5 ms·
No, your expected value is indeed positive over repeated iterations: (1.05^i)*w, where i is the number of iterations and w is the starting wealth. The intuitio
by ActivePattern 3y ago
No, your expected value is indeed positive over repeated iterations: (1.05^i)*w, where i is the number of iterations and w is the starting wealth.
The intuition for why this happens is that the losses of the majority are made up by the big gains of a minority. You can even see that after two iterations:
Case 1: two heads -- 225% wealth
Case 2: heads, tails -- 90% wealth
Case 3: tails, heads -- 90% wealth
Case 4: two tails -- 36% wealth
So, most people are losing money but a single lucky guy makes enough money that the expected value is still positive.
If you run a simulation with a low sample size, you might not see this effect because the "lucky guy" case is rare (and gets more rare with more iterations).
- petermcneeley 3y agoNote a single lucky guy is not enough. P of N heads is (1/2)^N but payout is (1.5)^N for this single lucky guy. (The question can be reformulated such that this could be the case.)
- mizzao 3y agoMmm, seems similar to the game of venture capital!
- jstanley 3y agoWow, great point! I had ran my simulation several times to check that it was reasonably stable and it gave me less than $1 average wealth each time, so I thought that was that. But just now I re-ran it with a much larger population and indeed the average wealth grows now instead of shrinking. Sadly I can't edit my comment above any more, but: > that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating! is indeed untrue. Thanks!
- rcxdude 3y agoYeah, which implies for any finite N eventually the average will collapse, once the lucky guy's luck runs out. The expected value prediction is only true over a long enough timescale for actually infinite N (I do wonder how this scales: I would intuitively expect that the time window scales with log(N), which implies even very large N will not last drastically longer, but I don't know if that's the case)