Do you agree with this paper?

ricko · Post by **ricko** » Wed Nov 14, 2012 1:08 pm

http://www.towerswatson.com/assets/pdf/8271/The-Irreversibility-of-Time.pdf

I see problem with the paradox formulation in this paper and in particular with the calculations in page 4 and the explanations in note 7. Do you also see a problem?

Specifically, I have problem with this statement: "

"Because it is a fair coin, one round will be heads and the other tails. "

What about two heads and two tails?

pj · Post by pj » Wed Nov 14, 2012 3:53 pm

I gave up when he has decided to take

the "ensemble average" of

six consecutive (why not five or seven?) dice throws

instead of one.

Evidently if someone proposes the same gamble with, say, 10 dollars. I would accept it right now.

ig · Post by ig » Wed Nov 14, 2012 4:55 pm

in theory of finance statement that price is random is a quite strong a general assumption. the remark that we do not know a price in a week does not equivalent to state that the price is random. to testify that in practice we need other assumptions which in turn could be verified. this basic of formalization it seems could not be reduced to fair coin dice experiments.

sfca · Post by **sfca** » Thu Nov 15, 2012 12:13 am

I think pj made an error to stop reading when he got to "ensemble average." At the end of the paper, it mentions this was put together by TAG, which stands for their Thinking Ahead Group. Big Smile

FatChoi · Post by **FatChoi** » Thu Nov 15, 2012 1:59 am

The title doesn't relate to anything in the text. Same for the subtitle even if it is true. The point they are making on page 4 is true but the argument is incomplete handwaving. The Appendix is botched, the arithmetic mean which they are calling the ensemble average should be

[img]/User%20Files/4166/Latex-Equation-11541.gif[/img]

while the geometric mean should be

[img]/User%20Files/4166/Latex-Equation-11542.gif[/img]

The investment analysis is various and in places a little embarrassing.

I'm not sure what Ole Peters is on about but I'm pretty sure that he's not using non-ergodicity the way I learned it.

athletico · Post by **athletico** » Wed Nov 21, 2012 2:38 am

I like their cited papers more. This one is a little awkwardly written but I think they basically got the point of Peters' paper on the St Petersburg paradox (say that three times fast).

The TAG paper didn't quite go in the direction I hoped. They describe the problem of interpreting a single-shot trial of the SPP lottery via an ensemble, but they don't clearly say why it matters so much. It's not merely a philosophical point, it's practical: my bid to enter a single-shot game will be different than my bid to enter a repeated game. That would be the way I would try to summarize Peters' paper (which TAG is cribbing from).

I agree with the TAG authors (and Peters) re utility functions: I've always felt (like Kelly) that the logarithm is the One True utility function. You don't have to agree to bet full Kelly, but it should be the starting point. So I liked that aspect of the paper; they derive the natural log as not merely a way out of the SPP, as Bernoulli did, but as THE way.

Btw the results in the appendix are right unless I am missing something... ensemble growth rate is mu (arithmetic returns so no Ito drift), time-average growth rate is mu - 0.5 * sig^2 ... this question is like deja vu....

athletico · Post by **athletico** » Sun Dec 02, 2012 1:21 am

Just saw this blog entry from The Hammock Physicist, which says exactly the same thing as the Ole Peters paper (and the paper the OP refers to) on the St Petersburg paradox. Non-ergodicity implies that one cannot replace the time-average with an ensemble average, and this is the key to resolving the paradox.