How Much Would You Pay To Play This Game? The St-Petersburg Paradox

By: ispeculatornew
Date posted: 02.17.2012 (5:00 am) | Write a Comment  (0 Comments)

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Today I wanted to write a lighter post about a theory or game that I read about in the Wall Street Journal a few days ago. They wrote about it in regards to Facebook, which I did not entirely understand for but that does not matter much. I had never heard about it and maybe you had not either. First, I will introduce this “game”. Suppose that you entered a casino that had a new game, from Wikipedia:

Consider the following game of chance: you pay a fixed fee to enter and then a fair coin is tossed repeatedly until a tail appears, ending the game. The pot starts at 1 dollar and is doubled every time a head appears. You win whatever is in the pot after the game ends. Thus you win 1 dollar if a tail appears on the first toss, 2 dollars if a head appears on the first toss and a tail on the second, 4 dollars if a head appears on the first two tosses and a tail on the third, 8 dollars if a head appears on the first three tosses and a tail on the fourth, etc. In short, you win 2k−1 dollars if the coin is tossed k times until the first tail appears

Basically, an investor would have a probability of earnings money that looks something like this:

50% of investors would win $1
25% of investors would win $2
12.5% of investors would win $4
6.25% of investors would win $8
3.125% of investors would win $16
Approx 1.56% of investors would win $32
Approx 0.78% of investors would win $64
Approx 0.39% of investors would win $128
Approx 0.20% of investors would win $256
Approx 0.10% of investors would win $512
Approx 0.05% of investors would win $1024

You can imagine how the list goes on. One of the interesting parts here is that the gains are actually unlimited. That is also why no casino would ever offer this game.

How Much Would YOU Pay To Play This Game?

I think it’s a very interesting question honestly. Why? The answer would tell a lot about your profile and in most cases I would argue that it says a lot about your investor profile. Why? Investors that like risk a lot will tend to pay a lot to play this game as they can see the expected value of this game as being very high. That being said, the large majority of investors would end up as losers in this game for a lucky few to come away with huge gains.

Interesting On Many Levels

First of all, as an investor, my first thought was to figure out the “expected valued” or “average gain” that an investor would expect. In this case, it is impossible to calculate. In fact, the “expected gain” is infinite since an investor could in theory hit 10, 20 or even 30 straight “faces” making $1024 for the first scenario but those gains move up quickly. For example, someone hitting 20 straight would win over $1 million…!

I would also think that our background would have a significant effect on how such a game is perceived.

I read that the average amount potential players were willing to pay to play was $20 or so… I would tend to pay higher. How about you?

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