 7, for n from 1 to 300.

Here is the Approximate averseness graph for the 7-round St. Petersburg Gamble with Entrance Fee 7, for n from 1 to 2000, using the Central Limit Theorem approximation.

Here is the Approximate averseness graph for the 11-round St. Petersburg Gamble with Entrance Fee 11, for n from 1 to 10000, using the Central Limit Theorem approximation.

Articles of Doron Zeilberger
Doron Zeilberger's Home Page
Lucy Martinez's Home Page