4 ms·
The expectation of a single flip is 'positive' ( > 1 ), the expectation of n flips is 'negative' ( < 1). My layman reading of this is that this 'appears' to be
by Sahhaese 7y ago
The expectation of a single flip is 'positive' ( > 1 ), the expectation of n flips is 'negative' ( < 1).
My layman reading of this is that this 'appears' to be a violation of martingale theory because this 'should' be a sub-martingale but martingale theory requires finite variance so I suspect the variance of this must go infinite as n tends to infinity?
- kgwgk 7y agoWhy do you say that the expectation of n flips is negative? For 2 flips there are 4 equiprobable cases with gains +125%, -10%, -10%, -64% and the expectation is 10.25%. The expected gain for n flips is 1.05^n-1
- pfortuny 7y agoAFAIK (am reading quickly) mean values of COMPOUND interest rates (which is what you are doing) are not useful. If you earn 10% and then lose 10% you are worse off than before. Start with 10: (11-1.1 < 10).
- kgwgk 7y agoThe definition of expectation is what it is :-) I agree that expectation may not be the most meaningful number here, in the long run is almost sure that we lose everything but there is a zero probability of infinite gains that makes up for it!
- gowld 7y ago1.05^n is > 1, with positive finite expectation at every finite time step. Why do you say almost sure to lose everything?
- kgwgk 7y agoFor any threshold (say $1, assuming you start with $1mn) and any confidence level (say 99%) there is a number of flips n such that you have over 99% probability of holding less than $1 after playing n times. On the other hand, the expected value of your wealth at that time is 1.05^n million dollars.
- mturmon 7y agoThis is well-stated. In fact, an even stronger statement is possible: If you start with $1MM, eventually you will fall below $1 and never go above $1 again no matter how long you keep playing. And all of that not just on a fraction of sequences, but with probability 1.
- kgwgk 7y agoTrue, as I said in a previous comment we have “almost sure” convergence to zero (i.e. the sequence will converge to zero with probability one).
- yorwba 7y agoEarning 50% and then losing 40% is equivalent to losing 10%. The naïve calculation you're arguing against would yield a gain of 10%. kgwgk didn't make that mistake.
- Sahhaese 7y agoYou're right, I screwed up the arithmetic making it look like something it's not. It's just a regular sub-martingale with 'positive' expectation as you say.