St. Petersburg paradox

When reading "Thinking, Fast and Slow" one of the things that struck me was that if you use the reference point for gains and losses as the point you end up at then you would have a pretty good model. For example if you have a hundred and lose 20 your loss is 20/80 but if you win 20 your gain is 20/120 [1]. The formulas is x/(x+worth)

I set up a python fiddle to mess about with this: http://pythonfiddle.com/st-petersburgh-paradox

One of the interesting things I found was that there is a power law for how much you would be willing to play the game. There are two ways to run the figures.

One is to never update your worth which produces (1,2.31)(2,5.80)(3,9.38)(4,12.78)(5,16.10)(6,19.43) - where (1,2.31) is an original worth of 10^1 and an expected value of 2.31.

The second is to update your worth after each winning bet: (1,1.94)(2,5.00)(3,8.48)(4,11.80)(5,15.11)(6,18.43)

In both instances a 10x increase in worth increase the expected value about 3.3-3.5. This may explain after you factor in the enjoyment of playing that it is rational for poorer people to play the lottery.

[1] One thing I didn't like about Prospect Theory is the steep curve for a loss this does not explain why people are so ready to buy insurance. If you use the above model if you could lose 90% of your value then -90/(-90+100) gives -9 and if you have to pay 1% then -1/(-1+100) gives -1/99 so you would do that for a 1/891 chance