Please see a description of the St Petersburg paradox in an earlier blog entry. Many of the philosophical questions related to probability can be illustrated with a game much simpler than the St Petersburg paradox. Let G be a game which pays a reward
50 times (2 to power 1,000,000,000,000,000,000,000,000)
dollars with probability
2 to power -1,000,000,000,000,000,000,000,000.
With the remaining probability, the reward is 0. The expected value of the reward is 50 dollars. Would you buy a ticket for 40 dollars to play this game?
Game G may seem to be completely different from St Petersburg paradox because everything about this game is finite - there is only one non-zero reward and its value is finite. However, a typical philosophical analysis of St Petersburg paradox touches upon the same problems that are raised by game G: What is the utility of a huge number of dollars? Is utility a bounded function? Can or should we use the frequency interpretation of probability? Would any casino offer this game? If yes, can such an offer be considered serious? If not, does it make sense to discuss the paradox? How does the paradox change if we replace the reward denominated in dollars into a reward denominated in utility units? Should we make a decision that maximizes the expected value of the gain? Should we make a decision that maximizes the expected value of utility?
I will argue in a later blog entry that game G described above is fundamentally different from the original St Petersburg paradox despite inspiring very similar philosophical questions.
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